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+{"text":"\\section{Introduction}\\label{sec:intro}\nConvolutional Neural Networks (CNNs) have demonstrated tremendous successes when a large volume of labeled data are available to train the models.\nAlthough a solid theory is still lacking, it is thought that both {\\em equivalence} and {\\em invariance} to image translations play a critical role in the success of CNNs \\cite{cohen2016group,cohen2018intertwiners,sabour2017dynamic,hinton2011transforming}, particularly for supervised tasks.\n\nSpecifically, while the whole network is trained in an end-to-end fashion,\na typical CNN model consists of two parts: the convolutional {\\em feature maps} of an input image through multiple convolutional layers, and the classifier of {\\em fully connected layers} mapping the feature maps to the target labels.\nIt is obvious that a supervised classification task requires the fully connected classifier to predict labels invariant to transformations. For training a CNN model, such a transformation invariance criterion is achieved by minimizing the classification errors on the labeled examples\naugmented with various transformations \\cite{krizhevsky2012imagenet}. Unfortunately, it is impossible to simply apply transformation invariance to learn an unsupervised representation without label supervision, since this would result in a trivial constant representation for any input images.\n\n\n\nOn the contrary, it is not hard to see that the representations generated through convolutional layers are equivariant to the transformations -- the feature maps of translated images are also shifted in the same way subject to edge padding effect \\cite{krizhevsky2012imagenet}. It it natural to generalize this idea by considering more types of transformations beyond translations, e.g., image warping and projective transformations \\cite{cohen2016group}.\n\nIn this paper, we formalize the concept of {\\em transformation equivariance} as the criterion to train an unsupervised representation. We expect it could learn the representations that are generalizable to unseen tasks without knowing their labels. This is contrary to the criterion of transformation invariance in supervised tasks, which aims to tailor the representations to predefined tasks and their labels.\nIntuitively, training a transformation equivariant representation is not surprising -- a good representation should be able to preserve the {\\em intrinsic} visual structures of images, so that it could {\\em extrinsically} equivary to various transformations as they change the represented visual structures. In other words, a transformation will be able to be decoded from such representations that well encode the visual structures before and after the transformation \\cite{zhang2019aet}.\n\n\n\nFor this purpose, we present a novel paradigm of Autoencoding Variational Transformations (AVT) to learn powerful representations that equivary against a generic class of transformations. We formalize it from an information-theoretic perspective by considering a joint\nprobability between images and transformations. This enables us to\nuse the mutual information to characterize the dependence between the representations and the transformations. Then, the AVT model can be trained directly by maximizing the mutual information in an unsupervised fashion without any labels. This will ensure the resultant representations contain intrinsic information about the visual structures that could be transformed extrinsically for individual images. Moreover, we will show that the representations learned in this way can be computed directly from the transformations and the representations of the original images {\\em without a direct access to the original samples}, and this allows us to generalize the existing linear Transformation Equivariant Representations (TERs) to more general nonlinear cases \\cite{qi2019learning}.\n\n\n\n\nUnfortunately, it is intractable to maximize the mutual information directly as it is impossible to  exactly evaluate the posterior of a transformation from the associated representation. Thus, we instead seek to maximize a variational lower bound of the mutual information by introducing a transformation decoder to approximate the intractable posterior.\nThis results in an efficient autoencoding transformation (instead of data) architecture by jointly encoding a transformed image and decoding the associated transformation.\n\n\nThe resultant AVT model disruptively differs from the conventional auto-encoders \\cite{hinton1994autoencoders,japkowicz2000nonlinear,vincent2008extracting} that seek to learn representations by reconstructing images.\nAlthough the transformation could be decoded from the reconstructed original and transformed images, this is a quite strong assumption as such representations could contain {\\em more than enough} information about both necessary and unnecessary visual details. The AVT model is based on a weaker assumption that the representations are trained to contain only the {\\em necessary} information about visual structures to decode the transformation between the original and transformed images.\nIntuitively, it is harder to reconstruct a high-dimensional image than decoding a transformation that has fewer degrees of freedom. In this sense, conventional auto-encoders tend to over-represent an image with every detail, no matter if they are necessary or not. Instead, the AVT could learn more generalizable representations by identifying the most essential visual structures to decode transformations, thereby yielding better performances for downstream tasks.\n\n\nThis remainder of this paper is organized as follows. In Section~\\ref{sec:related}, we will review the related works on unsupervised methods. We will formalize the proposed AVT model by maximizing the mutual information between representations and transformations in Section~\\ref{sec:formulation}. It is followed by the variational approach elaborated in Section~\\ref{sec:var}. Experiment results will be demonstrated in Section~\\ref{sec:exp} and we conclude the paper in Section~\\ref{sec:conc}.\n\n\\section{Related Works}\\label{sec:related}\nIn this section, we will review some related methods for training transformation-equivariant representations, along with the other unsupervised models.\n\n\\subsection{Transformation-Equivariant Representations}\nThe study of transformation-equivariance can be traced back to the idea of training capsule nets \\cite{sabour2017dynamic,hinton2011transforming,hinton2018matrix}, where the capsules are designed to equivary to various transformations with vectorized rather than scalar representations. However, there was a lack of explicit training mechanism to ensure the resultant capsules be of transformation equivariance.\n\nTo address this problem, many efforts have been made in literature  \\cite{cohen2016group,cohen2016steerable,lenssen2018group} to extend the conventional translation-equivariant convolutions to cover more transformations.\nFor example, group equivariant convolutions (G-convolution) \\cite{cohen2016group} have been developed to equivary to more types of transformations so that a richer family of geometric structures can be explored by the classification layers on top of the generated representations.\nThe idea of group equivariance has also been introduced to the capsule nets \\cite{lenssen2018group} by ensuring the equivariance of output pose vectors to a group of transformations with a generic routing mechanism.\n\nHowever, these group equivariant convolutions and capsules must be trained in a supervised fashion \\cite{cohen2016group,lenssen2018group} with labeled data for specific tasks, instead of learning unsupervised transformation-equivariant representations generalizable to unseen tasks. Moreover, their representations are restricted to be a function of groups, which limits the ability of training future classifiers on top of more flexible representations.\n\n\nRecently, Zhang et al.~\\cite{zhang2019aet} present a novel Auto-Encoding Transformation (AET) model by learning a representation from which an input transformation can be reconstructed. This is closely related to our motivation of learning transformation equivariant representations, considering the transformation can be decoded from the learned representation of original and transformed images. On the contrary, in this paper, we approach it from an information-theoretic point of view in a more principled fashion.\n\n\nSpecifically, we will define a joint probability over the representations and transformations, and this will enable us to train unsupervised representations by directly maximizing the mutual information between the transformations and the representations. We wish the resultant representations can generalize to new tasks without access to their labels beforehand.\n\n\n\\subsection{Other Unsupervised Representations}\n{\\noindent \\bf Auto-Encoders and GANs.}\nTraining auto-encoders in an unsupervised fashion has been studied in literature \\cite{hinton1994autoencoders,japkowicz2000nonlinear,vincent2008extracting}. Most auto-encoders are trained by minimizing the reconstruction errors on input {\\em data} from the encoded representations.\nA large category of auto-encoder variants have been proposed.\nAmong them is the Variational Auto-Encoder (VAE) \\cite{kingma2013auto} that maximizes the lower-bound of the data likelihood to train a pair of probabilistic encoder and decoder, while beta-VAE seeks to disentangle representations by introducing an adjustable hyperparameter on the capacity of latent channel to balance between the independence constraint and the reconstruction accuracy \\cite{higgins2017beta}.  Denoising auto-encoder \\cite{vincent2008extracting}\nseeks to reconstruct noise-corrupted data to\nlearn robust representation, while\ncontrastive Auto-Encoder \\cite{rifai2011contractive} encourages to learn representations invariant to small perturbations on data.\nAlong this line, Hinton et al.~\\cite{hinton2011transforming} propose capsule nets by minimizing the discrepancy between the reconstructed and target data.\n\nMeanwhile, Generative Adversarial Nets (GANs) have also been used to train unsupervised representations in literature.  Contrary to the auto-encoders, a GAN model generates data from the noises drawn from a simple distribution, with a discriminator trained adversarially to distinguish between real and fake data. The sampled noises can be viewed as the representation of generated data over a manifold, and one can train an encoder by inverting the generator to find the generating noise. This can be implemented by jointly training a pair of mutually inverse generator and encoder \\cite{donahue2016adversarial,dumoulin2016adversarially}. There also exist better generalizable GANs in producing unseen data based on the Lipschitz assumption on the real data distribution \\cite{qi2017loss,arjovsky2017wasserstein}, which can give rise to more powerful representations of data out of training examples  \\cite{donahue2016adversarial,dumoulin2016adversarially,edraki2018generalized}. Compared with the Auto-Encoders, GANs do not rely on learning one-to-one reconstruction of data; instead, they aim to generate the entire distribution of data.\n\n\n\n\n{\\noindent \\bf Self-Supervisory Signals.} There exist many other unsupervised learning methods using different types of self-supervised signals to train deep networks.\nMehdi and Favaro~\\cite{noroozi2016unsupervised} propose to solve Jigsaw puzzles to train a convolutional neural network.\nDoersch et al.~\\cite{doersch2015unsupervised} train the network by predicting the relative positions between sampled patches from an image as self-supervised information. Instead, Noroozi et al.~\\cite{noroozi2017representation} count features that satisfy equivalence relations between downsampled and tiled images, while Gidaris et al.~\\cite{gidaris2018unsupervised} classify a discrete set of image rotations to train deep networks. Dosovitskiy et al.~\\cite{dosovitskiy2014discriminative} create a set of surrogate classes by applying various transformations to individual images. However, the resultant features could over-discriminate visually similar images as they always belong to different surrogate classes.\nUnsupervised features have also been learned from videos by estimating the self-motion of moving objects between consecutive frames \\cite{agrawal2015learning}.\n\n\n\\section{Formulation}\\label{sec:formulation}\nWe begin with the notations for the proposed unsupervised learning of the transformation equivariant representations (TERs). Consider a random sample $\\mathbf x$ drawn from the data distribution $p(\\mathbf x)$. We sample a transformation $\\mathbf t$ from a distribution $p(\\mathbf t)$, and apply it to $\\mathbf x$, yielding a transformed image $\\mathbf t(\\mathbf x)$.\n\nUsually, we consider a distribution $p(\\mathbf t)$ of parameterized transformations, e.g., affine transformations with the rotations, translations and shearing constants being sampled from a simple distribution, and projective transformations that randomly shift and interpolate four corners of images. Our goal is to learn an unsupervised representation that contains as much information as possible to recover the transformation. We wish such a representation is able to compactly encode images such that it could equivary as the visual structures of images are transformed.\n\nSpecifically, we seek to learn an encoder that maps a transformed sample $\\mathbf t(\\mathbf x)$ to the mean $f_\\theta$ and variance $\\sigma_\\theta$ of a desired representation. This results in the following probabilistic representation $\\mathbf z$ of $\\mathbf t(\\mathbf x)$:\n\\begin{equation}\\label{eq:rep}\n\\mathbf z = f_\\theta(\\mathbf t(\\mathbf x))+\\sigma_\\theta(\\mathbf t(\\mathbf x)) \\circ \\epsilon\n\\end{equation}\nwhere $\\epsilon$ is sampled from a normal distribution $\\mathcal N(\\epsilon|\\mathbf 0, \\mathbf I)$, and $\\circ$ denotes the element-wise product. In this case, the probabilistic representation $\\mathbf z$ follows a normal distribution $p_\\theta(\\mathbf z|\\mathbf t, \\mathbf x)\\triangleq \\mathcal N\\left(\\mathbf z|f_\\theta(\\mathbf t(\\mathbf x)),\\sigma_\\theta^2(\\mathbf t(\\mathbf x))\\right)$ conditioned on the randomly sampled transformation $\\mathbf t$ and input data $\\mathbf x$. Meanwhile, the representation $\\mathbf {\\tilde z}$ of the original sample $\\mathbf x$ can be computed as a special case when $\\mathbf t$ is set to an identity transformation.\n\n\nAs discussed in Section~\\ref{sec:intro}, we seek to learn a representation $\\mathbf z$ equivariant to the sampled transformation $\\mathbf t$ whose information can be recovered as much as possible from the representation $\\mathbf z$.  Thus, the most natural choice to formalize this notion of transformation equivariance is the mutual information $I(\\mathbf t,\\mathbf z|\\mathbf {\\tilde z})$ between $\\mathbf z$ and $\\mathbf t$ from an information-theoretic perspective. The larger the mutual information, the more knowledge about $\\mathbf t$ can be inferred from the representation $\\mathbf z$.\n\nMoreover, it can be shown that the mutual information $I(\\mathbf t;\\mathbf z|\\mathbf {\\tilde z})$ is the lower bound of the joint mutual information $I(\\mathbf z;(\\mathbf t,\\mathbf {\\tilde z}))$ that attains its maximum value when $I(\\mathbf z;\\mathbf x|\\mathbf {\\tilde z},\\mathbf t)=0$. In this case, $\\mathbf x$ provides no additional information about $\\mathbf z$ once $(\\mathbf {\\tilde z},\\mathbf t)$ are given. This implies one can estimate $\\mathbf z$ directly from $(\\mathbf {\\tilde z},\\mathbf t)$ without accessing the original sample $\\mathbf x$, which generalizes the {\\em linear} transformation equivariance to {\\em nonlinear} case. For more details, we refer the readers to the long version of this paper \\cite{qi2019learning}.\n\n\nTherefore, we maximize the mutual information between the representation and the transformation to train the model\n$$\n\\max_{\\theta} I(\\mathbf t;\\mathbf z|\\mathbf {\\tilde z})\n$$\n\nUnfortunately, this maximization problem requires us to evaluate the posterior $p_\\theta(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})$ of the transformation, which is often difficult to compute directly. This makes it intractable to train the representation by directly maximizing the above mutual information. Thus, we will turn to a variational approach by introducing a transformation decoder $q_\\phi(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})$ with the parameter $\\phi$ to approximate $p_\\theta(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})$. In the next section, we will elaborate on this variational approach.\n\n\\section{Autoencoding Variational Transformations}\\label{sec:var}\nFirst, we present a variational lower bound of the mutual information $I(\\mathbf t;\\mathbf z|\\mathbf x)$ that can be maximized over $q_\\phi$ in a tractable fashion.\n\nInstead of lower-bounding data likelihood in other variational approaches such as variational auto-encoders \\cite{kingma2013auto}, it is more natural for us to maximize the lower bound of the mutual information \\cite{agakov2004algorithm} between the representation $\\mathbf z$ and the transformation $\\mathbf t$ in the following way\n\\[\n\\begin{aligned}\n&I(\\mathbf t;\\mathbf z|\\mathbf {\\tilde z}) = H(\\mathbf t|\\mathbf {\\tilde z}) - H(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})\\\\\n&=H(\\mathbf t|\\mathbf {\\tilde z}) +\\mathop\\mathbb E\\limits_{p_\\theta(\\mathbf t,\\mathbf z,\\mathbf {\\tilde z})} \\log p_\\theta(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})\\\\\n&= H(\\mathbf t|\\mathbf {\\tilde z})  + \\mathop\\mathbb E\\limits_{p_\\theta(\\mathbf t,\\mathbf z,\\mathbf {\\tilde z})} \\log q_\\phi(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})\\\\\n&+ \\mathop\\mathbb E\\limits_{p(\\mathbf z,\\mathbf {\\tilde z})} D(p_\\theta(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})\\|q_\\phi(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z}))\\\\\n&\\geq H(\\mathbf t|\\mathbf {\\tilde z})  + \\mathop\\mathbb E\\limits_{p_\\theta(\\mathbf t,\\mathbf z,\\mathbf {\\tilde z})} \\log q_\\phi(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z}) \\triangleq \\tilde I_{\\theta,\\phi}(\\mathbf t;\\mathbf z|\\mathbf {\\tilde z})\n\\end{aligned}\n\\]\nwhere $H(\\cdot)$ denotes the (conditional) entropy, and $D(p_\\theta(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})\\|q_\\phi(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z}))$ is the Kullback divergence between $p_\\theta$ and $q_\\phi$, which is always nonnegative.\n\nWe choose to maximize the lower variational bound $\\tilde I(\\mathbf t;\\mathbf z|\\mathbf {\\tilde z})$. Since $H(\\mathbf t|\\mathbf {\\tilde z})$ is independent of the model parameters $\\theta$ and $\\phi$, we simply maximize\n\\begin{equation}\\label{eq:var}\n\\max\\limits_{\\theta,\\phi}  \\mathop\\mathbb E\\limits_{p_\\theta(\\mathbf t,\\mathbf z,\\mathbf{\\tilde z})}\\log q_\\phi(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})\n\\end{equation}\nto learn $\\theta$ and $\\phi$ under the expectation over $p(\\mathbf t, \\mathbf z, \\mathbf {\\tilde z})$.\n\nThis variational approach differs from the variational auto-encoders \\cite{kingma2013auto}: the latter attempts to lower bound the data loglikelihood, while we instead seek to lower bound the mutual information here. Although both are derived based on an auto-encoder structure, the mutual information has a simpler form of lower bound than the data likelihood -- it does not contain an additional Kullback-Leibler divergence term, and thus shall be easier to maximize.\n\n\\subsection{Algorithm}\nIn practice, given a batch of samples $\\{\\mathbf x^{i}|i=1,\\cdots,n\\}$, we first draw a  transformation $\\mathbf t^i$ corresponding to each sample.\nThen we use the reparameterization (\\ref{eq:rep}) to generate the probabilistic representation $\\mathbf z^i$ with $f_\\theta$ and $\\sigma_\\theta$ as well as a sampled noise $\\epsilon^i$.\n\nOn the other hand, we use a normal distribution $\\mathcal N(\\mathbf t|d_\\phi(\\mathbf z,\\mathbf {\\tilde z}),\\sigma^2_\\phi(\\mathbf z,\\mathbf {\\tilde z}))$ as the decoder $q_\\phi(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})$, where the mean $d_\\phi(\\mathbf z,\\mathbf {\\tilde z})$ and variance $\\sigma^2_\\phi(\\mathbf z, \\mathbf {\\tilde z})$ are implemented by deep network respectively.\n\nWith the above samples, the objective (\\ref{eq:var}) can be approximated as\n\\begin{equation}\\label{eq:obj}\n\\max\\limits_{\\theta,\\phi} \\dfrac{1}{n} \\sum_{i=1}^n \\log \\mathcal N(\\mathbf t^i|d_\\phi(\\mathbf z^i,\\mathbf {\\tilde z}^i),\\sigma_\\phi(\\mathbf z^i,\\mathbf {\\tilde z}^i))\n\\end{equation}\nwhere\n\\[\n\\mathbf z^i = f_\\theta(\\mathbf t^i(\\mathbf x^i))+\\sigma_\\theta(\\mathbf t^i(\\mathbf x^i)) \\circ \\epsilon^i\n\\]\nand\n\\[\n\\mathbf {\\tilde z}^i = f_\\theta(\\mathbf x^i)+\\sigma_\\theta(\\mathbf x^i) \\circ \\tilde\\epsilon.\n\\]\nand $\\epsilon^i, \\tilde\\epsilon^i \\sim \\mathcal N(\\epsilon|\\mathbf 0, \\mathbf I)$, and $\\mathbf t^i\\sim p(\\mathbf t)$ for each $i=1,\\cdots,n$.\n\n\\subsection{Architecture}\n\n\nAs illustrated in Figure~\\ref{fig:avt}, we implement the transformation decoder $q_\\phi(\\mathbf t|\\mathbf z,\\mathbf {\\tilde z})$ by using\na Siamese encoder network with shared weights to represent the original and transformed images with $\\mathbf{\\tilde z}$ and $\\mathbf z$ respectively, where the mean $d_\\phi$ and the variance $\\sigma_\\phi^2$ of the sampled transformation are predicted from the concatenation of both representations.\n\n\n\n\n\n\nWe note that, in a conventional auto-encoder, error signals must be backpropagated through a deeper decoder to reconstruct images before they train the encoder of interest. In contrast, the AVT allows a shallower decoder to estimate transformations with fewer variables so that stronger training signals can reach the encoder before it attenuates remarkably. This can more sufficiently train the encoder to represent images in downstream tasks.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[c]{0.49\\textwidth}\n        \\includegraphics[width=\\textwidth]{Figure\/AVT.png}\n       \n    \\end{subfigure}\\\\\\vspace{-4mm}\n    \\caption{The architecture of the proposed AVT. The original and transformed images are fed through the encoder $p_\\theta$ where $\\mathbf 1$ denotes an identity transformation to generate the representation of the original image. The resultant representations $\\mathbf {\\tilde z}$ and $\\mathbf z$ of original and transformed images are sampled and fed into the transformation decoder $q_\\phi$ from which the transformation $\\mathbf t$ is sampled. }\\label{fig:avt}\n\\end{figure}\n\n\n\\vspace{-1mm}\n\\section{Experiments}\\label{sec:exp}\n\\vspace{-1mm}\nIn this section, we evaluate the proposed AVT model by following the standard protocol in literature.\n\n\\vspace{-1mm}\n\\subsection{CIFAR-10 Experiments}\n\\vspace{-1mm}\nWe evaluate the AVT model on the CIFAR-10 dataset.\n\\vspace{-2mm}\n\\subsubsection{Experiment Settings}\n\\vspace{-1mm}\n\n\n\\vspace{2mm}\n{\\noindent \\bf Architecture} For a fair comparison with existing models, the Network-In-Network (NIN) is adopted on the CIFAR-10 dataset for the unsupervised learning task \\cite{zhang2019aet}. \nThe NIN consists of four convolutional blocks, each containing three convolutional layers.\nThe AVT has two NIN branches, each of which takes the original and transformed images as its input, respectively. We average-pool and concatenate the output feature maps from the forth block of two branches to form a $384$-d feature vector. Then an output layer follows to output the mean $d_\\phi$ and the log-of-variance $\\log \\sigma_\\phi^2$ of the predicted transformation, with the logarithm scaling the variance to a real value.\n\nThe two branches share the same network weights, with the first two blocks of each branch being used as the encoder network to directly output the mean $f_\\theta$ of the representation. An additional $1\\times 1$ convolution followed by a batch normalization layer is added on top of the representation mean to output the log-of-variance $\\log\\sigma_\\theta^2$.\n\n\n\\vspace{2mm}\n{\\noindent\\bf Implementation Details} The AVT networks are trained by the SGD with a batch size of $512$ images and their transformed versions. Momentum and weight decay are set to $0.9$ and $5\\times 10^{-4}$, respectively. The model is trained for a total of $4,500$ epochs. The learning rate is initialized to $10^{-3}$. Then it is gradually decayed to $10^{-5}$ from $3,000$ epochs after it is increased to $5\\times 10^{-3}$ after the first $50$ epochs.\nThe previous research \\cite{zhang2019aet} has shown the projective transformation outperforms the affine transformation in training unsupervised models, and thus we adopt it to train the AVT for a fair comparison.\nThe projective transformation is composed of a\nrandom translation of the four corners of an image in both horizontal and vertical directions by $\\pm 0.125$ of its height and width, after it is scaled by a factor of $[0.8, 1.2]$ and rotated randomly with an angle from $\\{0^\\circ, 90^\\circ, 180^\\circ, 270^\\circ\\}$.\n\nDuring training the AVT model, a single representation is randomly sampled from the encoder $p_\\theta(\\mathbf z|\\mathbf t,\\mathbf x)$, which is fed into the surrogate decoder $q_\\phi(\\mathbf t|\\mathbf x, \\mathbf z)$. In contrast, to fully exploit the uncertainty of probabilistic representations in training the downstream classification tasks, five random samples are drawn and averaged as the representation of an image used by the classifiers. We found averaging randomly sampled representations outperforms only using the mean of the representation to train the downstream classifiers.\n\n\n\n\\vspace{-1mm}\n\\subsubsection{Results}\n\\vspace{-1mm}\n\n\n\n\n\n{\\noindent\\bf Comparison with Other Methods.} A classifier is usually trained upon the representation learned by an unsupervised model to assess the performance. Specifically, on CIFAR-10, the existing evaluation protocol \\cite{oyallon2015deep,dosovitskiy2014discriminative,radford2015unsupervised,oyallon2017scaling,gidaris2018unsupervised,zhang2019aet} is strictly followed by building a classifier on top of the second convolutional block.\n\nFirst, we evaluate the classification results by using the AVT features with both model-based and model-free classifiers.  For the model-based classifier, we follow \\cite{zhang2019aet} by training a non-linear classifier with three Fully-Connected (FC) layers -- each of the two hidden layers has $200$ neurons with batch-normalization and ReLU activations, and the output layer is a soft-max layer with ten neurons each for an image class. We also test a convolutional classifier upon the unsupervised features by adding a third NIN block whose output feature map is averaged pooled and connected to a linear soft-max classifier.\n\n\\begin{table}\n\\caption{Comparison between unsupervised feature learning methods on CIFAR-10. The fully supervised NIN and the random Init. + conv have the same three-block NIN architecture, but the first is fully supervised while the second is trained on top of the first two blocks that are randomly initialized and stay frozen during training.}\\label{tab01}\n\\centering\n \\begin{tabular}{l|c} \\toprule\nMethod&Error rate\\\\ \\midrule\nSupervised NIN \\cite{gidaris2018unsupervised} (Upper Bound)&7.20  \\\\\nRandom Init. + conv \\cite{gidaris2018unsupervised} (Lower Bound)&27.50  \\\\ \\midrule\nRoto-Scat + SVM \\cite{oyallon2015deep} &17.7 \\\\\nExamplarCNN \\cite{dosovitskiy2014discriminative} &15.7 \\\\\nDCGAN \\cite{radford2015unsupervised}&17.2 \\\\\nScattering \\cite{oyallon2017scaling}&15.3\\\\\nRotNet + non-linear \\cite{gidaris2018unsupervised}&10.94\\\\\nRotNet + conv \\cite{gidaris2018unsupervised}&8.84\\\\\nAET-affine + non-linear \\cite{zhang2019aet} &9.77\\\\\nAET-affine + conv \\cite{zhang2019aet} &8.05\\\\\nAET + non-linear \\cite{zhang2019aet} &9.41\\\\\nAET + conv \\cite{zhang2019aet} &7.82\\\\ \\midrule\nAVT + non-linear &\\textbf{8.96} \\\\\nAVT + conv &\\textbf{7.75} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{tab01} shows the results by the AVT and other models. It compares the AVT with both fully supervised and unsupervised methods on CIFAR-10. The unsupervised AVT with the convolutional classifier almost achieves the same error rate as its fully supervised NIN counterpart with four convolutional blocks ($7.75\\%$ vs. $7.2\\%$). This remarkable result demonstrates the AVT could greatly close the performance gap with the fully supervised model on CIFAR-10.\n\n\n\\begin{table}\n\\caption{Error rates of different classifiers trained on top of the learned representations on CIFAR 10, where $n$-FC denotes a classifier with $n$ fully connected layers and conv denotes the third NIN block as a convolutional classifier. Two AET variants are chosen for a fair direct comparison since they are based on the same architecture as the AVT and have outperformed the other unsupervised representations before \\cite{zhang2019aet}.}\\label{tab02}\n\\centering\n \\begin{tabular}{c|cccc} \\toprule\n   &1 FC&2 FC&3 FC&conv\\\\ \\midrule\nAET-affine \\cite{zhang2019aet} &17.16 &9.77 &10.16 &8.05\\\\\nAET-project \\cite{zhang2019aet}&16.65 &9.41 &9.92 &7.82 \\\\ \\midrule\n(Ours) AVT &\\textbf{16.19} &\\textbf{8.96} &\\textbf{9.55} &\\textbf{7.75}\\\\\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\nWe also evaluate the AVT when varying numbers of FC layers and a convolutional classifier are trained on top of unsupervised representations respectively in Table~\\ref{tab02}.\nThe results show that AVT can consistently achieve the smallest errors no matter which classifiers are used.\n\n\n\n\\begin{table}\n\\caption{The comparison of the KNN error rates by different models with varying numbers $K$ of nearest neighbors on CIFAR-10.}\\label{tab04}\n\\centering\n\\small\n \\begin{tabular}{c|ccccc} \\toprule\n$K$   &3&5&10&15&20\\\\ \\midrule\nAET-affine \\cite{zhang2019aet}&24.88&23.29&23.07&23.34&23.94\\\\ \\\nAET-project \\cite{zhang2019aet}&23.29&22.40&\\bf 22.39&23.32&23.73 \\\\\n(Ours) AVT &\\bf 22.46&\\bf 21.62&23.7&\\bf 22.16&\\bf 21.51 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n{\\noindent\\bf Comparison based on Model-free KNN Classifiers.}  We also test the model-free KNN classifier based on the averaged-pooled feature representations from the second convolutional block. The KNN classifier is model-free without training a classifier from labeled examples.  This enables us to make a direct evaluation on the quality of learned features.\nTable~\\ref{tab04} reports the KNN results with varying numbers of nearest neighbors.  Again, the AVT outperforms the compared representations when they are used to calculate $K$ nearest neighbors for classifying images.\n\n\n\\begin{table*}\n\\caption{Error rates on CIFAR-10 when different numbers of samples per class are used to train the downstream classifiers. A third convolutional block is trained with the labeled examples on top of the first two blocks of the NIN ($*$ the 13-layer network) pre-trained with the unlabeled data.\nWe compare with the fully supervised models that are trained with all the labeled examples from scratch.}\\label{tab03}\n\\centering\n \\begin{tabular}{c|ccccc} \\toprule\n   &20&100&400&1000&5000\\\\ \\midrule\nSupervised conv &66.34&52.74 &25.81 &16.53 &6.93\\\\\nSupervised non-linear &65.03&51.13 &27.17 &16.13 &7.92\\\\\\midrule\nRotNet + conv \\cite{gidaris2018unsupervised}&35.37 &24.72&17.16&13.57 &8.05 \\\\\nAET-project + conv \\cite{zhang2019aet} &\\bf 34.83&24.35 &16.28 &12.58 &7.82 \\\\\nAET-project + non-linear \\cite{zhang2019aet} &\\bf 37.13&25.19&18.32 & 14.27 &9.41 \\\\\n\\midrule\nAVT + conv &35.44&\\textbf{24.26} &\\textbf{15.97} &\\textbf{12.27} &\\textbf{7.75}\\\\\nAVT + non-linear &37.62&\\textbf{25.01} &\\textbf{17.95} &\\textbf{14.14} &\\textbf{8.96}\\\\\\midrule\nAVT + conv (13 layers)$^*$ &\\bf 26.2&\\bf 18.44&\\bf 13.56&\\bf 10.86 &\\bf 6.3\\\\\\bottomrule\n\\end{tabular}\n\\end{table*}\n\n{\\noindent\\bf Comparison with Small Labeled Data.} Finally, we also conduct experiments when a small number of labeled examples are used to train the downstream classifiers on top of the learned representations.\nThis will give us some insight into how the unsupervised representations could help with only few labeled examples.   Table~\\ref{tab03} reports the results of different models on CIFAR-10. The AVT outperforms the fully supervised models when only a small number of labeled examples ($\\leq 1000$ samples per class) are available. It also performs better than the other unsupervised models in most of cases. Moreover, if we adopt the widely used 13-layer network \\cite{laine2016temporal} on CIFAR-10 to train the unsupervised and supervised parts, the error rates can be further reduced significantly particularly when very few labeled examples are used.\n\n\n\n\n\n\\vspace{-1mm}\n\\subsection{ImageNet Experiments}\n\\vspace{-1mm}\nWe further evaluate the performance by AVT on the ImageNet dataset. The AlexNet is used as the backbone to learn the unsupervised features.\n\\vspace{-1mm}\n\\subsubsection{Architectures and Training Details}\n\\vspace{-1mm}\nTwo AlexNet branches with shared parameters are created with original and transformed images as inputs respectively to train unsupervised AVT. The $4,096$-d output features from the second last fully connected layer in two branches are concatenated and fed into the output layer producing the mean and the log-of-variance of eight projective transformation parameters. We still use SGD to train the network, with a batch size of $768$ images and the transformed counterparts, a momentum of $0.9$, a weight decay of $5\\times 10^{-4}$. The initial learning rate is set to $10^{-3}$, and it is dropped by a factor of $10$ at epoch 300 and 350. The AVT is trained for $400$ epochs in total. Finally, the projective transformations are randomly sampled in the same fashion as on CIFAE-10, and the unsupervised representations fed into the classifiers are the average over five sampled representations from the probabilistic encoder.\n\\vspace{-1mm}\n\\subsubsection{Results}\n\\vspace{-1mm}\n\\begin{table}\n\\caption{Top-1 accuracy with non-linear layers on ImageNet. AlexNet is used as backbone to train the unsupervised models. After unsupervised features are learned, nonlinear classifiers are trained on top of Conv4 and Conv5 layers with labeled examples to compare their performances. We also compare with the fully supervised models and random models that give upper and lower bounded performances. For a fair comparison, only a single crop is applied in AVT and no dropout or local response normalization is applied during the testing. }\\label{tab05}\n\\centering\n \\begin{tabular}{l|cc} \\toprule\nMethod&Conv4 &Conv5\\\\ \\midrule\nSupervised from \\cite{bojanowski2017unsupervised}(Upper Bound)&59.7&59.7  \\\\\nRandom from \\cite{noroozi2016unsupervised} (Lower Bound)&27.1 &12.0  \\\\ \\midrule\nTracking \\cite{wang2015unsupervised} &38.8&29.8 \\\\\nContext \\cite{doersch2015unsupervised} &45.6&30.4 \\\\\nColorization \\cite{zhang2016colorful}&40.7&35.2 \\\\\nJigsaw Puzzles \\cite{noroozi2016unsupervised}&45.3&34.6\\\\\nBIGAN \\cite{donahue2016adversarial}&41.9&32.2\\\\\nNAT \\cite{bojanowski2017unsupervised}&-&36.0\\\\\nDeepCluster \\cite{caron2018deep} &-&44.0\\\\\nRotNet \\cite{gidaris2018unsupervised}&50.0&43.8\\\\\nAET-project \\cite{zhang2019aet} &{53.2}&{47.0}\\\\\\midrule\n(Ours) AVT &\\textbf{54.2}&\\textbf{48.4}\\\\\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table*}\n\\caption{Top-1 accuracy with linear layers on ImageNet. AlexNet is used as backbone to train the unsupervised models under comparison. A $1,000$-way linear classifier is trained upon various convolutional layers of feature maps that are spatially resized to have about $9,000$ elements. Fully supervised and random models are also reported to show the upper and the lower bounds of unsupervised model performances. Only a single crop is used and no dropout or local response normalization is used during testing for the AVT, except the models denoted with * where ten crops are applied to compare results.}\\label{tab06}\n\\centering\n \\begin{tabular}{l|ccccc} \\toprule\nMethod&Conv1 &Conv2&Conv3&Conv4&Conv5\\\\ \\midrule\nImageNet labels(Upper Bound)&19.3&36.3&44.2&48.3&50.5  \\\\\nRandom (Lower Bound)&11.6 &17.1&16.9&16.3&14.1  \\\\\nRandom rescaled \\cite{krahenbuhl2015data}&17.5 &23.0&24.5&23.2&20.6  \\\\\n\\midrule\nContext \\cite{doersch2015unsupervised} &16.2&23.3&30.2&31.7&29.6 \\\\\nContext Encoders \\cite{pathak2016context}&14.1&20.7&21.0&19.8&15.5 \\\\\nColorization\\cite{zhang2016colorful}&12.5&24.5&30.4&31.5&30.3\\\\\nJigsaw Puzzles \\cite{noroozi2016unsupervised}&18.2&28.8&34.0&33.9&27.1\\\\\nBIGAN \\cite{donahue2016adversarial}&17.7&24.5&31.0&29.9&28.0\\\\\nSplit-Brain \\cite{zhang2017split}&17.7&29.3&35.4&35.2&32.8\\\\\nCounting \\cite{noroozi2017representation}&18.0&30.6&34.3&32.5&25.7\\\\\nRotNet \\cite{gidaris2018unsupervised}&18.8&31.7&38.7&38.2&36.5\\\\\nAET-project \\cite{zhang2019aet}&19.2&32.8&40.6&39.7&37.7\\\\\n\\midrule\n(Ours) AVT &\\bf 19.5&\\bf 33.6&\\bf 41.3&\\bf 40.3&\\bf 39.1\\\\\n\\bottomrule\n\\toprule\nDeepCluster* \\cite{caron2018deep} &13.4&32.3&41.0&39.6&38.2\\\\\nAET-project* \\cite{zhang2019aet} &19.3&35.4&44.0&43.6&42.4\\\\\n(Ours) AVT*&\\textbf{20.9}&\\textbf{36.1}&\\textbf{44.4}&\\textbf{44.3}&\\textbf{43.5}\\\\\\bottomrule\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Top-1 accuracy on the Places dataset. A $205$-way logistic regression classifier is trained on top of various layers of feature maps that are spatially resized to have about $9,000$ elements. All unsupervised features are pre-trained on the ImageNet dataset, and then frozen when training the logistic regression classifiers with Places labels. We also compare with fully-supervised networks trained with Places Labels and ImageNet labels, as well as with random models. The highest accuracy values are in bold and the second highest accuracy values are underlined.}\\label{tab07}\n\\centering\n \\begin{tabular}{l|ccccc} \\toprule\nMethod&Conv1 &Conv2&Conv3&Conv4&Conv5\\\\ \\midrule\nPlaces labels(Upper Bound)\\cite{zhou2014learning}&22.1&35.1&40.2&43.3&44.6 \\\\\nImageNet labels&22.7&34.8&38.4&39.4&38.7\\\\\nRandom (Lower Bound)&15.7 &20.3&19.8&19.1&17.5  \\\\\nRandom rescaled \\cite{krahenbuhl2015data}&21.4 &26.2&27.1&26.1&24.0  \\\\\n\\midrule\nContext \\cite{doersch2015unsupervised} &19.7&26.7&31.9&32.7&30.9 \\\\\nContext Encoders \\cite{pathak2016context}&18.2&23.2&23.4&21.9&18.4 \\\\\nColorization\\cite{zhang2016colorful}&16.0&25.7&29.6&30.3&29.7\\\\\nJigsaw Puzzles \\cite{noroozi2016unsupervised}&\\underline{23.0}&31.9&35.0&34.2&29.3\\\\\nBIGAN \\cite{donahue2016adversarial}&22.0&28.7&31.8&31.3&29.7\\\\\nSplit-Brain \\cite{zhang2017split}&21.3&30.7&34.0&34.1&32.5\\\\\nCounting \\cite{noroozi2017representation}&\\textbf{23.3}&\\textbf{33.9}&36.3&34.7&29.6\\\\\nRotNet \\cite{gidaris2018unsupervised}&21.5&31.0&35.1&34.6&33.7\\\\\n AET-project \\cite{zhang2019aet}& 22.1&32.9&\\underline{37.1}&\\underline{36.2}&\\underline{34.7}\\\\\\midrule\nAVT&22.3&\\underline{33.1}&\\textbf{37.8}&\\textbf{36.7}&\\textbf{35.6}\\\\\\bottomrule\n\\end{tabular}\n\\end{table*}\n\nTable~\\ref{tab05} reports the Top-1 accuracies of the compared methods on ImageNet by following the evaluation protocol in \\cite{noroozi2016unsupervised,zhang2017split,gidaris2018unsupervised,zhang2019aet}. Two settings are adopted for evaluation, where Conv4 and Conv5 mean to train the remaining part of AlexNet on top of Conv4 and Conv5 with the labeled data. All the bottom convolutional layers up to Conv4 and Conv5 are frozen after they are trained in an unsupervised fashion.\nFrom the results, in both settings, the AVT model consistently outperforms the other unsupervised models.\n\nWe also compare with the fully supervised models that give the upper bound of the classification performance by training the whole AlexNet with all labeled data end-to-end. The classifiers of random models are trained on top of Conv4 and Conv5 whose weights are randomly sampled, which set the lower bounded performance. By comparison, the AVT model further closes the performance gap to the full supervised models to $5.5\\%$ and $11.3\\%$ on Conv4 and Conv5 respectively. This is a relative improvement by $15\\%$ and $11\\%$ over the previous state-of-the-art AET model.\n\n\n\nMoreover, we also follow the testing protocol adopted in \\cite{zhang2019aet} to compare the models by training a $1,000$-way linear classifier on top of different numbers of convolutional layers in Table~\\ref{tab06}.  Again, the AVT consistently outperforms all the compared unsupervised models in terms of the Top-1 accuracy.\n\\vspace{-1mm}\n\\subsection{Places Experiments}\n\\vspace{-1mm}\nFinally, we evaluate the AVT model on the Places dataset. Table~\\ref{tab07} reports the results. Unsupervised models are pretrained on the ImageNet dataset, and a linear logistic regression classifier is trained on top of different layers of convolutional feature maps with Places labels. It assesses the generalizability of unsupervised features from one dataset to another. The models are still based on AlexNet variants. We compare with the fully supervised models trained with the Places labels and ImageNet labels respectively, as well as with the random networks. The AVT model outperforms the other unsupervised models, except performing slightly worse than Counting \\cite{zhang2017split} with a shallow representation by Conv1 and Conv2.\n\\vspace{-1mm}\n\\section{Conclusion}\\label{sec:conc}\n\\vspace{-1mm}\nIn this paper, we present a novel paradigm of learning representations by Autoencoding Variational Transformations (AVT) instead of reconstructing data as in conventional autoencoders. It aims to maximize the mutual information between the transformations and the representations of transformed images. The intractable maximization problem on mutual information is solved by introducing a transformation decoder to approximate the posterior of transformations through a variational lower bound. This naturally leads to a new probabilistic structure with a representation encoder and a transformation decoder. The resultant representations should contain as much information as possible about the transformations to equivary with them. Experiment results show the AVT representations set new state-of-the-art performances on CIFAR-10, ImageNet and Places datasets, greatly closing the performance gap to the supervised models as compared with the other unsupervised models.\n\n\n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Summary of Results.} \nLet $v$ be an indeterminate and $k$ a field of characteristic zero. Let\n$U$ be the quantized enveloping algebra defined over $k(v)$ with generators\n$K^{\\pm 1},E,F$ and relations \n\\begin{gather*}\n[E,F]=\\frac{K-K^{-1}}{ v-v^{-1}} , \\quad KEK^{-1}=v^2E,\\\\\n KFK^{-1}=v^{-2}F\\quad\\text{and} \\quad KK^{-1}=K^{-1}K=1.\n\\end{gather*}\nLet $\\mathbf U^0$ be the subalgebra generated by $K^{\\pm 1}$ and let $B$ be\nthe subalgebra generated by $\\mathbf U^0$ and $E$. More precisely we are\nfollowing the notation given in \\cite{MR96c:17022} where we take\n$I=\\{i\\}$, $i\\cdot i=2$, $Y=\\mathbb Z[I]\\cong \\mathbb Z$, \n$X=\\hom(\\mathbb Z[I], \\mathbb Z)\\cong \\mathbb Z$, $F=F_i$, $E=E_i$, and\n$K=K_i$.\n\nLet $R$ be the power series ring in $T-1$ with coefficients in $k(v)$ i.e. \n$$\nR=k(v)[[T-1]]:=\\lim_\\leftarrow \\frac{k(v)[T,T^{-1}]}{ (T-1)^i}.\n$$\nSet $\\mathcal K$ equal to the field of fractions of $R$. Let $s$ be the\ninvolution of $R$ induced by $T\\rightarrow T^{-1}$ i.e. the involution that sends\n$T$ to $T^{-1}=1\/(1+(T-1))=\\sum_{i\\geq 0}(-1)^i(T-1)^i$. Let the subscript\n$R$ denote the extension of scalars from $k(v)$ to $R$ , e.g. $\\ru =\nR\\otimes_{k(v)} \\mathbf U$. For any representation $(\\pi,A)$ of $\\ru$ we can\ntwist the representation in two ways by composing with automorphisms of\n$\\ru$. The first is $\\pi \\circ (s\\otimes 1)$ while the second is $\\pi\n\\circ (1\\otimes \\Theta)$ for any automorphism $\\Theta$ of $\\mathbf U$. We designate the corresponding\n$\\ru$-modules by $A^s$ and $A^{\\Theta}$. Twisting the action by both $s$\nand $\\Theta$ we obtain the composite $(A^s)^{\\Theta}=(A^{\\Theta})^s$ which\nwe denote by\n$A^{s\\Theta}$. \n\n\n\nLet $m$ denote the homomorphism of $\\ru^0$ onto $R$ with $m(K)=T$. For\n$\\lambda \\in \\mathbb Z$ let $m+\\lambda$ denote the homomorphism of\n$\\mathbf U^0$ to\n$R$ with $(m+\\lambda )(K)=Tv^\\lambda$. We use the additive notation\n$m+\\lambda$ to indicate that this map originated in the classical setting\nfrom an addition of two algebra homomorphisms. It however is not a sum of\ntwo homomorphisms but rather a product. Let\n$R_{m+\\lambda}$ be the corresponding $_RB$-module and define the Verma\nmodule\n \\begin{equation}\n_RM(m+\\lambda )=\\ru\\otimes_{_RB} R_{m+\\lambda-\\rho} .\n\\end{equation}\n \n\n\n\nLet $\\rho_1:\\mathbf U\\to \\mathbf U$ be the algebra isomorphism\ndetermined by the assignment \n\\begin{equation}\\label{rhoone}\n\\rho_1(E)=-vF,\\quad \\rho_1(F)=-v^{-1}E,\\quad\n\\rho_1(K)=K^{-1}. \n\\end{equation}  \nDefine also an algebra\nanti-automorphism $\\brho:\\mathbf U\\to \\mathbf U$ by \n\\begin{equation}\n\\label{antiauto}\n\\brho(E)=vKF,\\quad \\brho(F)=vK^{-1}E,\\quad\n\\brho(K_\\mu)=K_\\mu. \n\\end{equation}\n These maps are related through the antipode $S$ of  $\\mathbf U$ by\n$\\brho=\\brho_1S$. \n\nFor $\\ru$-modules $M,N$ and $\\mathcal F$, let $\\mathbb P(M,N)$ and $\\mathbb\nP(M,N,\\mathcal F)$ denote the space of $R$-bilinear maps of $M\\times N$ to $R$\nand $\\mathcal F$ respectively, with the following invariance condition: \n\\begin{equation}\\label{correction}\n\\sum x_{(1)}*\\phi(Sx_{(3)}\\cdot a,\\varrho(x_{(2)})b)=\\mathbf e(x) \n\\phi(a,b)\n\\end{equation} where $\\Delta\\otimes 1\\circ \\Delta (x)=\\sum x_{(1)}\\otimes\nx_{(2)}\\otimes x_{(3)}$,  $\\mathbf e:\\mathbf U\\to k(v)$ is the counit and $*$\ndenote the action twisted by $\\rho_1$; in other words $x*n:=\\rho_1(x)n$.  If we let\n$\\hom_\\ru(A,B)$ denote the set of module $\\ru$-module homomorphisms, then one\ncan check on generators of $\\ru$ that\n$\\mathbb P(M,N,\\mathcal F)\\cong \\hom_\\ru (M\\otimes_R N^{\\rho_1},{_R\\mathcal\nF}^\\brho)$ (see \\cite[3.10.6]{MR96m:17029}).  Formula \\eqnref{correction} corrects an\nerror in  \\cite[6.2.2]{MR96c:17022}. Let $\\mathbb P(N)=\\mathbb\nP(N,N)$ denote the\n$R$-module of invariant forms on $N$. \n\nFor the rest of the introduction we let $M$ denote the $\\ru$ Verma\nmodule with highest weight $Tv^{-1}$ ; i.e. $M=M(m)$ and let $\\mathcal\nF$ be any finite dimensional $\\mathbf U$-module.  A natural parameterization for\n$\\mathbb P(M\\otimes {_R\\mathcal F})$ was given in \\cite{MR96c:17022}.\nFix an invariant form $\\phi_M$ on $M$ normalized as in  ( .). For each\n$\\ru$-module homomorphism $\\beta:{_R\\mathcal E}\\otimes_R\\mathcal\nF^{\\rho_1}\\rightarrow\\ru$ define what we call the {\\it induced form}\n$\\chi_{\\beta,\\phi_M}$ by the formula, for\n$e\\in_ R\\mathcal E,f\\in _R\\mathcal F, \\ m,n\\in N$,\n\\begin{equation}\\label{inducedform}\n\\chi_{\\beta,\\phi_M}(m\\otimes e,n\\otimes f)=\\phi_M( m,\\beta(e\\otimes f)*n).\n\\end{equation}\n \n\n\n\\begin{prop}\nSuppose $\\beta:{_R\\mathcal F}\\otimes {_R\\mathcal F^{\\rho_1}}\\rightarrow \\ru$ is a module\nhomomorphism with $\\ru$ having the adjoint action.    Then $M\\otimes\n{_R\\mathcal F}$ decomposes as the $\\chi_{\\beta,\\phi_M}$-orthogonal sum of\nindecomposable $\\ru$-modules.\n\\end{prop}\n\nWe now begin the description of our main result:\nRecall from \\cite{MR96c:17022} we define a {\\it cycle} ( for $A$ ) to be a pair $(A,\\Psi)$\nwhere\n$A$ is a $\\mathbf U$ (or $_R\\mathbf U$)  module and $\\Psi$ is a module\nhomomorphism  \n\\begin{equation}\n\\Psi : A_\\pi^{sT_{-1}'} \\\n\\rightarrow \\ A .\n\\end{equation} \nHere $A_\\pi$ is defined to be $A_F\/\\iota A$ where $\\iota:A\\to A_F$ is the canonical embedding of the module $A$ into its localization $A_F$ with respect to root vector $F$.  Note that modules of the form $A_\\pi^{sT_{-1}'}$ above appear naturally in other mathematical work (see \\cite{MR2032059} and \\cite{MR2074588}) besides our own (see \\cite{MR96c:17022}).\nWe  choose a homomorphism $\\Psi : M_F\\rightarrow M$ and set\n\\begin{equation}\n\\bar \\Psi:=\\Psi \\otimes sT_{1}''\\circ L^{-1}:(M_\\pi\\otimes \\mathcal E)^{sT_{-1}'}\\to\nM\\otimes \\mathcal E.\n\\end{equation}\n(The linear map $L$ is defined in Lusztig's book - see also \\eqnref{L}.)\nLet $\\iota:\\mathbb P(M\\otimes \\mathcal E, N\\otimes\n\\mathcal F)\\to\\hom_\\ru(M\\otimes \\mathcal E\\otimes (N\\otimes\n\\mathcal F)^{\\rho_1},R)$ be the canonical isomorphism with $\\iota(\\chi)(a\\otimes b)=\\chi(a,b)$.  Note that $a\\otimes b\\in\nM\\otimes\n\\mathcal E\\otimes (N\\otimes\n\\mathcal F)^{\\rho_1}$ on the left hand side, while $(a,b)\\in M\\otimes \\mathcal E\\times N\\otimes\n\\mathcal F$ on the right hand side.  Define $\\chi\\mapsto \\chi^\\sharp$ in $\n\\text {\\rm End}\\,(\\mathbb P(M\\otimes\n\\mathcal E, N\\otimes\n\\mathcal F))$ by\n\\begin{equation}\n\\iota(\\chi^\\sharp)(\\bar\\Psi(a)\\otimes \\bar\\Psi(b)):=\ns\\circ\\iota(\\chi_\\pi)\\circ L(a\\otimes b)\n\\end{equation}\nfor $a\\in(M\\otimes \\mathcal E)_\\pi^{sT_{-1}'}$,\n$b\\in (N\\otimes\\mathcal  F)^{s T_{-1}'\\rho_1}_\\pi$ \nand $\\chi\\in\\mathbb P(M\\otimes \\mathcal E,N\\otimes \\mathcal F)$. \n\nLet  $\\mathcal F_m$\nand\n$\\mathcal F_n$ be $X$-admissible finite dimensional $\\mathbf U$-modules\ngiven in \\secref{CG} with basis $u^{(m)}_k$, $0\\leq k\\leq m$.  Fix a homomorphism\n$\\beta:_R\\mathcal F_m\\otimes _R\\mathcal F_n^{\\rho_1}\\to {_RF(\\mathbf U)}$ which has the\nform \\eqnref{factorization}\n\\begin{equation}\n\\beta=\\sum_{m,n,k}r^{m,n}_{k}\\beta^{m,n}_{2k}\n\\end{equation}\nwhere $r^{m,n}_{k}\\in R$, and $\\beta^{m,n}_{2r}$ is defined by \n\\begin{equation}\n\\beta^{m,n}_{2r}(u^{(m+n-2q)})=\\delta_{2r,m+n-2q}E^{(r)}\nK^{-r}.\n\\end{equation}\nOur main symmetry result on induced forms is \\thmref{firstinvariance}:\n\\begin{thm}\nLet $M$ be the Verma module of highest weight $Tv^{-1}$ (so that $\\lambda=0$) and assume that $\\beta:_R\\mathcal F_m\\otimes _R\\mathcal F_n^{\\rho_1}\\to\n{_RF(\\mathbf U)}$  has the form \\eqnref{factorization}.\nIf\n$\\phi$ is a $\\ru$-invariant pairing on $M$ satisfying $s\\circ\n\\phi_{\\pi}\\circ L=\\phi\\circ (\\Psi\\otimes\n\\Psi)$, then \n\\begin{equation}\n\\chi _{\\beta,\\phi}^\\sharp = \\chi_{s\\beta ,\\phi}\\ .\n\\end{equation} \n\\end{thm}\n\nMost of the results in sections 1-7 are used in the proof of this theorem. In sections 8 and 9 we give a taste of how one can use induced forms to get information on filtrations of modules.  We plan to pursue this in future work.  \n\nLet $\\Pi=\\{\\alpha,\\beta\\}$ be the set of simple roots and $\\gamma\\in\\Pi$ for $ \\mathfrak g=\\mathfrak{sl}(3)$ or $\\mathfrak{sp}(4)$.  In the last section we give examples of how one can relate the Shapovalov form for $U_v(\\mathfrak g)$, to the Shapovalov form on a reductive subalgebra $U(\\mathfrak a)$ generated $E_\\alpha,F_\\alpha,K_\\gamma$, $\\gamma\\in\\Pi$.  In particular we explicitly describe the coefficients $r^{m,n}_{k}$ for particular $\\mathbf\\beta$ that appear in the study of these Shapovalov forms.  We will expand on this study in future work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{$q$-Calculus.}\n\\subsection{Definitions}\nAs many before us have done, for $m\\in\\mathbb Z$ we define\n\\begin{align*} \n[m]&:=\\frac{v^m-v^{-m}}{v-v^{-1}},  \\\\\n[m]_{(n)}&:=[m]\\cdot[m-1]\\cdots [m-n+1] \\\\\n[m]!&:=[m]_{(m)}, \\quad [0]!:=1  \\\\\n\\qbinom{m}{ n}&=\\begin{cases}\\frac{[m]_{(n)}}{[n]!}&\\quad \\text{for} \\quad n\\geq 0 \\\\ \n  0 &\\quad {\\rm if}\\quad  n<0 . \n\\end{cases}\n\\end{align*}\nFor $j\\geq 0$, Gauss' versions of the Binomial Theorem are \n\\begin{equation}\\label{gaussbinomial}\n\\prod_{l=1}^{j}(1-zv^{2(l-1)})=\\sum_{k=0}^{j}(-1)^k\\qbinom{j}{ k}v^{k(j-1)}z^k\n\\end{equation}\nand \n\\begin{equation}\\label{gaussbinomial2}\n\\prod_{l=1}^{j}(1-zv^{2(l-1)})^{-1}=\\sum_{k=0}^{\\infty}(-1)^k\\qbinom{-j}{ k}v^{k(j-1)}z^k\n\\end{equation}\nSee \\cite[1.157, 1.158]{MR96m:17032}.\nFor $r\\in \\mathbb Z$ define\n\\begin{gather}\\label{quantumbinom}\n[T;r]:=\\frac{v^{r}T- v^{-r}T^{-1}}{ v-v^{-1}},   \\\\ \n[T;r]_{(j)}:=[T;r][T;r-1]\\cdots [T;r-j+1],\n\\quad[T;r]_{(0)}:= [T;r]^{(0)}:=1,\n\\quad\\text{if}\\quad j>0,\\notag\\\\ \n[T;r]^{(j)}:=[T;r+1]\\cdots [T;r+j]\n\\notag \\\\ \\notag \\\\\n\\qbinom{T;r}{ j}:=\\begin{cases}[T;r]_{(j)}\/[j]!\n& \\text{if}\\quad j\\geq 0 \\\\\n0&  \\text{if}\\quad j<0.\\end{cases}\n\\end{gather}\nNote that $[T;\\lambda]^{(k)}$ is invertible in $R$ for $\\lambda\\geq\n0$ and $[T;\\lambda+1]_{(k)}$ is invertible provided\n$\\lambda+1\\geq  k$ or $\\lambda<0$ ($k\\geq 0$). On the other hand $[T;\\lambda+1]_{(k)}$ is divisible by $T-1$ and not by $(T-1)^2$ for $k>\\lambda+1$ due to\nthe fact that $[T,r]$ is divisible by exactly $(T-1)^i$ where $i=0$ if $r\\neq 0$ and $i=1$ for $r= 0$.  A useful related computation is \n\\begin{equation}\\label{matrixentry}\n[r]![T;r]_{(r)}^{-1}\n\\equiv \\left(1+\\left(r-\\frac{2}{\\{1\\}}\n\\sum_{k=1}^r\\frac{v^{k}}{[k]}\\right)(T-1)\\right)\\mod (T-1)^2\n\\end{equation}\nwhere $\\{1\\}=v-v^{-1}$.  \nIndeed \n\\begin{align*}\n[T;k]^{-1}\n&=T[k]^{-1}\\left(\n\\sum_{i\\geq 0}\\left(\\frac{v^{k}}{1-v^k}\\right)^i(T-1)^i\\right)\\left(\n\\sum_{j\\geq 0}(-1)^j\\left(\\frac{v^{k}}{1+v^k}\\right)^j (T-1)^j\\right)\n\\end{align*}\nThis implies \\eqnref{matrixentry} above.\n\n\\subsection{Identities}\nTwo useful formulae for us will be\n\n\\begin{equation}\\label{Ma1}\n\\qbinom{s-u}{ r}=\\sum_p\n(-1)^pv^{\\pm (p(s-u-r+1)+ru)}\\qbinom{u}{ p}\\qbinom{s-p}{\nr-p}\n\\end{equation}\n\\begin{equation}\\label{Ma2}\n\\qbinom{u+v+r-1}{ r}=\\sum_p\nv^{\\pm (p(u+v)-ru)}\\qbinom{u+p-1}{ p}\\qbinom{v+r-p-1}{r-p}\n\\end{equation}\nwhich come from \\cite[1.160a, 1.161a ]{MR96m:17032}, respectively.\n\nWe have a yet another variant of the Binomial Theorem:\n\n\\begin{lem}\\label{thirdbinomial}\n\\begin{align*}\n v^{k(n+1-k)} [T;2k-n-1]_{(k)}&=\\sum_{j=0}^k (-1)^jv^{j(n-2k+1)}\n                T^{-j}\\qbinom{k}{j} [n-k+j]_{(j)} [T;k]_{(k-j)}\\ \n\\end{align*}\n\\end{lem}\n\\begin{proof}  \n\nThe  proof follows from an application of Gauss' binomial theorem and \\eqnref{Ma1}\n\\end{proof}\n\n\n\\begin{lem}[Chu-Vandermonde formula]\\label{chuvandermonde}  For integers $k$ and $r$ with $0\\leq k\\leq r$ we have\n\\begin{equation}\n\\sum_{l=0}^{k}(-1)^{l}\n  v^{  l (r -k+1) }\n    \\left[\\begin{matrix}T;k \\\\ l \\end{matrix}\\right]\n    \\left[\\begin{matrix}T;r+k-l \\\\ k-l \\end{matrix}\\right]=v^{-k^2}T^{-k}\\qbinom{r}{k}\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}  \nThe proof is obtained by using the Taylor series expansion in $T$ together with Gauss' binomial theorem.\n \\end{proof}\n\n\\section{${{\\mathbf U}}$-algebra Automorphisms and Intertwining Maps.} \n\n\\subsection{} Following Lusztig, \\cite[Chapter 5]{MR94m:17016}, we let\n$\\mathcal C'$ denote the category whose objects are $\\mathbb Z$- graded\n$\\mathbf U$-modules $M=\\oplus _{n\\in\\mathbb Z}M^n$ such that \n\n\\begin{enumerate}[(i)]\n\\item  $E,F$ act locally nilpotently on $M$,\n\\item  $Km=v^nm$ for all $m\\in M^n$.\n\\end{enumerate}\n\n\nFix $e=\\pm 1$ and let $M\\in \\mathcal C'$.  Define Lusztig's automorphisms\n$T_{e}',T_{e}'':M\\to M$ by\n\\begin{equation}\\label{defofT}\nT_{e}'(m):=\\sum_{a,b,c;a-b+c=n}(-1)^bv^{e(-ac+b)}F^{(a)}E^{(b)}F^{(c)}m, \n\\end{equation}\n and\n\n\\begin{equation}\nT_{e}''(m):=\\sum_{a,b,c;-a+b-c=n}(-1)^bv^{e(-ac+b)}E^{(a)}F^{(b)}E^{(c)}m\n\\end{equation}\nfor $m\\in M^n$.  In the above $E^{(a)}:=E^a\/[a]!$ is the $a$th {\\it\ndivided power of $E$}.\n\nLusztig defined automorphisms $T_{e}'$ and $T_{e}''$ on $\\mathbf U$ by\n$$\nT'_e(E^{(p)})=(-1)^pv^{ep(p-1)}K^{ep}F^{(p)},\\quad \\quad\nT'_e(F^{(p)})=(-1)^pv^{-ep(p-1)}E^{(p)}K^{-ep}\n$$\nand\n$$\nT''_{-e}(E^{(p)})=(-1)^pv^{ep(p-1)}F^{(p)}K^{-ep},\\quad \\quad\nT''_{-e}(F^{(p)})=(-1)^pv^{-ep(p-1)}K^{ep}E^{(p)}.\n$$\n\nOne can check on generators that \n\\begin{equation}\\label{Tandrho}\n\\rho_1\\circ T_{-1}'=T_{-1}'\\circ\n\\rho_1.\n\\end{equation}\n  \nIf\n$M$ is in\n$\\mathcal C'$,\n$x\\in \\mathbf U$ and $m\\in M$, then we have\n\n\\begin{equation}\n\\Theta (x\\cdot m) =\\Theta(x) \\Theta m \n\\end{equation}\nfor $\\Theta=T_{e}'$ or $\\Theta=T_{e}''$ (see \\cite[37.1.2]{MR94m:17016}). The last identity can be interpreted to say that\n$\\Theta$ and\n$\\Theta\\otimes s$ are intertwining maps;\n\\begin{equation}\n\\Theta:M \\rightarrow M^\\Theta\\quad \\quad \\quad \\quad\n\\Theta\\otimes s\\ :{_R}M\\rightarrow {_R}M^{\\Theta\\otimes s}.\n\\end{equation}\nTo simplify notation we shall sometimes write $s\\Theta$\nin place of $\\Theta\\otimes s$. \\smallskip\n\nWe now describe the explicit action of $\\Theta$ on $M$.\n\\begin{lem}[{\\cite[Prop. 5.2.2]{MR94m:17016}}]\\label{symmetries} Let\n$m\\geq 0$ and\n$j,h\\in [0,m]$ be such that $j+h=m$.\n\\begin{enumerate}[(a)]\n\\item If $\\eta\\in M^m$ is such that $E\\eta=0$, then\n$T_{e}'(F^{(j)}\\eta)=(-1)^jv^{e(jh+j)}F^{(h)}\\eta$.\n\\item If $\\zeta\\in M^{-m}$ is such that $F\\zeta=0$, then\n$T_{e}''(E^{(j)}\\zeta) =(-1)^jv^{e(jh+j)}E^{(h)}\\zeta$.\n\\end{enumerate}\n\\end{lem}\n\n\nLet $F(\\mathbf U)$ denote the ad-locally finite submodule of $\\mathbf U$.  We know\nfrom \\cite{MR95e:17017} that $F(\\mathbf U)$ is tensor\nproduct of harmonic elements $\\mathcal H$ and the center $Z(U)$.  Here\n$\\mathcal H=\\oplus_{m\\in\\mathbb Z} \\mathcal H_{2m}$ and $\\mathcal\nH_{2m}=\\text{ad}\\, \\mathbf U (EK^{-1})$.\n\n \n \n There is another category that\nwe will need and it is defined as follows: Let\n$M$ be a\n$\\ru$-module.  One says that\n$M$ is {\\it\n$_R\\mathbf U^0$-semisimple} if $M$ is the direct sum of $R$-modules\n$M^{\\mu}$ where $K$ acts by $T \\ v^\\mu$, $\\mu\\in\\mathbb Z$; i.e. by\nweight $m+\\mu$.  Then $\\mathcal C_{R}$ denotes the category of $\\ru$-modules\n$M$ for which $F$ acts locally nilpotently and  $M$ is\n$_R\\mathbf U^0$-semisimple.\n\nFor $M$ and $N$ two objects in $\\mathcal C'$ or one of them is in \n$_R\\mathcal C$,\nLusztig defined\nthe linear map\n$L:M\\otimes N\\to M\\otimes N$ given by\n\\begin{equation}\\label{L}\nL(x\\otimes y)=\\sum_n(-1)^nv^{-n(n-1)\/2}\\{n\\}F^{(n)}x\\otimes E^{(n)}y\n\\end{equation} \nwhere $\\{n\\}:=\\prod_{a=1}^n(v^a-v^{-a})$ and $\\{0\\}:=1$. \nOne can show\n$$ \nL^{-1}(x\\otimes y)=\\sum_nv^{n(n-1)\/2}\\{n\\}F^{(n)}x\\otimes E^{(n)}y.\n$$ \n\n\n\n\\begin{lem}[\\cite{MR94m:17016}]\\label{LLemma1}\nLet $M$ and $N$ be two objects in $\\mathcal C'$.  Then\n$ T_{1}'' L(z)=(T_{1}''\\otimes T_{1}'')(z)$ for all $z\\in\nM\\otimes N$. \n\\end{lem}\n\n\n\n\n\\begin{lem}\\label{LLemma2}\nLet $M$ be a module in $\\mathcal C_R$ and $N$ a module in\n$\\mathcal C'$.  Then for $x\\in M^t$ and $y\\in N^s$ we have\n\\begin{align*}\nFL(x\\otimes y) &=L(x\\otimes Fy+v^sFx\\otimes y) \\\\\nEL(x\\otimes y) &=L(Ex\\otimes y+v^{-t}T^{-1}x\\otimes Ey)\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}  Let $\\bar{}: k(v)\\to k(v)$ be the $\\mathbb Q$-algebra\nisomorphism given by sending $v$ to $v^{-1}$. More over let\n$\\bar {}:\\mathbf U\\to\n\\mathbf U$ denote the unique\n$\\mathbb Q$-algebra homomorphism defined by \n$$\n\\bar E=E, \\quad \\bar F=F,\\quad \\bar K_\\mu=K_{-\\mu},\\quad \\overline{fu}=\\bar f\\bar\nu,\\quad f\\in k(v),\\,u\\in\\mathbf U.\n$$\nFrom\n\\cite[Theorem 4.1.2]{MR94m:17016}, we have\n$\\Delta(u)L=L\\overline{\\Delta(\\bar u)}$ for any $u\\in \\mathbf U$.  Let\n$x\\in M^t$ and\n$y\\in\n\\mathcal E^s$.  In particular we deduce\n\\begin{align*}\nFL(x\\otimes y) &=L\\overline{\\Delta(F)}(x\\otimes y)=L(v^sFx\\otimes y+x\\otimes Fy).\n\\end{align*}\nSimilarly\n$EL(x\\otimes y) \n  =L (Ex\\otimes y+v^{-t}T^{-1}x\\otimes Ey)$.\n\\end{proof}\n\n\\begin{cor}\\label{Lisomorphism}\nLet $M$ be a module in $\\mathcal C_R$ and $\\mathcal E$ a module in\n$\\mathcal C'$.  Then $L$ defines an isomorphism of\nthe $\\mathbf U$-module\n$M^{T_{-1}'}\\otimes \\mathcal E^{T_{-1}'}$ onto $(M\\otimes\\mathcal\nE)^{T_{-1}'}$.\n\\end{cor}\n\n\n\\begin{proof}  The proof follows from a direct calculation on weight vectors in $M$ and $\\mathcal E$.\n\\end{proof} \n\n\\noindent{\\bf Remark:}  It seems that one should be able to prove\n\\corref{Lisomorphism} through the use of \\lemref{LLemma1}, however one\nmust take into account that\n$T_e'$ may not be defined on $M$. \n\nSet \n\\begin{equation}\\label{mathcalLinv}\\index{mathcalLinv}\n\\mathcal L^{-1}=\\sum_p(-1)^pv^{3\\frac{p(p-1)}{\n2}}\\{p\\}E^{(p)}K^pF^{(p)},\n\\end{equation}\nand\n\\begin{align}\\label{mathcalL}\n\\mathcal L\n&=\\sum_p\n    v^{-3\\frac{p(p-1)}{2}}\\{p\\}E^{(p)}K^{-p}F^{(p)}.\n\\end{align}\nand note that $\\mathcal L$ and $\\mathcal L^{-1}$ are well\ndefined operators on lowest weight modules.  These maps are similar to some operators defined by Kashiwara.  See for example the \n$\\Pi_t$ in \\cite[16.1]{MR94m:17016}.\n\n\\begin{lem} Suppose that $M$ and $N$ be highest weight\nmodules with $\\psi_M:M^{sT'_{-1}}_\\pi\\to M$, $\\psi_N:N^{sT'_{-1}}_\\pi\\to\nN$ are $\\ru$-module homomorphisms and \n$\\phi$ a $\\rho$-invariant form on $M\\times N$.  Then\n\\begin{equation}\\label{Ll}\n\\phi\\circ(\\psi_M\\otimes \\psi_N)\\circ L^{-1}=\\phi\\circ\n(\\psi_M\\otimes \\psi_N) \\circ (\\mathcal  L^{-1}\\otimes 1).\n\\end{equation}\n\n\n\\end{lem}\n\n\\begin{proof} Let $m\\in M_{\\pi}$, $n\\in N_{\\pi}^{\\rho_1}$ and let $*$\ndenote the action on  $N_{\\pi}^{\\rho_1}$ so that $x*n=\\rho_1(x)n$ where\njuxtaposition means the action on $N_\\pi$. Recall\n$L^{-1}$ is defined on $M_\\pi\\otimes_R N_\\pi^{\\rho_1}$, but one can view\nit as an $\\ru$-module homomorphism from $ (M_\\pi\\otimes_R\nN_\\pi^{\\rho_1})^{sT_{-1}'}$ to $ M_\\pi^{sT_{-1}'}\\otimes_R\n(N_\\pi^{sT_{-1}'})^{\\rho_1}$.  Hence the left hand side of \\eqnref{Ll}\nis in $\\text {\\rm Hom}\\,_\\ru((M_\\pi\\otimes_R N_\\pi^{\\rho_1})^{sT'_{-1}},R)$.\nNow\n\\begin{align*}\n\\phi\\circ(\\psi_M\\otimes \\psi_N)\\circ L^{-1}&(m\\otimes\nn)=\\sum_{p\\geq 0}v^{\\frac{p(p-1)}{2}}\\{p\\}\n               \\phi(\\psi_M(F^{(p)}m), \\psi_N(E^{(p)}*n)) \\\\\n&=\\sum_{p\\geq 0}v^{\\frac{p(p-1)}{2}}\\{p\\}\n               \\phi(\\rho(T''_1\\rho_1\n                E^{(p)})\\psi_N(F^{(p)}m),\\psi(n))\\\\\n&=\\sum_{p\\geq 0}v^{\\frac{p(p-1)}{2}}\\{p\\}\n               \\phi(\\psi_M(T'_{-1}\\rho T''_1\\rho_1\n                (E^{(p)})F^{(p)}m),\\psi_N(n))\\\\\n&=\\phi\\left(\\psi_M\\left(\n             \\mathcal  L^{-1}m\\right),\\psi_N(n)\\right)\\\\\n\\end{align*}\nsince\n$T'_{-1}\\rho T''_1\\rho_1( E^{(p)})\n=(-1)^pv^{p(p-1)}E^{(p)}K^p$.\n\\end{proof}\nA similar argument shows that if that $\\mathcal F$ and $\\mathcal E$ are finite dimensional $\\mathbf U$-modules\nwith $T''_1:\\mathcal F^{T'_{-1}} \\to \\mathcal F$, $T''_1:\\mathcal E^{T'_{-1}}\\to\n\\mathcal E$, $\\mathbf U$-module homomorphisms and \n$\\phi$ a $\\rho$-invariant form on $\\mathcal F\\times \\mathcal E$, then\n\\begin{equation}\\label{L2}\n\\phi\\circ(T''_1\\otimes T''_1)\\circ L^{-1}=\\phi\\circ\n(T''_1\\otimes T''_1) \\circ (\\mathcal  L^{-1}\\otimes 1).\n\\end{equation}   \nObserve that $\\rho(\\mathcal  L^{-1})=\\mathcal  L ^{-1}$.\n\nOn the other hand\n\\begin{equation}\\label{L3}\n\\phi\\circ L=\\phi\\circ\n(\\mathcal  L\\otimes 1).\n\\end{equation}   \nIndeed\n\\begin{align*}\n\\phi\\circ L(m\\otimes n)&=\\sum_{p\\geq 0}(-1)^pv^{\\frac{-p(p-1)}{2}}\\{p\\}\n               \\phi(F^{(p)}m, E^{(p)}*n) \\\\\n&=\\sum_{p\\geq 0}(-1)^pv^{\\frac{-p(p-1)}{2}}\\{p\\}\\phi(\\rho(\\rho_1 E^{(p)})(F^{(p)}m),n)\\\\\n&=\\sum_{p\\geq 0}v^{\\frac{-3p(p-1)}{2}}\\{p\\}\\phi( E^{(p)}K^{-p}F^{(p)}m),n).\n\\end{align*}\nSimilarly we have\n$\\rho(\\mathcal L)=\\mathcal L$.  Observe that $\\mathcal L=\\overline{\\mathcal L^{-1}}$ where $\\bar{\\enspace}:\\mathbf U \\to \\mathbf U$ is the automorphism of $\\mathbb\nQ$-algebras defined by\n$\\overline{E}=E$, $\\overline{F}=F$,\n$\\overline{K}=K^{-1}$ and $\\overline{fx}=\\overline{f}\\overline{x}$ for $f\\in k(v)$, $x\\in\\mathbf U$.  Here $\\overline{v}=v^{-1}$ with\n$\\bar{\\enspace}$ fixing $\\mathbb Q$.  See \\cite[3.1.12]{MR94m:17016} for more details. \n\nMoreover we can prove that $\\mathcal  L^{-1}$ is an\nintertwining map i.e. $\\mathcal  L^{-1}:M^{T_1'}_\\pi\\to\nM^{T_{-1}'}_\\pi$ is a module  homomorphism for $M$ a highest weight module:\n\n\\begin{lem}\\label{TandL} $T'_{-1}(u)\\mathcal  L^{-1}=\\mathcal  L^{-1}T_1'(u)$ as\noperators on $M_\\pi$ for all $u\\in{_R\\mathbf U}$.  From this we also get\n$T'_1( u)\\mathcal  L=\\mathcal  LT_{-1}'( u)$ for all $u$.\n\\end{lem}\n\\begin{proof}  The proof follows from a direct calculation on generators of $\\mathbf U$.\n\\end{proof}\n\n\n\n\n\n\n\\section{Invariant Forms and Liftings} \n\n\\subsection{} Elements of $\\mathbb P(M,N)$ are called {\\it invariant pairings} and for $M=N\n$, set\n$\\mathbb P(M) =\\mathbb P(M,M)$ and call the elements {\\it invariant forms} on\n$M$.\n\n\\begin{lem}  \\begin{enumerate}\n\\item For $\\ru$-modules $M,N$ and $\\mathcal F$, \n\\begin{equation}\n\\mathbb P(M,N,\\mathcal F)\\cong \\hom_\\ru (M\\otimes_R N^{\\rho_1},{_R\\mathcal\nF}^\\brho)\n\\end{equation}\n\\item If $\\mathcal F=R$ is the trivial $\\ru$-module, then the condition $\\phi\\in\\mathbb P(M,N)$, is equivalent to\n\\begin{equation}\n\\phi(u\\,a,b)=\\phi(a,\\rho(u)b)\n\\end{equation}\nfor all $a\\in M$, $b\\in N$ and $u\\in\\mathbf U$.\n\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof} The first statement follows from \\cite[3.10.6]{MR96m:17029} and the second follows from a computation using the\ngenerators of $\\mathbf U$.\n\\end{proof}\n\n\\begin{lem}\nLet $M$ and $N$ be finite dimensional $\\mathbf U$-modules in\n$\\mathcal C'$ and   $\\phi$ an invariant pairing. Then, for $m\\in {_RM},n\\in {_R N}$, $\n\\phi(T_{e}'' m,T_{-e}'' n)=\\phi(m,n)$.\n\\end{lem}\n\n\\begin{proof} We may assume $M$ and $N$ are irreducible with highest weight\n$\\nu$. Using the basis in (2.1), \n$h+j=\\nu$, and the invariance of $\\phi$ we get\n\\begin{align*}\\phi(T_{e}'' F^{(j)}\\eta,T_{-e}'' F^{(j)}\\eta)&=\n\\phi(F^{(h)}\\eta,F^{(h)}\\eta)\n       \n        =\\phi(F^{(j)}\\eta,F^{(j)}\\eta)\n\\end{align*} where we have used \\cite[Cor.\n3.1.9]{MR94m:17016}, in the third line.\n\\end{proof}\n\nNote that if $\\phi(\\eta,\\eta)=1$, then for $0\\leq j\\leq \\nu$, the calculation required for the\nproof above shows that \n\\begin{equation}\\label{normalizedforms}\n\\phi(F^{(j)}\\eta,F^{(j)}\\eta)=v^{j^2-\\nu j}{\\qbinom \\nu j}\n\\end{equation}\n\n\n\nFor the proof of some future results we must be explicit about the \ndefinition of the \"$R$-matrix\" $_f\\mathcal R:A\\otimes\n\\mathcal E\\to \\mathcal E\\otimes A$:   Recall a\n$\\mathbf U$-module\n$M$ is said to be {\\it integrable} if for any $m\\in M$ and all $i\\in I$, there exists a\npositive integer\n$N$ such that $E^{(n)}_im=0=F^{(n)}_im$ for all $n\\geq N$ , and $M=\\oplus\n_{\\lambda\\in X}M^\\lambda$ where for any\n$\\mu\\in Y, \\lambda\\in X$ and $m\\in M^\\lambda$ one has\n$K_\\mu m=v^{\\angles{\\mu ,\\lambda}}m$. Let $R$ denote a commutative algebra over the ring\n$\\mathbb Q(v)[T^{\\pm 1}]$ (such as in the introduction) and let\n$f:X\\times X\\to R$ be a function such that  \n\\begin{align*}\nf(\\zeta,\\zeta'\\pm i')&=f(\\zeta,\\zeta')v^{ \\mp \\angles{i,\\zeta}(i\\cdot i\/2)}  \\\\\nf(\\zeta\\pm i',\\zeta')&=f(\\zeta,\\zeta')v^{ \\mp \\angles{i,\\zeta'}(i\\cdot i\/2)}T^{ \\mp(i\\cdot i)\/2}\n\\end{align*}\nfor all $\\zeta,\\zeta'\\in X$ and all $i\\in I$ (see [L, 32.1.3] or\n[Ja, 3.15]). \n\nSuch a function $f$ exists: Let $H$ denote a set of coset representatives of $X\/\\mathbb Z[I]$,  let $c:H\\times H\\to \\mathbb Z$ denote an arbitrary function and set \n$$\nf(h+\\nu,h'+\\nu'):=v^{c(h,h')-\\sum_i\\nu_i\\langle i,h'\\rangle (i\\cdot i)\/2-\\sum_i\\nu_i'\\langle i,h\\rangle (i\\cdot i)\/2-\\nu\\cdot \\nu'}T^{-\\sum \\nu_i(i\\cdot i)\/2}\n$$\nfor $h,h'\\in H$ and $\\nu,\\nu'\\in \\mathbb Z[I]$.  \n\n\n\n\\begin{thm}{[L, 32.1.5], or [Ja, 3.14]} If $\\mathcal E$ is an\nintegrable\n$\\ru $ module and $A\\in \\mathcal C_{R}$, then for each $f$ satisfying (3.1.1), there\nexists an isomorphism $_f\\mathcal R:A\\otimes\n\\mathcal E\\to \\mathcal E\\otimes A$. \\end{thm}\n\n\nThe map $\\tau:A\\otimes B\\to B\\otimes A$ for any two modules $A$ and $B$ denotes the\ntwist map\n$\\tau(a\\otimes b)=b\\otimes a$.   Define $\\prod _f:\\in\\text {\\rm End}\\,_R({_R\\mathcal E}\\otimes{_R\\mathcal F}\\otimes_R M)$ by \n$\\prod_f(e\\otimes e'\\otimes  m)= f(\\lambda,\\lambda')e\\otimes e'\n\\otimes  m$ for $m\\in M^{\\lambda'}$ and\n$e\\otimes e'\\in (\\mathcal E\\otimes{_R\\mathcal F})^{\\lambda}$.  Lastly we define\n$\\chi\\in\n\\text {\\rm End}\\,_R({_R\\mathcal E\\otimes{_R\\mathcal F}}\\otimes_R M)$ by\n$$\n\\chi(e\\otimes e'\\otimes m)=\n\\sum_\\nu\\sum_{b,b'\\in\n\\mathbf B_\\nu}p_{b,b'}b^-(e\\otimes e')\\otimes {b'}^+m\n$$ \nwhere $p_{b,b'}=p_{b',b}\\in R$, and $\\mathbf B_\\nu$ is a subset of\n$\\mathfrak f$.   Then $_f\\mathcal R$ is\ndefined to be equal to $\\chi\\circ \\prod_f\\circ \\tau$.  The proof that it is an\n$\\mathbf U$-module homomorphism is almost exactly the same as in [L, 32.1.5] or [Ja,\n3.14], which the exception that one must take into account that $M$ is in the\ncategory\n$\\mathcal C_R$ instead of $_R\\mathcal C'$. \n\n\n\n(The above formula for 6.2.1 corrects\nthe corresponding 6.2.1 in the earlier version of [CE]. ) \n\n\n\n In the case that $I=\\{i\\}$ and $X=Y=\\mathbf Z$ with $i=1\\in Y$, $i'=2\\in X$, the linear map $L$ (see \\eqnref{L}) coincides with $\\chi$ (Lusztig uses the notation $\\Theta$ where we use $\\chi$ - see \\cite[4.1.4]{MR94m:17016}).  Moreover since $X=\\mathbb Z\\supset 2\\mathbb Z=\\mathbf Z[i]$ there are only two cosets in $X\/\\mathbb Z[i]$.   In this case the function $f$ above can be defined by\n\\begin{equation}\\label{fexample}\n f(h+\\nu,h'+\\nu'):=v^{-\\sum_i\\nu_i\\langle i,h'\\rangle (i\\cdot i)\/2-\\sum_i\\nu_i'\\langle i,h\\rangle (i\\cdot i)\/2-\\nu\\cdot \\nu'}T^{-\\sum \\nu_i(i\\cdot i)\/2}\n\\end{equation}\nfor all $h,h'=0,1$ and $\\nu,\\nu'\\in 2\\mathbb Z$.  So for example $_f\\mathcal R^{-1}$ is given by \n\\begin{align}\\label{commutativity} \n_f\\mathcal R^{-1}(e\\otimes m)&= \\mathbf s\\circ \\Pi_f^{-1}\\circ L^{-1}(e\\otimes m)  \\\\\n&=\\sum_{n\\geq 0} v^{n(n-1)\/2}\\{n\\} \\mathbf s\\circ \\Pi_f^{-1}F^{(n)}e\\otimes E^{(n)}m \\notag \\\\\n&=\\sum_{n\\geq 0} f(\\lambda-2n,\\lambda'+2n)^{-1}v^{n(n-1)\/2}\\{n\\}E^{(n)}m\\otimes F^{(n)}e\\notag \\\\\n&=f(\\lambda,\\lambda')^{-1}\\sum_{n\\geq 0} v^{n(\\lambda-\\lambda')-2n^2+n(n-1)\/2}T^{-n}\\{n\\}E^{(n)}m\\otimes  F^{(n)}e\\notag\n\\end{align}\nfor $e\\in \\mathcal E^{\\lambda}$ and $m\\in M^{\\lambda'}$.\n\n\nLet $\\mathcal E$ and $\\mathcal F$ be finite dimensional $\\mathbf U$-modules and \n$\\tau: _R\\mathcal E\\otimes _R\\mathcal F^{\\rho_1}  \\rightarrow \\mathbf U$, a\n$\\mathbf U$-module homomorphism into $\\mathbf U$, where $\\mathbf U$ is a module under\nthe adjoint action.\nSuppose $\\phi$\nis a pairing of $M$ and $N$.  Define $\\psi_{\\tau,\\phi}$ to be  the\ninvariant pairing of $ M\\otimes _R\\mathcal E$ and $N\\otimes _R\\mathcal F$ defined\nby\nthe formula, for $e\\in \\mathcal E ,f\\in  \\mathcal F,m\\in M,$ and $n\\in N$, \n\\begin{equation}\\label{inducedpairing}\n\\psi_{\\tau,\\phi} (m\\otimes e,n \\otimes f)=\n\\phi( m,\\tau(e\\otimes f)*n)\\ ,  \n\\end{equation}\nHere  ${\\mathcal F}^{\\rho_1}$ is a twist of the representation\n$\\mathcal F$ by $\\rho_1$.  We call the pairing $\\psi_{\\tau,\\phi}$ the\n{\\it pairing induced} by\n$\\tau$ and $\\phi$. In the cases when $M,N$ and $\\phi$ are fixed we write\n$\\psi_\\tau$ in place of $\\psi_{\\tau,\\phi}$ and say this pairing is induced\nby $\\tau$.\n\n\n\n\n\\subsection{} \nA result from \\cite{MR96c:17022} shows that in the setting of Verma\nmodules the collection of maps $\\tau$ is a natural set of parameters for\ninvariant forms. \n\n\\begin{prop}\nSuppose $\\mathbf U$ is of finite type and $\\mathcal E$ and\n$\\mathcal F$ are finite dimensional $\\mathbf U$-modules. Let  $M$ be an\n$_R\\mathbf U$-Verma module  and $\\phi$ the Shapovalov form on $M$.\nThen\nevery invariant pairing of\n$M\\otimes _R\\mathcal E$ and $M\\otimes _R\\mathcal F$ is induced by $\\phi$.\n\\end{prop}\n\n\n\n\n\\subsection{}\n\n\n\n\\begin{thm}[Lifting Theorem] \nLet $A$ and\n$B$ be modules in $\\mathcal C_{R}$ and $\\phi\\in {\\mathbb P}_{\\brho }(A,B)$. Then\n$\\phi$ uniquely determines an invariant form\n$\\phi_F\\in  {\\mathbb P}_{\\brho}(A_F,B_F)$ which is determined by the following\nproperties:\n\\begin{enumerate}\n\\item $\\phi_F$ vanishes on the subspaces $\\iota A\\times B_F$ and\n$A_F\\times \\iota B$ .\n\\item For each $\\mu \\in \\mathbb Z$ with $\\mu+1= r\\in \\mathbb N$, and any vectors $a\\in A$ and $b\\in B$ both of weight $m+\\epsilon\\mu$ with\n$\\epsilon\\in\n\\{1,s\\}$ and  $E\\ a=E\\ b=0$ , \n\\begin{equation}\\label{lift}\n\\ \\phi_F(F^{-1}a ,F^{-1} b)=  v^{-r+1}\\frac{\\iota \\epsilon[T;0]}{  \\iota\n\\epsilon[T;r-1]\\ }\\ \\\n\\phi(a,b).\n\\end{equation}\n\\end{enumerate}\n\\end{thm}\n\n\n\\begin{prop}\\label{invariance}  The form\n$\\phi_F$ induces an $\\brho$-invariant bilinear map on\n$A_\\pi \\times B_\\pi$ which we denote by\n$\\phi_\\pi$.\n\\end{prop}        \n\n\\subsection{} At times the subscript notation for lifted forms will be\ninconvenient and so we shall also use the symbol $loc$ for the localization\nof both forms and modules. We write $loc(\\phi)$ and $loc(A)$ in place of\n$\\phi_F$ and $A_F$. \n\nFor invariant forms we find that induction and localization commute in the\nfollowing sense.\n\\begin{prop}[{\\cite[Prop 7.5]{MR96c:17022}}]  Suppose $A$ and $B$ are objects in\n$\\mathcal C_{R}$. Then $\\phi_F$ and $\\phi_\\pi$ are $\\ru$-invariant; i.e.        $\\phi_F\\  \\in {\\mathbb\nP}_\\brho(A_F,B_F)\\  ,\\\n\\phi_\\pi\\in {\\mathbb P}_\\brho(A_\\pi,B_\\pi)\\ .$\n\\end{prop}\n\n\\section{Quantum Clebsch-Gordan decomposition}\\label{QCG decomposition}\n\n\n\\subsection{Bases and Symmetries} For\n$m\\in\\mathbb Z$, let\n$\\mathcal F_m$\\label{fm} denote the finite dimensional irreducible module of\nhighest weight\n$v^m$ with highest weight vector $u^{(m)}$.  For $k$ any non-negative\ninteger set\n$u^{(m)}_k=F^{(k)}u^{(m)}$ and\n$u^{(m)}_{-1}=0$.\n\nIn particular \n$$\nT_1''(u^{(m)}_j)=(-1)^{m-j}v^{(m-j)(j+1)}u^{(m)}_{m-j}\n$$\nand\n\\begin{equation}\\label{irred}\nK^pu^{(m)}_j=v^{p(m-2j)}u^{(m)}_j,\\enspace  F^{(p)}u^{(m)}_j=\n{\\qbinom {p+j} j}u^{(m)}_{j+p},\\enspace\n E^{(p)}u^{(m)}_j=\\qbinom{m+p-j}{\np} u^{(m)}_{j-p}.\n\\end{equation}\n\n\n\\begin{lem}[Clebsch-Gordan, \\cite{MR96e:17041},\n\\cite{MR90m:17022}] \\label{CG} For any two non-negative integers\n$m$ and $n$, there is an isomorphism of\n$\\mathbf U$-modules\n$$\n\\mathcal F_{m+n}\\oplus \\mathcal\nF_{m+n-2}\\oplus \\cdots \\oplus \\mathcal F_{|m-n|}\\cong\\mathcal F_m\\otimes\n\\mathcal F_n.\n$$\nMoreover the isomorphism may be defined on highest weight vectors by\n\\begin{equation}\n\\varPhi(u^{(m+n-2p)})=\\sum_{k=0}^{p}(-1)^kv^{(k-p)(m-p-k+1)}\\frac{[n-p+k]_{(k)}}{[m]_{(k)}}u_k^{(m)}\\otimes u_{p-k}^{(n)}\n\\end{equation}\nwhere $0\\leq p\\leq \\min\\{m,n\\}$.\n\\end{lem}\n\n\\begin{proof}\nFirst note $|m-n|\\leq m+n-2p\\leq m+n$ so that $0\\leq p\\leq\\min\\{m,n\\}$.\n The element $\\varPhi(u^{(m+n-2p)})$ certainly has the right weight and it is straightforward to check\n that $E\\varPhi(u^{(m+n-2p)})=0$.\n\\end{proof}\n\n\\begin{cor} For any two non-negative integers $m$, $n$, and $p\\leq\\min\\{m,n\\}$,\n\\begin{align*}\n\\varPhi(u^{(m+n-2p)}_{m+n-2p})\n&=v^{p(n- m)}\\sum_{k=0}^{p}(-1)^{p+k}v^{-k(n+k-2p+1)}\\frac{[n+k-p]_{(k)}}{[m]_{(k)}}\n  u_{m-k}^{(m)}\\otimes u_{n+k-p}^{(n)}.\n\\end{align*}\n\n\\end{cor}\n\n\\begin{lem}  The map $\\varphi:\\mathcal F_m^{\\rho_1}\\to \\mathcal F_m$ given\nby $\\varphi(u^{(m)}_k)=(-v)^{-k}u^{(m)}_{m-k}$ is an isomorphism.\n\\end{lem}\n\\begin{proof} This is a straightforward calculation on weight vectors and generators of $\\mathbf U$. \n\n\\end{proof}\n\n\n\\begin{cor}\\label{CG2} Let\n$m$ and $n$ be two non-negative integers. Then there is an isomorphism of\n$\\mathbf U$-modules\n$$\n\\mathcal F_{m+n}\\oplus \\mathcal\nF_{m+n-2}\\oplus \\cdots \\oplus \\mathcal F_{|m-n|}\\cong\\mathcal F_m\\otimes\n\\mathcal F_n^{\\rho_1}.\n$$\nMoreover for $m\\geq n$, this isomorphism can defined on highest weight vectors by\n\\begin{align*}\n\\Phi(u^{(m+n-2p)})&=\\sum_{k=0}^{p}(-1)^{n-p}\\frac{[n-p+k]![m-k]!}{[n-p]![m]!}\nv^{(k-p)( 2 + m)+p^2 -k^2  + n}u_k^{(m)}\\otimes\nu_{n-p+k}^{(n)}\n\\end{align*}\n\n\n(the action on the second factor $u^{(n)}_l$ is twisted by the\nautomorphism $\\rho_1$).\n\\end{cor}\n\n\\begin{proof}  This follows from the isomorphism $\\phi:\\mathcal\nF_n^{\\rho_1}\\to\\mathcal F_n$ which sends\n$u^{(n)}_{n+k-p}$ to $(-v)^{p-n-k}u^{(n)}_{p-k}$.\n\\end{proof}\n\nSince we have two basis $\\{u_i^{(m)}\\otimes u_j^{(n)}|\\,0\\leq i\\leq\nm,\\enspace 0\\leq j\\leq n\\}$, and $\\{u_k^{(m+n-2p)}|\\,0\\leq p\\leq\nn,\\enspace 0\\leq k\\leq m+n-2p\\}$ of $\\mathcal F_m\\otimes \\mathcal\nF_n$, we can relate them by the {\\it quantum Clebsch-Gordan\ncoefficients} or {\\it quantum $3j$-symbols};\n$$\nu_k^{(m+n-2p)}=\n\\sum_{0\\leq i\\leq m,0\\leq j\\leq n,i+j=p+k}\n\\left[\\begin{matrix} m & n & m+n-2p\n\\\\ i & j & k \\end{matrix}\\right]u_i^{(m)}\\otimes u_j^{(n)}\n$$\n\nConsider now the $\\rho$-invariant forms \\eqnref{normalizedforms} on\n$\\mathcal F_m$ and\n$\\mathcal F_n$, both denoted by $(,)$, normalized so that their\nhighest weight vectors have norm $1$.  Define the symmetric invariant\nbilinear form on $\\langle,\\rangle$ on $\\mathcal F_m\\otimes \\mathcal\nF_n$ given by the tensor product of the two forms (the resulting\npairing is $\\rho$-invariant). Assume that the forms on $\\mathcal F_m$ and $\\mathcal F_n$ are normalized so that their highest weight vectors\n$u^{(m)}$ and $u^{(n)}$ have norm $1$. In this case\n\\begin{align*}\n\\langle u^{(m+n-2p)},u^{(m+n-2p)}\\rangle\n=\nv^{p\\,\\left(2\\,p - 2\\,m\n-1\\right)}\\frac{[n]![m+n-p+1]![m-p]!}{[m]![p]![m+n-2p+1]![n-p]!}.\n\\end{align*}\nwhere we have used formula \\eqnref{Ma2} for $p\\leq \\min\\{m,n\\}$.\n\n\nThe same proof that gave us \\eqnref{normalizedforms} now implies for $p\\leq\\min\\{m,n\\}$,\n\\begin{align*}\n||u^{(m+n-2p)}_k||^2:=v^{p\\,\\left(2\\,p - 2\\,m -1\\right)-(m+n-2p-k) k}\n\\frac{\\left[\\begin{matrix} n \\\\  p\n\\end{matrix}\\right]\\left[\\begin{matrix} m+n-p+1\n \\\\  p \\end{matrix}\\right]\\left[\\begin{matrix} m+n-2p \\\\ k\\end{matrix}\\right]}{\\left[\\begin{matrix} m \\\\  p\n\\end{matrix}\\right]}\n\\end{align*}\n\n\n\\begin{prop}[\\cite{MR90m:17022}]\n\\begin{enumerate}[(i).]\n\\item The basis $\\{u_k^{(m+n-2p)}\\}$\nof \\hbox{$\\mathcal F_m\\otimes{\\mathcal F_n}$} is orthogonal.\n\\item For $0\\leq i\\leq m$, and $0\\leq j\\leq n$, \n\\begin{align*}\n&u_i^{(m)}\\otimes u_j^{(n)} \\\\%&=v^{i(i-m) +j(j-n)} \n&=v^{mj+ni-2ij} \n   {\\qbinom m i}{\\qbinom n j} \\\\\n&\\quad \\times \\sum_{p=0}^{\\min\\{m,n\\}}\\frac{[i+j-p]![m+n-i-j-p]!}{v^{p(m+n-p)}||u^{(m+n-2p)}||^{2}[m+n-2p]!}\n  \\left[\\begin{matrix} m & n & m+n-2p\n  \\\\ i & j & i+j-p \\end{matrix}\\right]u_{i+j-p}^{(m+n-2p)}.\n\\end{align*}\n\\end{enumerate}\n\\end{prop}\n\nIn \\hbox{$\\mathcal F_m\\otimes{\\mathcal F_n}^{\\rho_1}$} \n(recall $\\phi(u^{(n)}_j)=(-v)^{-j} u^{(n)}_{n-j}$ )\n\\begin{align}\n&u^{(m)}_i\\otimes u_j^{(n)} \\\\\n&=(-1)^jv^{m(n-j)+ni-2i(n-j)-j} \n   {\\qbinom m i}{\\qbinom{n}{n-j}}\\notag \\\\\n&\\quad \\times \\sum_{p=0}^{\\min\\{m,n\\}}\\frac{v^{-p(m+n-p)}}{||u^{(m+n-2p)}||^{2}} \\notag\\\\\n&\\qquad\\times\\frac{[n+i-j-p]![m-i+j-p]!}{[m+n-2p]!}\n  \\left[\\begin{matrix} m & n & m+n-2p\n  \\\\ i & {n-j} & n+i-j-p \\end{matrix}\\right]u_{n+i-j-p}^{(m+n-2p)}.\\notag\n\\end{align}\n\n\\begin{proof}  First we observe that due to fact that different weight\nspaces having different weights we get\n$$\n\\langle u_k^{(m+n-2p)},u_l^{(m+n-2p)}\\rangle=0,\n$$\nfor $k\\neq l$.  Now suppose $p\\neq q$.  Then the pairing $(,)$ on\n$\\mathcal F_m\\otimes \\mathcal F_n$ induces a module homomorphism from\n$\\mathcal F^{(m+n-2p)}\\to ((\\mathcal F^{(m+n-2q)})^\\rho)^*$.  Since\nthese two irreducible modules are not isomorphic for $p\\neq q$, we must\nhave\n$$\n\\langle u_k^{(m+n-2p)},u_l^{(m+n-2q)}\\rangle=0,\n$$\nfor all $k$ and $l$.\n\nThe second formula follows from the fact that if we write \n$$\nu_i^{(m)}\\otimes u_j^{(n)}=\\sum_{p=\\frac{\\left|m-n\\right|}{2}}^{\\frac{m+n}{2}}c^{ij}_p\\,u_{i+j-p}^{(m+n-2p)},\n$$\nthen $k=i+j-p$,\n\\begin{align*}\nc^{ij}_p\\langle u_k^{(m+n-2p)},u_k^{(m+n-2p)}\\rangle \n&=\\langle u_i^{(m)}\\otimes u_j^{(m)},u_k^{(m+n-2p)}\\rangle \\\\\n&=\\left[\\begin{matrix} m & n & m+n-2p \\\\ i & j & k \\end{matrix}\\right](u_i^{(m)},u_i^{(m)})(u_j^{(n)}, u_j^{(n)})\\\\\n&=v^{i(i-m)+j(j-n)}\\left[\\begin{matrix} m & n & m+n-2p \\\\ i & j & k \\end{matrix}\\right]{\\qbinom m i}{\\qbinom n j}\n\\end{align*}\n\\end{proof}\n\n\\begin{lem} For $m$ and $p$ non-negative integers, one has\n\\begin{equation}  \n\\mathcal L^{-1}u^{(m)}_{m-p}=v^{2p(p-1-m)}u^{(m)}_{m-p}\n\\end{equation}\nand\n\\begin{equation}  \n\\mathcal Lu^{(m)}_{m-p}=v^{-2p(p-1-m)}u^{(m)}_{m-p}.\n\\end{equation}\nThese two equations explain the labeling of $\\mathcal L^{-1}$ and $\\mathcal L$ i.e. they really are inverses when restricted to finite dimensional\n$X$-admissible modules. \n\\end{lem}\n\\begin{proof}  This follows directly from \\lemref{TandL} \n\\end{proof}\n\\begin{cor} Suppose $m$, $n$, $k$ and $p$ are non-negative integers with $c^{m,n}_{i,j}\\in k(v)$ and\n$$ \n\\bar u=\\sum_{i+j=p+k}c^{m,n}_{i,j}u^{(m)}_i\\otimes u^{(n)}_j.\n$$\nThen\n\\begin{equation}  \nT''_1\\mathcal L^{-1}\\bar u=\\sum_{i+j=p+k}d^{m,n}_{i,j}u^{(m)}_i\\otimes u^{(n)}_j,\n\\end{equation}\nwhere \n\\begin{align}\\label{key}\nd^{m,n}_{r,s}&=(-1)^{r+s}v^{r(r - m-1 )+s(s - n-1)} \\\\ \n&\\qquad \\times\\sum_{p\\geq 0}(-1)^pv^{-\\frac{p(p-1)}{2}+ p(2p+ m - n - 2r + 2s )}\\left\\{p\\right\\}\nc^{m,n}_{m-r+p,n-s-p}\\qbinom{r}{p}\\qbinom{n-s}{p}\\notag\n\\end{align}\n\n\\end{cor}\n\\begin{proof}  Due to the invariance of the form $\\langle\\,,\\,\\rangle$ we get\n\\begin{align*}\n&d_{r,s}^{m,n}v^{r(r-m)+s(s-n)}\\qbinom{m}{r}\\qbinom{n}{s}=d_{r,s}^{m,n}(u^{(m)}_r,u^{(m)}_r)(u^{(n)}_s,u^{(n)}_s) \\\\\n&=\\langle T''_1\\mathcal L^{-1}\\bar u, u^{(m)}_r\\otimes u^{(n)}_s\\rangle  \n=\\langle T''_1\\mathcal L^{-1}\\bar u, T''_1T'_{-1}(u^{(m)}_r\\otimes u^{(n)}_s)\\rangle   \\\\\n&=\\langle L(\\mathcal L^{-1}\\bar u\\otimes T'_{-1}(u^{(m)}_r\\otimes u^{(n)}_s)\\rangle  \\qquad \\text{by \\eqnref{L2}} \\\\\n&=\\langle \\bar u,T'_{-1}(u^{(m)}_r\\otimes u^{(n)}_s)\\rangle \\qquad \\text{by \\eqnref{L3}}   \\\\\n&=\\langle \\bar u,L(T'_{-1}(u^{(m)}_r)\\otimes T'_{-1}(u^{(n)}_s))\\rangle   \\\\\n&=(-1)^{r+s}v^{r(2r-2m-1)+s(2s-2n-1)} \\\\ \n&\\qquad \\times\\sum_{p\\geq 0}(-1)^pv^{-\\frac{p(p-1)}{2}+ p(2p+ m - n - 2r + 2s )}\\left\\{p\\right\\}\nc^{m,n}_{m-r+p,n-s-p}\\qbinom{r}{p}\\qbinom{n-s}{p}\\qbinom{m}{r}\\qbinom{n}{s}.\n\\end{align*}\n\n\n\n\\end{proof}\n\\section{Basis and the Intertwining map $\\mathcal L$}\n\\subsection{A Basis} For $s\\geq 1$,\nand any lowest weight vector $\\eta$ of weight $Tv^{\\lambda+\\rho}$, set\n\\begin{equation}\\label{fminusketa}\nF^{(-k)}\\eta:=T'_{-1}(F^{(k)})\\eta\n=v^{k(k-1)}F^{-k}K^{k}[K;-1]^{(k)}\\eta\n=v^{k(\\lambda+k)}T^k[T;\\lambda]^{(k)}F^{-k}\\eta\n\\end{equation}\n\nas\n$$\nT'_{-1}(E^{(k)})=(-1)^kv^{-k(k-1)}K^{-k}F^{(k)},\\quad\n\\quad T'_{-1}(F^{(k)})=(-1)^kv^{k(k-1)}E^{(k)}K^{k}.\n$$\n\\begin{lem}\\label{binomiallemma} Suppose $r,s\\in\\mathbb Z$, $s>\nr\\geq 0$, $\\zeta$ is a highest weight vector of weight\n$Tv^{\\lambda-\\rho}$ and $\\eta$ is a lowest weight vector of\nweight $Tv^{\\lambda+\\rho}$.  Then\n\\begin{equation}\\label{oldandnew}\nE^{(r)}F^{(s)}\\zeta\n=\\qbinom{T;\\lambda-\\rho+r-s}{ r}F^{(s-r)}\\zeta,\n\\quad F^{(r)}F^{(s)}\\zeta\n=\\qbinom{r+s}{ r}F^{(s+r)}\\zeta, \n\\end{equation}\nand\n\\begin{align}\nF^{(r)}F^{(-s)}\\eta\n&=v^{r(\\lambda+2s-r)}T^r\n \\qbinom{T;\\lambda+s}{ r} F^{(r-s)}\\eta, \\\\\nE^{(r)}F^{(-s)}\\eta\n&=(-1)^rv^{-r(\\lambda+r+2s)}T^{-r}\n \\qbinom{r+s}{ r}\n  F^{(-r-s)}\\eta, \n\\end{align}\n\n\\end{lem}\n\n$n=\\lambda-\\rho$.\n\n\nDefine indexing sets $ I_\\lambda$ and $ I_{-\\lambda}$ by $ I_\\lambda =\n\\lbrace n-2,n-4,...\\rbrace$, $ I_{-\\lambda} = \\lbrace -n,-n-2,...\\rbrace $ where $n=\\lambda+1$.  \nOne should compare the previous result with\n\\begin{lem}[\\cite{MR96c:17022}, 2.2]\\label{locstructure} Now for\nintegers $j \\in  I_\\lambda$ (resp.\n$ I_{-\\lambda}$) set $ k_j = {\\frac{n-2-j}{ 2}}$ and $l_j = {\\frac{-n-j}{2}}\n$ and define basis vectors for $M(m+\\lambda)$ and $M(m-\\lambda)$ by \n$w_{\\lambda,j} = F ^{k_j} \\otimes 1_{m+\\lambda-\\varrho}$ and $w_{-\\lambda,j} = F ^{l_j}\\otimes 1_{m-\\lambda\n-\\rho}$. The action of $\\ru$ is given by \n$$ \nKw_{\\lambda,j} = Tv^{\\lambda-1-2k_j}w_{\\lambda,j}\n\\quad ,\\quad F w_{\\lambda,j} = w_{\\lambda,j-2}\\ \\  , \n$$ \n$$ Kw_{-\\lambda,j} =Tv^{-\\lambda-1-2l_j}w_{-\\lambda,j} \\quad ,\\quad  F\nw_{-\\lambda,j} = w_{-\\lambda,j-2} \\ , \n$$ \n$$ Ew_{\\lambda,j} =  [k_j][T; -l_j] w_{\\lambda,j+2}\\quad ,\\quad    Ew_{-\\lambda,j} = [l_j][T; -k_j]\nw_{-\\lambda,j+2}. \n$$\n\n\\end{lem}   \n\n\n\n\\subsection{} The\narticles \\cite{MR96c:17037} and\n\\cite{MR96c:17022} study noncommutative localization of highest\nweight modules. This article may be viewed as an extension of what\nwas begun there. For any $\\mathbf U$-module $A$ let $A_F$ denote the\nlocalization of\n$A$ with respect to the multiplicative set in $\\mathbf U$ generated by $F$. If\n$F$ acts without torsion on $A$ (we shall assume this throughout) then $A$\ninjects into $A_F$ and we have the short exact sequence of $\\mathbf U$-modules:\n$0\\rightarrow A\\rightarrow A_F\\rightarrow A_\\pi \\rightarrow 0$. \n\\begin{lem}[Mackey] \\label{Mackey}  Let $M=\\ru\\otimes _{_RB}R_{m+\\lambda-\\rho}$ be a Verma module of highest weight\n$m+\\lambda-\\rho$ (see \\cite[1.7]{MR96c:17022}) and suppose that $\\mathcal E$ is a $X$-admissible (hence a free $R$-module of\nfinite rank). Then $M_\\pi\\otimes_R\\mathcal E$ is generated as an $\\ru$-module by $Rm\\otimes \\mathcal E$ where $m$ is a\nlowest weight vector in $M_\\pi$.  \n\\end{lem}\n\\begin{proof}  First note that $M\\otimes_R \\mathcal E$ can be identified with its image in $M_F\\otimes R\\mathcal E$ due to the\nfact that it is torsion free with respect to the action of $F$.  The Lemma then follows from the series of isomorphisms:\n\\begin{align*}\nM_\\pi\\otimes_R\\mathcal E&\\cong (M_F\\otimes_R\\mathcal E)\/(M\\otimes_R\\mathcal E)\\cong (M\\otimes_R\\mathcal E)_F\/(M\\otimes_R\\mathcal E) \\\\\n\t\t\t&\\cong (\\ru\\otimes_{_RB}(R_{m+\\lambda-\\rho}\\otimes_R\\mathcal E))_F\/\n   (\\ru\\otimes_{_RB}(R_{m+\\lambda-\\rho}\\otimes_R\\mathcal E)) \\\\\n\t\t\t&= (\\ru\\otimes_{_RB}(R_{m+\\lambda-\\rho}\\otimes_R\\mathcal E))_\\pi.\n\\end{align*}\nThe first isomorphism follows as tensoring the short exact sequence \n$$\n0\\to M\\to M_F\\to M_\\pi\\to 0\n$$\nwith $\\mathcal E$ (a finite rank free $R$-module), leads to a short exact sequence.  The second isomorphism comes from\n\\cite[Theorem 3.9]{MR96c:17022}) where it is implemented by $1\\otimes e\\mapsto 1\\otimes 1\\otimes e$ where $e\\in\\mathcal E$. \nThe third isomorphism comes from Mackey's Isomorphism Theorem and it is implemented by $1\\otimes 1\\otimes e\\mapsto\n1\\otimes 1\\otimes e$.\n\\end{proof}\n\n\\subsection{Highest and Lowest Weight Decompositions.}\n\n\n\\begin{lem}\\label{highestlowestwtvectors}  Suppose $n-2p=r$ with $n$ and $r$ nonnegative integers.\n Then \n \\begin{equation}\nw_{r,r-1}=\\sum_{0\\le k\\le p}\\frac{[n-p+k]_{(k)}v^{-k^2}}{T^{-k}\n[T^{-1};k]_{(k)}} F^{(k)}w_{0,-1}\\otimes u^{(n)}_{p-k}.\n\\end{equation}\nis a highest weight vector of weight $Tv^{r-1}$ in $M\\otimes \\mathcal F$\nand \n\\begin{equation}\nw_{-r,-r-1}=\\sum_{0\\le k\\le n-p}\\frac{[p+k]_{(k)}v^{-k^2}}{T^{-k}\n[T^{-1};k]_{(k)}} F^{(k)}w_{0,-1}\\otimes u^{(n)}_{n-p-k}.\n\\end{equation}\nis a highest weight vector of weight $Tv^{-r-1}$ in $M\\otimes \\mathcal F$.  In $M_\\pi\\otimes \\mathcal F$,\n \\begin{equation}\nm_{r,r+1}=\\sum_{0\\le k\\le n-p}(-1)^k\\frac{[p+k]_{(k)}v^{-k(-r+2k+1)}}{T^k\n[T;k]_{(k)}} F^{(-k)}m_{0,1}\\otimes u^{(n)}_{p+k},\n\\end{equation}\nis a lowest weight vector of lowest weight $Tv^{r+1}$.  Moreover\n \\begin{equation}\nm_{-r,-r+1}=\\sum_{0\\le k\\le p}(-1)^k\\frac{[n-p+k]_{(k)}v^{-k(r+2k+1)}}{T^k[T;k]_{(k)}}F^{(-k)}m_{0,1}\\otimes u^{(n)}_{n-p+k},\n\\end{equation}\nis a lowest weight vector in $M_\\pi\\otimes \\mathcal F$ of\nlowest weight $Tv^{-r+1}$\n\\end{lem}\n\nObserve that the second identity can be obtained from the first by replacing $p$ with $n-p$.\nThis implies that if we set\n \\begin{equation*}\\label{sigma}\n\\sigma=\\sigma(\\epsilon,n,p)=\\begin{cases} p &\\quad\\text{ if }\\quad \\epsilon =1, \\\\\n  n-p &\\quad \\text{if} \\quad \\epsilon =-1,\\end{cases}\n\\end{equation*}\nand\n \\begin{align*}\na_{\\epsilon, n, p}(k)=(-1)^k\\frac{[\\sigma+k]_{(k)}}{T^k[T;k]_{(k)}},\n\\end{align*}\nthen\n \\begin{equation}\nm_{\\epsilon r,\\epsilon r+1}=\\sum_{0\\le k\\le n-\\sigma}v^{-k(-\\epsilon r+2k+1)}a_{\\epsilon, n, p}(k)F^{(-k)}m_{0,1}\\otimes\nu^{(n)}_{\\sigma+k},\n\\end{equation}\n\n\\begin{proof}  The first equality follows using \\eqnref{irred} and \\lemref{binomiallemma}.\n\nOn the other hand a direct calculation shows $Fm_{r,r+1}=0$.\nUsing \n\\begin{align*}\nm_{-r,-r+1}&=\\sum_{0\\le k\\le p}A_k F^{(-k)}m_{0,1}\\otimes u^{(n)}_{n+k-p}\n\\end{align*}\nand $Fm_{-r,-r+1}=0$ we get\n$$\nA_k =(-1)^k\\frac{v^{-k(r+2k+1)}[n+k-p]_{(k)}}{T^k[T;k]_{(k)}}A_0.\n$$\n\\end{proof}\nNote that by the definition of $\\mathcal L$, one has\n\\begin{align}\n\\mathcal L(m_{r,r+1})=m_{r,r+1},\\quad \n\\mathcal L(m_{-r,-r+1})=m_{-r,-r+1}.\n\\end{align}\n\\begin{cor}\nSuppose $n-2p=r$ with $n$ and $r$ nonnegative integers. If we let $m_{r,r+1}$ (resp. $m_{-r,-r+1}$) denote a lowest weight vector in $M_\\pi\\otimes\n\\mathcal F$ of lowest weight $Tv^{r+1}$ (resp.   $Tv^{-r+1}$),  then\n \\begin{align}\nL^{-1}(m_{r,r+1})&= v^{-2(n-p)(p+1)}\\sum_{s=0}^{n-p}(-1)^s\\frac{v^{s(1-r)}[p+s]_{(s)}}{T^s[T,s]_{(s)}}F^{(-s)}m_{0,1}\\otimes\n  u^{(n)}_{p+s},\\label{needlater}  \\\\  \nL^{-1}(m_{-r,-r+1})&=v^{-2p(n-p+1)}\\sum_{s=0}^{p}\\frac{v^{s(1+r)}[n-p+s]_{(s)}}{T^s[T,s]_{(s)}}F^{(-s)}m_{0,1}\\otimes\nu^{(n)}_{n-p+s}.\n\\end{align}\n\\end{cor}\nAs in the previous lemma the second identity can be obtained from the first by replacing $p$ with $n-p$.\nUsing the above definition of $\\sigma$ and $a_{\\epsilon, n, p}(k)$ we get\n \\begin{align*}\nL^{-1}(m_{\\epsilon r,\\epsilon  r+1})&= v^{-2(n-\\sigma)(\\sigma+1)}\\sum_{s=0}^{n-\\sigma}(-1)^sv^{s(1-\\epsilon\nr)}a_{\\epsilon, n, p}(k)F^{(-s)}m_{0,1}\\otimes u^{(n)}_{\\sigma+s}.\n\\end{align*}\n\\begin{proof}  Define coefficients $B_s$ and $C_s$ through\n \\begin{align*}\nL^{-1}(m_{r,r+1})  \n&= \\sum_{s=0}^{n-p}B_sF^{(-s)}m_{0,1}\\otimes u^{(n)}_{p+s},\\quad \\text{and}  \\\\  \nL^{-1}(m_{-r,-r+1})  \n&= \\sum_{s=0}^{p}C_sF^{(-s)}m_{0,1}\\otimes u^{(n)}_{s+n-p}.\n\\end{align*} \nNow \n\\begin{align*}\n0&=L^{-1}(Fm_{\\epsilon r,\\epsilon r+1})\n=-(K\\otimes K)(T'_{-1}(E)\\otimes 1+K^{-1}\\otimes T'_{-1}(E))L^{-1}(m_{\\epsilon r,\\epsilon r+1})\n\\end{align*}\nThus\n \\begin{align*}\n0&= -\\sum_{s=1}^{n-p}\\left(B_s[T;s]+B_{s-1}T^{-1}v^{1-n+2p}[p+s]\\right)F^{(1-s)}m_{0,1}\\otimes u^{(n)}_{p+s}    ,\n\\end{align*} \nand\n \\begin{align*}\n0&= -\\sum_{s=1}^{p}\\left(C_s[T;s]+C_{s-1}T^{-1}v^{n-2p+1}[n-p+s]\\right)F^{(1-s)}m_{0,1}\\otimes u^{(n)}_{s+n-p}.\n\\end{align*}\n\nHence\n\\begin{align*}\nB_s&=-B_{s-1}\\frac{v^{1-n+2p}[p+s]}{T[T,s]}=(-1)^s\\frac{v^{s(1-r)}[p+s]_{(s)}}{T^s[T,s]_{(s)}}B_{0} \\\\\nC_s&=-C_{s-1}\\frac{v^{n-2p+1}[n-p+s]}{T[T,s]}=(-1)^s\\frac{v^{s(r+1)}[n-p+s]_{(s)}}{T^s[T,s]_{(s)}}C_{0}.\n\\end{align*}\nIn particular $B_{n-p}=(-1)^{n-p}\\frac{v^{(n-p)(1-r)}[n]_{(n-p)}}{T^{n-p}[T;n-p]_{(n-p)}}B_{0}$.  On the other hand\n\\begin{align*}\n&L^{-1}(m_{r,r+1})=\n  \\sum_{0\\le k\\le n-p}(-1)^k\\frac{[p+k]_{(k)}v^{k(r-2k-1)}}{T^k\n  [T;k]_{(k)}} L^{-1}\\left(F^{(-k)}m_{0,1}\\otimes u^{(n)}_{p+k}\\right) \\\\\n&= \\sum_{s=0}^{n-p}\\sum_{k=s}^{n-p}(-1)^k\\frac{[p+k]_{(k)}}{\n[T;s]_{(s)}[k-s]!}\\qbinom{n-p-s}{k-s}v^{k(r-2k-1)+\\frac{(k-s)(k-s-1)}{2}+(k-s)(k+s)}\\\\\n&\\hskip 100pt \\times \\{k-s\\}T^{-s}F^{(-s)}m_{0,1}\\otimes u^{(n)}_{p+s}. \n\\end{align*}\nA very similar calculation shows\n\\begin{align*}\nL^{-1}&(m_{-r,-r+1}) \\\\\n&= \\sum_{s=0}^{p}\\sum_{k=s}^{p}(-1)^k\\frac{[n-p+k]_{(k)}}{\n[T;s]_{(s)}[k-s]!}\\qbinom{p-s}{k-s}v^{-k(r+2k+1)+\\frac{(k-s)(k-s-1)}{2}+(k-s)(k+s)}\\\\\n&\\hskip 100pt \\times \\{k-s\\}T^{-s}F^{(-s)}m_{0,1}\\otimes\nu^{(n)}_{n-p+s}  \\\\  \n\\end{align*}\n\nWhen $s=n-p$ we get $F^{(-s)}m_{0,1}\\otimes\nu^{(n)}_{p+s}$ has a coefficient\n$$\nB_{n-p}=(-1)^{n-p}\\frac{[n]_{(n-p)}v^{(n-p)(-n-1)}}{T^{n-p}\n[T;n-p]_{(n-p)}}=(-1)^{n-p}\\frac{v^{(n-p)(1-r)}[n]_{(n-p)}}{T^{n-p}[T;n-p]_{(n-p)}}B_{0}\n$$\nThus $B_0=v^{(n-p)(r-n-2)}=v^{-2(n-p)(p+1)}$, \n$$\nB_s=(-1)^s\\frac{v^{s(1-r)-2(n-p)(p+1)}[p+s]_{(s)}}{T^s[T,s]_{(s)}},\n$$\nSimilarly setting $s=p$, we get\n$$\nC_p=(-1)^p\\frac{[n]_{(p)}}{T^p[T;p]_{(p)}}v^{-p(r+2p+1)}=(-1)^p\\frac{[n]_{(p)}}{T^p[T;p]_{(p)}}v^{p(r+1)}C_{0}\n$$\nso that $C_0=v^{-2p(r+p+1)}=v^{-2p(n-p+1)}$.\n\nThis proves the two identities.\n\\end{proof}\n\\begin{cor}\\label{link}  For $0\\leq s\\leq n-p$,\n\\begin{align}\n&v^{s(1-r)+2(p-n)(p+1)}[p+s]_{(s)} \\\\\n&=\\sum_{k=s}^{n-p}(-1)^{k+s}\\frac{[p+k]_{(k)}}{[k-s]!}\\qbinom{n-p-s}{k-s}v^{k(r-2k-1)+\\frac{(k - s)(3k + s-1) }{2}}\\{k-s\\}.\\notag\n\\end{align}\n\\end{cor}\n\n\n\\section{Maps into the Harmonics.}  \n\\subsection{Harmonics}  We know from \\cite{MR95e:17017} that\n$F(\\mathbf U)\\cong \\mathcal H\\otimes Z(\\mathbf U)$ where $\\mathcal\nH=\\oplus_{n\\in\\mathbb N}\\mathcal L_{2n}$\\label{mathcalL2n} is the space of {\\it\nharmonics}, and\n$\\mathcal L_{2n}\\cong\n\\mathcal F_{2n}$.\n\nFor $m$, $n$, $r$ integers with $m+n$ even, $|m-n|\\leq 2r\\leq m+n$, we let\n$\\beta^{m,n}_{2r}:\\mathcal F_m\\otimes\\mathcal F_n^{\\rho_1}\\to\n\\mathcal H$, be the\n$\\mathbf U$-module homomorphism determined by \n\\begin{equation}\n\\label{betamn}\n\\beta^{m,n}_{2r}(u^{(m+n-2q)})=\\delta_{2r,m+n-2q}E^{(r)}\nK^{-r}.\n\\end{equation}\nso that $\\text {\\rm im}\\, \\beta^{m,n}_{2r}=\\mathcal L_{2r}$.\n\n\n\\begin{prop}\\label{betaonabasis}Suppose $\\eta$ is a lowest weight\nvector of weight\n$Tv^{\\lambda+\\rho}$, \nand\n$\\max\\{|\\frac{m-n}{2}|,|\\frac{m-n}{2}-i+j|\\}\\leq r\\leq \\frac{m+n}{2}$ \nwith $0\\leq i\\leq m$ and $0\\leq j\\leq n$.    Then\n\\begin{align*}\n\\rho_1\\Big(\\beta^{m,n}_{2r}(&u^{(m)}_i\\otimes u^{(n)}_{j})\n \\Big) F^{(-c)}\\eta\\\\\n&=(-1)^jv^{m(n-j)+ni-2i(n-j)-j+r^2-\\frac{(m+n)^2}{4}} \n   {\\qbinom m i}{\\qbinom{n}{j}} \\\\ \n&\\quad \\times \n  \\frac{\\left[\\begin{matrix} m & n & 2r\n  \\\\ i & {n-j} & \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}{||u^{(2r)}||^2\\left[\\begin{matrix} 2r\n  \\\\  \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}\n(v^{2\\lambda+2+i-j+\\frac{n-m}{2}+4c}T^2)^{j-i+\\frac{m-n}{2}}\\\\ \n&\\quad \\times \n \\sum_{l=0}^{i-j+\\frac{n-m}{2}+r} (-1)^{r-l}\n  v^{l\\left(r+j-i+\\frac{m-n}{2}+1 \\right) } \\\\\n&\\hskip 50pt \\times \\qbinom{i-j+\\frac{n-m}{2}+c}{ i-j+\\frac{n-m}{2}+r-l}\n \\qbinom{T;\\lambda+l+c}{ r}\n \\qbinom{l+c}{ l}\n  F^{(j-i+\\frac{m-n}{2}-c)}\\eta \\end{align*}\nand\n\\begin{align*}\n\\rho_1\\Big(T_{-1}'&\\beta^{m,n}_{2r}(u^{(m)}_i\\otimes u^{(n)}_{j})\n \\Big) F^{(-c)}\\eta\\\\\n&=(-1)^jv^{m(n-j)+ni-2i(n-j)-j+r^2-\\frac{(m+n)^2}{4}} \n   {\\qbinom m i}{\\qbinom{n}{j}} \\\\ \n&\\quad \\times ||u^{(2r)}||^{-2}\n  \\frac{\\left[\\begin{matrix} m & n & 2r\n  \\\\ i & {n-j} & \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}{\\left[\\begin{matrix} 2r\n  \\\\  \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}  (v^{ 2\\,c\n+\\lambda}T)^{i-j+\\frac{n-m}{2}}\\\\ \n&\\quad\\times  \\sum_{l=0}^{i-j+\\frac{n-m}{2}+r}(-1)^{r-l}\n  v^{  l (r +j-i+\\frac{m-n}{2}+1) }\\\\\n&\\hskip 50pt \\times \n    \\left[\\begin{matrix}T;\\lambda+c \\\\ l \\end{matrix}\\right]\n    \\left[\\begin{matrix}T;\\lambda+r+c-l \\\\ i-j+\\frac{n-m}{2}+r-l \\end{matrix}\\right]\n    \\left[\\begin{matrix}r+c-l \\\\ r \\end{matrix}\\right]\n  F^{(i-j+\\frac{n-m}{2}-c)}\\eta  \n\\end{align*}\n\n\\end{prop}\n\n\n\\begin{proof}\nFrom \\cite{MR96m:17029}, 4.18(5), we have for any $a,b\\in\\mathbb N$,\n$$\n\\text{ad}\\,F^{(a)}(E^{(b)}K^{-b})=\\sum_{m=0}^a(-1)^{a-m}\n  v^{- \\left( a-1 \\right) \\left(  a-m \\right)}\n  F^{(m)} E^{(b)}K^{-b}F^{(a-m)}K^{a}.\n$$\nThis implies \n\\color{black}\n\n\\begin{align*}\n\\rho_1&\\left(\\text{ad}\\,F^{(a)}(E^{(b)}K^{-b})\\right) F^{(-c)}\\eta \\\\\n&=\\sum_{m=0}^a(-1)^{a-m}\n  v^{- \\left( a-1 \\right) \\left(  a-m \\right)}\n \\rho_1\\left( F^{(m)} E^{(b)}K^{-b}F^{(a-m)}K^{a}\\right) F^{(-c)}\\eta \\\\\n&=(v^{2\\lambda+2+a-b+4c}T^2)^{b-a}\n \\sum_m (-1)^{b-m}\n  v^{m\\left(2b-a+1 \\right) }\\\\\n&\\hskip 50pt \\times \n \\qbinom{a-b+c}{ a-m}\n \\qbinom{T;\\lambda+m+c}{ b}\n \\qbinom{m+c}{ m}\n  F^{(b-c-a)}\\eta ,\n\\end{align*}\n\nand since $\\rho_1\\circ T_{-1}'=T_{-1}'\\circ \\rho_1$, we get\n\\begin{align*}\n\\rho_1&\\left(T_{-1}'\\text{ad}\\,F^{(a)}(E^{(b)}K^{-b})\\right)\nF^{(-c)}\\eta \\\\\n&=\\sum_{m=0}^a(-1)^{a-m}\n  v^{- \\left( a-1 \\right) \\left(  a-m \\right)}\n T_{-1}'\\rho_1\\left( F^{(m)} E^{(b)}K^{-b}F^{(a-m)}K^{a}\\right)\n  F^{(-c)}\\eta \\\\ \n&=\n (-v^{ 2\\,c +\\lambda}T)^{a-b}\\sum_{m}(-1)^{a-m}\n  v^{  m (2b -a+1) }  \\\\\n&\\hskip 50pt \\times \n    \\left[\\begin{matrix}T;\\lambda+c \\\\ m \\end{matrix}\\right]\n    \\left[\\begin{matrix}T;\\lambda+b+c-m \\\\ a-m \\end{matrix}\\right]\n    \\left[\\begin{matrix}b+c-m \\\\ b \\end{matrix}\\right]\n  F^{(a-c-b)}\\eta\n\\end{align*}\n\\color{black}\nwhere we have used \\eqnref{binomiallemma} and the calculation\n\\begin{align*}\nT_{-1}'(&E^{(m)})T_{-1}'(F^{(b)})T_{-1}'(K^{b})\nT_{-1}'(E^{(a-m)})T_{-1}'(K^{-a})\n\\\\\n&=(-1)^{b+a}v^{a(a+1) + b(b -1- 2m)}F^{(m)} E^{(b)}F^{(a-m)}\n\\end{align*}\nBy the Clebsch-Gordan decomposition \\lemref{CG}\n\\color{black}\n\\begin{align*}\n\\rho_1&\\left(\\beta^{m,n}_{2r}(u^{(m)}_{i}\\otimes\nu^{(n)}_{j})\\right)\nF^{(-c)}\\eta  \\\\\n&=(-1)^jv^{m(n-j)+ni-2i(n-j)-j+r^2-\\frac{(m+n)^2}{4}} \n   {\\qbinom m i}{\\qbinom{n}{j}} \\\\ \\\\\n&\\quad \\times ||u^{(2r)}||^{-2}\n  \\frac{\\left[\\begin{matrix} m & n & 2r\n  \\\\ i & {n-j} & \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}{\\left[\\begin{matrix} 2r\n  \\\\  \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}\\\\   \\\\\n&\\quad \\times \n(v^{2\\lambda+2+i-j+\\frac{n-m}{2}+4c}T^2)^{j-i+\\frac{m-n}{2}}\\\\   \\\\\n&\\quad \\times \n \\sum_{l=0}^{i-j+\\frac{n-m}{2}+r} (-1)^{r-l}\n  v^{l\\left(r+j-i+\\frac{m-n}{2}+1 \\right) }\n \\qbinom{i-j+\\frac{n-m}{2}+c}{ i-j+\\frac{n-m}{2}+r-l}\n  \\\\\n&\\hskip 50pt \\times \\qbinom{T;\\lambda+l+c}{ r}\n \\qbinom{l+c}{ l}\n  F^{(j-i+\\frac{m-n}{2}-c)}\\eta \\\\ \n\\end{align*}\nfor $\\max\\{|\\frac{m-n}{2}|,|\\frac{m-n}{2}-i+j|\\}\\leq r\\leq \\frac{m+n}{2}$,\nand\n\\begin{align*}\n\\rho_1&\\left(T_{-1}'\\beta^{m,n}_{2r}(u^{(m)}_{i}\\otimes\nu^{(n)}_{j})\\right)\nF^{(-c)}\\eta  \\\\\n&=(-1)^jv^{m(n-j)+ni-2i(n-j)-j+r^2-\\frac{(m+n)^2}{4}} \n   {\\qbinom m i}{\\qbinom{n}{j}} \\\\ \\\\\n&\\quad \\times ||u^{(2r)}||^{-2}\n  \\frac{\\left[\\begin{matrix} m & n & 2r\n  \\\\ i & {n-j} & \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}{\\left[\\begin{matrix} 2r\n  \\\\  \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}  (v^{ 2\\,c\n+\\lambda}T)^{i-j+\\frac{n-m}{2}}\\\\   \\\\\n&\\quad\\times  \\sum_{l=0}^{i-j+\\frac{n-m}{2}+r}(-1)^{r-l}\n  v^{  l (r +j-i+\\frac{m-n}{2}+1) }\n    \\left[\\begin{matrix}T;\\lambda+c \\\\ l \\end{matrix}\\right]\n    \\left[\\begin{matrix}T;\\lambda+r+c-l \\\\ i-j+\\frac{n-m}{2}+r-l \\end{matrix}\\right]\n  \\\\\n&\\hskip 50pt \\times    \\left[\\begin{matrix}r+c-l \\\\ r \\end{matrix}\\right]\n  F^{(i-j+\\frac{n-m}{2}-c)}\\eta  \n\\end{align*}\n\\end{proof}\n\n\n\n\\begin{cor}\\label{firstbetacor}  If $m=n$ and $i=j$, then we get\n\\begin{align*}\n\\rho_1&\\left(\\beta^{m,n}_{2r}(u^{(m)}_{i}\\otimes\nu^{(m)}_i)\\right)\nF^{(c)}\\zeta  \\\\\n&=(-1)^iv^{2i(i-m-1)+r^2} \n   {\\qbinom m i}^2 \\frac{\\left[\\begin{matrix} m & m & 2r\n  \\\\ i & {m-i} & r \\end{matrix}\\right]}\n   {||u^{(2r)}||^2\\left[\\begin{matrix} 2r\n  \\\\  r \\end{matrix}\\right]} \\\\\n&\\quad \\times \\sum_{l=0}^{r} (-1)^{l-r}\n  v^{( l-r)\\left(r-1\\right) }\n \\\\\n&\\hskip 10pt \\times   \\qbinom{T;\\lambda-\\rho-c+r-l}{r-l}\\qbinom{l+c}{r}\n\\qbinom{T;\\lambda-\\rho-c}{l} F^{(c)}\\zeta.\n\\end{align*}\n\\end{cor}\n\n\n\\begin{cor}\\label{secondbetacor} Let $m$, $n$ be non-negative integers and\n$\\max\\{|\\frac{m-n}{2}|,|\\frac{m-n}{2}-i+j|\\}\\leq r\\leq \\frac{m+n}{2}$ with $0\\leq\ni\\leq m$ and\n$0\\leq j\\leq n$.    Then\n\\begin{align*}\n\\rho_1\\Big(\\beta^{m,n}_{2r}&(u^{(m)}_i\\otimes u^{(n)}_{j})\n \\Big) u^{(k)}_s\\\\\n&=(-1)^{i+\\frac{n-m}{2}}v^{m(n-j)+ni-2i(n-j)-j+r^2-\\frac{(m+n)^2}{4}+(j-i-\n\t\\frac{n-m}{2})(k-2s+1)} \n   {\\qbinom m i}{\\qbinom{n}{j}} \\\\  &\\quad \\times ||u^{(2r)}||^{-2}\n  \\frac{\\left[\\begin{matrix} m & n & 2r\n  \\\\ i & {n-j} & \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}{\\left[\\begin{matrix} 2r\n  \\\\  \\frac{n-m}{2}+r+i-j \\end{matrix}\\right]}\\\\  &\\quad \\times \n \\sum_{q=0}^{i-j+\\frac{n-m}{2}+r}(-1)^{ q}\n  v^{- \\left( i-j+\\frac{n-m}{2}-r-1 \\right) q}\\qbinom{k+q-s}{q}\\qbinom{r+s-q}{s-q}\n  \\\\ &\\hskip 100pt \\times   \\qbinom{k+i-j+\\frac{n-m}{2}-s}{i-j+\\frac{n-m}{2}+r-q}\nu^{(k)}_{s-i+j-\\frac{n-m}{2}}\n\\end{align*}\n\\end{cor}\n\n\\begin{proof}  The calculation is very similar to the proof of the previous proposition and so omitted.\n\n\\end{proof}\n\\section{Symmetry Properties of Induced Forms} \n\n\n\\subsection{Twisted action of $R$.}  We shall twist by an automorphism\nof \n$\\ru$ in the setting of $\\ru$-modules. Let $\\theta$ be an automorphism of\n$\\ru$. Then for any $\\ru$-module\n${_R\\mathcal E}$ define a new $\\ru $-module ${_R\\mathcal E}^\\theta$ with set equal to\nthat of ${_R\\mathcal E}$ and action given by : for $e\\in {_R\\mathcal E}$ and $x\\in\\ru $\nthe action of $x$ on $e$ equals $\\theta(x)e$. For any\n$\\ru$-module $A$, let $A^{s\\theta}$ denote the module with action $\\diamond $ on\n$A^{s\\theta}$ defined as follows: For\n$a\\in A$ and $x\\in\\ru$, \n$$ x\\diamond a= s\\theta (x)\\ a\\ .\n$$\n\\smallskip \n\n\n\\begin{lem}\nSuppose  $\\phi$ is a $\\brho$-invariant $R$-valued pairing of $\\ru $-modules\n$A$ and $B$.  Then $s\\circ \\phi$ is a $\\brho$-invariant pairing of\n$A^s$ and $B^s$.  Furthermore if either $A$ or $B$ is $\\text{ad}\\,F$ locally finite, then $s\\circ \\phi\\circ L$ a\n$\\rho_\\pi$-invariant pairing of\n$A^{sT'_{-1}}$ and\n$B^{sT'_{-1}}$. Also\n$\\phi$ and $\\phi\\circ L$ respectively are $\\brho$-invariant\npairing of these two pairs taking values\nin the $R$-module $R^s$.\n.\n\\end{lem}\n\n\n\\begin{proof} By \\corref{Lisomorphism} the map $L:A^{T'_{-1}}\\otimes B^{T'_{-1}}\\to (A\\otimes B)^{T'_{-1}}$ is\nan\n$\\ru$-module homomorphism. Now we know that $\\phi\\in\\text {\\rm Hom}\\,_{\\ru}(A\\otimes B^{\\rho_1},R)$ is module homomorphism and so\nafter twisting the action we get $\\phi\\in\\text {\\rm Hom}\\,_{\\ru}((A\\otimes B^{\\rho_1})^{T_{-1}'},R)$.  Composing with $s$ we\nhave $s\\circ\\phi\\circ L\\in \\text {\\rm Hom}\\,_{\\ru}(A^{T_{-1}'}\\otimes B^{\\rho_1T_{-1}'},R^s)$ is a $\\mathbf U$-module homomorphism\nthat is $s$-linear.  To make it $R$-linear we twist the action on $A^{T_{-1}'}$ and $B^{\\rho_1T_{-1}'}$ by $s$. \nIndeed let\n$\\sharp$ denote the action of\n$R$ twisted by\n$s$. For\n$r\\in R,a\\in A^{T_{-1}'}$ and\n$b\\in B^{\\rho_1T_{-1}'}$, since $s^2 = 1$ , $ s\\circ\\phi(r\\sharp a,b) = s\\circ\\phi(s(r)a,b)\n= r\\phi(a,b) = s\\circ\\phi(a,r\\sharp b).$ \n\\end{proof} \n\nRecall from \\cite{MR96c:17022} we define a {\\it cycle} ( for $A$ ) to be a pair $(A,\\Psi)$\nwhere\n$A$ is a $\\mathbf U$ (or $_R\\mathbf U$)  module and $\\Psi$ is a module\nhomomorphism  \n\\begin{equation}\n\\Psi : A_\\pi^{sT_{-1}'} \\\n\\rightarrow \\ A .\n\\end{equation} \nWe say that the cycle is {\\it nondegenerate} whenever $\\Psi$ is an\nisomorphism and that $A$ {\\it admits a nondegenerate cycle} when such a\npair exists with $\\Psi$ an isomorphism.\n\nThroughout the following, suppose \n$$\n\\Psi:(A_\\pi)^{sT_{-1}'}\\to A,\n$$ \nis a \nnondegenerate cycle.  For example if we return to the setting of\nsection one and set\n$M$ equal to the Verma module with highest weight\n$Tv^{-1}$: i.e.\n$M=M(m)$, then $M_\\pi^{sT_{-1}'}$ is isomorphic to $M$\nitself.  Then we  choose maps $\\Psi : M_F\\rightarrow M$. For $a\\in M, x\\in \\ru, \\\n\\Psi(sT_{-1}'(x)\\cdot a)=x\\cdot \\Psi(a)$. More precisely \n\n{\\begin{lem}\\label{vermacycle} Let $m$ be a lowest weight vector in $M_\\pi$ of weight\n$Tv$ and $\\Psi(m)$ the highest weight vector in $M$ of weight\n$Tv^{-1}$ where $F^{-1}\\Psi(m)\\equiv m\\mod \\iota M$.  The map\n$\\Psi:(M_\\pi)^{sT_{-1}'}\\to M$ given by\n$$\n\\Psi(F^{-k}m)=\\frac{(-1)^kv^{-k^2}T^k}{[k]![T;-1]_{(k)}}F^{k}\\Psi(m)\n$$\nfor $k\\geq 0$ is an $R$-linear $\\mathbf U$-module isomorphism.\n\\end{lem}}\nThe above can be rewritten as \n$$\n\\Psi(F^{(-k)}m)=F^{(k)}\\Psi(m).\n$$\n\n\n\\begin{proof}  The proof is straight forward and so omitted.\n\\end{proof}\n\nSet\n\\begin{equation}\n\\bar \\Psi:=\\Psi \\otimes sT_{1}''\\circ L^{-1}:(M_\\pi\\otimes \\mathcal E)^{sT_{-1}'}\\to\nM\\otimes \\mathcal E.\n\\end{equation}\nLet $\\iota:\\mathbb P(M\\otimes \\mathcal E, N\\otimes\n\\mathcal F)\\to\\hom_\\ru(M\\otimes \\mathcal E\\otimes (N\\otimes\n\\mathcal F)^{\\rho_1},R)$ be the canonical isomorphism with $\\iota(\\chi)(a\\otimes b)=\\chi(a,b)$.  Note that $a\\otimes b\\in\nM\\otimes\n\\mathcal E\\otimes (N\\otimes\n\\mathcal F)^{\\rho_1}$ on the left hand side, while $(a,b)\\in M\\otimes \\mathcal E\\times N\\otimes\n\\mathcal F$ on the right hand side.  Define $\\chi\\mapsto \\chi^\\sharp$ in $\n\\text {\\rm End}\\,(\\mathbb P(M\\otimes\n\\mathcal E, N\\otimes\n\\mathcal F))$ by\n\\begin{equation}\\label{twistedpairing}\n\\iota(\\chi^\\sharp)(\\bar\\Psi(a)\\otimes \\bar\\Psi(b)):=\ns\\circ\\iota(\\chi_\\pi)\\circ L(a\\otimes b)\n\\end{equation}\nfor $a\\in(M\\otimes \\mathcal E)_\\pi^{sT_{-1}'}$,\n$b\\in (N\\otimes\\mathcal  F)^{s T_{-1}'\\rho_1}_\\pi$ \nand $\\chi\\in\\mathbb P(M\\otimes \\mathcal E,N\\otimes \\mathcal F)$. \nRecall that when one evaluates $L$ on the\nright hand side, one has\n\\begin{equation*}\nL(a\\otimes b)=\\sum_n(-1)^nv^{-n(n-1)\/2}\\{n\\}F^{(n)}a\\otimes E^{(n)}b\n\\end{equation*} \nwhere $F^{(n)}a$ is evaluated using the untwisted action and $E^{(n)}b$ is evaluated with the action twisted only by\n$\\rho_1$.    Note also that as a linear map\n$L\\in\\text{End}\\,((M\\otimes\n\\mathcal E)_\\pi\\otimes(N\\otimes\n\\mathcal F)_\\pi^{\\rho_1})$ is well defined as $F$ acts locally nilpotently on $\n(M\\otimes \\mathcal E)_\\pi$. We can view $L:(M\\otimes \\mathcal\nE)_\\pi^{sT_{-1}'}\\otimes (N\\otimes \\mathcal\nF)_\\pi^{ sT_{-1}'\\rho_1}\\to  ((M\\otimes \\mathcal E)_\\pi\\otimes\n(N\\otimes \\mathcal F)^{\\rho_1}_\\pi)^{sT_{-1}'}$ as a module isomorphism (see\n\\eqnref{Tandrho} and \n\\corref{Lisomorphism}). Then\n$s\\circ \\iota(\\chi_\\pi)\\circ L:(M\\otimes \\mathcal\nE)_\\pi^{sT_{-1}'}\\otimes (N\\otimes \\mathcal\nF)_\\pi^{sT_{-1}'\\rho_1}\\to \nR$ is a module homomorphism.  Indeed $\\iota(\\chi_\\pi)\\in\\hom_\\ru((M\\otimes \\mathcal E)_\\pi\\otimes (N\\otimes\n\\mathcal F)^{\\rho_1}_\\pi,R)$, which implies $\\iota(\\chi_\\pi)\\in\\hom_\\ru(((M\\otimes \\mathcal E)_\\pi\\otimes (N\\otimes\n\\mathcal F)^{\\rho_1}_\\pi)^{sT'_{-1}},R^s)$.  This gives us $\\iota(\\chi_\\pi)\\circ L\\in\\hom_\\ru((M\\otimes \\mathcal\nE)_\\pi^{sT'_{-1}}\\otimes (N\\otimes\n\\mathcal F)^{\\rho_1sT'_{-1}},R^s)$.\n\nIf we suppress the map $\\iota$, then we can write \\eqnref{twistedpairing} as\n\\begin{equation*}\n\\chi^\\sharp(\\bar\\Psi(a), \\bar\\Psi(b)):=\ns\\circ \\chi_\\pi\\circ L(a\\otimes b)\n\\end{equation*}\nwere we view $a\\in(M\\otimes \\mathcal E)_\\pi^{sT_{-1}'}$ with $b\\in(N\\otimes \\mathcal F)_\\pi^{sT_{-1}'}$ on the left hand side\nof the equality and  $b\\in(N\\otimes \\mathcal F)_\\pi^{sT_{-1}'\\rho_1}$ on the right hand side.\n\n\nMore explicitly  we can show that $\\chi^\\sharp\\in\n\\mathbb P(M\\otimes \\mathcal E, N\\otimes\n\\mathcal F)$ by the following calculation for $x\\in \\ru$:\n\\begin{align*}\n\\sum\\chi^\\#(S(x_{(2)})\\bar\\Psi(a),\\brho(x_{(1)})\\bar\\Psi(b))&=\n\\sum\\chi^\\#(\\bar\\Psi(T_{-1}' S(x_{(2)})a),\\bar\\Psi(\nT_{-1}' \\brho(x_{(1)})b)) \\\\ \n  &=\\sum s\\circ\\chi_\\pi\\circ L(T_{-1}' S(x_{(2)})a\\otimes \nT_{-1}' \\brho(x_{(1)})b) \\\\ \n  &= s\\circ\\chi_\\pi(T_{-1}' S(x)L(a\\otimes b)) \\\\ \n  &= \\mathbf e(x)s\\circ\\chi_\\pi\\circ L(a\\otimes b)  \\\\\n  &=\\mathbf e(x)\\chi^\\#(\\bar\\Psi(a),\\bar\\Psi(b))\n\\end{align*}\nwhere the third equality is from \\corref{Lisomorphism}, and the fourth\nequality is due to the fact that $\\chi_\\pi$ is $\\rho$-invariant. \n\n\\begin{lem}  Let $\\phi$ be the Shapovalov form on the Verma module $M$ of highest weight $Tv^{-1}$ and $\\Psi$ the cycle in \\lemref{vermacycle}.  Then\n$$\ns\\circ\\phi_\\pi\\circ L=\\phi\\circ\\Psi\\otimes \\Psi.\n$$\nIn other words $\\phi^\\sharp=\\phi$.\n\\end{lem}\n\n\\begin{proof}  Let $w$ be a fixed highest weight vector of $M$ of highest weight $Tv^{-1}$ such that $\\Psi(m)=w$ with $m$ a lowest weight \nvector in $M_\\pi$.\nRecall $\\phi^\\#(\\Psi(a),\\Psi(b))=s\\circ\\phi_\\pi\\circ L(a\\otimes b)$ for\n$a,b\\in M_\\pi$ and that the argument above shows $\\phi^\\#$ is $\\rho$-invariant.  Observe now\n\\begin{align*}\n\\phi^\\#(w,w)&=\\phi^\\#(\\Psi(m),\\Psi(m))=s\\circ\\phi_\\pi\\circ L(m\\otimes m) \\\\\n&=s\\circ\\phi_\\pi(m,m) \\quad \\text{as $m$ is a lowest weight vector} \\\\\n&=\\phi(\\Psi(m),\\Psi(m))=\\phi(w,w) \\quad  \\text{by \\eqnref{lift}}.\n\\end{align*} Since\n$\\phi^\\#$ and\n$\\phi$ agree on the generator $w=\\Psi(m)$ of $M$, the\n$\\rho$-invariance property proves that they agree everywhere.\n\n\\end{proof}\n\n\nIf $\\phi$ is a $\\rho$-invariant form on $\\mathcal F\\times \\mathcal E$, where  $\\mathcal F$ and $\\mathcal E$ are two finite dimensional $\\mathbf U$-modules in\nthe category\n$\\mathcal C'$ we define in a similar manner the $\\rho$-invariant form $\\phi^\\sharp$ on $\\mathcal F\\times \\mathcal E$ given by $\n\\iota(\\phi^\\sharp)(T''_1(a)\\otimes T''_1(b)):=\n\\iota(\\phi)\\circ L(a\\otimes b)$.  If $\\mathcal F=\\mathcal E$ is irreducible with a highest weight vector $u^{(m)}$, then\n\\begin{align}\\label{finaltouch}\n\\iota(\\phi^\\sharp)( u^{(m)}\\otimes u^{(m)})&=\\iota(\\phi^\\sharp)(T_1''(u^{(m)}_m)\\otimes T''_1(u^{(m)}_m)) \\\\\n&=\\iota(\\phi)(u^{(m)}_m\\otimes\nu^{(m)}_m)=\\iota(\\phi)(u^{(m)}\\otimes u^{(m)})\\notag\n\\end{align}\nThe second equality comes from the definition of $L$ and last equality is due to \\eqnref{normalizedforms}.  Thus\n$\\phi^\\sharp=\\phi$.  This implies that if\n$\\mathcal F=\\oplus_i\\mathcal F_{n_i}$ with\n$n_i$ distinct and $\\mathcal E=\\oplus_j\\mathcal F_{m_j}$ with\n$m_j$ distinct nonnegative integers, then $\\phi=\\sum_{i,j,n_i=m_j}\\phi_i$ where $\\phi_i$ is a $\\rho$-invariant nondegenerate form on $\\mathcal F_{n_i}$ and\nzero on the other summands $\\mathcal F_{n_k}$, $k\\neq i$, and we still have $\\phi^\\sharp=\\phi$.  \n\nFor four finite dimensional $X$-admissible $\\mathbf U$-modules $\\mathcal E$, $\\mathcal F$, $\\mathcal M$ and $\\mathcal N$ and invariant form $\\phi$ on $\\mathcal\nM\\times \\mathcal N$ we can define for each\n$\\mathbf U$-module homomorphism $\\beta:{\\mathcal E}\\otimes \\mathcal\nF^{\\rho_1}\\rightarrow\\mathbf U$ the {\\it induced form}\n$\\phi_{\\beta}$ by the formula, for\n$e\\in\\mathcal E,f\\in \\mathcal F, m\\in\\mathcal M,n\\in \\mathcal N$,\n\\begin{equation}\n\\phi_{\\beta}(m\\otimes e,n\\otimes f)=\\phi( m,\\rho_1(\\beta(e\\otimes f))n).\n\\end{equation}\nThis is similar to the definition \\eqnref{inducedform}.\n \n\\begin{lem}\\label{tediouslemma}\nFor $m$, $n$, $\\sigma$, $l$ and $r$ non-negative integers with $0\\leq \\sigma\\leq n$ and $|m-n|\\leq 2r\\leq m+n$, one has\n\\begin{align}\n&v^{\\frac{2n+n^2-(2\\,m+m^2)}{4}+n- 2\\,\\sigma +r(r+1)}\\\\ \n&\\quad \\times\\sum_{k=0}^{ n-\\sigma}  (-1)^{k}v^{-\\frac{k(k+2l+3)}{2}}(v-v^{-1})^{k} \\notag \\\\\n&\\hskip 50pt \\times \\qbinom{r-k}{l}\\frac{[r+k]![\\sigma+k]_{(k)}}{[k]!}\n     {\\qbinom{n}{\\sigma+k}}  \n      \\left[\\begin{matrix} m & n & 2r\n     \\\\ \\frac{m-n}{2}+\\sigma & {n-k-\\sigma} & r-k \\end{matrix}\\right]\\notag \\\\ \\notag \\\\\n&=(-1)^{\\frac{m+n}{2}+r}\\sum_{k=0}^{n-\\sigma}v^{\\frac{k\\left(3+k-2l+4\\sigma-2n+2r\\right)}{2}}\n   (v-v^{-1})^{k}  \\notag\\\\\n&\\hskip 70pt \\times \\qbinom{r-k}{l-k}\\frac{[r+k]![\\sigma+k]_{(k)}}{[k]!}\n     {\\qbinom{n}{\\sigma+k}}\n   \\left[\\begin{matrix} m & n & 2r\n  \\\\ \\frac{m+n}{2}-\\sigma & {\\sigma+k} & r+k \\end{matrix}\\right].\\notag\n\\end{align}\n\n\\end{lem}\n\n\\begin{proof}  The result follows from a rather tedious calculation using $\\phi=(\\enspace,\\enspace)$, the normalized nondegenerate form on $\\mathcal\nF_{a}$, and \n\\corref{secondbetacor}.\n\\end{proof}\n\n\n\\subsection{} In this section we suppose $\\mathcal F_m$\nand\n$\\mathcal F_n$ are the $X$-admissible finite dimensional $\\mathbf U$-modules\ngiven in \\secref{CG}.  For any homomorphism\n$\\beta:_R\\mathcal F_m\\otimes _R\\mathcal F_n^{\\rho_1}\\to {_RF(\\mathbf U)}$ which has the\nform\n\\begin{equation}\\label{factorization}\n\\beta=\\sum_{m,n,k}r^{m,n}_{k}\\beta^{m,n}_{2k}\n\\end{equation}\nwhere $r^{m,n}_{k}\\in R$, observe that\n$s\\circ \\beta$ has the same form.\n\n\n\\begin{thm}\\label{firstinvariance}\nLet $M$ be the Verma module of highest weight $Tv^{-1}$ (so that $\\lambda=0$) and assume that $\\beta:_R\\mathcal F_m\\otimes _R\\mathcal F_n^{\\rho_1}\\to\n{_RF(\\mathbf U)}$  has the form \\eqnref{factorization}.\nIf\n$\\phi$ is a $\\ru$-invariant pairing on $M$ satisfying $s\\circ\n\\phi_{\\pi}\\circ L=\\phi\\circ (\\Psi\\otimes\n\\Psi)$, then \n\\begin{equation}\\label{hardprop}\n\\chi _{\\beta,\\phi}^\\sharp = \\chi_{s\\beta ,\\phi}\\ .\n\\end{equation} \n\\end{thm}\n\n\\begin{proof}  Since $\\chi _{\\beta,\\phi}^\\sharp$ is $R$-linear in $\\beta$ we can\nreduce to the case that $\\beta=a\\beta^{m,n}_{2r}$ for some $a\\in R$.   \nLet us choose basis $\\{F^{(-k)}m_{0,1}|\\,k\\in \\mathbb N\\}$, $\\{u_i^{(m)}|\\,0\\leq i\\leq\nm\\}$,\n$\\{u_i^{(n)}|\\,0\\leq i\\leq n\\}$, respectively for $M=N$,\n$\\mathcal F_m$ and $\\mathcal F_n$, where $m_{0,1} $ is a lowest weight vector of weight $Tv$ that generates $M_\\pi$, and $u^{(m)}$ and $u^{(n)}$\nare highest weight vectors of weights $v^m$ and $v^n$.  \n\nNow the element\n$m_{0,1}\\otimes u^{(m)}$ generates $(M\\otimes \\mathcal F_m)_\\pi^{sT'_{-1}}$\n(see \\lemref{Mackey}), so we can reduce the proof of the identity further to\nshowing that\n\\begin{equation}\\label{firstreduction}\n\\chi _{\\beta,\\phi}^\\sharp(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\\bar\\Psi(F^{(-k)}\\zeta \\otimes\nu_i^{(n)}))=\n\\chi_{s\\beta ,\\phi}(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\\bar\\Psi(F^{(-k)}\\zeta \\otimes\nu_i^{(n)}))\n\\end{equation}\nfor all $k\\in \\mathbb N$ and $0\\leq j\\leq n$.  Indeed since both sides are\ninvariant, if $w\\in (M\\otimes \\mathcal F_m)_\\pi^{sT'_{-1}}$, then there\nexists $u\\in \\ru$ such that $w=u\\bar\\Psi(m_{0,1}\\otimes u^{(m)})$ and hence\n\\begin{align*}\n\\chi _{\\beta,\\phi}^\\sharp(w,\\bar\\Psi(F^{(-k)}m_{0,1} \\otimes u^{(n)}_j))\n&=\\chi_{\\beta,\\phi}^\\sharp(u\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),F^{(-k)}m_{0,1}\n\\otimes u^{(n)}_j) \\\\  \n&=\\chi_{\\beta,\\phi}^\\sharp(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\\,\n   \\rho(u)\\bar\\Psi(F^{(-k)}m_{0,1} \\otimes u^{(n)}_j)) \\\\ \n&= \\chi_{s\\beta ,\\phi}(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\n \\rho(u)\\bar\\Psi(F^{(-k)}m_{0,1} \\otimes u^{(n)}_j)) \\\\ \n&= \\chi_{s\\beta ,\\phi}(u\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\n   \\bar\\Psi(F^{(-k)}m_{0,1} \\otimes u^{(n)}_j))  \\\\\n&= \\chi_{s\\beta ,\\phi}(w,\n   \\bar\\Psi(F^{(-k)}m_{0,1} \\otimes u^{(n)}_j)).\n\\end{align*}\nHere the third equality would be true as $\\rho(u)\\bar\\Psi(F^{(-k)}m_{0,1} \\otimes u^{(n)}_j)$ is a linear combination of other $\\bar\\Psi(F^{(-l)}m_{0,1}\n\\otimes u^{(n)}_s)$.\n Recall that $\\mathcal K$ is the field of fractions of $R$.  By \\cite[Lemma 4.2]{MR96c:17022,MR96c:17037}, to prove \\eqnref{firstreduction}, we need to\nshow\n\\begin{align}\\label{secondreduction}\n_{\\mathcal K}\\chi _{\\beta,\\phi}^\\sharp(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\\bar\\Psi&(F^{(-k)}m_{0,1} \\otimes\nu_i^{(n)}))= \\\\\n&{_{\\mathcal K}\\chi_{s\\beta ,\\phi}}(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\\bar\\Psi(F^{(-k)}m_{0,1} \\otimes\nu_i^{(n)}))\\notag\n\\end{align}\nNow $_{\\mathcal K}M_\\pi\\otimes \\mathcal E_n$ is a direct sum of lowest weight irreducible Verma modules with lowest weight vector $m_{\\epsilon\nr',\\epsilon r'+1}$ of weight $Tv^{\\epsilon r'+1}$, where\n$\\epsilon=\\pm 1$ and $r'=n-2p$, $0\\leq p\\leq n$.  From this fact we get that $F^{(-k)}m_{0,1} \\otimes u^{(n)}_j$ is a $\\mathcal K$-linear combination\nof \n$E^{(i)}m_{\\epsilon r',\\epsilon r'+1}$.\nHence we need to show\n\\begin{align*}\n_{\\mathcal K}\\chi _{\\beta,\\phi}^\\sharp(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\\bar\\Psi&(E^{(i)}m_{\\epsilon r',\\epsilon r'+1}))= \\\\\n&{_{\\mathcal K}\\chi_{s\\beta ,\\phi}}(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}),\\bar\\Psi(E^{(i)}m_{\\epsilon r',\\epsilon r'+1}))\n\\end{align*}\nfor all $r'$, $\\epsilon$ and $i$ with $\\epsilon r'+2i=m$.  Using the invariance of the two forms and the fact that $m_{0,1}$ is a lowest weight vector,\nproving the above identity is equivalent to verifying\n\\begin{equation*}\n_{\\mathcal K}\\chi _{\\beta,\\phi}^\\sharp(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}_i),\\bar\\Psi(m_{\\epsilon r',\\epsilon r'+1}))=\n{_{\\mathcal K}\\chi_{s\\beta ,\\phi}}(\\bar\\Psi(m_{0,1}\\otimes u^{(m)}_i),\\Psi(m_{\\epsilon r',\\epsilon r'+1})).\n\\end{equation*}\n\nRecall the definition of $\\sigma(\\epsilon,n,p)$ and $a_{\\epsilon, n, p}(k)$ from \\eqnref{sigma}.  Then\n\\begin{align*}\n\\chi_{\\beta,\\phi}^\\sharp(&\\bar\\Psi(m_{0,1} \\otimes u^{(m)}_i),\n  \\bar\\Psi(m_{\\epsilon r',\\epsilon r'+1}))=\ns\\circ\\chi_{\\beta,\\phi_\\pi}\n   \\circ L(m_{0,1} \\otimes u^{(m)}_i,m_{\\epsilon r',\\epsilon r'+1})\\\\\n&= \\sum_q(-1)^qv^{-\\frac{q(q-1)}{2}}\\{q\\}s\\circ\\chi_{\\beta,\\phi_\\pi}(\n   F^{(q)}(m_{0,1} \\otimes u^{(m)}_i),\n  E^{(q)}*(m_{\\epsilon r',\\epsilon r'+1})) \\\\\n&= s\\circ\\chi_{\\beta,\\phi_\\pi}(m_{0,1} \\otimes u^{(m)}_i,\n   \\mathcal L\n   (m_{\\epsilon r',\\epsilon r'+1})) \\\\\n&= s\\circ\\chi_{\\beta,\\phi_\\pi}(m_{0,1} \\otimes u^{(m)}_i,m_{\\epsilon r',\\epsilon r'+1}) \\\\\n&= s\\circ\\phi_\\pi(m_{0,1} , \\sum_k a_{\\epsilon, n,p}(k)(\\epsilon)\\rho_1(\\beta(u^{(m)}_i\\otimes\n  u^{(n)}_{k+\\sigma}))F^{(-k)}m_{0,1} ) \\\\\n&= \\phi(\\Psi(\\mathcal L^{-1}m_{0,1} ), \n   \\Psi(\\sum_k a_{\\epsilon, n,p}(k)\\rho_1(\\beta(u^{(m)}_i\\otimes\n  u^{(n)}_{k+\\sigma}))F^{(-k)}m_{0,1} )) \\\\ \t\n&= \\phi\\left(\\Psi(m_{0,1} ), \n   \\Psi\\left(\\sum_k a_{\\epsilon,n,p}(k)\\rho_1(\\beta(u^{(m)}_i\\otimes\n  u^{(n)}_{k+\\sigma}))F^{(-k)}m_{0,1}\n  \\right)\\right)\t\n\\end{align*}\n \nThe seventh equality is due to the fact that $\\mathcal L^{-1}m_{0,1}=m_{0,1}$.   We may assume that $\\sigma-i+\\frac{m-n}{2}=0$, \nas distinct root spaces are orthogonal. On the other hand\n\\begin{align*}\n\\chi_{s\\beta,\\phi}&(\\bar\\Psi(m_{0,1} \\otimes u^{(m)}_i), \n  \\bar\\Psi (m_{\\epsilon r,\\epsilon r+1}))  \\\\\n&= v^{-2(n-\\sigma)(\\sigma+1)}\\sum_{s=0}^{n-\\sigma}v^{s(1-\\epsilon (n-2p))}a_{\\epsilon, n, p}(k) \\\\\n&\\hskip 30pt \\times \\phi\\left(\\Psi(m_{0,1} ),\n\\Psi\\left(\\left(\\rho_1\\circ\nT'_{-1}\\circ\\beta\n  \\right)\\left(T''_1(u^{(m)}_i)\\otimes T''_1(u^{(n)}_{\\sigma+k}\\right)\\right) F^{(-k)}m_{0,1})\\right)\n\\end{align*} \nWe used the reduction to $s\\beta =\\beta$ and the fact that $\\rho_1\\circ T'_{-1}=T'_{-1}\\circ \\rho_1$. Consequently the two sides of \\eqnref{hardprop} are\nequal provided\n\\begin{align}\n\\sum_{k=0}^{n-\\sigma} &a_{\\epsilon,n,p}(k)\\rho_1(\\beta(u^{(m)}_i\\otimes\n  u^{(n)}_{k+\\sigma}))F^{(-k)}m_{0,1} \\\\\n&= v^{-2(n-\\sigma)(\\sigma+1)}\\sum_{k=0}^{n-\\sigma}v^{s(1-\\epsilon (n-2p))}a_{\\epsilon, n, p}(k)\\notag \\\\\n&\\hskip 50pt\\times\\left(\\rho_1\\circ\nT'_{-1}\\circ\\beta\n  \\right)\\left(T''_1(u^{(m)}_i)\\otimes T''_1(u^{(n)}_{\\sigma+k})\\right) F^{(-k)}m_{0,1}.\\notag\n\\end{align}\n\nWe will now expand out the two sides and eventually show that they are equal.\nWe begin with a calculation for the left hand side. \n\\begin{align*}\n\\rho_1\\Big(\\beta^{m,n}_{2r}(&u^{(m)}_i\\otimes u^{(n)}_{k+\\sigma})\n \\Big) F^{(-k)}m_{0,1}\\\\\n&=(-1)^{\\sigma}v^{k(2k+ 2i - m )  +(m- i)n - \\frac{(m+n)^2}{4} +  (2i-1- m)\\sigma + r(1 + 2r)} \n   {\\qbinom m i}{\\qbinom{n}{k+\\sigma}} \\\\ \n&\\quad \\times ||u^{(2r)}||^{-2}\n  \\left[\\begin{matrix} m & n & 2r\n  \\\\ i & {n-k-\\sigma} &r-k \\end{matrix}\\right]\\frac{[r-k]![r+k]!}{ [2r]!}T^{2k}\n  \\qbinom{T;r}{ r}\\qbinom{r}{k}m_{0,1}\n\\end{align*}\nThis implies\n\\begin{align*}\n&\\sum_k a_{\\epsilon,n,p}\\rho_1(\\beta(u^{(m)}_i\\otimes\n  u^{(n)}_{k+\\sigma}))F^{(-k)}m_{0,1}\n   \\\\\n&=(-1)^{\\sigma}\\frac{v^{(2i-1- m)\\sigma + r(1 + 2r)+(m- i)n - \\frac{(m+n)^2}{4}}}{||u^{(2r)}||^2}\n{\\qbinom m i} \n[T;r]_{(r)}\\\\ \n&\\quad \\times\\sum_{k=0}^{\\min\\{n-\\sigma,r\\}}  (-1)^k \n  v^{-k} \n   {\\qbinom{n}{k+\\sigma}}\n  \\frac{[\\sigma+k]_{(k)}}{\n[T;k]_{(k)}}\\frac{[r+k]!}{ [k]![2r]!}\\left[\\begin{matrix} m & n & 2r\n  \\\\ i & {n-k-\\sigma} & r-k \\end{matrix}\\right]T^{k}m_{0,1}\n\\end{align*}\n\nby \\lemref{highestlowestwtvectors} and \\corref{betaonabasis}. \nThe right hand side is \n\\begin{align*}\n&v^{-2(n-\\sigma)(\\sigma+1)}\\sum_{k=0}^{n-\\sigma}(-1)^k\\frac{v^{k(1-n+2\\sigma)}[\\sigma+k]_{(k)}}{T^k[T,k]_{(k)}} \\\\\n&\\hskip 100pt\\times \\left(\\rho_1\\circ T'_{-1}\\circ\\beta\n  \\right)\\left(T''_1(u^{(m)}_i)\\otimes T''_1(u^{(n)}_{\\sigma+k})\\right) F^{(-k)}m_{0,1}  \\\\ \n&\\quad\\times  \\sum_{l=0}^{k+r}(-1)^{r-l}\n  v^{  l (r -k+1) }\n    \\left[\\begin{matrix}T;k \\\\ l \\end{matrix}\\right]\n    \\left[\\begin{matrix}T;r+k-l \\\\ r+k-l \\end{matrix}\\right]\n    \\left[\\begin{matrix}r+k-l \\\\ r \\end{matrix}\\right]m_{0,1}  \\\\ \n&=(-1)^{m-i+r}\\frac{v^{i(i-1)+ m - 2\\,n + m\\,n + \\sigma( \\sigma+2 - m - n ) + r^2 - \\frac{(m+n)^2}{4}}}{||u^{(2r)}||^2}{\\qbinom{m}{i}}\n\\left[T; r \\right]_{(r)}\\\\   \n&\\quad \\times\\sum_{k}(-1)^k\\frac{[\\sigma+k]_{(k)}\n  v^{k\\left(1 + 2\\,i - m \\right)}}{T^k\n[T;k]_{(k)}}\n   {\\qbinom{n}{\\sigma+k}} \n  \\frac{[r+k]!}{ [k]![2r]!}\\left[\\begin{matrix} m & n & 2r\n  \\\\ m-i & {\\sigma+k} & r+k \\end{matrix}\\right]\n    m_{0,1}  \\\\\n\\end{align*}\nwhich follows from the Chu-Vandermonde formula, \\lemref{chuvandermonde}.\n\nThus to prove they are equal we need for $2\\sigma-2i+m-n=0$ and $|m-n|\\leq 2r\\leq m+n$, that\n\\begin{align*}\n&v^{(2\\sigma-n-1)\\sigma + r(1 + r)+ \\left(\\frac{m+n}{2}-\\sigma\\right)n }\\\\ \n&\\quad \\times\\sum_{k=0}^{n-\\sigma}  (-1)^k \n  v^{-k} \n  \\frac{[\\sigma+k]_{(k)}}{[r]!}\n     {\\qbinom{n}{\\sigma+k}}\\qbinom{r}{k}\\frac{[r-k]![r+k]!}{[2r]!}\n\\\\\n&\\hskip 50pt\\times \\left[\\begin{matrix} m & n & 2r\n  \\\\ \\frac{m-n}{2}+\\sigma & {n-k-\\sigma} & r-k \\end{matrix}\\right][T;r]_{(r-k)}T^{k}\\\\ \n&=(-1)^{\\frac{m+n}{2}+r}\\sum_{k=0}^{\\min\\{r, n-\\sigma\\}}(-1)^kv^{k\\left(1 + 2\\sigma-n \\right)}\\frac{[\\sigma+k]_{(k)}}{[r]!}\n     {\\qbinom{n}{\\sigma+k}}\\qbinom{r}{k}\\frac{[r-k]![r+k]!}{[2r]!}\t\\\\\n&\\hskip 50pt \\times\\left[\\begin{matrix} m & n & 2r\n  \\\\ \\frac{m+n}{2}-\\sigma & {\\sigma+k} & r+k \\end{matrix}\\right][T;r]_{(r-k)}T^{-k}.\n\\end{align*}\nIn order to prove that these are equal we need to expand in powers of $T$ and then show that the coefficients are\nequal:\n\\begin{align*}\n&v^{\\frac{2\\,n+n^2-(2m+m^2)}{4}+ n-2\\sigma + \\frac{3r(r+1)}{2}}\\\\ \n&\\quad \\times\\sum_{l\\geq 0}(-1)^lv^{-l(r+1)}\\sum_{k=0}^{n-\\sigma}  (-1)^{k} \n \\frac{[r+k]!}{[n-\\sigma-k]![k]!}\\left[\\begin{matrix} m & n & 2r\n  \\\\ \\frac{m-n}{2}+\\sigma & {n-k-\\sigma} & r-k \\end{matrix}\\right] \\\\ \n&\\hskip 100pt \\times\\frac{v^{-\\frac{k(k+2l+3)}{2}}}{ (v-v^{-1})^{r-k}} \\qbinom{r-k}{l}T^{r-2l}  \\\\ \n&=(-1)^{\\frac{m+n}{2}+r}v^{\\frac{r(r+1)}{2}}\\sum_{l\\geq 0}(-1)^lv^{-l(r+1)}\\sum_{k=0}^{ n-\\sigma}\n \\frac{v^{\\frac{k\\left(3+k-2l +\n4\\sigma-2n+2r\\right)}{2}}}{(v-v^{-1})^{r-k}} \\frac{[r+k]!}{[n-\\sigma-k]![k]!} \\\\\n&\\hskip 150pt \\times\n  \\left[\\begin{matrix} m & n & 2r\n  \\\\ \\frac{m+n}{2}-\\sigma & {\\sigma+k} & r+k \\end{matrix}\\right]\\qbinom{r-k}{l-k}T^{r-2l} \\\\\n\\end{align*}\nThis reduces to the follwing identity between Clebsch-Gordan coefficients that must hold for all $m$, $n$ with $m+n$ even, $0\\leq l\\leq r$, $|m-n|\\leq 2r\\leq\nm+n$,\n$0\\leq \\sigma\\leq n$:\n\\begin{align}\n&v^{\\frac{2n+n^2-(2\\,m+m^2)}{4}+n- 2\\,\\sigma +r(r+1)}\\label{byinduction}\\\\ \n&\\quad \\times\\sum_{k=0}^{ n-\\sigma}  (-1)^{k}v^{-\\frac{k(k+2l+3)}{2}}(v-v^{-1})^{k}\\frac{[\\sigma+k]_{(k)}[r+k]!}{[k]!}\\notag \\\\\n&\\hskip 100pt \\times     {\\qbinom{n}{\\sigma+k}}\\qbinom{r-k}{l} \n      \\left[\\begin{matrix} m & n & 2r\n     \\\\ \\frac{m-n}{2}+\\sigma & {n-k-\\sigma} & r-k \\end{matrix}\\right]\\notag \\\\ \\notag \\\\\n&=(-1)^{\\frac{m+n}{2}+r}\\sum_{k=0}^{n-\\sigma}v^{\\frac{k\\left(3+k-2l+4\\sigma-2n+2r\\right)}{2}}\n   (v-v^{-1})^{k}\\frac{[\\sigma+k]_{(k)}[r+k]!}{[k]!}  \\notag\\\\ \n&\\hskip 100pt \\times     {\\qbinom{n}{\\sigma+k}}\n  \\qbinom{r-k}{l-k} \\left[\\begin{matrix} m & n & 2r\n  \\\\ \\frac{m+n}{2}-\\sigma & {\\sigma+k} & r+k \\end{matrix}\\right].\\notag\n\\end{align}\n\nThe above identity follows from \\lemref{tediouslemma}.\n\n\\end{proof}\n\n\\subsection{}\\label{decompositionsection} Let us return to the setting of section one.  Fix a finite dimensional  $X$-admissible \n $\\mathbf U$-module $\\mathcal F$  with highest weight $v^n$ and let $M$ be\nthe Verma module of highest weight $Tv^{-1}$.  Recall from \\cite[{\\S 2}]{MR96c:17022} the\nmodules $P(m+\\lambda):=P_{m+\\lambda}$. Then we have the decomposition \n$M\\otimes_R\\mathcal F =\\sum_iP(m+i)$ where the sum is over the nonnegative\nweights of $\\mathcal F$ and  by convention we set $P(m)=M(m)$. Set\n$P_i=P(m+i)$ and following the notation of \\cite[3.6]{MR96c:17037} let $\\mathbb Z_i$ equal the\nset of integers with the opposite parity to $i$. For $j\\in \\mathbb Z_\\iota$ , set\n$z^i_j = w_{\\lambda,j} + w_{-\\lambda,j}$. Then for $i\\in \\mathbb N^*$, the set $\\{ [T;0] w_{\\lambda,j} :j\\in\n\\mathbb Z_i \\} \\cup \\{ z^i_j : j\\in \\mathbb Z_\\iota \\}$ is an $R$ basis for the \nlocalization $P_{i,F}$. Also the action of ${_R{\\mathbf U}}$ is given by the formulas\nin \\lemref{locstructure} as well  as the formulas : for all indices\n$j\\in\n\\mathbb Z_\\lambda$,\n\\begin{gather*}\\label{Prelations}\nK_\\mu z_j^i = T\\, v^{\\angles{\\mu, s\\lambda-\\rho-l_ji'}}z_j^i, \\\\ F\nz_j^i = z_{j-2}^i, \\\\ E z_j = [l_j][T;-k_j] z_{j+2}^i +[k_j-l_j][T;0] w_{\\lambda ,j+2}.\n\\end{gather*}\n\n\n\nFix a positive weight $v^r$ of $\\mathcal F$ and let $P=P_r$. Set \n$\\mathfrak L$ equal to\nthe\n$m-r$-th weight space of $P$. Then $\\mathfrak L$ is a free rank two $R$-module with\nbasis $\\{z^r_{-r-1},[T;0] w_{r,-r-1}\\}$. Define an $s$-linear map\n$\\Gamma$ on $\\mathfrak L$ and constants $a_{\\pm r}$ by the formula: \n\\begin{equation}\\label{ar}\n\\Gamma ([T;0] w_{\\epsilon r,-r-1}) = \\bar\\Psi([T;0] w_{\\epsilon r,r+1}) = a_{\\epsilon r}\\ [T;0]\nw_{-\\epsilon r,-r-1}.\n\\end{equation}\n\n\\def\\overline{\\overline}\n\nThis s-linear map $\\Gamma$ is the mechanism by which we analyze the symmetries\nwhich arise through the exchange of\n$\\mathfrak L\\cap ([T;0]\\cdot M(m+r))$ and $\\mathfrak L\\cap ([T;0]\\cdot M(m-r))$. This is a\nfundamental calculation for all which follows. Set $\\overline{\\Gamma}=[r]!\\\n\\Gamma$.\n\n\\begin{lem}\\label{exchange} Let $\\epsilon=\\pm 1$.  For $a,b\\in \\mathfrak L\\cap M(m+\\epsilon r)$, we have: \n$$\n\\chi^\\sharp(\\Gamma a,\\Gamma b)=\\frac{1}{ T^r [r]![T^{-\\epsilon};r]_{(r)}}\\ s\\chi(a,b)\\quad\nand \\quad\n\\chi^\\sharp(\\overline\\Gamma a,\\overline\\Gamma b)=u_\\epsilon\\ s\\chi(a,b),\n$$  where $u_\\epsilon$ is a unit and $u_\\epsilon\\equiv 1 \\mod (T-1)$. \n\n\\end{lem}\n\\begin{proof} Using the defining identity for localization\n\\cite[4.3.2]{MR96c:17022}, we obtain, for any invariant form\n$\\chi$ on\n$P$ and\n$\\epsilon=\\pm 1$,\n\\begin{align*}\na_{\\epsilon r}^2\\ \\chi^\\sharp ([T;0] w_{-\\epsilon r,-r-1},&[T;0]\nw_{-\\epsilon r,-r-1})\n        = \\chi^\\sharp (\\Gamma [T;0] w_{\\epsilon r,-r-1},\\Gamma [T;0]\nw_{\\epsilon r,-r-1}) \\\\\n        &=s\\chi_{_F}([T;0] w_{\\epsilon r,r+1}, [T;0]\nw_{\\epsilon r,r+1})  \\\\\n        &=\\frac{1}{T^r [r]![T^{-\\epsilon};r]_{(r)}} s\\chi([T;0]\nw_{\\epsilon r,-r-1}, [T;0] w_{\\epsilon r,-r-1}) \\ .\n\\end{align*}  This implies the lemma.\n\\end{proof} \n\n\\subsection{} Now we turn to the delicate calculation of the constant\n$a_{\\pm r}$.\n\n\n\n\\begin{lem}\\label{delicate} We may choose a basis for $P_r$ satisfying the relations in\n\\eqnref{locstructure} and \\eqnref{Prelations}, \ndependent only on the cycle $\\bar\\Psi$, and such that the\nconstants $a_{\\pm r}$ are uniquely determined by the three relations:\n$$ \na_{-r}=s\\ a_r\\ , \\quad a_r^2=\\frac{1}{\n[r]![T^{-1};r]_{(r)}}\\quad and\n\\quad a_r\\equiv \\frac{ (-1)^{r+1}}{ [r]!}\n\\mod\\ T-1\\ .\n$$\n\\end{lem}\n\\begin{cor}\\label{matrixcor} For $\\epsilon=\\pm 1$,\n$\\overline\\Gamma ([T;0] w_{\\epsilon r,-r-1}] = b_{r,\\epsilon} [T;0] w_{-\\epsilon r,-r-1}\\ $,  where\n$b_{r,\\epsilon}$ is the unit determined by conditions: \n$$ \nb_{r,\\epsilon}^2=\\frac{[r]!}{ [T^{-\\epsilon};r]_{(r)}}\\quad and\n\\quad b_{r,\\epsilon}\\equiv -1-\\epsilon\\alpha(r)(T-1)\\ \\mod\\ (T-1)^2\\ \n$$ \nwhere $\\displaystyle{\\alpha(r)=\\frac{r}{2}-\\{1\\}^{-1}\\sum_{s=1}^r\n\\frac{v^s}{[s]}}$ (see \\eqnref{matrixentry}).  Moreover $\\overline\\Gamma$ induces a $k(v)$-linear map on\n$\\mathfrak L\/\n\\ (T-1)\\cdot\n\\mathfrak L$ given by the matrix\n\\begin{equation}\n\\begin{pmatrix}\n 1&-\\alpha(r)\\{1\\} \\\\  0&1\n\\end{pmatrix}.\n\\end{equation}\n  Moreover, if\n$x_\\epsilon\\in M(m+\\epsilon r)$ and $[T;0]\\cdot x_\\epsilon$ is an $R$-basis vector for $\\mathfrak L\\cap\nM(m+\\epsilon r)$, then $\\{[T;0]\\cdot x_\\epsilon,x_\\epsilon +\\overline \\Gamma x_ \\epsilon\\}$ is an $R$-basis\nfor $\\mathfrak L$ and $x_\\epsilon+\\overline \\Gamma x_\\epsilon$ generates the\n$\\ru $-submodule $P_r$.\n\\end{cor}\n\\begin{proof} The first identity is a direct consequence of the\nlemma. As for the second set\n$z=z^r_{-r-1}$ and $w_\\pm=w_{\\pm r,-r-1}$ and let $\\pi$ denote the\nprojection $\\pi:\\mathfrak L\\rightarrow \\mathfrak L\/\\ (T-1)\\cdot \\mathfrak L$.  Then $\\mathfrak L$ has $R$-basis $\\{[T;0]\nw_+,z\\}$ and since $z=w_++w_-$, we obtain $\\pi([T;0] w_+)= -\\pi([T;0] w_-)$.\nSo $\\overline \\Gamma([T;0] w_+)=b_{r,1}[T;0] w_-\\equiv -b_{r,1}[T;0] w_+\\equiv [T;0]\nw_+$ mod \n$(T-1)\\cdot \\mathfrak L$. This gives the first column in the matrix.\n\nFrom the $s$-linearity of $\\overline\\Gamma$ and the congruence for the unit \n$b_{r,\\epsilon}$ we obtain: \n\\begin{align*}\n\\overline\\Gamma z&=-b_{r,1}w_--b_{r,-1}w_+\\equiv z +\\alpha(r)((T-1) w_--(T-1) w_+) \\\\\n&\\equiv z-2\\alpha(r)(T-1) w_+\n\\equiv z- \\alpha(r)\\{1\\}[T;0] w_+ \\mod (T-1)\\cdot \\mathfrak L\n\\end{align*} \n\nTo prove the identity for $\\alpha(r)$ just solve $\\displaystyle{b_{r,\\epsilon}^2=\\frac{[r]!}{ [T^{-\\epsilon};r]_{(r)}}}$ using \\eqnref{matrixentry}.\nFinally a short calculation shows that the transition matrix from the basis\n$\\{z,[T;0]w_\\epsilon\\}$ to \n$\\{w_\\epsilon +\\overline \\Gamma w_\\epsilon,[T;0]w_\\epsilon\\}$ is \n\\begin{equation}\\label{transistionmatrix}\n \\begin{pmatrix}\n1+[T;0]\\overline u_\\epsilon &0\\\\-\\overline u_\\epsilon&1\n\\end{pmatrix} \n\\end{equation}\nwhere $\\overline u_\\epsilon$ is the unit\n$$\n(-b_{r,\\epsilon}-1)\/[T;0]\\equiv \\epsilon\\frac{\\alpha(r)T(v-v^{-1})}{ 1+T}\\equiv \\frac{\\epsilon}{\n2}\\alpha(r)\\{1\\}\\mod (T-1).\n$$  If $[T;0]\\cdot x$ is\na basis vector of $\\mathfrak L\\cap M(m+\\epsilon r)$ then\n$x=uw_\\epsilon$ for some unit $u$ and the transition matrix takes the\nform as in \\eqnref{transistionmatrix},  with the unit $\\overline u_\\epsilon$ replaced by\nthe element of $R$ equal to\n$(-(su)b_{r,\\epsilon}-u)\/[T;0]$.  The determinant of this matrix is a\nunit which implies we have a basis.  This completes the proof of the\ncorollary.\n\\end{proof}\n\n\n\n\n\n\n\n\\begin{proof} We begin with any basis satisfying \\lemref{locstructure} and \\eqnref{Prelations}\nand let $a_{\\pm r}$ be given by \\eqnref{ar}. Since the vector $z^r_{r+1}$\nprojects to a cyclic vector of its generalized eigenspace in \n$M_\\pi\\otimes_R \\mathcal F$, we find that\n$\\Psi(z^r_{r+1})=-a_rw_{-r,-r-1}-a_{-r}w_{r,-r-1}$ is a cyclic vector of\n$P$. \nThus both\n$a_{\\pm r}$ are units and $a_r\\equiv a_{-r}\\mod T-1$.\nAny automorphism $\\kappa$\nis given by multiples of the identity $\\beta\\cdot 1$ on $M(m+r)$ and\n$\\gamma\\cdot 1$ on $M(m-r)$ with both $\\beta$ and $\\gamma$  units and\n$\\beta\\equiv \\gamma\\mod T-1$. Set $w_\\epsilon=w_{\\epsilon r,-r-1}$.  Then $\\Gamma(\\kappa\nw_+)=\\Gamma(\\beta w_+)=s\\beta\\Gamma (w_+)= -\\frac{s\\beta}{ \\gamma}a_r\\kappa\nw_-$ and similarly,\n$\\Gamma(\\kappa w_-)= -\\frac{s\\gamma}{ \\beta }a_{-r}\\kappa w_+$. From the\npreceding remark\n$\\frac{sa_r}{ a_{-r}}\\equiv 1\n\\mod T-1$ and so we set $s\\gamma$ equal to the square root of this quotient\nwhich is also congruent to $1\\mod T-1$. Put $\\beta=1$. If $\\kappa$ is the\nautomorphism of $P$ corresponding to this choice of $\\beta$ and $\\gamma$,\nthen  applying $\\kappa$ to the original basis of $P$ gives a basis for which\nthe constants $a_{\\pm r}$ satisfy the  first identity of the lemma $\na_{-r}=s\\ a_r$. \n\n\n\nFix a form $\\phi_M$ on $M$ with $\\phi_M=\\phi_M^\\sharp$ and let $\\phi_{\\mathcal F}$ be an invariant form on \n$\\mathcal F$. Set\n$\\phi=\\phi_M\\otimes \\phi_{\\mathcal F}$. Using the invariance (\\propref{invariance}) for $\\phi_\\pi$\nwe now check that\n $\\phi=\\phi^\\sharp$.  As in the proof of \\thmref{firstinvariance} we need only check on certain basis vectors:\n\\begin{align*}\n\\phi^\\sharp(&\\bar\\Psi(m_{0,1}\\otimes u^{(n)}),\\bar \\Psi(F^{(-i)}m_{0,1}\\otimes u^{(n)}_i))= s\\phi_\\pi\\circ L(m_{0,1}\\otimes u^{(n)}\\otimes \nF^{(-i)}m_{0,1}\\otimes u^{(n)}_i) \\\\\n&=\\sum_pv^{-p(p-1)\/2}\\{p\\} \\\\ \n&\\qquad \\sum_{p=r'+r''}v^{p-r'r''} s[\\phi_{M,\\pi}(m_{0,1},F^{(r')}F^{(-i)}m_{0,1})\n  \\phi_{\\mathcal F}( u^{(n)}_p,K^{-r'}F^{(r'')}u^{(n)}_i) ] \\\\\n&\\hskip 100pt \\text{by \\cite[Theorem 5.1]{MR96c:17022}}\\\\ \n&=v^{2\\,i^2 - i\\,n}\\sum_pv^{\\frac{-p(p-3)}{2} + p( i - n)+p^2}\n  \\{p\\}\\qbinom{p}{p-i}\\qbinom{n}{p}T^{-i}\\qbinom{T^{-1};i}{i}s[\\phi_{M,\\pi}(m_{0,1},m_{0,1}) ] \\\\ \n&\\hskip 100pt\\text{ by \\lemref{binomiallemma}  and \\eqnref{normalizedforms}} \\\\ \n&=v^{2\\,i^2 - i\\,n}\n\\sum_pv^{\\frac{p(p+3)}{2}+p(i-n)}\\{p\\}\\qbinom{n-i}{p-i}T^{-i}\\qbinom{n}{i}\\qbinom{T^{-1};i}{i}\\phi_{M}(\\Psi(m_{0,1}),\\Psi(m_{0,1})).\n\\end{align*}\nOn the other hand\n\\begin{align*}\n&\\phi(\\bar\\Psi(m_{0,1}\\otimes u^{(n)}),\\bar\\Psi(F^{(-i)}m_{0,1}\\otimes u^{(n)}_i)) \\\\\n&=v^{\\frac{i(i-1)}{2}+i^2+2n}\\{i\\}T^{-i}\\qbinom{n}{i}\\qbinom{T^{-1};i}{i}\\phi_M(\\Psi(m_{0,1}),\\Psi(m_{0,1}))\\phi_{\\mathcal F}\n  (u^{(n)},u^{(n)}).\n\\end{align*}\nNow if we use \\corref{link} with $p=0$ and replace $k$ by $p$ and $s$ by $i$ we get\n\\begin{equation*}\nv^{i(1-r)-2n}\\{i\\}=\\sum_{p=i}^{n}(-1)^{p+i}v^{p(r-2p-1)+\\frac{(p - i)(3p + i-1) }{2}}\\{p\\}\\qbinom{n-i}{p-i}.\n\\end{equation*}\nUsing the automorphism of $k(v)$ induced by $v\\mapsto v^{-1}$ we get\n\\begin{align*}\nv^{2n-i(1-r)}\\{i\\\n&=v^{\\frac{i^2-i}{2}}\\sum_{p=i}^{n}v^{ \\frac{p^2+3p}{2} +  p\\,\\left(i - r\\right)}\\{p\\}\\qbinom{n-i}{p-i}.\n\\end{align*}\nTaking $r=n$ then \n\\begin{equation*}\nv^{2i^2-i\\,n }\\sum_pv^{\\frac{p(p+3)}{2}+p(i-n)}\\{p\\}\\qbinom{n-i}{p-i\n=v^{2n+i^2+\\frac{i^2-i}{2}}\\{i\\}.\n\\end{equation*}\nThis completes the proof that $\\phi=\\phi^\\sharp$.\n\n\nNow from \\eqnref{ar}, \n\\begin{align*}\na_{\\epsilon r}^2\\ \\phi^\\sharp(w_{-\\epsilon r,-r-1},w_{-\\epsilon r,-r-1}) & =\ns\\phi_\\pi(w_{\\epsilon r,r+1}, w_{\\epsilon r,r+1}) \\\\\n&=\\frac{1}{ [r]!T^r\\ [T^{-\\epsilon};r]_{(r)}}s\\phi(w_{\\epsilon r,-r-1}\n,w_{\\epsilon r,-r-1}).\n\\end{align*}\n\nUsing this identity twice with $\\phi=\\phi^\\sharp$ we obtain: \n\\begin{equation}\na_r^4=(a_r\\ sa_{-r})^2=\\left(\\frac{1}{ [r]!\\\n[T^{-1};r]_{(r-1)}}\\right)^2.\n\\end{equation}\nThis gives the\nsecond identity of \\lemref{delicate} up to a sign. The correct sign is implied by the third\nidentity which we now prove.\n\\end{proof} \n\n\n\\subsection{} To verify the correct choice of sign for the third identity we\nshall need some preliminary lemmas. Let $M^\\prime$ denote the span of all\nthe weight subspaces of $M$ other than the highest weight space. Let $\\delta$\ndenote the projection of $M\\otimes_R \\mathcal F$ onto\n$w_{0,-1}\\otimes \\mathcal F$ with kernel $M'\\otimes_R \\mathcal F$. Define constants\n$c_\\pm$ by the relations:\n $\\delta(w_{r,r-1})\\equiv c_+ w_{0,-1}\\otimes u^{(n)}_k\\ \\mod \\ M'\\otimes\n_R\\mathcal F$  and\n$\\delta(w_{ -r,-r-1})\\equiv c_- w_{0,-1}\\otimes u^{(n)}_l\\ \\mod \n\\ M'\\otimes_R\\mathcal F$ where $n-2p-1=r-1$ and $n-2l-1=-r-1$. For any integer\n$t$ set $z_t=w_{0,-1}\\otimes u^{(n)}_t$. In a similar fashion define the\nprojection\n$\\delta^\\vee$ of $M_F\\otimes _R\\mathcal F$ onto $w_{0,1}\\otimes _R\\mathcal F$ with\nkernel \n$M^\\vee\\otimes _R\\mathcal F$ and $M^\\vee$ equal to the span of all weight\nsubspaces in $M_F$ for weights other than $m+1$. \n\n\n\n\\begin{lem}\\label{lastlemma} For $r$ a non-negative integer\n\\begin{align}\n\\delta^\\vee(w_{r,r+1})&=c_+\\ \\frac{v^{(n-k)(k+1)}\n[T^{-1};2k-n-1]_{(p)}}{ [T^{-1};p]_{(p)}} \\ w_{0,1} \\otimes u^{(n)}_p, \\\\\n  -\\frac{c_-}{ [l]!}&\\equiv \\frac{c_+}{ [p]!}\\mod T-1, \\notag \\\\\n\\text{and}\n\\quad\n\\delta(\\bar\\Psi w_{r,r+1})&\\equiv c_-\\ \\frac{(-1)^{r+1}v^{2l(l-r+1)}}{ [r]!} z_l\\mod\n(T-1)\\cdot P_r. \\notag\n\\end{align}\n\n\\end{lem}\n\\begin{proof} Let $\\mathcal F$ be an $\\mathbf U$-module that is $X$-admissible and finite dimensional, then we have the expansion (see\n\\cite[3.9.1]{MR96c:17022})\n\\begin{equation}\nF^{-s}\\otimes a \\otimes e \\ \\mapsto \\  \\sum_{j \\in  \\mathbb N}(-1)^j\n\\qbinom{j+s-1}{ s-1}_i v^{-j(j+s)}\\  F^{-j-s}a \\otimes F^jK^{j+s}e \\ ,\n\\end{equation}\n\nRecall from \\lemref{highestlowestwtvectors} \n\\begin{equation}\nw_{r,r-1}=c_+\\sum_{0\\le j\n\\le p}\\frac{[n-p+j]_{(j)}v^{-j^2}}{T^{-j}\n[T^{-1};j]_{(j)}} F^{(j)}w_{0,-1}\\otimes u^{(n)}_{p-j}\n\\end{equation}\nfor $r=n-2p$.\n Then \n\\begin{align*}\n&F^{-1}w_{r,r-1} \\\\\n&=c_+\\sum_{j=0}^pv^{-j^2+(j+1)(n-2p+2j)}\\frac{[n-p+j]_{(j)}}{T^{-j}\n  [T^{-1};j]_{(j)}[j]![p-j]!} \\\\\n&\\hskip 100pt\\times \\Big(\\sum_{l \\in  \\mathbb N}(-1)^l v^{-l(l+1)} [p-j+l]!  F^{j-l-1}w_{0,-1}\\otimes u^{(n)}_{p-j+l}\\Big).\n\\end{align*}\nSet $u= w_{0,1}\\otimes u^{(n)}_p$.\n\\begin{align*}\n\\delta^\\vee(F&^{-1}w_{r,r-1})\n        =\\frac{{c_+} v^{(n-p)(p+1)}}{ [T^{-1};p]_{(p)}}\\\n[T^{-1};2p-n-1]_{(p)}\\\n\\cdot u .\n\\end{align*}  Here we have used the binomial identity given in \\lemref{thirdbinomial}.  This implies \n\\begin{align*}\nw_{r,r+1}=F^{-1}w_{r,r-1}\n        &=\\frac{{c_+} v^{(n-p)(p+1)}}{ [T^{-1};p]_{(p)}}[T^{-1};2p-n-1]_{(p)}m_{r,r+1} .\n\\end{align*}  \nThus by \\eqnref{needlater}\n\\begin{align*}\n\\bar\\Psi(w_{r,r+1}\n&=(-1)^{n-p} \\frac{{sc_+} v^{-(n-p)(p+1)}}{[T;p]_{(p)}}[T;2p-n-1]_{(p)} \\\\\n&\\quad \\quad \\times \\sum_{s=0}^{n-p}\\frac{v^{s(1-r)+(n-p-s)(p+s+1)}[p+s]_{(s)}}{T^{-s}[T^{-1},s]_{(s)}}\n   F^{(s)}w_{0,-1}\\otimes u^{(n)}_{n-p-s}   \\\\\n&=(-1)^{n-p} \\frac{{sc_+} }{[T;p]_{(p)}}[T;2p-n-1]_{(p)} \\\\\n&\\quad \\quad \\times \\sum_{s=0}^{n-p}\\frac{v^{-s^2}[p+s]_{(s)}}{T^{-s}[T^{-1},s]_{(s)}}\n   F^{(s)}w_{0,-1}\\otimes u^{(n)}_{n-p-s}  \\\\\n&\\color{red}=(-1)^{n-p} \\frac{{sc_+} }{[T;p]_{(p)}}[T;2p-n-1]_{(p)} w_{-r,-r-1}\\color{black}\n\\end{align*}\n\nIn $P_r$ we know from the basis that $F^rw_{r,r-1}\\equiv -w_{-r,-r-1}\\mod\n\\ (T-1)\\cdot P_r$.  Since the action of\n$F$ commutes with $\\delta$,\nwe obtain: $[p]!\\ {c_-}\\equiv -[l]!\\ {c_+} \\mod\\\nT-1.$ \nFor the last identity of the Lemma apply $\\delta$ to the equation above. We get \n\\begin{align*}\n\\delta(\\bar\\Psi w_{r,r+1})\n&\\equiv(-1)^{r+1}c_-\\ \\frac{1}{[r]!}\\ z_l \\end{align*}  \nHere we use $r+p=l=n-p$ and $[T;r]\\equiv [r]\\mod (T-1)$.\n\n\\end{proof}\n\nWe now return to the proof of the congruence. Since $\\Psi(w_{r,r-1})=-a_r\nw_{-r,-r-1}$  we can calculate the constant $a_r$ as the ratio of\n$\\delta(\\Psi(w_{r,r+1}))$ and $\\delta( w_{-r,-r-1})$.  From \\eqnref{lastlemma} we find the ratio\nis congruent to $\\frac{(-1)^{r+1}}{ [r]!}\\mod T-1$. This completes the proof\nof \\lemref{delicate}.\n\n\\subsection{} Recall from (3.3) the category $_R\\mathcal C_i$ and note\nthat any module $N$ in the category is the direct sum of generalized\neigenspaces for the Casimir element \\cite{MR96d:17015} in the sense that \n$N=\\sum N^{(\\pm r)}$ where the sum is over $\\mathbb N$ and $N^{(\\pm r)}$\ncontains all highest  weight vectors in $N$ with weights $m+r-1$ and\n$m-r-1$. Note that $N^{(\\pm r)} $ need not be generated by its highest\nweight vectors. The decomposition in \\secref{decompositionsection}, $M\\otimes_R\\mathcal F\\equiv \\sum\nP_i$ where the sum is over the nonnegative weights of $_R\\mathcal F$ is such a\ndecomposition. In this case $(M\\otimes _R\\mathcal F)^{(\\pm i)}= P_i$.  The\nCasimir element $\\Omega_0$ of $_R\\mathbf U$ by \n$$\n\\Omega_0 =F E + \\frac{vK-2+v^{-1} K^{-1}}{( v-v^{-1})^2}.\n$$   \nLet $N^{(r)}$ (resp. $N^{(-r)}$) denote the submodule of $N$ where the\nCasimir element acts by the scaler \n$$ c(\\lambda)=\\frac{v^{r-1}T-2+v^{-r+1}T^{-1}}{( v-v^{-1})^2\n\n\\quad \\text{\\rm resp.}\\quad  c(s\\lambda)= \\frac{v^{-r+1}T-2+v^{r-1} T^{-1}}{ (\nv-v^{-1})^2\n$$\n\n\\subsection{} We now turn to the general case where $\\mathcal F$ is a finite\ndimensional ${{\\mathbf U}}$-module but not   necessarily irreducible. We extend the\ndefinition of the\n$s$-linear maps $\\Gamma$ and $\\overline \\Gamma$ defined in \\eqnref{ar} as follows. Decompose\n$M\\otimes_R \\mathcal F$  into generalized eigenspaces for the Casimir\n$(M\\otimes_R \\mathcal F)^{(\\pm r)}$ and let $\\mathfrak L^r$ denote the $m-r-1$ weight\nsubspace of\n$(M\\otimes_R \\mathcal F)^{(\\pm r)}$. Then set $\\mathbb L=\\sum \\mathfrak L^r$. Decompose\n$\\mathcal F=\\sum \\mathcal F_{n_j}$  into irreducible ${_R{\\mathbf U}}$ modules.  Then $M\\otimes_R\n\\mathcal F=\\sum M\\otimes_R \\mathcal F_{n_j}$ and so we obtain $s$-linear  extensions\nalso denoted $\\Gamma$ and \n$\\overline \\Gamma$ from $\\mathbb L\\cap (M\\otimes_R \\mathcal F_j)$ to all of $\\mathfrak L$.  Set $(M\\otimes _R\\mathcal F)^{(+)}:\n=\\oplus_{r\\in\\mathbb N^*} M\\otimes _R\\mathcal F^{(r)}$\n\n\n\\begin{prop} Suppose $\\phi$ is any invariant form on\n$M\\otimes_R \\mathcal F$ with $\\phi = \\pm \\phi^\\sharp$. Let $\\{w_j, j\\in J\\}$ be\nan $R$-basis for the\n highest weight space of $(M\\otimes_R \\mathcal F)^{(0)}$ and\n$\\{u_i, i\\in I\\}$ a basis of weight vectors for the $E$-invariant weight\nspaces of  weight $m+t$ for $t<-1$. Set\n$M_j$ equal to the ${_R{\\mathbf U}}$-module generated by $w_j$ and $Q_l$ the\n${_R{\\mathbf U}}$-module generated by $(T-1)^{-1}(u_l+\n\\Gamma(u_l))$. Then $M\\otimes_R \\mathcal F=\\sum_jM_j\\oplus \\sum_lQ_l$ where each\n$M_j\\cong M(m)$ and  if $u_l$ has weight $m-t$, then $Q_l\\cong P(m+t)$.\nMoreover, if the basis vectors\n$w_j$ and $u_l$ are $\\phi$-orthogonal then the sum is an orthogonal sum of\n${_R{\\mathbf U}}$-modules.\n\n\\end{prop}\n\\begin{proof}\n Since   $M\\otimes_R\n\\mathcal F=\\sum M\\otimes_R \\mathcal F_{j}$ we may apply \\corref{matrixcor} to each\nsummand to obtain an\n$R$-basis of weight vectors $\\{[T,0]\\cdot x_i\\}$ for $\\mathfrak L\\cap (M\\otimes\n_R\\mathcal F)^{(+)}$ for which $\\{[T,0]\\cdot x_i,x_i+\\overline\\Gamma x_i\\}$ is a basis for\n$\\mathfrak L$ and the\n$\\ru$-module generated by $\\mathfrak L$ is the direct sum of the submodules\ngenerated by the vectors $x_i+\\overline\\Gamma x_i$. If $x_i$ has weight\n$m+t_i-1$ then $\\ru\\cdot(x_i+\\overline\\Gamma x_i)\\cong P(m+t_i)$ and\nthe intersection of this module with $\\mathfrak L$ has $R$-basis $\\{[T;0]\\cdot\nx_i,x_i+\\overline\\Gamma x_i\\}$. \n\nNow let $A$ denote the transition matrix from the basis \n$\\{[T;0]\\cdot u_i, i\\in I\\}$ to the basis $\\{[T;0]\\cdot x_i, i\\in I\\}$.\nThen the determinant of $A$ is a unit of $R$. The block matrix\n\\begin{equation}\n\\begin{pmatrix}\n sA&0\\\\mathbb B&A\n\\end{pmatrix}\n  \\ ,\\quad \\text{for}\\  [T;0]\\cdot B=A-sA\n\\end{equation}\nis the transition matrix from $\\{u_i+\\overline\\Gamma u_i ,[T;0]\\cdot\nu_i\\}$ to $\\{x_i+\\overline\\Gamma x_i ,[T;0]\\cdot\nx_i\\}$. The determinant is a unit and so the former set is a basis\nof $\\mathfrak L$. From this we conclude $M\\otimes_R\\mathcal F=\\sum_jM_j\\oplus\n\\sum_iQ_i$ and each $\\mathfrak L\\cap Q_i$ has $R$-basis $\\{u_i+\\overline\\Gamma u_i,[T;0]\\cdot\nu_i\\}$. So if\nthe basis vectors\n$w_j$ and $u_i$ are $\\phi$-orthogonal then the sum $\\mathfrak L=\n\\sum_j\\mathfrak L\\cap M_j\\oplus\n\\sum_i\\mathfrak L\\cap Q_i$ is an\northogonal sum. It follows that $M\\otimes_R\\mathcal F=\\sum_jM_j\\oplus\n\\sum_iQ_i$ is an orthogonal sum.\n\\end{proof}\n\n\n\\subsection{Diagonalizing forms.}\n\nFix a non-zero form $\\phi$ on $A$, a free $R$-module of rank $n$. Choose an\ninteger $d_1$ with $\\phi(A,A)=R\\pi^{d_1}$ and choose vectors $a_1$ and\n$a_1^\\prime$ with $\\phi( a_1, a_1^\\prime)= \\pi^{d_1}$\n\n\\begin{lem} Suppose the form $\\phi$ is not zero. Set $A_1=\\{a\\in A |\n\\phi(a,a_1^\\prime)=0\\}$ and\n$A_1^\\prime=\\{a\\in A | \\phi(a_1,a)=0\\}$. Then we have direct sum\ndecompositions\n$$R a_1\\oplus A_1\\cong A\\cong R a_1^\\prime\\oplus A_1^\\prime\\ .$$\nMoreover if $\\phi$ is symmetric we may choose a unit of $R$, $u_1$ with\n$u_1 a_1= a_1^\\prime$.\n\\end{lem}\n\\begin{proof} For $a\\in A$ define\n$$P(x)= x-\\frac{\\phi(x,a_1^\\prime)}{\\phi(a_1,a_1^\\prime)} a_1\\ ,$$\n$$P^\\prime(x)= x-\\frac{\\phi(a_1,x)}{\\phi(a_1,a_1^\\prime)} a_1^\\prime\\ .$$\nThen $P$ (resp. $P^\\prime$) is the projection of $A$ onto $A_1$ (resp.\n$A_1^\\prime$). We have\n$$x= P(x)+\\frac{\\phi(x,a_1^\\prime)}{\\phi(a_1,a_1^\\prime)} a_1\\ \\quad,\\quad\nx=P^\\prime(x)+\\frac{\\phi(a_1,x)}{\\phi(a_1,a_1^\\prime)} a_1^\\prime\\ .$$\nSince $a_1\\notin A$ and $a_1^\\prime\\notin A^\\prime$,\nthese are the\ndesired decompositions.\nNow suppose that $\\phi$ is symmetric. \nConsider the line between $a_1$\nand $a_1^\\prime$,  $b_t=t a_1+\n(1-t)a_1^\\prime$.\nThen $\\phi(b_t,b_t)=t^2 \\phi(a_1,a_1)\n+2t(1-t)\\phi(a_1,a_1^\\prime)+(1-t)^2\\phi(a_1^\\prime,a_1^\\prime)$. Since $t^2,\n2t(1-t),$ and $(1-t)^2$ are linearly independent if we are not in\ncharacteristic two  we can find an open set\nof $t$ in the base field so that we have equality of ideals\n$R\\phi(b_t,b_t)=\\phi(A,A)$.    Alternatively first suppose $\\phi(a_1,a_1)=u_2\\pi^{d_1}$ with $u_2$ a unit.\nThen we have $\\phi(a_1,u_2^{-1}a_1)=\\pi^{d_1}$ and we have proved the remaining statement.  Similarly $\\phi(a_1',a_1')=u_2'\\pi^{d_1}$ with $u_2'$ a unit\nleads to the same conclusion.  Lastly we suppose $\\phi(a_1,a_1)=u_2\\pi^{d_2}$ and $\\phi(a_1',a_1')=u_2'\\pi^{d_2'}$ with $d_2,d_2'>d_1$.\nThen\n\\begin{align*}\n\\phi(b_t,b_t)&=t^2 \\phi(a_1,a_1)\n+2t(1-t)\\phi(a_1,a_1^\\prime)+(1-t)^2\\phi(a_1^\\prime,a_1^\\prime) \\\\\n&=(2t(1-t)+t^2 u_2\\pi^{d_2-d_1}+(1-t)^2u_2'\\pi^{d_2'-d_1})\\pi^{d_1} \\\\\n&=u_t\\pi^{d_1} \n\\end{align*}\nwhere $u_t$ is a unit for $t\\neq 0,1$.\n\n From this we find a unit $u_1\\in R$ such that\n$ \\phi(b_t,u_1 b_t) =u_1 \\phi(b_t,b_t)=\\pi^{d_1}$.\n\\end{proof}\n\n\nDirectly as a corollary to this lemma we have\n\n\\begin{cor}  There exist integers $m\\leq n$ and $d_i, 1\\leq\ni\\leq m$ and two bases for $A$, $\\{a_i|1\\leq i\\leq n\\}$ and $\\{a^\\prime_i|1\\leq i\\leq n\\}$ with the\nfollowing property\n$$\n\\phi(a_i,a_j^\\prime)=\\begin{cases}\\delta_{i,j}\\ \\pi^{d_i} & \\ i\\leq m,  \\\\\n0& \\ m< i\\leq n.\\end{cases}$$\n\n\nMoreover if $\\phi$ is symmetric we may choose the bases so that\n$a_i^\\prime=u_i a_i$ for some choice of units $u_i, 1\\le i\\le n$.\n\\end{cor}\n\n\n\\begin{proof}  If the form is zero then any two bases will suffice. So\nassume $\\phi$ is not zero and apply the lemma obtaining vectors $a_1$\n, $a_1^\\prime$ and submodules  $A_1$ , $A_1^\\prime$. Proceed inductively and\nassume for some\n$t$ with $ 1\\le t\\le n-1$ and $m'\\leq m$, we have chosen integers $d_i$ and vectors\n$\\{a_i\\}$ and\n$\\{a_i^\\prime\\}$ which satisfy the conditions \n$$\\phi(a_i,a_j^\\prime)=\\delta_{i,j}\\ \\pi^{d_i} \\ \\ i\\leq t$$\n$$ \\phi(a_i,a_j^\\prime)=\\delta_{i,j}\\ \\pi^{d_i} \\ \\ i\\leq m' $$\n$$\\phi(a_i,a_j^\\prime)=0\\  \\ ,\\ m< i\\leq t \\ .$$\n Also assume that for\n  $A_t=\\{a\\in A |\n\\phi(a,a_i^\\prime)=0, 1\\le i\\le t\\}$ and\n$A_t^\\prime=\\{a\\in A | \\phi(a_i,a)=0 , 1\\le i\\le t\\}$, we have direct\nsum decompositions\n$$\\sum_{1\\le i\\le t} R a_i\\oplus A_t\\cong A\\cong \\sum_{1\\le i\\le t} R\na_i^\\prime\\oplus A_t^\\prime\\ .$$\nNow choose $a_{t+1}$ and $a_{t+1}^\\prime$ so that\n$\\phi(a_{t+1},a_{t+1}^\\prime)=\\phi(A_t,A_t^\\prime)$. Then applying the lemma\ngives the following decomposition: $R a_{t+1}\\oplus A_{t+1}\\cong A_t$, \n$ R a_{t+1}^\\prime\\oplus A_{t+1}^\\prime\\ \\cong A_t^\\prime$. This completes the\ninductive step which in turn proves the corollary.\n\n\\end{proof}\n\n\n\\begin{cor}  Suppose $\\phi$ is symmetric. There exist integers\n$m\\leq n$ and\n$d_i$ and units $u_i$ of $R, 1\\leq i\\leq m$ and a basis for $A$, $\\{a_i\\}$ ,\nsuch that\n$\\phi$ is represented by the diagonal $n\\times n$ matrix $S$ with entries:\n$$S_{i,i}=u_i\\ \\pi^{d_i}\\quad  \\text{for}\\quad i\\leq m,\\quad \\text{and }\\quad S_{i,i}=0\\quad i>m  $$\n$$ S=\\text{diag}(u_1\\pi^{d_1},\\dots ,u_m\\pi^{d_m},0,\\dots, 0) $$\n\\end{cor}\n\nNote that we cannot get rid of the units in the symmetric form case since doing\nso would require a square root of each unit $u_i$. These square roots may not\nlie in the ring $R$.\n\n\\begin{prop}\nSuppose $\\phi$ is any invariant  symmetric  form on\n$M\\otimes_R \\mathcal F$ with $\\phi = \\pm \\phi^\\sharp$. Then $M\\otimes_R \\mathcal F$\nadmits an orthogonal decomposition with each summand an indecomposable\n$\\ru$-module and isomorphic to $M$ or some $P(m+t)$ for $t\\in\\mathbb N^*$.\n\\end{prop}\n\n\\begin{proof}   Since $R$ is a discrete valuation ring we may choose\nan orthogonal $R$-basis for the free $R$-module $\\mathfrak L\\cap (M\\otimes_R\\mathcal F)^{(+)}$.\n\\end{proof}\n\n\n\n\n\\section{Filtrations}\n\n\\subsection{} We continue with the notation of the previous section. So\n$\\phi$ is an invariant form on\n$M\\otimes_R \\mathcal F=\\sum_iP_i$. For any $R$-module $B$ set $\\overline\nB=B\/(T-1)\\cdot B$ and for any  filtration $B=B_0\\supset B_1\\supset\n...\\supset B_r$, let $\\overline B=\\overline B_0\\supset \\overline B_1\\supset\n...\\supset\n\\overline B_r$ be the induced filtration of $\\overline B$, with $\\overline\nB_i=( B_i+(T-1)\\cdot B)\/ (T-1)\\cdot B$.\nNow $\\phi$ induces a filtration on $M\\otimes_R \\mathcal F$ by\n\\begin{equation}\\label{filtration}\n(M\\otimes_R \\mathcal F)^i=\\{v\\in  M\\otimes_R \\mathcal F| \\phi(v,M\\otimes_R \\mathcal\nF)\\subset (T-1)^i\\cdot R\\}.\n\\end{equation}\n\n\n\n\\subsection{}\n Let notation be as in (8.3) with $P=P_r$ and $\\mathfrak L$ equal to the\n$m-r-1$ weight subspace of $P$. Suppose $P=A_0\\supset A_1\\supset ...\\supset\nA_t=0$ is a filtration. Then since the $U$-module $P\/(T-1)\\cdot P$ contains only the\ntwo proper subspaces $M(\\pm r)$ we can\nchoose constants $a\\ge b \\ge c$,  so that\n\\begin{align}\n\\overline P = \\overline A_0&=\\cdots=\\overline A_a \\\\\n&\\cup  \\notag\\\\\nM(r)\\cong \\overline A_{a+1}&=\\cdots =\\overline A_b \\notag  \\\\\n &\\cup \\notag \\\\\n M(-r)\\cong \\overline A_{b+1}&=\\cdots = \\overline A_c \\notag \\\\\n&\\cup   \\notag\\\\\n\\overline A_{c+1}&=0\\ .\\notag\n\\end{align}\nIn this case we say that the filtration is of {\\it type} $(a,b,c)$.   When $a=b=c$ we use the convention $\\overline P=M(0)$. \n\nRecall from \\secref{decompositionsection} the bases $w_{\\pm r,-r-1}$ of the $m-r-1$ weight space of $M(\\pm r)$\nand the basis $\\{ z_{r,-r-1}= w_{r,-r-1}+w_{-r,-r-1}\\ ,\\ [T;0] w_{r,-r-1} \\}$ of the\n$m-r-1$ weight space of $P$. Fix $r$ and for convenience set\n\\begin{equation}\nw_+=w_{r,-r-1}\\ ,\\ w_-=w_{-r,-r-1} \\ ,\\ z=w_+ +w_- \\ .\n\\end{equation}\n\nLet $\\phi_\\pm$ denote the Shapovalov form on the Verma module $M_{\\pm r}$ and\nnormalized by the identities\n\\begin{equation}\n\\phi_+(w_+,w_+)=1  \\quad \\quad  \\phi_-(w_-,w_-)=1\\ .\n\\end{equation}\nNow choose constants $b_\\pm\\in R$ with $\\phi|_P=b_+\\phi_+ +b_- \\phi_-\\ .$ Then the\nrestriction of $\\phi$ to $\\mathfrak L$ is given by the matrix $\\mathcal M$ with respect\nto the basis $\\{z,[T;0]w_+\\}$\n$$\\mathcal M=\n\\begin{pmatrix}\n\\phi(z,z)& \\phi(z, [T;0] w_+)\\\\ \\phi([T;0] w_+,z) & \\phi([T;0] w_+,[T;0] w_+)\n\\end{pmatrix}\n=\n \\begin{pmatrix}\nb_+ +b_- & \\  [T;0] b_+ \\\\  \\ [T;0] b_+ & \\ [T;0]^2 b_+\n\\end{pmatrix}\n$$\n\nDefine the order of elements $a\\in  \\mathcal K$ by $ord(a)=n$ if $a\\in R(T-1)^n$ and\n$a\\notin R(T-1)^{n+1}$. For a matrix define the order to be the minimum of the orders\nof the matrix entries. For any matrices $\\mathcal A$ and $\\mathcal B$ with entries in\n$R$, $ord (\\mathcal A \\mathcal B) \\ge ord \\mathcal A$ and if $det \\mathcal A$ is a\nunit then $ord (\\mathcal A \\mathcal B) = ord (\\mathcal B)$.\n\nThe type of the filtration on $P$ can now be determined easily by the constants\n$b_\\pm$. The result separates into three cases.\n\n\n\\begin{lem}\\label{formonP} \\begin{enumerate}[(a).]\n\\item Suppose $ord(b_+)=ord(b_-)< ord(b_++b_-)$. Then the\nfiltration on $P$ is of type $(a,a,a)$ with $a= ord(b_+)+1$.\n\\item Suppose $ord(b_++b_-)=ord(b_+)\\leq  ord(b_-)$. Then the\nfiltration on $P$ is of type\n$( ord(b_+), ord(b_+)+1, ord(b_-)+2 )$.\n\\item Suppose $ord(b_++b_-)=ord(b_-)\\leq  ord(b_+)$. Then the\nfiltration on $P$ is of type $( ord(b_-), ord(b_+)+1, ord(b_+)+2 )$.\n\\end{enumerate}\n\n\n\\end{lem}\n\\begin{proof} Let $(a,b,c)$ designate the type of filtration on $P$ induced by the\nform $\\phi$.  Diagonalize $\\mathcal M$ as follows. Choose invertible $R$-valued\n$2\\times 2$ matrices\n$\\mathcal U$ and $\\mathcal V$ and integers $d_1\\leq d_2$ so that\n$$\\mathcal U \\mathcal M \\mathcal V = \\begin{pmatrix} (T-1)^{d_1} & \\ \\\\  \\ &\n(T-1)^{d_2}\n\\end{pmatrix} = \\mathcal D\\ .  $$\nThen $ord(\\mathcal M)= ord(\\mathcal D)=d_1$ and $d_2=ord(det(\\mathcal D))-d_1=\nord(det(\\mathcal M))-ord(\\mathcal M)$. We conclude $a=d_1= ord(\\mathcal M)$ and so\nfrom the form of $ \\mathcal M$ above we get the formulas for $a$ in the three cases.\nSimilarly\n$c=d_2= ord(det(\\mathcal M))-ord(\\mathcal M)$ and in the three cases of the lemma this\ntranslates to the formulas in the lemma.\n\nNext we determine $b$. The highest weight space of $P$ is a free rank one $R$-module.\nSo $b= ord(\\phi([T;0] w_{r,r-1} , [T;0] w_{r,r-1}))$. Then for some unit $u\\in R$,\n$$\\phi([T;0] w_{r,r-1}\\ ,\\ [T;0] w_{r,r-1})=\\phi(u E^r z\\ ,\\ [T;0] w_{r,r-1})=\n\\phi(u z\\ ,\\ F^r [T;0] w_{r,r-1}) = \\ $$\n$$\\phi(u z\\ ,\\  [T;0] w_{r,-r-1}) = u [T;0] b_+\\ .$$\nSo the form restricted to the highest weight space has order $ord(b_+)+1$. This proves\n$b= ord(b_+)+1$ and completes the proof of the lemma.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Examples}    In this section we will roughly follow the notation in \\cite{MR94m:17016}.  \nA pair $(\\Pi,(\\enspace,\\enspace))$ where $\\Pi$ is a finite set and $(\\enspace,\\enspace)$ denotes a\nsymmetric bilinear form on the free abelian group $\\mathbb Z[\\Pi]$ with\nvalues in $\\mathbb Z$ is called a {\\it Cartan datum} if  \n\\begin{equation}\n(\\alpha, \\alpha)\\in\\{2,4,6,\\dots\\}\\text{ for  any } \\alpha \\in \\Pi ;\\quad 2\n\\frac{(\\alpha,\\beta)}{ (\\alpha,\\alpha)}\\in\\{0,-1,-2,\\dots\\}  \\text{ for any }  \\alpha\\neq \\beta\n\\in \\Pi.,\n\\end{equation}\nIf $P\\in \\Bbb Q(v)$ is a\nrational function then $P_\\alpha$ denotes $P(v_\\alpha)$ where $v_\\alpha=v^{(\\alpha,\\alpha)\/2}$\\label{Gaussintegers}.\n\\subsection{Verma modules for the quantum group $U_v(\\mathfrak{sl}(3))$.} Let \n\\[\n\\Phi=\\{\\pm \\alpha,\\pm\\beta,\\pm(\\alpha+\\beta)\\}\n\\]\n be the root system of $\\mathfrak{sl}_3(\\mathbb C)$ with $\\Pi=\\{\\alpha,\\beta\\}$ a set of simple roots, $W$ the Weyl group, and $(\\enspace,\\enspace)$  the unique  $W$-invariant form defined on $\\Phi$ with $(\\gamma,\\gamma)=2$ for all $\\gamma\\in \\Phi$.  Let $\\varpi_\\gamma$ denote a fundamental weight with respect to $\\Pi$, $\\gamma\\in \\Pi$, and let the weight lattice of $\\Phi$ be $\\Lambda=\\sum_{\\gamma\\in\\Pi}\\mathbb Z\\varpi_\\gamma$\\label{Lambda}. \n \n  \\color{black} The quantum enveloping algebra $U_v(\\mathfrak{sl}_3(\\mathbb C))$ is\n defined to be the associative algebra over $\\mathbb Q[v,v^{-1}]$ with generators $E_\\gamma$, $F_\\gamma$, $K_\\gamma$ and $K^{-1}_\\gamma$, ($\\gamma\\in \\Pi$) subject to the relations \n\\begin{gather*}\nK_\\gamma K^{-1}_\\gamma=K_\\gamma^{-1}K_\\gamma=1, \\quad K_\\gamma K_\\nu =K_\\nu K_\\gamma \\tag{R1}\\\\\nK_\\gamma E_\\nu K^{-1}_\\gamma=v^{(\\gamma,\\nu)}E_\\nu ,\\quad\nK_\\gamma F_\\nu K^{-1}_\\gamma=v^{-(\\gamma,\\nu)}F_\\nu \\tag{R2} \\\\\n[E_\\gamma,F_\\nu]=\\delta_{\\gamma,\\nu} \\frac{K_\\gamma-K_{-\\gamma}}{v_\\gamma-v^{-1}_\\gamma} \\tag{R3}\\\\\nE_\\gamma^2 E_\\nu -[2] E_\\gamma E_\\nu E_\\gamma+ E_\\nu E_\\gamma^2=0  \\quad \\text{ for }\\gamma\\neq \\nu\\tag{R4} \\\\\nF_\\gamma^2 F_\\nu -[2] F_\\gamma F_\\nu F_\\gamma+ F_\\nu F_\\gamma^2=0 \\quad \\text{ for }\\gamma\\neq \\nu   \n\\end{gather*}\nwhere $\\gamma,\\nu\\in \\Pi$.\nOne also sets $\\rho=\\varpi_\\alpha+\\varpi_\\beta=\\alpha+\\beta$, $U=U_v(\\mathfrak{sl}(3)))$ and let $U^+$ (resp. $U^-$) denote the subalgebra of $U$ generated by $E_\\nu$ (resp. $F_\\nu$) with $\\nu\\in \\Phi$.   Moreover let $U_v(\\mathfrak a)$\\label{ua} denote the subalgebra generated by $E_\\alpha,F_\\alpha, K_\\gamma$ with $\\gamma\\in \\Pi$.  We will now consider $M(-\\varpi_\\alpha)$ which has two Verma submodules $M(-2\\varpi_\\beta)$ and $M(-2\\varpi_\\alpha-\\varpi_\\beta)$ due to the fact that $s_\\beta(-\\varpi_\\alpha+\\rho)-\\rho=-\\varpi_\\alpha-\\beta=-2\\varpi_\\beta$ and $s_{\\alpha}(-2\\varpi_\\beta+\\rho)-\\rho=-2\\varpi_\\alpha-\\varpi_\\beta$.\n\n  The subalgebra $U^-$ generated by $F_\\gamma$, $\\gamma\\in\\Pi$, has a basis of the form \n $$\nT_\\alpha T_\\beta (F_\\alpha^{(m)})T_\\alpha(F^{(n)}_\\beta)F_\\alpha^{(p)} =T_\\alpha \\left(T_\\beta (F_\\alpha^{(m)})F^{(n)}_\\beta\\right)F_\\alpha^{(p)} \n$$\nwhere $m,n,p\\in\\mathbb N$ and $T_\\alpha=T''_{1,1}, T_\\beta=T''_{2,1}$ (in Lusztig's notation, see \\cite{MR94m:17016} and \\cite[Theorem 8.24]{MR96m:17029}).\nBy \\cite[8.16.(6)]{MR96m:17029} $T_\\alpha T_\\beta(F_\\alpha)=F_\\beta$, so we get\n\\begin{equation*}\nT_\\alpha T_\\beta (F_\\alpha^{(m)}K_\\alpha^m)= F_\\beta^{(m)}K_\\beta^{m},\n\\end{equation*}\nand\n\\begin{equation}\\label{calc1}\nT_\\beta(F_\\alpha K_\\alpha)=\\left(F_\\alpha F_\\beta -vF_\\beta F_\\alpha\\right)K_{\\beta+\\alpha}.\n\\end{equation}\n\nOn the other hand\n\\begin{align*}\n\\text{ad}\\,(F_\\alpha)(F_\\beta K_\\beta) =(F_\\alpha F_\\beta -vF_\\beta F_\\alpha)K_{\\alpha+\\beta}\n\\end{align*}\n\n\n\n In the notation of section \\secref{QCG decomposition}, we have under the adjoint action on $U$,\n\\begin{equation}\\label{F1}\n u^{(1)}=F_\\beta K_\\beta ,\\quad u^{(1)}_1=\\text{ad}\\,(F_\\alpha)(F_\\beta K_\\beta) \n\\end{equation}\n and $\\text{ad}\\,(F_\\alpha^{(2)})(F_\\beta K_\\beta)=0$ by \\cite[Lemma 4.18]{MR96m:17029} so that the set $\\{u^{(1)}$, $u^{(1)}_1\\}$ spans a copy of $\\mathcal F_1$ inside $U^-$.\n \n\nLet $M(\\lambda)$ denote the Verma module for $U_\\nu(\\mathfrak{sl}(3))$ with highest weight $\\lambda\\in \\Lambda$  defined by \n$$\nM(\\lambda):=U\/\\left( \\sum_{\\nu\\in \\Pi}UE_\\nu +\\sum_{\\nu \\in\\Pi} U(K_\\nu- T^{\\nu\\cdot\\nu\/2}v^{(\\lambda, \\nu)})\\right).\n$$\nLet $\\mathbf 1$ denote the image of $1$ in the quotient $M(\\lambda)$.  Set\n$$\nM(\\lambda)^\\nu:=\\{m\\in M(\\lambda)\\,|\\, K_\\gamma m=T^{\\nu\\cdot\\nu\/2} v^{(\\gamma,\\lambda)}\\text{ for all }\\gamma\\in\\Pi\\}.\n$$\nThe following is a quantum analogue of a result Shapovalov's: \n\\begin{prop}[{\\cite[Proposition 19.1.2]{MR94m:17016}}]  For any $\\lambda\\in \\Lambda$ there exists a unique symmetric bilinear form $M(\\lambda)\\times M(\\lambda)\\to R$ such that \n\\begin{enumerate}\n\\item $(\\mathbf 1,\\mathbf 1)=1$;\n\\item $(ux,y)=(x,\\varrho(u)y)$ for all $x,y\\in M(\\lambda)$ and $u\\in\\mathbf U$.\n\\end{enumerate}\nMoreover $(x,y)=0$ if $x\\in M(\\lambda)^\\nu$ and $y\\in M(\\lambda)^{\\nu'}$ with $\\nu\\neq \\nu'\\in \\Lambda$ .\n\\end{prop}\n\n\nSince the Shapovalov form is $\\varrho$-invariant it must be an induced form when restricted to a $U_\\nu(\\mathfrak a)$-summand isomorphic to $ U_\\nu(\\mathfrak a)\\mathbf 1\\otimes\\mathcal F_n$.  Here $\\mathcal F_n$ is any ad-invariant irreducible summand of $\\mathbf U$.  To illustrate what is going on in the paper we use the copy of $\\mathcal F_1$ given in \\eqnref{F1}, taking into account that we need to use the $_f\\mathcal R^{-1}:\\mathcal F_n \\otimes U_\\nu(\\mathfrak a)\\mathbf 1\\to U_\\nu(\\mathfrak a)\\mathbf 1\\otimes\\mathcal F_n$ (we will use $f$ as defined in \\eqnref{fexample}).  More precisely let us determine $\\boldsymbol{\\beta}:\\mathcal F_1\\otimes \\mathcal F_1^{\\rho_1}\\to U_v(\\mathfrak a)$ satisfying \n\\begin{equation}\\label{shap}\n(um,u'n)=\\chi_{\\boldsymbol\\beta,\\phi_M}({_f\\mathcal R}^{-1}(u\\otimes m), {_f\\mathcal R}^{-1}(u'\\otimes n))\n\\end{equation}\nwhere $u,u'\\in \\mathcal F_1$, $m,n\\in M: =U_\\nu(\\mathfrak a)\\mathbf 1$ and $\\phi_M$ is the Shapovalov form on the Verma $U_v(\\mathfrak a)$-module of highest weight $Tv^{-1}$ with respect to $K_\\alpha$ ($T$ with respect to $K_\\beta$) and is normalized so that $\\phi_M(\\mathbf 1,\\mathbf 1)=1$.  Note that the linear map $\\boldsymbol{\\beta}$ has a bold font to distinguish it from the root $\\beta$.\nSince\n$$\n\\mathcal F_1\\otimes \\mathcal F_1^{\\rho_1}\\cong\\mathcal F_{0}\\oplus \\mathcal F_2,\n$$\nthe problem then is to find $r_i\\in\\mathbb C[T]$ with\n$$\n\\boldsymbol{\\beta}=r_0\\boldsymbol{\\beta}^{1,1}_0+r_2\\boldsymbol{\\beta}^{1,1}_2\n$$\nwhere \n$$\n\\boldsymbol{\\beta}^{1,1}_{2r}(u^{(a)})=\\delta_{2r,a}E_\\alpha^{(r)}K_\\alpha^{-r}.\n$$\nFirst of all (for $\\mathbf 1$ the highest weight vector) using \\eqnref{antiauto}, \n$$\n(F_\\beta K_\\beta \\mathbf 1,F_\\beta K_\\beta \\mathbf 1)=T^2(\\mathbf 1,\\brho(F_\\beta)F_\\beta\\mathbf 1)=T^2v (\\mathbf 1, K_\\beta^{-1} E_\\beta F_\\beta \\mathbf 1)=vT[T;0].\n$$\nNow we use $f$ as \\eqnref{fexample}\n\\begin{align*}\n_f\\mathcal R^{-1}(F_\\beta K_\\beta \\otimes \\mathbf 1)\n&=f(1,-1)^{-1}\\mathbf 1 \\otimes F_\\beta K_\\beta.\n\\end{align*}\nwhich has as a consequence \n\\begin{align*}\n(F_\\beta K_\\beta\\mathbf 1 ,F_\\beta K_\\beta\\mathbf 1 )&=\\chi_{\\boldsymbol\\beta,\\phi_M}(\\mathbf s\\circ \\Pi_f^{-1}(F_\\beta K_\\beta\\otimes \\mathbf 1 ),\\mathbf s\\circ \\Pi_f^{-1}(F_\\beta K_\\beta\\otimes \\mathbf 1)) \\\\\n&=f(1,-1)^{-2}\\phi_M(\\mathbf 1, \\rho_1\\boldsymbol\\beta(F_\\beta K_\\beta\\otimes  F_\\beta K_\\beta)\\mathbf 1).\n\\end{align*}\n\nSo by \\corref{firstbetacor} we get\n\\begin{align}\\label{relation1}\nf(1,-1)^{-2}\\rho_1\\boldsymbol{\\beta}(F_\\beta K_\\beta\\otimes F_\\beta K_\\beta)\\mathbf 1\n&=f(1,-1)^{-2}\\left(\\frac{r_0}{[2]}+\\frac{v r_2}{[2]}[T,-1]\\right)\\mathbf 1 \n=vT[T;0]\\mathbf 1 \n\\end{align}\n\nNext by \\eqnref{calc1}\n\\begin{align*}\n(\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\mathbf 1,\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\mathbf 1)\n&=vT [T;0]+v^2T^3[T;-1].\n\\end{align*}\nOn the other hand \\corref{firstbetacor} gives us\n\\begin{align}\nf(-1,-1)^{-2}\\rho_1\\left(r_0\\boldsymbol{\\beta}^{1,1}_0+r_2\\boldsymbol{\\beta}^{1,1}_2\\right)&(\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\otimes \\text{ad}F_\\alpha(F_\\beta K_\\beta))\\mathbf 1  \n\\label{relation2} \\\\\n&= \n\\left(vT [T;0]+v^2T^3[T;-1]\\right) \\mathbf 1\\notag\n\\end{align}\n\nUsing the equations \\eqnref{relation1} and \\eqnref{relation2} we finish the determination of $\\boldsymbol\\beta$:\n$$\n \\boldsymbol{\\beta}=f(1,-1)^{2}\\left(T^2(T+T^{-1})[T;0]+vT^5[T;-1])\\boldsymbol{\\beta}^{1,1}_0-T^2(\\{1\\}  [T;0] +T^3)\\boldsymbol{\\beta}^{1,1}_2\\right).\n$$\n\n\n\n\n \nHere however we have only determined one $\\boldsymbol \\beta$ for one $U_\\nu(\\mathfrak a)$-summand of $M(-\\varpi_\\alpha)$.  In future we plan to investigate the other $\\boldsymbol\\beta$ that appear.\n\n\n\n\n\n\\subsection{Verma Modules for $U_\\nu(\\mathfrak{sp}(4))$}\nLet \n\\[\n\\Phi=\\{\\pm \\alpha,\\pm\\beta,\\pm(\\alpha+\\beta),\\pm (2\\alpha+\\beta)\\}\n\\]\n be the root system of type $B_2$ with $\\Pi=\\{\\alpha,\\beta\\}$ a set of simple roots, $\\alpha$ short, $\n \\beta$ long, $W$ the Weyl group, and $(\\enspace,\\enspace)$  the unique  $W$-invariant form defined on $\\Phi$ with $(\\alpha, \\beta)=-2$, $(\\alpha,\\alpha)=2$ and $(\\beta,\\beta)=4$,\n so that\n $\n \\langle \\alpha,\\check\\beta\\rangl\n =-1,\n  \\langle \\beta,\\check\\alpha\\rangl\n  =-2.\n $\n Let $\\varpi_\\gamma$ denote a fundamental weight with respect to $\\Pi$, $\\gamma\\in \\Pi$, and let the weight lattice of $\\Phi$ be $\\Lambda=\\sum_{\\gamma\\in\\Pi}\\mathbb Z\\varpi_\\gamma$.   Then $q_\\alpha=q$ and $q_\\beta=q^2$.  \n \n The quantum enveloping algebra $U_v(\\mathfrak{sp}(4))$ is\n defined to be the associative algebra over $\\mathbb Q[v,v^{-1}]$ with generators $E_\\gamma$, $F_\\gamma$, $K_\\gamma$ and $K^{-1}_\\gamma$, ($\\gamma\\in \\Pi$) subject to the relations (R1)-(R3) but with a different Serre relation:\n\\begin{gather*}\nE_\\gamma^3 E_\\nu -[3]_\\gamma E_\\gamma^2 E_\\nu E_\\gamma+[3]_\\gamma E_\\gamma E_\\nu E_\\gamma^2-E_\\nu E_\\gamma^3=0  \\quad \\text{ for }\\gamma\\neq \\nu \\\\\nF_\\gamma^3 F_\\nu -[3]_\\gamma F_\\gamma^2 F_\\nu F_\\gamma+[3]_\\gamma F_\\gamma F_\\nu F_\\gamma^2-F_\\nu F_\\gamma^3=0   \\quad \\text{ for }\\gamma\\neq \\nu \n\\end{gather*}\nwhere $\\gamma,\\nu\\in \\Pi$.\n$U=U_v(\\mathfrak{sp}(4)))$ and let $U^+$ (resp. $U^-$) denote the subalgebra of $U$ generated by $E_\\nu$ (resp. $F_\\nu$) with $\\nu\\in \\Phi$.   Moreover let $U_v(\\mathfrak a)$ denote the subalgebra generated by $E_\\alpha,F_\\alpha, K_\\gamma$ with $\\gamma\\in \\Pi$.  Let $\\rho_1:\\bold U\\to \\bold U$ be the algebra isomorphism determined by\nthe assignment \n\\begin{equation}\n\\rho_1(E_\\gamma)=-v_\\gamma F_\\gamma,\\quad \\rho_1(F_\\gamma)=-v_\\gamma^{-1}E_\\gamma,\\quad\n\\rho_1(K_\\gamma)=K_{\\gamma}^{-1}\n\\end{equation}\nfor all $\\gamma\\in\\Pi$.   Define also an algebra\nanti-automorphism $\\brho:\\bold U\\to \\bold U$ by \n\\begin{equation} \\label{antiauto2}\n\\brho(E_\\gamma)=v_\\gamma K_\\gamma F_\\gamma,\\quad \\brho(F_\\gamma)=v_\\gamma \nK_{\\gamma}^{-1}E_\\gamma,\\quad \\brho(K_\\gamma)=K_\\gamma. \n\\end{equation}\nThese maps are related through the antipode $S$ of  $\\bold U$\nby $\\brho=\\rho_1S$ .\n\nWe will now consider (a $U_\\nu(\\mathfrak a)$-submodule of) $M(-\\varpi_\\alpha+\\varpi_\\beta)$ which has three Verma submodules $M(-\\varpi_\\alpha+\\varpi_\\beta-2\\beta)$, $M( -\\varpi_\\alpha+\\varpi_\\beta-4\\alpha-2\\beta)$, and $M(-\\varpi_\\alpha+\\varpi_\\beta-4\\alpha-4\\beta)$ due to the fact that \n$$\ns_\\beta(-\\varpi_\\alpha+\\varpi_\\beta+\\rho)-\\rho\n=-\\varpi_\\alpha+\\varpi_\\beta-2\\beta,\n$$\n$$\ns_{\\alpha}(-\\varpi_\\alpha+\\varpi_\\beta-2\\beta+\\rho)-\\rho\n=-\\varpi_\\alpha+\\varpi_\\beta-4\\alpha-2\\beta,\n$$\n$$\ns_{\\beta}(-\\varpi_\\alpha+\\varpi_\\beta-2\\beta-4\\alpha+\\rho)-\\rho\n=-\\varpi_\\alpha+\\varpi_\\beta-4\\alpha-4\\beta.\n$$\n\n\n\nNote that \n\\begin{align}\n\\text{ad}\\,(F_\\alpha)(F_\\beta K_\\beta) &=(F_\\alpha F_\\beta -v^2F_\\beta F_\\alpha)K_{\\alpha+\\beta}\\text{ and }\\label{calc4}  \\\\\n\\text{ad}\\,(F_\\alpha^{(2)})(F_\\beta K_\\beta) \n&=\\Big(F_\\alpha^{(2)} F_\\beta -vF_\\alpha F_\\beta F_\\alpha +v^2F_\\beta F_\\alpha^{(2)}\\Big)K_{2\\alpha+\\beta}.\n\\end{align}\n\n\\color{black}\n In the notation of section \\secref{QCG decomposition}, we have under the adjoint action on $U$,\n\\begin{equation}\\label{F2}\n u^{(2)}=F_\\beta K_\\beta ,\\quad u^{(2)}_1=\\text{ad}\\,(F_\\alpha)(F_\\beta K_\\beta) ,\\quad u^{(2)}_2=\\text{ad}\\,(F_\\alpha^{(2)})(F_\\beta K_\\beta) \n\\end{equation}\n and $\\text{ad}\\,(F_\\alpha^{(3)})(F_\\beta K_\\beta)=0$ by \\cite[Lemma 4.18]{MR96m:17029} so that the set $\\{u^{(2)}, u^{(2)}_1, u^{(2)}_2\\}$ spans a copy of $\\mathcal F_2$ inside $U^-$.\nLet us determine $\\boldsymbol{\\beta}:\\mathcal F_2\\otimes \\mathcal F_2^{\\rho_1}\\to U_v(\\mathfrak a)$ satisfying \n\\begin{equation}\n(um,u'n)=\\chi_{\\boldsymbol\\beta,\\phi_M}({_f\\mathcal R}^{-1}(u\\otimes m), {_f\\mathcal R}^{-1}(u'\\otimes n))\n\\end{equation}\nwhere $u,u'\\in \\mathcal F_1$, $m,n\\in M: =U_\\nu(\\mathfrak a)\\mathbf 1$ and $\\phi_M$ is the Shapovalov form on the Verma $U_v(\\mathfrak a)$-module of highest weight $v^{-1}T$ with respect to $K_\\alpha$ ($v^2T$ with respect to $K_\\beta$  - this is because $(\\varpi_\\beta,\\beta)=2$ and $(\\varpi_\\alpha,\\alpha)=1$) and is normalized so that $\\phi_M(\\mathbf 1,\\mathbf 1)=1$.  \nSince\n$$\n\\mathcal F_2\\otimes \\mathcal F_2^{\\rho_1}\\cong\\mathcal F_{0}\\oplus \\mathcal F_2\\oplus\n\\mathcal F_4\n$$\nthe problem then is to find $r_i\\in\\mathbb C[T]$ with\n$$\n\\boldsymbol{\\beta}=r_0\\boldsymbol{\\beta}^{2,2}_0+r_2\\boldsymbol{\\beta}^{2,2}_2\n+r_4\\boldsymbol{\\beta}^{2,2}_4\n$$\nwhere \n$$\n\\boldsymbol{\\beta}^{2,2}_{2r}(u^{(a)})=\\delta_{2r,a}E_\\alpha^{(r)}K_\\alpha^{-r}.\n$$\nFirst of all (for $\\mathbf 1$ the highest weight vector) using \\eqnref{antiauto2}, \n$$\n(F_\\beta K_\\beta \\mathbf 1,F_\\beta K_\\beta \\mathbf 1)=v^2T^4(\\mathbf 1,\\brho(F_\\beta)F_\\beta\\mathbf 1)=T^4v^4 (\\mathbf 1, K_\\beta^{-1} E_\\beta F_\\beta \\mathbf 1)=v^3T^2[T^2;1]_\\beta.\n$$\nNext\nfrom \\corref{firstbetacor} (with $m=n=2$) we get\n\\begin{align*}\n\\Big(r_0\\boldsymbol{\\beta}^{1,1}_0&+r_2\\boldsymbol{\\beta}^{2,2}_2+r_4\\boldsymbol{\\beta}^{2,2}_4\\Big)((\\text{ad}\\,F_\\alpha^{(i)})F_\\beta K_\\beta\\otimes (\\text{ad}\\,F_\\alpha^{(i)})F_\\beta K_\\beta) \\\\\n&=(-1)^iv^{2i(i-3)}{\\qbinom 2 i}^2  \\left(\\sum_{k=0}^2r_{2k}\n   \\frac{\\qbinom{T;-1}{k}}{||u^{(2k)}||^2\\left[\\begin{matrix} 2k\n  \\\\  k \\end{matrix}\\right]} \\left[\\begin{matrix} 2 & 2 & 2k\n  \\\\ i & {2-i} & k \\end{matrix}\\right]\n\\right)\n\\end{align*}\n\nNow we use $f$ as \\eqnref{fexample}\n\\begin{align*}\n_f\\mathcal R^{-1}(F_\\beta K_\\beta \\otimes \\mathbf 1)\n&=f(2,-1)^{-1}\\mathbf 1 \\otimes F_\\beta K_\\beta.\n\\end{align*}\nwhich has as a consequence \n\\begin{align*}\n(F_\\beta K_\\beta\\mathbf 1 ,F_\\beta K_\\beta\\mathbf 1 )&=\\chi_{\\boldsymbol\\beta,\\phi_M}(\\mathbf s\\circ \\Pi_f^{-1}(F_\\beta K_\\beta\\otimes \\mathbf 1 ),\\mathbf s\\circ \\Pi_f^{-1}(F_\\beta K_\\beta\\otimes \\mathbf 1)) \\\\\n&=f(2,-1)^{-2}\\phi_M(\\mathbf 1, \\rho_1\\boldsymbol\\beta(F_\\beta K_\\beta\\otimes  F_\\beta K_\\beta)\\mathbf 1).\n\\end{align*}\n\n\nFrom this simple calculation and \\corref{firstbetacor} we have\n\\begin{align}\\label{relation3}\nf(2,-1)^{-2}\\rho_1\\boldsymbol{\\beta}(F_\\beta K_\\beta\\otimes F_\\beta K_\\beta)\\mathbf 1\n&=f(2,-1)^{-2}\\left(\\frac{r_0}{[3]}+\n\\frac{ v^2\\,r_2}{[4]}[T;-1]+\\frac{v^4 [2]\\,r_4}{[4][3]}\\qbinom{T;-1}{2}\\right)\\mathbf 1 \\\\\n&=v^4T^2[T^2;1]_\\beta\\mathbf 1 \n\\end{align}\n\nNext by \\eqnref{calc4}\n\\begin{align*}\n(\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\mathbf 1,&\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\mathbf 1)\n=((F_\\alpha F_\\beta -v^2F_\\beta F_\\alpha )K_{\\alpha+\\beta}\\mathbf 1,(F_\\alpha F_\\beta -v^2F_\\beta F_\\alpha )K_{\\alpha+\\beta}\\mathbf 1)\\\\\n&=vT^2[2][T^2;1]_\\beta +v^6T^5[T;-1]. \n\\end{align*}\nOn the other hand\n\\begin{align*}\n(\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\mathbf 1 ,\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\mathbf 1 )\n&=f(0,-1)^{-2}\\phi_M(\\mathbf 1, \\rho_1\\boldsymbol\\beta(\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\otimes  \\text{ad}F_\\alpha(F_\\beta K_\\beta))\\mathbf 1).\n\\end{align*}\nAgain from \\corref{firstbetacor} we obtain \n\n\\begin{align}\\label{relation4}\nf(2,&-1)^{-2}v^2T^{-2}\\rho_1\\left(r_0\\boldsymbol{\\beta}^{2,2}_0+r_2\\boldsymbol{\\beta}^{2,2}_2+r_4\\boldsymbol{\\beta}^{2,2}_4\\right)(\\text{ad}F_\\alpha(F_\\beta K_\\beta)\\otimes \\text{ad}F_\\alpha(F_\\beta K_\\beta))\\mathbf 1  \n\\\\\n&=f(2,-1)^{-2}v^2T^{-2}\\left(\\frac{r_0v^{-2}[2]}{[3]}+\\frac{r_2v^{-1} \n   [2]\\{1\\}}\n   {\\left[4\\right]} \\qbinom{T;-1}{1}-r_4v^{-1}[2]\\qbinom{4}{2}^{-1}\\qbinom{T;-1}{2}\\right)\\mathbf 1\n\\notag \\\\ \n&= \n\\left(vT^2[2][T^2;1]_\\beta +v^6T^5[T;-1]\\right) \\mathbf 1\\notag\n\\end{align}\nThirdly we have by a rather tedious calculation and  \\lemref{binomiallemma}\n\\begin{align*}\n(\\text{ad}F_\\alpha^{(2)}&(F_\\beta K_\\beta)\\mathbf 1,\\text{ad}F_\\alpha^{(2)}(F_\\beta K_\\beta)\\mathbf 1)\n\\\\\n&=T^4\\Big(v^4\\left(T^{-2}-T^{-1}[T,-1]+\\qbinom{T;-1}{2}\\right)[T^2,1]_\\beta  \\\\\n &\\quad +v^5(T^{-1}-v[T;-2])[T,-1][T^2;2]_\\beta  +v^{6} \\qbinom{T;-1}{ 2}[T^2;3]_\\beta\\Big) .\n\\end{align*}\n\nUsing \\corref{firstbetacor}  we get\n\\begin{align}\\label{relation5}\nf(-2,-1)^{-2}\n&\\rho_1\\left(r_0\\boldsymbol{\\beta}^{2,2}_0+r_2\\boldsymbol{\\beta}^{2,2}_2+r_4\\boldsymbol{\\beta}^{2,2}_4\\right)(\\text{ad}F_\\alpha(F_\\beta^{(2)} K_\\beta)\\otimes \\text{ad}F_\\alpha^{(2)}(F_\\beta K_\\beta))\\mathbf 1  \\\\ \n&=f(2,-1)^{-2}v^4T^{-4}\\left(\\frac{v^{-2}r_0}{[3]} -\n    \\frac{r_2v^{-2} }\n   {[4]} \\qbinom{T;-1}{1}+r_4v^{-4}\\left[\\begin{matrix} 4\n  \\\\  2 \\end{matrix}\\right]^{-1} \\qbinom{T;-1}{2}\\right) \\notag\n\\\\ \n&=T^4\\Big(v^4\\left(T^{-2}-T^{-1}[T,-1]+\\qbinom{T;-1}{2}\\right)[T^2,1]_\\beta  \\notag\\\\\n &\\quad +v^5(T^{-1}-v[T;-2])[T,-1][T^2;2]_\\beta  +v^{6} \\qbinom{T;-1}{ 2}[T^2;3]_\\beta\\Big)  \\notag\n\\end{align}\n\nFrom equations \\eqnref{relation3}, \\eqnref{relation4}  and \\eqnref{relation5} we determine the coefficients of $\\boldsymbol\\beta$:\n\\begin{align*}\n\\begin{pmatrix} r_0 \\\\ r_2[T,-1] \\\\ r_4 \\qbinom{T;-1}{2} \\end{pmatrix} =f  \\begin{pmatrix}\n v^{-2} & \\frac{v^2}{[2]} & v^4 \\\\\n \\frac{[2]}{v^2} & v\\{1 \\}& -v^2[2]\\\\\n 1 & -v & 1\n\\end{pmatrix}\n\\begin{pmatrix}v^4T^2[T^2;1]_\\beta   \\\\ \n\tvT^4[2][T^2;1]_\\beta +v^6T^7[T;-1]   \\\\ \n\tT^{11}\\Big(Tv^5 \\{1\\}+[T^3,1]_\\beta+T^4v^4[T,-1]_\\beta\\Big) \n\\end{pmatrix} \\end{align*}\nwhere $f=f(2,-1)^2$.\nHere however we have only determined the $\\boldsymbol \\beta$ for one $U_\\nu(\\mathfrak a)$-summand of $M(-\\varpi_\\alpha+\\varpi_\\beta)$.  In future we plan to determine the coefficients for the other $\\boldsymbol\\beta$ that appear.\n\n\\printindex\n\\begin{theindex}\n\\item $\\rho_1$,\\hfill\\pageref{rhoone}\n\\item $\\varrho$,\\hfill\\pageref{antiauto}\n\\item $\\beta^{m,n}_{2r}$,\\hfill \\pageref{betamn}\n\\item $F^{(-k)}\\eta$,\\hfill \\pageref{fminusketa}\n\\item $U_v(\\mathfrak a)$,\\hfill\\pageref{ua}\n\\item $\\mathcal F_m$,\\hfill \\pageref{fm}\n\\item $L$, $L^{-1}$, \\hfill\\pageref{L}\n\\item $\\mathcal L$,  \\hfill\\pageref{mathcalLinv}\n\\item $\\mathcal L_{2n}$, \\hfill \\pageref{mathcalL2n}\n\\item $\\Lambda$, \\hfill\\pageref{Lambda}\n\\item $[n]_\\alpha$, \\hfill\\pageref{Gaussintegers}\n\\item $[T;r]^{(j)},\n\\qbinom{T;r}{ j}$,\\hfill\\pageref{quantumbinom}\n\\end{theindex}\n\\def$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDespite years of study, we still do not know how the seeds of supermassive black holes formed.  Few if any of the pathways in Martin Rees's famous flow chart (Begelman \\& Rees 1978) can be ruled out, but none of the routes is particularly well understood, either.  What we do know is that some very massive ($> 10^9 M_\\odot$) black holes had to exist by $z\\sim 6$ in order to explain early quasars (Fan 2006).  If the seeds of these black holes were the remnants of massive stars, then they must have grown by Eddington-limited accretion for most of the time since their formation, or else much of their growth was due to mergers.  A second possibility is that the seeds formed by such a rapid accumulation of matter that it may be considered to be a direct collapse.  I will focus on the latter possibility in this paper.\n\n\\section{Direct collapse}\n\nThe Pop III star formation processes we have heard about at this conference result from the infall of gas at rates $\\sim 10^{-4} - 10^{-2} M_\\odot$ yr$^{-1}$.  What would happen if the infall rate were much higher?  The entropy of matter laid down by gravitational infall onto a growing central mass increases with time.  At low inflow rates, however, nuclear ignition halts the contraction of the core and raises the entropy in the interior, leading to a high-entropy object --- a star. If the inflow rate is high enough, however, the core will be so tightly bound by the time nuclear reactions start that the energy release will be insufficient to halt core contraction. In this case, we are left with an object with a low-entropy core and a high-entropy envelope.  This is the situation that can lead to the direct formation of a black hole, without a stellar precursor. This situation should apply when the infall rate exceeds a few tenths of a solar mass per year (Begelman et al.~2006), although more work needs to be done to refine this estimate.  \n\n\\subsection{Inflow rate}\n\nThe conditions under which such high inflow rates might occur are very uncertain.  Such rapid infall would almost certainly be driven by gravitational torques, which could be local (Gammie 2001) or global (as in the ``bars within bars\" mechanism: Shlosman et al.~1989).  The ``natural\" gravitational inflow rate is given by $\\dot M \\sim v^3\/G \\sim 0.1 (v\/10 \\ {\\rm km \\ s}^{-1})^3 M_\\odot$ yr$^{-1}$, where $v$ represents the internal velocity dispersion ($\\sim$ turbulent or sound speed) for a locally unstable thin disk and the orbital speed for a globally unstable (fully self-gravitating) system. This means that very large inflow rates are possible in dark matter haloes with velocity dispersions exceeding about 10 km s$^{-1}$, which have masses exceeding $\\sim 10^8-10^9 M_\\odot$ and become common at redshifts $\\sim 10-15$.  Global gravitational instabilities could occur in a significant fraction of such haloes if the gas is unable to cool much below the virial temperature ($\\sim 10^4$ K), which requires both that they have been spared significant metal enrichment and that H$_2$ formation is suppressed (Bromm \\& Loeb 2003; Begelman et al.~2006).  However, recent calculations by Wise \\& Abel (2007, and these proceedings) suggest that molecular hydrogen formation is inevitable. This may have the additional effect that many more haloes formed Pop III stars at higher redshifts, making it harder to avoid metal enrichment and possibly depleting the supply of gas available for infall as haloes merge. \n\nRapid infall may be easier in more massive haloes. There will be a larger disparity between the thermal temperature of the gas and the virial temperature, but presumably the gas will form a multiphase structure with turbulent velocities (perhaps driven by stellar energy sources) dominating the internal energy --- much like the interstellar medium.  Simulations are beginning to elucidate these structures, but we have a long way to go before we understand the tradeoffs between large scale inflow and in situ fragmentation and star formation.  Inflow may be stimulated by the large-scale gravitational torques associated with mergers. a key element in the simulations of black-hole growth and fuelling (e.g., Di Matteo et al.~2005).  Finally, we note that such inflow rates must be possible, because they are required to power quasars.  The open question is whether they can also occur when the black hole is absent, or very small.  For the remainder of this article we will assume that the answer is affirmative.   \n\n\\subsection{Black hole formation}\n\nOnce the mass of accumulated gas exceeds a few solar masses, radiation pressure dominates the envelope.  The core, with gas pressure comparable to radiation pressure, maintains a roughly constant mass $\\sim 10 M_\\odot$.  The boundary between dynamic infall and quasistatic contraction is close to the radius where infalling gas liberates energy at the Eddington limit.  For a constant infall rate of $\\dot M = 0.1 \n\\dot m_{-1} M_\\odot  {\\rm yr}^{-1}$, this radius turns out to be constant, $R_{\\rm env} \\sim 1 \\dot m_{-1}$ AU.  The core shrinks under the increasing pressure of the envelope, $R_{\\rm core} \\sim R_{\\rm env}\/ m_{\\rm env}$, where $m_{\\rm env} \\sim 0.1 \\dot m_{-1} t_{\\rm yr}$ is the envelope mass in solar units.  \n\nOne can sketch the likely interior structure of the envelope using scaling arguments.  Since the interior is undergoing Kelvin--Helmholtz contraction, the density profile adjusts so that the diffusion timescale $t_{\\rm diff} \\sim (\\rho \\kappa r^2 \/c ) (T\/r|\\nabla T|)$ is of order the elapsed time at all radii.  This means that the specific entropy of any mass shell, $s(M)$, declines slowly (logarithmically) with time, and is never far below the value it had when the mass shell was added. In the radiation pressure-dominated envelope, $s(M) \\propto p_r\/p_g \\propto M(r)^{1\/2}$, where $p_r \\propto T^4$ and $p_g \\propto \\rho T$ are the radiation and gas pressure, respectively.  There are then two possible scaling laws for the envelope structure.  If $T\/r | \\nabla T| \\sim O(1)$, then $\\rho(r) \\propto r^{-2}$, $T(r) \\propto r^{-1\/2}$ (for uniform opacity $\\kappa$), and the envelope joins smoothly onto the core. This was the result presented in Begelman et al.~(2006). The other possible scaling law \nhas $ T\/r |\\nabla T| \\gg 1$, i.e., nearly isothermal structure.  The density profile in the envelope is then $\\rho(r) \\propto r^{-1\/2}$, and there is a large jump in density and temperature going from the envelope to the core.  The latter structure seems to be the one that emerges naturally from ``nearly self-similar\" models for the core+envelope, which neglect rotation.  Given the manner in which the mass accumulates, though, rotation is bound to be important. Moreover, even a small amount of it can affect the structure dramatically, increasing the binding energy of the gas (which is otherwise very weakly bound since the mean equation of state has $\\gamma\\approx 4\/3$) and protecting it from pulsational instabilities.  Including the effects of rotation will require the modeling of angular momentum transport and dissipation, but these complications are not prohibitive.  It will be interesting to see which of the scalings applies when rotation is included. \n\nFortunately, the behavior of the core is insensitive to the structure of the envelope.  Since the core is largely dominated by gas pressure, its temperature must track the virial temperature, $T_{\\rm core} \\propto m_{\\rm env}$. Rapid nuclear burning starts in the core when the envelope mass is $\\sim 100 M_\\odot$ and the density is not that different from the cores of main sequence stars, but the system passes through this phase so quickly ($t_{\\rm evol} \\sim 10^3 \\dot m_{-1}^{-1}$ yr) that the energy release has little effect.  By the time the envelope mass reaches several thousand solar masses, the core temperature has climbed to a few $\\times 10^8$ K.  Because of the extremely deep potential well created by the envelope, the energy released by nuclear reactions at this stage is unable to unbind the gas (especially if rotation has increased its binding energy) and the core proceeds to the temperatures ($\\sim 5 \\times 10^8$ K) at which runaway neutrino losses occur.  At this point the core loses pressure support and collapses to form a $\\sim 10-20 M_\\odot$ black hole.     \n\n\\section{Post-BH evolution}\n\n\\subsection{Super-Eddington accretion}\n\nAt the time of its formation, the black hole is embedded in an envelope of more than a hundred times its mass.  If it were limited to accreting at the Eddington limit for the mass of the black hole, the pressure of the escaping radiation would have essentially no effect on the envelope.  In fact, the pressure of the envelope should be able to drive the accretion rate up to the point where the liberated luminosity approaches the Eddington limit {\\it for the mass of the envelope}.  Relative to the mass of the black hole, the accretion is super-Eddington by a factor $ M_{\\rm env}\/M_{\\rm BH}$.\n\nAccretion inside the massive envelope (which itself continues to grow at a rate $\\dot M$) can lead to very rapid growth of the black hole. The Eddington ratio for black hole growth is \n\\begin{equation}\n{\\dot M_{\\rm BH} \\over \\dot M_E} \\sim 3\\times 10^3 \\left( {\\dot m_{-1}\\over \\varepsilon_{-1}} \\right)^{1\/2}  \\left( { M_{\\rm BH}\\over M_\\odot} \\right)^{-1\/2} , \n\\end{equation}       \nwhere $0.1\\varepsilon_{-1}$ is  the accretion efficiency.  For an initial black hole mass of $10M_\\odot$, the e-folding time is a thousandth of the Salpeter time, or only $\\sim 10^4$ yr, and super-Eddington growth appears possible up to black hole masses $> 10^6 M_\\odot$.  However, this estimate does not take fully into account the back reaction of black hole growth on the envelope.  We will see below that the era of rapid growth is limited to much smaller (but still interesting!) black hole masses. \n \n\\subsection{Quasistars}\n\nThe energy liberated by accretion has to escape.  Since the accretion flow is rotating, some of it could exhaust through a low-density funnel. But much of it  presumably percolates through the accretion flow and envelope.  Thus, the accreting black hole provides an energy source for the envelope, which is therefore a kind of star-like object which we have dubbed a ``quasistar\" (Begelman et al.~2006).  Since there is a limit to the outward energy flux that the accreting gas can carry, the accretion rate is regulated at a fraction $\\sim \\varepsilon^{-1} (c_s\/ c)^2$ of the Bondi rate, where $c_s \\sim (p_r\/\\rho)^{1\/2}$ is the sound speed at the Bondi radius (Gruzinov 1998; Blandford \\& Begelman 1999; Narayan et al.~2000; Quataert \\& Gruzinov 2000).  The accretion rate, and associated energy flux, can be expressed in terms of the temperature and density deep within the quasistar, but outside the black hole's sphere of influence.  The energy flux must also equal the Eddington limit for the entire quasistar, $L_E(M_*)$ (where we now use $M_*$ to denote the mass of the quasistar) --- the interior structure adjusts so that this is satisfied.  Since the Eddington limit is lower inside the quasistar (where the enclosed mass is lower), the quasistar's interior is strongly convective and can be modeled by an $n=3$ polytrope.  (Strictly speaking, one should use a ``loaded polytrope\" model [Huntley \\& Saslaw 1975], taking into account the mass of the black hole, but the loaded and unloaded models converge in the region of interest when $M_* \\gg M_{\\rm BH}$.) The interior structure of a quasistar is shown schematically in Fig.~1.\n\n\\begin{figure}\n  \\includegraphics[height=.1\\textheight]{Begelmanfig1}\n  \\caption{Schematic illustration of the quasistar, from Begelman et al.~(2007). A seed black hole of mass $M_{BH}$ accretes gas from a massive, radiation pressure-supported envelope at a rate set by conditions outside the Bondi radius. The luminosity liberated by the accretion process is transported convectively in the inner regions of the envelope, with a transition to a radiative zone where convection becomes inefficient.  We illustrate rotational flattening and ongoing disk accretion at a fraction of a Solar mass per year.}\n\\end{figure}\n\nAccording to the polytropic relations, the density and temperature outside the Bondi radius are uniquely related to the mass and radius ($R_*$) of the quasistar.  Expressing the accretion rate in terms of these quantities and setting the luminosity equal to $L_E(M_*)$, we can express $R_*$ in terms of $M_*$ and $M_{\\rm BH}$:\n\\begin{equation} \nR_* \\sim 4 \\times 10^{14}  \\alpha^{2\/5} m_{\\rm BH}^{4\/5} m_*^{-1\/5}  \\ {\\rm cm}  ,\n\\end{equation} \nwhere the masses are expressed in solar units and $\\alpha <1$ parametrizes the efficiency of angular momentum and energy transport inside the Bondi radius (Begelman et al.~2007).  The photospheric temperature is\n\\begin{equation} \nT_{\\rm ph} \\sim 10^3 \\alpha^{-1\/5} m_{\\rm BH}^{-2\/5} m_*^{7\/20}  \\ {\\rm K} .  \n\\end{equation}\n\nSince the black hole growth rate is proportional to $M_*$, while the quasistar mass increases linearly with time, $M_{\\rm BH} \\propto M_*^2$ at late times.  This implies that the quasistar's radius grows with time, and its photosphere becomes cooler.  (The result holds even if the quasistar mass is fixed.) The appearance of a quasistar differs dramatically from that of the pre-black hole envelope.  Shortly after the black hole forms the envelope expands to $\\sim 100$ AU (from $\\sim 1$ AU), and the interior temperature drops to $\\sim 10^6$ K, quenching all nuclear reactions. A quasistar resembles a red supergiant, except that it is radiation pressure supported and its energy source is accretion.  In some respects it is reminiscent of a very massive Thorne-\\.Zytkow (1977) object, but with crucial differences. Besides being dominated by radiation pressure and powered exclusively by accretion, quasistars have a distributed rather than shell-like energy source, which cannot be regulated by the slow settling that characterizes accretion onto a neutron star.  As a result, quasistars come to grief if their photospheric temperatures get too low.\n\n\\begin{figure}\n  \\includegraphics[height=.5\\textheight]{Begelmanfig2}\n  \\caption{The shaded areas show forbidden photospheric temperatures of Pop III quasistars, as a function of envelope and black hole mass, from Begelman et al.~(2007). The lighter shaded region is computed assuming that $\\alpha =0.1$, the darker shaded region is for $\\alpha =0.05$, while the dashed line is for an analytic ``toy\" opacity model.  Superimposed on the figure are evolutionary tracks for an accretion rate onto the envelope of $\\dot{M}_* $= 1 $\\ M_{\\odot} \\ {\\rm yr}^{-1}$ and $\\dot{M}_*$= 0.1$ \\ M_{\\odot}\\ {\\rm yr}^{-1}$.  The evolution pushes the quasistar into the forbidden region of parameter space, where it evaporates.}\n\\end{figure}\n\nBegelman et al.~(2007) computed quasistar models using Pop III opacities from Mayer \\& Duschl (2005).  As in red giants with standard abundances, there is a minimum photospheric temperature associated with a sharp drop in opacity. Because of the absence of metals, the ``opacity crisis\" occurs at a somewhat higher temperature --- around 4000 K --- than for metal-enriched compositions.  Like the Hayashi track (Hayashi 1961; Hayashi \\& Hoshi 1961) for red giants and convective protostars, the minimum temperature of quasistars arises from the impossibility of matching the convective interior to the radiative zone and photosphere.  The details are somewhat different, however, because radiation pressure dominates in quasistars, whereas the convective zones of ordinary red giants are gas pressure-dominated and resemble $n=3\/2$ polytropes.  As a quasistar crosses into the forbidden zone, the flux escaping from the convective interior exceeds the Eddington limit and prevents the quasistar from maintaining hydrostatic equilibrium. Fig.~2 shows the forbidden zone in the $M_*-M_{\\rm BH}$ plane, along with representative evolutionary tracks.  \n\nUnlike ordinary red giants reaching the Hayashi track, quasistar photospheres cannot hover stably at close to the minimum temperature.  Fixing the photospheric temperature, we find that the Eddington ratio scales as \n\\begin{equation}\n{L\\over L_E} \\propto M_*^{-7\/9} M_{\\rm BH}^{8\/9}\n\\end{equation}\nThus, growth of the black hole exacerbates the dynamical imbalance, as does partial evaporation of the envelope\\footnote{Since the black hole's growth rate is proportional to $M_*$, rapid accretion onto the envelope also would not be able to stabilize the quasistar for very long.} --- the opacity crisis is an unstable situation.  We predict that quasistars entering the forbidden zone must evaporate rather quickly.\n\nThere is a complication due to a bump in the opacity from bound-free transitions, which creates a narrow region in the radiative zone where the luminosity is super-Eddington. A similar problem arises in models of Luminous Blue Variables (e.g., Owocki et al.~2004), and remains unresolved.  However, it does not seem likely that this feature alone will lead to catastrophic mass loss in the case of quasistars.\n \n\\section{Discussion}\n\nThe rapid infall of gas in galactic nuclei or pregalactic haloes provides a means for forming seed black holes and rapidly growing them into the intermediate mass regime.  For Pop III quasistars with inflow rates $\\sim 0.1-1 M_\\odot$ yr$^{-1}$, and simple assumptions about parameters like $\\alpha$, the black holes could reach masses $\\sim 10^3-10^4 M_\\odot$ before the quasistar evaporates.  The quasistar masses could be as large as $10^5-10^6 M_\\odot$. Metal-rich quasistars could reach somewhat lower temperatures, but it is not clear whether this implies larger black hole masses, because the run of opacity in the radiative zone would be more complex and dust formation might lead to enhanced mass loss.\n\nAre quasistars detectable? If so, they would be seen at their most massive, shortly before they evaporate. Spectrally, they would resemble 2000--4000 K blackbodies, depending on metallicity, and in the Pop III limit their spectra would be featureless.  Since they radiate at the Eddington limit corresponding to the opacity at the convective--radiative transition --- which is close to that of electron scattering --- their luminosities could reach $10^{43}-10^{44}$ erg s$^{-1}$.  However, their short lifetimes ($\\sim 10^5-10^6$ yr) would make them fairly rare.\n \nQuasistars can exist only when the envelope mass greatly exceeds the mass of the embedded black hole.  Therefore, they are unlikely to form in the nuclei of galaxies that already possess a supermassive black hole --- in this case, a period of rapid infall (e.g., following a merger) would presumably trigger a quasar outburst instead.  However, low-redshift quasistars might conceivably form in galactic nuclei in which the black hole had been ejected due to three-body interactions, or in which a black hole had never formed.  The earliest plausible sites of quasistar formation would have been pregalactic haloes with virial temperatures exceeding $10^4$ K.  These would have been most common at redshifts $\\sim 6-15$.  The spectra of Pop III quasistars at these redshifts would peak at about 10$\\mu$m, but they might be marginally detectable by {\\it James Webb Space Telescope} on the Wien tail at $3-5\\mu$m.  If direct collapse in $10^4$ K haloes is a principal route for forming supermassive black hole seeds, there could be as many as 1--10 per {\\it JWST} field, but identifying them would be an extreme challenge. \n\n\n\n\\begin{theacknowledgments}\nThe work described here was done in collaboration with Marta Volonteri, Elena Rossi, Martin Rees, and Phil Armitage, and was supported in part by NSF grant AST-0307502, NASA's Beyond Einstein Foundation Science Program grant NNG05GI92G, and a University of Colorado Faculty Fellowship.  The author thanks the Director of the Institute of Astronomy and the Master and Fellows of Trinity College, Cambridge, for their hospitality during a sabbatical visit. \n\\end{theacknowledgments}\n\n \n \n\\bibliographystyle{aipproc}  \n\n \n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nD(irichlet)-branes have many faces. In string theory,\nthey arise as nonperturbative dynamic objects, allowing\nstrings to end on them and carrying  R(amond)-R(amond)\ncharge \\cite{Pol}. In the conformal field theory\nformulation, a D$p$-brane is a $p$-dimensional\nhyperplane in target space on which strings\nsatisfy the Dirichlet boundary conditions\n\\cite{DLP}. In the low energy field theory limit\n(supergravity), it appears as a soliton-like background\nwith nontrivial R-R antisymmetric tensor field, solving\nclassical equations of motion (see a recent review\n\\cite{LP} and references therein). The low energy\ndynamics of  parallel D-branes, due to strings stretched\nbetween them, can be described by a dimensionally\nreduced supersymmetric Yang-Mills theory on their\nworld volume \\cite{Wit}, which happens to\ndescribe a quantum space in the sense of\nnon-commutative geometry \\cite{HW}.\n\nIn the M(atrix) model \\cite{BFSS} for M theory,\nwhich is conjectured to unify all known perturbative\nstring theories, the D0-branes are treated as\nfundamental microscopic degrees of freedom. The\nSYM quantum mechanics, which was originally\nthought to be the low-energy theory of $N$\nD0-branes, is promoted in the large $N$ limit\nto the status of the fundamental light-cone\ndynamics of M theory. As dimensionally reduced\n$U(N)$ SYM theory, its field content matches\nthe lowest modes of open strings ending on D0-branes.\nThus, in M(atrix) theory, everything else appears\nas a collective (bound) state of D0-branes. In\nparticular, a multiple parallel D-brane background is\nrealized as a block-diagonal matrix \\cite{BFSS},\neach block represented by a topologically nontrivial\ngauge field configuration \\cite{BSS,GRT} on a\nD-brane volume. In this paper we study the dynamics\nof D-branes by introducing and examining off-diagonal\nblocks, that are supposed to correspond to strings\nstretched between D-branes. One of the advantages\nof the M(atrix) theory is that it provides a unifying\nframework for explicitly dealing with both D-brane\nbackgrounds and strings stretched between them.\n\nPreviously Berkooz and Douglas \\cite{BD} have\nconsidered the background of a longitudinal M5-brane,\nwhich wraps around the (invisible) 11-th direction that\ndefines the light-cone to give rise to a D4-brane\nin \\mbox{\\II A\\hspace{.2em}} language. They bypassed the question of\nexplicitly representing the D4-brane in matrix form,\nbut rather proposed a modified M(atrix) theory by\nintroducing by hand additional dynamical variables\nthat are supposed to correspond to the massless\nmodes of open strings stretched between the D4-brane\nand the D0-branes (called 0-4 strings).\nIt was shown that integrating out the extra variables\nleads to the correct gravitational field of an\nM5-brane. Later Dijkgraaf, Verlinde and\nVerlinde \\cite{DVVa} showed that if one\nintegrates out the off-diagonal blocks in the\n$U(N)$ matrix fields with two diagonal\nblocks for a D4-brane and a D0-brane\nrespectively, one can also recover the\ngravitational field of a longitudinal M5-brane.\nBased on this result, one may be tempted to\nidentify the extra fields introduced in Ref.\n\\cite{BD} with the above-mentioned off-diagonal\nblocks. However, there is a mismatch for the\nquantum numbers: the extra bosonic field in\nRef. \\cite{BD} is a spinor of the $SO(4)$\nin the 4-brane directions, in accordance with\nstring theory \\cite{Pol1}, while the bosonic\noff-diagonal block is an $SO(4)$ vector.\nResolving this puzzle was part of the\nmotivation for this paper.\n\nAnother related, unsettled issue is how to obtain\nthe 32 additional fermions in the heterotic matrix\ntheory, which is the M(atrix) theory compactified\non $S^1\/Z_2$. First it was suggested to add\nthese fermions by hand to cancel anomalies in\nthe 1+1 dimensional field theory \\cite{KR,BSS1,BM}.\nLater Horava \\cite{Hor} proposed that they are\nzero modes of the off-diagonal blocks that\ncorrespond to 0-8 strings. However, there is\na puzzle of why these fermions are invariant\nunder surviving supersymmetries. A better\nunderstanding of the 0-8 strings in the M(atrix)\ntheory should help resolve this problem.\n\nIn this paper we study the spectrum of the\noff-diagonal blocks in M(atrix) theory that are\nsupposed to correspond to $p$-$p'$ strings in the\nbackground of a D$p$-brane and a D$p'$-brane.\nIn particular we show that the spectrum\nof zero modes for the off-diagonal blocks matches\nthe massless spectrum of $p$-$p'$ strings.\nSince the string theory results about $p$-$p'$ string\nspectrum are most directly seen in the\nNeveu-Schwarz-Ramond formalism, while the\nM(atrix) description of type \\mbox{\\II A\\hspace{.2em}} theory\n\\cite{Motl,BS,DVV} is in the Green-Schwarz\nformalism, it is nontrivial to check if their\npredictions agree. Moreover, note that D-brane\ncharges and supersymmetry do not give a complete\ncharacterization for parallel D-brane configurations\nin M(atrix) theory. The study of the zero-modes\nof  off-diagonal blocks will provide more\ninformation on proper identification of D-brane\nbackgrounds, and on their dynamical behavior\nas well, such as R-R charge and stability etc.\n\nIn this paper we will refer to configurations\nin M theory by their names in the \\mbox{\\II A\\hspace{.2em}} theory\nthat is related to the M theory through\ncompactification of the (invisible) eleventh\ndimension. Hence a D0-brane is a Kaluza-Klein\nmode of a graviton, a D2-brane an M-membrane,\na D4-brane a longitudinal M5-brane \\cite{Town}.\nIt is unclear what D6 and\nD8-branes in \\mbox{\\II A\\hspace{.2em}} really correspond to in M theory,\nbut they are needed to give various D-branes\nunder compactifications.\n\nWe will review related results in string theory in\nSec.\\ref{review} and M(atrix) description of\nD-brane configurations in Sec.\\ref{bundle}.\nIn Sec.\\ref{eom} we derive the equations of motion\nfor the bosonic and fermionic zero modes\nof the off-diagonal blocks, which we will use\nto find the zero modes, and explain how to derive their\nsupersymmetry transformations and low-energy\neffective action, for 0-2, 0-4, 0-6 and 0-8 strings\nrespectively in Sec.\\ref{02}-\\ref{0608}. Sec. \\ref{04}\nalso includes a discussion on the application of\nthe off-diagonal zero modes to the matrix description\n\\cite{BD} of longitudinal fivebranes. More discussions\non the physical implications of our results can be found\nin Sec. \\ref{0608} and in Sec. \\ref{Disc} .\n\n\\section{Review of $p$-$p'$ Strings} \\label{review}\n\nIn this section we briefly review the results in\nstring theory on $p$-$p'$ strings \\cite{Pol1}.\nFirst we consider an open string connecting a\nD$p$-brane and a D$p'$-brane parallel to each\nother. Since we are using \\mbox{\\II A\\hspace{.2em}} language,\nboth $p$ and $p'$ are even integers. Assuming\nthat $p' \\geq p$. In directions $0,1,\\cdots,p$,\nwhere the two D-branes overlap, the bosonic fields\n$X$ have Neumann boundary conditions on\nboth ends. In directions $p+1,\\cdots,p'$,\nthey have Dirichlet boundary condition on the\n$p$-brane and Neumann condition on the $p'$-brane.\nIn the rest directions $p'+1,\\cdots,9$, the open string\nhas Dirichlet conditions on both ends.\n\nThere will be unbroken supersymmetries for a\nsystem of parallel D$p$-branes and D$p'$-branes,\nif and only if the number, $\\nu$, of the directions\nin which the bosonic sector has DN or ND boundary\nconditions is 0,4 or 8. (Note that $\\nu=p'-p$ for a\nparallel D$p$- and D$p'$-brane.)\n\nThe Ramond sector of the $p$-$p'$ string\nhas the same kind of boundary conditions\nas the bosonic part. It always offers a massless\nfermionic $SO(1,9-(p'-p))$ Weyl spinor\n(after GSO projection) for the directions with\nNN or DD boundary conditions. The NS sector has\nthe opposite kind of boundary conditions.\nOnly when $(p'-p)=4$ will there be a massless\n$SO(p'-p)$ bosonic Weyl spinor for the directions\nwith ND or DN boundary conditions.\n\nSince we can always use T-duality to switch a\nD$p$-brane to a D0-brane, we only need to\nconsider four types of open strings: the 0-2, 0-4,\n0-6 and 0-8 strings.  In summary, the massless\nspectrum for a 0-$p$ string consists of\nonly a fermionic $SO(1,9-p)$ Weyl spinor,\nexcept that when $p=4$ there is in addition\na bosonic $SO(4)$ Weyl spinor. Below we are\ngoing to verify this spectrum of massless fermionic\nand bosonic modes in M(atrix) theory. (Though\nit is amusing to note that in M(atrix) theory, the\nbosonic off-diagonal blocks that are supposed\nto correspond to 0-4 strings are $SO(4)$ vectors!)\n\n\\section{M(atrix) Description of D-Brane Configurations}\n\\label{bundle}\n\nThe action of the M(atrix) model is \\cite{BFSS}\n\\begin{equation} \\label{action}\nS=\\int dt Tr(\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\n-\\frac{1}{2}\\bar{\\Psi}\\Gamma^{\\mu}[X_{\\mu},\\Psi]),\n\\end{equation}\nwhere $\\mu, \\nu=0,1,\\cdots,9$,\n$F_{\\mu\\nu}=[X_{\\mu}, X_{\\nu}]$\nand $X_0=-iD_0=-i(\\frac{\\partial}{\\partial t}+A_0)$.\n$X_{\\mu}$ and $\\Psi_{\\alpha}$ are Hermitian $N\\times N$ matrices.\nThe dynamical and kinematical SUSY transformations\nare respectively \\cite{BSS}\n\\begin{eqnarray}\n\\delta X_{\\mu}&=&i\\bar{\\epsilon}\\Gamma_{\\mu}\\Psi, \\quad \\mu=0,1,\\cdots,9, \\label{dyn1}\\\\\n\\delta\\Psi&=&(D_0 X_i)\\Gamma^{0i}\\epsilon+\\frac{i}{2}[X_i, X_j]\\Gamma^{ij}\\epsilon,\n\\quad i,j=1,2,\\cdots,9, \\label{dyn2}\n\\end{eqnarray}\nand\n\\begin{equation} \\label{kin}\n\\delta X_{\\mu}=0, \\quad \\delta\\Psi=\\tilde{\\epsilon},\n\\end{equation}\neach with 16 generators.\n\nThe configuration of a D$p$-brane in M(atrix) theory\nis given by big (infinite dimensional) matrices\ngiving the appropriate $p$-brane charge \\cite{BSS}:\n\\footnote{Only the 2-brane and 4-brane charges are defined\nin the SUSY algebra. The 6 and 8-brane charges are\nextrapolations of those.}\n\\begin{equation}\ntr(\\epsilon_{\\mu_1,\\cdots,\\mu_p}[X_{\\mu_1},X_{\\mu_2}]\\cdots\n[X_{\\mu_{p-1}},X_{\\mu_p}]).\n\\end{equation}\nWe can choose the $X$'s to satisfy\n\\begin{equation} \\label{ZZ}\n[X_{2n-1},X_{2n}]=F_{(2n-1)(2n)}\n\\end{equation}\nfor $n=1,2,\\cdots,p\/2$\nwith $F_{\\mu\\nu}$ being constant $K\\times K$ matrices.\nThe fermionic partner is taken to be zero.\n\nThere are two ways to realize this physical setting\nin the M(atrix) theory.\nTake the D2-brane as an example.\nOne way is to set $X_1=R_1 P$, $X_2=R_2 Q$,\nwhere $[P,Q]=i2\\pi\/N$ \\cite{BFSS}.\n$P$ and $Q$ can in turn be realized as\n$P=-i(2\\pi\/N)\\frac{\\partial}{\\partial\\sigma}$ and $Q=-\\sigma$\nthrough an angle parameter $\\sigma\\in[0,2\\pi)$.\nAnother way is to first compactify the M(atrix) model\non a torus with radii $R_i, i=1,2$,\nand then take the limit $R_i\\rightarrow\\infty$\nif one wishes.\nA D$p$-brane configuration corresponds to\na gauge field configuration with certain\ntopological charge \\cite{BSS}\n(the $k$-th Chern character $Q_k=\\int trF^k$ for $k=p\/2$.)\non the dual torus which becomes infinitesimal\nin the large radii limit.\nThe $X$ matrices in the $p$-brane directions\nbecome ($(-i)$ times) the covariant derivatives.\nFor a D2-brane it can be taken as, say,\n$X_1=-i2\\pi R_1\\frac{\\partial}{\\partial\\sigma_1}$\nand $X_2=-i2\\pi R_2\\frac{\\partial}{\\partial\\sigma_2}-R_2\\sigma_1$.\n\nFor our purpose the difference between the two descriptions\nis only a scaling in the derivatives.\nFor simplicity in notation\nwe choose to use the latter description in this paper.\n\nA static D$p$-brane configuration (\\ref{ZZ})\npreserves half of the total SUSY if and only if\nthe $F$'s are proportional to the unit matrix,\nin which case 16 linear combinations of\nthe dynamical and kinematical SUSY are preserved \\cite{BSS}.\nThese states contain D0, D2,..., D$(p-2)$-branes\nin addition to the D$p$-brane.\nThe kinematical SUSY (\\ref{kin}) is never preserved by itself.\nThe condition for part of the dynamical SUSY to be preserved is\n\\begin{equation} \\label{FF}\nF_{12}-\\sum_{i=2}^{p\/2}\\varepsilon_i F_{(2i-1)(2i)}=0\n\\end{equation}\nfor some $\\varepsilon_i=\\pm 1$.\nIt preserves $1\/2^{(p\/2-1)}$ of the dynamical SUSY\nparametrized by $\\epsilon$ satisfying\n\\begin{equation}\n\\Gamma^{12}\\Gamma^{(2i-1)(2i)}\\epsilon=\\varepsilon_i\\epsilon, \\quad i=2,3,\\cdots,p\/2.\n\\end{equation}\nBecause $tr(F_{12}^2)\\neq 0$ it follows from (\\ref{FF})\nthat any D6 or D8-brane configuration with unbroken dynamical SUSY\nmust always include D4-branes.\nA discussion on general bound states from\nthe low energy D-brane point of view can be found in \\cite{GR}.\n\nIf all $F_{\\mu\\nu}$'s in (\\ref{ZZ})\nare proportional to the unit matrix,\nthey define a natural complex\nstructure on the dual torus.\nIt can be used to view the dual torus $T^p$ as\ncomposed of $p\/2$ complex tori $T^2$.\nA D$p$-brane with unit $p$-brane charge can be realized by\na $U(K)$ gauge field with twisted boundary conditions.\nThis is analogous to how one defines\na long string \\cite{DVV,BS}\nin the conjugacy class of length $K$.\nThe unit D$p$-brane charge means a twisted bundle\nwith the minimal topological charge on each $T^2$.\n\nAn explicit construction of\nthe minimal twisted bundle\nof the fundamental representation of $U(K)$\nis given in Ref.\\cite{GRT}.\nThere the gauge fields can be chosen as $A_1=0$\nand $A_2=-i(\\sigma_1\/2\\pi K){\\bf 1}$,\nwhere we use $\\sigma_i$ as the coordinates on $T^2$\nnormalized to range between $0$ and $2\\pi$\nand ${\\bf 1}$ is the unit matrix.\nThe field strength $F_{12}$ is then $1\/2\\pi K$.\nThe quasi-periodic boundary conditions on $A$ are \\cite{GRT}:\n\\begin{eqnarray}\n&A_{\\mu}(2\\pi,\\sigma_2)=\\Omega_1(\\sigma_2)A_{\\mu}(0,\\sigma_2)\\Omega_1^{-1}(\\sigma_2)\n+\\Omega_1(\\sigma_2)\\partial_{\\mu}\\Omega_1^{-1}(\\sigma_2), \\\\\n&A_{\\mu}(\\sigma_1,2\\pi)=\\Omega_2(\\sigma_1)A_{\\mu}(\\sigma_1,0)\\Omega_2^{-1}(\\sigma_1)\n+\\Omega_2(\\sigma_1)\\partial_{\\mu}\\Omega_2^{-1}(\\sigma_1),\n\\end{eqnarray}\nwhere $\\Omega_1$ and $\\Omega_2$ can be chosen as\n\\begin{equation}\n\\Omega_1(\\sigma_2)=q^{\\sigma_2\/2\\pi} U, \\quad \\Omega_2(\\sigma_1)=V,\n\\end{equation}\nwhere $q=e^{i2\\pi\/K}$, $U_{ij}=q^{i}\\delta_{ij}$ and\n$V_{ij}=\\delta_{i+1,j}$ with $i,j=0,\\cdots,K-1\\; (mod \\; K)$.\n$U$ and $V$ satisfy $UV=q^{-1}VU$.\nIt can be checked that $\\Omega_1(2\\pi)\\Omega_2(0)=\\Omega_2(2\\pi)\\Omega_1(0)$.\nThis is in contrast with the twisted bundle of $SU(K)\/{\\bf Z}_K$\nwhere one has $\\Omega_1(2\\pi)\\Omega_2(0)=\\Omega_2(2\\pi)\\Omega_1(0)Z$\nfor some element $Z$ in the center ${\\bf Z}_K$ of $SU(K)$ \\cite{tH}.\n\nThe bundle in fundamental representation has the corresponding\nboundary conditions:\n\\begin{equation}\n\\phi(2\\pi,\\sigma_2)=\\Omega_1(\\sigma_2)\\phi(0,\\sigma_2), \\quad\n\\phi(\\sigma_1,2\\pi)=\\Omega_2(\\sigma_1)\\phi(\\sigma_1,0).\n\\end{equation}\nNote that consistency of the boundary conditions\nrequires\n$$\\Omega_1(2\\pi)\\Omega_2(0)=\\Omega_2(2\\pi)\\Omega_1(0).$$\nA section of the bundle has the general form of \\cite{GRT}\n\\begin{equation} \\label{section}\n\\phi_j(\\sigma_1,\\sigma_2)=\\sum_{m\\in{\\bf Z}}\n\\hat{\\phi}(\\sigma_2\/2\\pi+j+mK)q^{(\\sigma_2\/2\\pi+j+mK)\\sigma_1\/2\\pi},\n\\quad j=0,1,\\cdots,K-1,\n\\end{equation}\nfor an arbitrary function $\\hat{\\phi}$ for which the series converges.\n\nSince this is the D-brane analogue of a long string in\nthe conjugacy class of length $K$,\nthis gauge field configuration\nis identified with a single D2-brane\ninstead of $K$ D2-branes.\nHere $K$ gets interpreted as the longitudinal\nmomentum carried by the single D2-brane, as can be\nseen by examining its light-cone energy.\n\nIt is essential that the gauge group is $U(K)$\ninstead of $SU(K)$.\nAlthough there are twisted $SU(K)\/{\\bf Z}_K$ bundles\nin the adjoint representation \\cite{tH}\nwith the same topological charge,\nthere is no corresponding vector bundle in\nthe fundamental representation,\nbecause the element $Z$ acts nontrivially on\nthe fundamental representation while\nit acts trivially on the adjoint.\nNote that the presence of anything other than the D4-branes\nintroduces off-diagonal blocks in\nthe fundamental representation.\nHence, for instance,\nalthough one can use two copies of\nthe twisted $SU(2)$ bundle on $T^2$\nwith (anti-)self-duality\nto construct pure D4-brane states\npreserving half of the dynamical SUSY,\nat this moment it is unclear how to describe their\ninteraction with other D-branes.\n\n\\section{Equations of Motion} \\label{eom}\n\nConsider a D$0$-brane very close to a D$p$-brane.\nWe decompose the matrix fields into the block form:\n\\begin{equation} \\label{block}\nX_{\\mu}=\\left(\\begin{array}{cc}\n                Z_{\\mu} & y_{\\mu} \\\\\n                y^{\\dagger}_{\\mu} & x_{\\mu}\n             \\end{array}\\right); \\quad\n\\Psi=\\left(\\begin{array}{cc}\n              \\Theta & \\theta \\\\\n              \\th^{\\dagger} & \\psi\n           \\end{array}\\right),\n\\end{equation}\nwhere $Z_{\\mu}$ represents the D$p$-brane\nand $x_{\\mu}$ the D$0$-brane.\nThe generalization to many D$p$-branes and D0-branes\nis straightforward.\nWhile the $Z$'s are realized as covariant derivatives,\nthe $x$'s in general can have nontrivial\ncoordinate dependence on the dual torus,\nbut when we take the limit of infinite radii,\nonly coordinate-independent states can have\nfinite energy and remain coupled to the theory.\n(We allow infinite energy for $Z$ because\nit is just the energy for the D$p$-brane.)\n\nFor simplicity we choose the coordinates of spacetime\nsuch that $x_{\\mu}=0$ and $Z_a=0, a=p+1,\\cdots,9$.\nThe D$p$-brane is parallel to the directions $1,2,\\cdots,p$\nand the D$0$-brane is right on top of it.\nThe diagonal part is taken as the background configuration.\n\nWhen putting the two D-branes together\nas in (\\ref{block})\nand set the off-diagonal parts to be zeros,\none can check easily that part of the supersymmetry is\npreserved only if $p$ is $0,4$ or $8$.\n\nTo count the number of zero modes,\nor equivalently to count the dimension of\nthe moduli space for this background,\nit is easier to consider the perturbation of this\nbackground and keep only the lowest order terms\nto obtain linear differential equations for the\nperturbative fields $y$ and $\\theta$.\nIn this way we count the dimension of the tangent space\non the moduli space.\nOne may also introduce perturbations in\nthe diagonal blocks for fluctuations on the D$p$-brane\nand deviations of the D0-brane from the origin,\nbut here we are for the time being only\ninterested in the off-diagonal blocks $y$ and $\\theta$\nsince they represent the $p$-$p'$ strings.\nThe perturbations of the diagonal blocks can be studied\nin the same way we study the off-diagonal part.\nTo the lowest order in perturbation,\nthe perturbative diagonal and off-diagonal parts\nare not correlated,\nhence we can treat the off-diagonal ones alone.\n\nPlugging the expression of the matrix fields\n(\\ref{block}) into the action of the M(atrix) model (\\ref{action}),\nwe find\n\\begin{equation}\nS=\\int dt (L_Z+L_x+L_y),\n\\end{equation}\nwhere $L_Z$ and $L_x$ are of the same form as (\\ref{action})\nexcept that we replace ($X,\\Psi$) by ($Z,\\Theta$) and ($x,\\psi$),\nrespectively.\n$L_y=L_B+L_F$ with\n\\begin{eqnarray}\nL_B&=&\\frac{1}{2}|(Z_{\\mu}y_{\\nu}-Z_{\\nu}y_{\\mu}+y_{\\mu}x_{\\nu}-y_{\\nu}x_{\\mu})|^2\n+y_{\\mu}^{\\dagger}[Z_{\\mu}, Z_{\\nu}]y_{\\nu}\n-[x_{\\mu}, x_{\\nu}]y_{\\mu}^{\\dagger}y_{\\nu}\\nonumber\\\\\n&&-y_{\\mu}^{\\dagger}y_{\\nu}y_{\\mu}^{\\dagger}y_{\\nu}\n+\\frac{1}{2}y_{\\mu}^{\\dagger}y_{\\mu}y_{\\nu}^{\\dagger}y_{\\nu}\n+\\frac{1}{2}y_{\\mu}^{\\dagger}y_{\\nu}y_{\\nu}^{\\dagger}y_{\\mu}\n\\end{eqnarray}\nand\n\\begin{equation}\nL_F=-\\theta^{\\dagger}\\Gamma^0\\Gamma^{\\mu}(Z_{\\mu}\\theta-\\theta x_{\\mu})\n+\\theta^{\\dagger}\\Gamma^0\\Gamma^{\\mu}(\\Theta y_{\\mu}-y_{\\mu}\\psi)\n+(y_{\\mu}^{\\dagger}\\Theta-\\psi y_{\\mu}^{\\dagger})\\Gamma^0\\Gamma^{\\mu}\\theta.\n\\end{equation}\nFor more than one D0-branes the $x$'s are matrices\nand we need to take traces for these formulas.\n\n>From the action\none can derive the equations of motion for $y$ and $\\theta$.\nSince the Hamiltonian for a time-independent background\nin the temporal gauge ($A_0=0$) is minimized\nby time-independent $y$,\nwe look for time-independent solutions for $y$ and $\\theta$.\nIgnoring the time derivative and higher order terms,\nwe find\n\\begin{eqnarray}\n&D^{\\mu}(D_{\\mu}y_{\\nu}-D_{\\nu}y_{\\mu})+[D_{\\nu}, D_{\\mu}]y^{\\mu}=0,\n\\quad &\\mu, \\nu=1,\\cdots,p,\\label{eom1-y}\\\\\n&D^{\\mu}D_{\\mu}y_a=0, \\quad &a=p+1,\\cdots,9,\\label{ya}\n\\end{eqnarray}\nwhere $D_{\\mu}=iZ_{\\mu}$ are\ncovariant derivatives on the dual torus $T^p$ as\n$D_{\\mu}=2\\pi R_{\\mu}(\\frac{\\partial}{\\partial\\sigma^{\\mu}}+A(\\sigma))$,\n$\\mu=1,\\cdots,p$.\nEq.(\\ref{eom1-y}) has to be supplemented by\nthe gauge-fixing condition\n\\begin{equation} \\label{eom2-y}\nD_{\\mu}y^{\\mu}=0.\n\\end{equation}\nUsing (\\ref{eom2-y}), eq.(\\ref{eom1-y}) can be written as\n\\begin{equation} \\label{eom3-y}\nD_{\\mu}D^{\\mu}y_{\\nu}+2[D_{\\nu},D_{\\mu}]y^{\\mu}=0.\n\\end{equation}\n\nThe equation of motion for $\\theta$ is\n\\begin{equation}\\label{eom-th}\n\\Gamma^{\\mu}D_{\\mu}\\theta=0.\n\\end{equation}\n\nIn terms of the covariant exterior derivative $d_A$,\nits dual $d_A^*$, the Hodge dual $*$\nand the projection $P=\\frac{1}{2}(1-*)$ (so $P^2=P$),\neqs.(\\ref{eom1-y}) and (\\ref{eom2-y}) now read\n\\begin{eqnarray}\n& d_A^* Pd_A y=0, \\\\\n& d_A^* y=0,\n\\end{eqnarray}\nwhere $y=y_{\\mu}d\\sigma^{\\mu}$.\nThese equations are formally the same as those\nfor the instanton zero modes,\nwhich correspond to perturbations\nof the $Z$'s above.\nThe only difference is that the perturbations of $Z$\nis in the adjoint representation of $U(K)$,\nwhile $y$ is in the fundamental representation.\n\nBecause we are considering the Euclidean torus,\nthe inner product $\\langle\\cdot|\\cdot\\rangle$\ndefined by integration on the torus\nand the trace of matrices is positive definite.\nHence $\\langle y|d_A^* Pd_A y\\rangle=0$ implies that\n\\begin{equation} \\label{PdAy}\nPd_A y=0.\n\\end{equation}\nIn addition, eq.(\\ref{ya}) implies that\n$\\langle D_{\\mu}y_a|D^{\\mu}y_a\\rangle=0$\nand so $D_{\\mu}y_a=0$,\nwhich means that the topological charge vanishes\nunless $y_a=0$.\nThus we conclude that $y_a=0$ for $a=p+1,\\cdots,9$.\n\n\\section{0-2 Strings} \\label{02}\n\nLet $Z_1$ and $Z_2$ be realized as $U(K)$ covariant derivatives\non the dual torus as $Z_i=-i D_i$ with\n$D_i=\\frac{\\partial}{\\partial\\sigma_i}+A_i$ as given in Sec.\\ref{bundle}\nso that\n\\begin{equation}\n[Z_1,Z_2]=if{\\bf 1},\n\\end{equation}\nwhere $f=2\\pi R_1 R_2\/K$.\nFor simplicity we are considering unslanted torus\nwith radii $R_1=R_2=1\/2\\pi$.\nIt is straightforward to generalize to slanted tori\nwith arbitrary radii.\nLet $z=(\\sigma_1+i\\sigma_2)\/2\\pi$, $\\bar{z}=(\\sigma_1-i\\sigma_2)\/2\\pi$\nbe the complex coordinates on $T^2$,\nand let ${\\cal D}=D_1-iD_2$ and $\\bar{\\cD}=D_1+iD_2=-{\\cal D}^{\\dagger}$,\nthen $[{\\cal D},\\bar{\\cD}]=2f{\\bf 1}$.\nIt follows that\n\\begin{equation}\nD^2={\\cal D}\\bar{\\cD}-f=\\bar{\\cD}{\\cal D}+f,\n\\end{equation}\nwhere $D^2=D_1^2+D_2^2$.\nNote that the algebra of $\\bar{\\cD}$ and $-{\\cal D}$\nis the canonical commutation relation\nfor annihilation and creation operators scaled by $2f$.\nTherefore the spectrum of ${\\cal D}\\bar{\\cD}$ is $\\{0,-2f,-4f,\\cdots\\}$\nand the spectrum of $D^2$ is\n\\begin{equation} \\label{spec}\n\\{-f,-3f,-5f,\\cdots\\}.\n\\end{equation}\n\nThe fermionic zero modes satisfy (\\ref{eom-th}),\nwhich gives $(D_1+\\Gamma^1\\Gamma^2 D_2)\\theta=0$, so that\n\\begin{equation}\n{\\cal D}\\theta_+=0, \\quad \\bar{\\cD}\\theta_-=0,\n\\end{equation}\nwhere $\\theta_{\\pm}$ are the two Weyl components of $\\theta$\nsatisfying $i\\Gamma^1\\Gamma^2\\theta_{\\pm}=\\pm\\theta_{\\pm}$.\nBecause $\\langle\\theta_+|\\bar{\\cD}{\\cal D}\\theta_+\\rangle=\\langle\\theta|(D^2-f)\\theta\\rangle<0$\nfor any $\\theta_+\\neq 0$,\nwe must have $\\theta_+=0$.\nThe solution of $\\theta_-$ is obviously the vacuum state\nannihilated by $\\bar{\\cD}$.\n\nOne can easily get the explicit expression of the vacuum\nas a section of the twisted bundle\nusing the explicit construction in Sec.\\ref{bundle}.\nAnother way is to note that\nthe equation $\\bar{\\cD}\\phi=0$ has the general solution of\n$\\phi =\\exp (-\\frac{\\pi}{4K}(\\bar{z}^2+2z\\bar{z}))f(z)$,\nwhere $f(z)$ is an arbitrary holomorphic function.\nFor $\\phi$ to be a section of the twisted bundle,\nwe need to impose the quasi-periodic\nboundary conditions on $\\phi$.\nOne then sees that\n$f(z)$ is related to the third elliptic theta function $\\vartheta_3$ and\nthe solution is\n\\begin{equation} \\label{theta}\n\\phi_k(\\sigma_1,\\sigma_2)=\\exp\\{\\pi\/K[2i(\\sigma_1\/2\\pi)(\\sigma_2\/2\\pi+k)-(\\sigma_2\/2\\pi+k)^2]\\}\n\\vartheta_3(q|\\pi(z+ik)),\n\\end{equation}\nwhere $q=\\exp (-\\pi K)$ and $\\phi_k$ ($k=0,1,\\cdots,K-1$)\ngives a section on the vector bundle in the fundamental representation.\nApplying the creation operator ${\\cal D}$ to the vacuum\none obtains other eigenstates of the operator $D^2$.\n\nObviously the zero mode of $\\theta_-$ is just given by\nthe solution (\\ref{theta}).\nThe fermionic zero mode is an $SO(2)$ Weyl spinor\nwith negative chirality.\n\nThe equations of motion (\\ref{eom3-y}) for $y_{\\mu}$ are\n\\begin{equation}\n(D^2-2f)y=0, \\quad (D^2+2f)\\bar{y}=0,\n\\end{equation}\nwhere $y=y_1+iy_2$, $\\bar{y}=y_1-iy_2$.\nThe constraint (\\ref{eom2-y}) is\n\\begin{equation}\n{\\cal D} y+\\bar{\\cD}\\bar{y}=0.\n\\end{equation}\nSince the spectrum of $D^2$ is given by (\\ref{spec}),\nwe see that there is no solution for $y$, $\\bar{y}$,\nhence there is no bosonic zero mode.\n\n\\section{0-4 Strings} \\label{04}\n\nWe decompose the 10 dimensional $\\gamma$-matrices as\n\\begin{eqnarray}\n\\Gamma^0&=&i\\sigma_2\\otimes{\\bf 1}\\otimes{\\bf 1}, \\\\\n\\Gamma^{\\mu}&=&\\sigma_1\\otimes\\gamma^{\\mu}\\otimes{\\bf 1}, \\quad \\mu=1,\\cdots,4,\\\\\n\\Gamma^a&=&\\sigma_1\\otimes\\gamma^5\\otimes\\gamma^a, \\quad a=5,\\cdots,9, \\\\\n\\Gamma^{10}&=&\\sigma_3\\otimes{\\bf 1}\\otimes{\\bf 1},\n\\end{eqnarray}\nwhere the $\\sigma_i$'s are the Pauli matrices satisfying\n$\\sigma_1\\sigma_2=i\\sigma_3$, the $\\gamma^{\\mu}$'s ($\\mu=1,\\cdots,4$)\nare the $\\gamma$-matrices\nfor $SO(4)$, the $\\gamma^a$'s ($a=5,\\cdots,9$) are\n$SO(5)$ $\\gamma$-matrices.\nCorresponding to this decomposition,\na 10-dimensional spinor $\\theta_{i\\alpha\\beta}$ has three indices,\nwhere $i=\\pm$, $\\alpha,\\beta=1,\\cdots,4$.\nFor Weyl spinors with positive (negative) chirality\none has $i=+$ ($i=-$).\nSince all spinors in this theory are\n10-dimensional Weyl spinors with positive chirality,\nwe will omit the index $i$ in the following.\nWe consider the case where the gauge field for\nthe D4-brane background is self-dual,\nso that half of the dynamical SUSY\nwith parameter $\\epsilon$ satisfying\n\\begin{equation} \\label{eps4}\n\\Gamma^1\\Gamma^2\\Gamma^3\\Gamma^4\\epsilon=\\epsilon\n\\end{equation}\nis preserved.\n\nThe number of zero modes for the case of $T^4$\nfor (anti-)self-dual gauge field configurations\ncan be determined using\nthe index theorem \\cite{ASch,Baal}.\nThe number of spinorial zero modes is found to be $\\alpha_V k$,\nwhere $\\alpha_V$ is the Dynkin index for the representation $V$\nof the fermions,\nand $k$ is the instanton number.\nThe number of vectorial zero modes is $2\\alpha_V k$.\nFor any $U(K)$, the number of\nspinorial and vectorial zero modes\nin the fundamental representation\nare $k$ and $2k$, respectively.\nThe general formula \\cite{ASch}\nfor spinorial and vectorial zero modes in\narbitrary representation $V$ on a 4-manifold $M$ are\n$\\alpha_V k+\\frac{1}{8}dimV\\tau(M)$ and\n$2\\alpha_V k-\\frac{1}{2}dimV(\\chi(M)+\\tau(M))$,\nrespectively, where\n$\\tau(M)=\\frac{1}{96\\pi^2}\\int\\epsilon^{\\mu\\nu\\alpha\\beta}R_{\\mu\\nu\\l\\rho}R_{\\alpha\\beta}^{\\l\\rho}dv$\nis the signature of $M$ and\n$\\chi(M)=\\frac{1}{128\\pi^2}\\int\\epsilon^{\\mu\\nu\\alpha\\beta}\\epsilon^{\\l\\rho\\gamma\\delta}\nR_{\\mu\\nu\\l\\rho}R_{\\alpha\\beta\\gamma\\delta}dv$ is the Euler characteristic.\n\nWhile according to the the index theorem\nthe number of zero modes is independent of\nthe details of\n(anti-)self-dual gauge field configurations,\nhere we give for example an explicit construction of a twisted bundle\nfor the cases with $R_1 R_2=R_3 R_4$.\nOn each $T^2$ factor of $T^4$, one can construct\na twisted $U(K)$ bundle as in Sec.\\ref{bundle}.\nWhen putting them together, we obtain a $U(K^2)$ bundle\nwith unit instanton number:\n$\\frac{1}{8\\pi^2}\\int tr(F^2)=1$.\nUnlike the case of twisted $SU(K)\/{\\bf Z}_K$ bundles which\ncan have fractional instanton numbers,\nfor $U(K)$ the instanton number are always integral \\cite{GR}.\nA section on the twisted $U(K^2)$ bundle on $T^4$ has\nthe general form of a linear combination of\nproducts of sections on each $T^2$:\n$\\phi_j(\\sigma_1,\\sigma_2)\\phi_k(\\sigma_3,\\sigma_4)$,\nwhere $\\phi$ is defined by (\\ref{section})\nfor $j, k=0,1,\\cdots,(K-1)$.\nIndices $j$ and $k$ compose an index for\nthe fundamental representation of $U(K^2)$.\nIn general one can also consider a $U(K_1)$ and $U(K_2)$ bundle\non the two $T^2$ factors, respectively,\nand obtain a $U(K_1 K_2)$ bundle on $T^4$.\n\nBecause the supersymmetry is not completely broken,\nthe solution of fermionic zero modes can be used to\nobtain the solution of bosonic zero modes.\nThe solution of the fermionic and bosonic zero modes\ncan be obtained explicitly by considering\n$T^4$ as $T^2\\times T^2$ and using the methods in Sec.\\ref{02}.\nLet the $SO(4)$ spinor satisfying (\\ref{eom-th})\nbe denoted by $\\theta^0$.\nIt is easy to see that the fermionic zero mode satisfies\n$i\\Gamma^1\\Gamma^2\\theta^0=i\\Gamma^3\\Gamma^4\\theta^0=-\\theta^0$,\nwhich implies that $\\theta^0$ is of negative chirality\nas an $SO(4)$ Weyl spinor:\n$\\Gamma^1\\Gamma^2\\Gamma^3\\Gamma^4\\theta^0=-\\theta^0$.\n(If the gauge field is anti-self-dual,\nthe zero mode will be a Weyl spinor with positive chirality.)\nFor a single D4-brane there is only one fermionic zero mode,\nwhich is given by the product of the solutions (\\ref{theta})\non each $T^2$ factor in $T^4$.\n\nThe general solution of fermionic zero modes\ncan then be written as $\\theta_{\\dot{\\r}\\beta}=\\theta^0_{\\dot{\\r}}\\chi_{\\beta}$\n(and $\\theta_{\\rho\\beta}=0$),\nwhere $\\dot{\\r}=1,2$ ($\\rho$) is the index for an $SO(4)$ Weyl spinor\nwith negative (positive) chirality\nand $\\chi$ is an $SO(5)$ spinor for the directions\n$5,\\cdots,9$.\nThus $\\chi$ represents the massless fermionic mode\nfrom the Ramond sector of the 0-4 string.\n\nSince the equations of motion for $y$ and $\\theta$\nare supersymmetric,\nthe bosonic zero mode can be obtained by\nSUSY transformation \\cite{Zum} as\n\\begin{equation}\ny^{\\mu}=iv^{\\dagger}_{\\rho}\\gamma^{\\mu}_{\\rho\\dot{\\r}}\\theta^0_{\\dot{\\r}},\n\\end{equation}\nwhere $v_{\\rho}$ is an $SO(4)$ Weyl spinor\nwith positive chirality.\nThis comes from the SUSY transformation of $y$:\n$\\delta y_{\\mu}=i\\bar{\\epsilon}\\Gamma_{\\mu}\\theta$.\nWhen one replaces in this transformation $\\theta_{\\alpha\\beta}$ by\nthe zero mode $\\theta^0_{\\dot{\\r}}$,\n$\\delta y$ will satisfy the equations of motion of $y$\nfor any $\\epsilon_{\\rho}$ in the SUSY preserved by\nthe background (\\ref{eps4}).\nIt follows that the $y$ given by the above expression\nis a zero mode of $y$.\nSince $\\theta^0$ is a function (bosonic),\n$v$ is a bosonic variable.\nIt matches the massless bosonic field from the NS sector\nof the 0-4 string.\nHere it is amusing to see how supersymmetry dictates\nthe zero modes of a field $y$ in vector representation\nto be described by a variable $v$ in spinor representation.\nThe index theorem \\cite{ASch} assures\nus that these\nare all the zero modes in the theory,\ngiving precisely the massless spectrum of 0-4 strings.\nThe supersymmetry transformation between $\\chi$\nand $v$ is induced from the SUSY transformation\nbetween $\\theta$ and $y$ by factoring out the common factor\nof $\\theta^0$.\nUp to first order perturbation,\nthe SUSY transformation of $\\theta$ is:\n$\\delta\\theta=\\frac{1}{2}(D_{A}y_{B}-D_{B}y_{A})\\Gamma^{AB}\\epsilon$,\nwhere $A,B=0,1,\\cdots,9$.\nUsing (\\ref{PdAy}), (\\ref{ya}) and (\\ref{eps4}),\none finds\n\\footnote{Our notation is slightly different from\nthat of Ref.\\cite{BD}.}\n\\begin{eqnarray}\n& \\delta v_{\\rho}=\\chi^{\\dagger}_{\\alpha}\\epsilon_{\\rho\\alpha}, \\\\\n& \\delta \\chi_{\\alpha}=2i(D_0 v^{\\dagger}_{\\rho})\\epsilon_{\\rho\\alpha}.\n\\end{eqnarray}\n\nThe instanton connection lies in $SU(2)_R\\in SO(4)$ which\nis supposed to be the global R-symmetry for the action of 0-4 strings.\nField $v$ carries the fundamental index of $SU(2)_R$.\nLet $\\tau^i$ denote generators of the R-symmetry group. There are two\npossible $SU(2)_R$ invariant D-terms, $\\sum_i |v^+\\tau^i v|^2$ and\n$|v^+v|^2$. The two terms are different when there are more than one\nD0-branes, in which case only the first is actually present\nin the action \\cite{Pol1}.\nThese D-terms are expected to arise from the $F^2$ term in\nthe Super Yang-Mills theory. Upon expanding this term in $y$ one\nfinds $tr |y_\\mu y_\\nu^+ -y_\\nu y_\\mu^+|^2$ and $|y_\\mu^+y_\\nu\n-y_\\nu^+y_\\mu|^2$. For a given instanton background, since $SU(2)_R$\nis broken explicitly, these terms do not give those $SU(2)_R$\ninvariant D-terms. Only after averaging over the moduli space does\none expect that the symmetry $SU(2)_R$ is restored. However, we do not\nknow how to rule out the $U(1)$ D-term $|v^+v|^2$.\n\nThe above discussion easily generalizes to the case of instanton\nnumber $k$. There are $2k$ zero modes for $y_\\mu$, and can be\ninterpreted as the fundamental of $U(k)\\times SU(2)_R$, where\n$U(k)$ is the gauge group associated to $k$ coincident D4-branes.\n\nIn ref. \\cite{BD}, an action describing M(atrix) theory\nof a longitudinal 5-brane is proposed. Since a longitudinal\n5-brane in M-theory corresponds to a D4-brane in type \\mbox{\\II A\\hspace{.2em}}\nstring theory, some extra dynamical variables corresponding\nto 0-4 strings were needed and were introduced by hand.\nTheir quantum numbers are exactly the same as those of the\nvariables $v$ and $\\chi$ that we have discussed above.\nThus, it is natural to identify the additional variables\nintroduced by Berkooz and Douglas \\cite{BD} with the degrees\nof freedom associated with the off-diagonal zero modes. We have\nverified that indeed the action of the latter naturally derives\nfrom the fundamental M(atrix) model action, and it agrees with\nthe action postulated in ref. \\cite{BD}, with a possible\n$U(1)$ D-term as we mentioned above. (In the derivation,\nthe coefficient of each term in the action is determined by\nan integral of a product of the zero mode solutions $\\theta^0$.\nWe have not been able to calculate all coefficients; presumbly\nthey are uniquely determined by the surviving supersymmetry.)\n\nIn addition to the variables $v$ and $\\chi$, the action\nin ref. \\cite{BD} has included also fields describing\nfluctuations of the longitudinal fivebrane background,\nwhich in our approach correspond to fluctuations residing\nin the diagonal blocks. In principle one can consider\nfluctuations of all blocks in the matrix fields for a\ngiven background, and then solve the exact (nonlinear)\nequations of motion. The parameters analogous to $v$ and\n$\\chi$ above for the general solutions\ncorrespond to the massless modes of\nthe whole system of ($p'$-$p$)-branes.\nIn the above we have only solved the linearized\nequations of motion for the off-diagonal blocks.\nThe supersymmetry derived from our solutions will only\nhold to the lowest order in perturbation.\nIf one solves the exact nonlinear equations of motion,\none should be able to derive the exact SUSY transformation\namong the zero mode parameters.\n\nIn the above we have only considered the case with\nvanishing distance between the D0-brane and the D4-brane.\nWhen we pull the D0-brane away from the D4-brane,\nthe zero modes will gain masses proportional to the distance.\nBut we expect that the number and representation\nof the lowest energy modes will remain the same\nas the zero modes. The proposal of Ref.\\cite{BD}\ncontains only the lowest energy modes and\ntherefore should be viewed as a low energy effective theory.\n\n\\section{0-6 Strings and 0-8 Strings} \\label{0608}\n\nThe case of 0-6 strings and 0-8 strings\ncan be studied in a similar fashion\nas the 0-2 and 0-4 strings.\nTo generalize the consideration for 0-2 and 0-4 strings\nto 0-$p$ strings for $p=2,4,6,8$,\nwe choose the gauge field configuration for the D$p$-brane\nto be $p\/2$ copies of the gauge field configuration\non $T^2$ described in Sec.\\ref{02},\nthat is,\n\\begin{equation}\n[Z_{2i-1},Z_{2i}]=if{\\bf 1}, \\quad i=1,\\cdots,p\/2,\n\\end{equation}\nwhere $f=1\/2\\pi K$.\nThis defines a twisted $U(K^{p\/2})$ bundle with\nunit $p$-brane charge:\n$\\frac{1}{k!(2\\pi)^k}\\int tr(F^k)=1$ for $k=p\/2$.\n\nWe focus our attention on the first copy of $T^2$.\nLet $y=y_1+iy_2$ and $\\bar{y}=y_1-iy_2$.\nThe equations of motion for them are\n$(D^2-2f)y=0$ and $(D^2+2f)\\bar{y}=0$,\nwhere $D^2=\\sum_{\\mu=1}^{p}D_{\\mu}^2$ for a D$p$-brane.\nSince the spectrum of $D_1^2+D_2^2$ is shown to be\n$\\{-f,-3f,-5f,\\cdots\\}$ in Sec.\\ref{02},\nthe spectrum of $(D^2+2f)y$ is\n$\\{-(p\/2-2)f,-p\/2f,\\cdots\\}$ and\nthe spectrum of $(D^2-2f)$ is purely negative for any $p$.\nIt then follows that there is a zero mode for $y$\nonly if $p=4$.\n\nThe equation of motion for the fermionic mode\nis decomposed into $p\/2$ equations for a D$p$-brane:\n$(D_{2i-1}+\\Gamma^{2i-1}\\Gamma^{2i}D_{2i})\\theta=0$, $i=1,\\cdots,p\/2$.\nObviously the solution of $\\theta$ is simply\nthe product of the solution (\\ref{theta}) for each\ncopy of $T^2$ and\nit is of negative chirality on each $T^2$\nso that $\\Gamma^1\\cdots\\Gamma^p\\theta=i^{p\/2}\\theta$.\nThe index theorem \\cite{AS}\n\\begin{equation}\nind(E,D)=(-1)^{m(m+1)\/2}\\int_M ch(\\oplus_r(-1)^r E_r)\n\\left.\\frac{Td(TM^{{\\bf C}})}{e(TM)}\\right|_{vol}\n\\end{equation}\ncan be used to show that there is\nonly one fermionic zero mode if one can show that\nthere is no zero mode of the opposite chirality:\n$\\Gamma^1\\cdots\\Gamma^p\\theta=-i^{p\/2}\\theta$.\nIndeed one can consider the spectrum of the Dirac operator\nsquared $(\\Gamma^{\\mu}D_{\\mu})^2=D^2+\\Gamma^{\\mu\\nu}[D_{\\mu},D_{\\nu}]$.\nThe spectrum of $D^2$ is given above and the spectrum\nof the second term is $\\{-(p\/2)f,-(p\/2-2)f,\\cdots,(p\/2)f\\}$.\nIt follows that the zero mode must have negative\nchirality on each $T^2$.\n\nThe result is therefore that for a 0-$p$ string\nthere is always a single fermionic zero mode\nand there is no bosonic zero mode except for the 0-4 string.\nThis is in agreement with the results of string theory.\n\nIn Sec.\\ref{04} we showed that the SUSY property of\nthe zero modes of a 0-4 string\nfollows from that of the off-diagonal blocks.\nThe SUSY transformation of the zero mode for a 0-8 string\ncan also be derived from the SUSY of SYM.\nNow let us show that the bosonic zero modes derived\nfrom the fermionic zero modes using the SUSY transformation\nas in Sec.\\ref{04} merely vanish.\nNote that the $SO(1,9)$ symmetry is decomposed into\n$SO(1,1)\\times SO(4)\\times SO(4)$ for the 0-8 string,\nwhere the D8-brane has two D4-branes with it.\nThe $\\Gamma$ matrices can be taken as in Sec.\\ref{04}.\nA 10-dimensional spinor $\\theta_{\\pm\\alpha\\beta}$\nhas three indices corresponding to the three factors\nof orthogonal group.\nThe SUSY preserved by the D8-brane background\nis parametrized by $\\epsilon$ with positive or negative chirality\non both factors of $SO(4)$;\nand the zero mode of $\\theta$ has negative chirality\nfor both $SO(4)$.\nSince a given $\\Gamma$-matrix can change the chirality of only one\nof the two copies of $SO(4)$,\nthe SUSY transformation $\\delta y_{\\mu}=i\\bar{\\epsilon}\\Gamma_{\\mu}\\theta$\nvanishes for $\\theta$ being the zero mode\nand does not give nontrivial solutions to $y$.\n\nIt is easy to see that the fermionic zero mode is given by\n$\\theta_{+\\dot{\\s}\\dot{\\r}}=\\chi_+\n\\lambda^0_{\\dot{\\s}\\dot{\\r}},$\nwhere the $\\lambda^0$ is the zero mode solution on $T^8$.\nThe SUSY transformation of the fermionic zero mode\nis trivial ($\\delta\\chi=0$) because all $y$'s vanish.\nThis agrees with the proposal of Horava \\cite{Hor}\nto interpret the zero modes as the extra fermions\nneeded in the heterotic matrix theory \\cite{DF,KR,BM}.\n\nIf the gauge field configuration for a D4-brane is not\n(anti-)self-dual,\nit is found that \\cite{Baal} the configuration is not stable\nbecause of the existence of negative energy states\nin the perturbation of the gauge fields.\nTherefore all states tend to decay into an (anti-)self-dual state\nwith the same topological charge.\nIn our consideration of the off-diagonal blocks $y$,\nthe spectrum of the operator $(-D^2\\pm 2f)\/2$\ncorresponds to the energy of states on the 0-$p$ strings.\nFor the 0-2 string the lowest energy of $y$ is $-f<0$\nand it signifies the instability of the system.\nThis is consistent with the fact that the 0-brane\ntends to distribute uniformly\nover the D2-brane \\cite{Tay1} to form a bound state.\nFor D4-branes corresponding to (anti-)self-dual\nconfigurations the lowest energy of $y$ is 0,\nbut otherwise there would be states with negative energy\nequal to $-|f_1-f_2|$ where $F_{12}=if_1$ and $F_{34}=if_2$.\nIn general for a 0-$p$ string the lowest energy is\nthe minimum of $\\{(\\sum_{i=1}^{p\/2}f_i-2f_j)\/2\\;|\\;j=1,\\cdots,p\/2\\}$.\nWhile there are D2-branes inside the D4, D6 and D8-brane\nconfigurations we considered,\nthe interaction between the D$p$-brane and D0-brane\nincludes the attraction from the D2-brane and\nrepulsion from the D6.\n(The D0-brane is marginally bound to a pure D4-brane.) \\cite{ASJ}\nIf the lowest energy is positive, zero or negative,\nit means that the configuration is stable, marginally stable\nor unstable, respectively. In the cases of D6 and D8-branes,\nthe negative modes are due to the D2-branes inside the higher\nbranes. Take D6-brane as an example. Let $f_i>0$ and\n$f_1=f_2$, then there is a D4-brane wrapping around the\nfirst two tori. If $f_3>2f_1$, there is a negative mode\nof energy $2f_1-f_3$. Apparently, the attractive force due to\nthe D2-brane on the third torus overcomes the repulsive force\nof the D6-brane.\n\nGenerically for D$p$-branes there is a Fock space ${\\cal H}_i$\nfor each $T^2$, where $-{\\cal D}_i\/\\sqrt{2f_i}$ and $\\bar{\\cD}_i\/\\sqrt{2f_i}$\nact as the creation and annihilation operators.\nAfter imposing the constraint (\\ref{eom2-y}),\nthe spectrum of $y_{\\mu}$ is found to be\n\\begin{equation}\n\\{(\\sum_{j=1}^{p\/2}f_j-2f_i)\/2,\\; (\\sum_{j=1}^{p\/2}(2n_j+1)f_j+2f_i)\/2\\;|\n\\; i=1,\\cdots,p\/2;\\; n_j=0,1,2,\\cdots\\}.\n\\end{equation}\n\n\\section{Discussions} \\label{Disc}\n\nIn this paper, we have presented a general framework\nand a systematic analysis for the zero modes in the\noff-diagonal blocks in M(atrix) theory. More concretely,\nwe have shown how to determine the number of zero\nmodes by index theorem and surviving supersymmetry,\nand moreover we have determined the quantum numbers\nof the zero modes, including the chirality of the\nfermion zero modes. These quantum numbers are nontrivial,\nand crucial for us to show the agreement with string\ntheory predictions on open $p-p'$ strings stretching\nbetween D-branes, providing one more check for\nM(atrix) theory. Previously in Refs.\n\\cite{Aharony,Lif1,Lif2}, in the middle of computing\nthe effective potential between a D0- and Dp-brane,\nthe energy levels of the off-diagonal block have\nbeen determined using a slightly different\nrepresentation for the Dp-brane. But the zero\nmodes were not mentioned and identified, and their\nquantum numbers were not studied.\n\nNow let us discuss the significance in M(atrix) theory\nof the zero modes residing in the off-diagonal blocks.\nFirst we have shown in Sec. \\ref{04} that for the case\nof a longitudinal fivebrane, the degrees of freedom\nassociated with the off-diagonal zero modes naturally\nprovide the extra degrees of freedom put in by hand by\nBerkooz and Douglas, ref. \\cite{BD}. And we have\nchecked that the action they postulated are derivable\nfrom the M(atrix) theory action, with a possible\n$D$-term. Indeed, in this case, besides the right\ntopological number (or brane charge), the correct\ncounting of zero modes we found in Sec. \\ref{04}\nis crucial for justifying our identification\nof a longitudinal 5-brane with proper instanton\nconfiguration on $T^4$ rather than on $S^4$. Also\nthe correct number of zero modes is crucial for\na check of the correct tension and R-R\ncharge for the longitudinal 5-brane. It\nis argued in \\cite{BD} that upon integrating\nout 0-4 strings the long range force\nbetween a longitudinal 5-brane and a\nprobe supergraviton is generated.\nIf we had a different number of zero modes\nwe would obtain a gravitational field with a\ndifferent magnitude for the 5-brane. Also\nas shown in Ref. \\cite{BD}, the R-R charge\nof a longitudinal 5-brane manifests itself in the\nDirac quantization of a membrane moving\nin its background.\nBy realizing the membrane as a collection of\nD0-branes, the zero modes on the 0-4 strings\nwould induce fields on the membrane.\nIt is the fermion zero\nmode $\\chi$ induced on the membrane\nthat is responsible for generating the\nBerry phase.\nIn fact, by T-duality the induced zero mode\nis related to the zero mode on a 0-6 string.\nOur results in Sec.\\ref{0608} provide the proof\nfor the existence of a single chiral zero mode\nnecessary for the correct Berry phase.\nHad we had two zero modes,\nwe would have generated twice the correct\nBerry phase, and therefore twice the R-R charge.\nAs pointed out in Sec.6, in the background of\n$k$ instantons, there are $k$ fermionic zero\nmodes. The Berry phase is then $k$ times\nas large, and this signals that there are $k$\nunits of R-R charge.\n\nUpon compactifying on a 5-torus $T^5$, instanton\nstrings will appear in the spectrum. These are\npart of constituents of some 5D black holes\n\\cite{LM,DVVa}. A 5D black hole is described\nby a long instanton-string carrying momentum.\nProbing the black hole with a supergraviton,\none expects that the corresponding static potential\nas well as the velocity dependent force are generated\nby integrating out the off-diagonal blocks. This is\nshown to leading order in Ref. \\cite{JM}, where\nthe full $5+1$ massive modes are integrated out.\nIt is an interesting question whether the relevant\nterms can be generated by integrating out only the\nzero modes discussed in Sec.\\ref{04}.\n\nIt should be interesting to compare our result with\nthe work of Ref.\\cite{Doug}. There a D5-brane is\ninterpreted as an instanton inside 9-branes. The\nprobe is a D1-brane. The 1-5 string sector is\nconstructed with the D-brane technology.\nA (0,4) sigma model in an instanton background\n\\cite{Witten} results from integrating out the\nmassive 1-5 strings. There are two differences\nbetween the case under discussion and Ref.\n\\cite{Doug}. First, it is crucial for us to work\nwith $T^4$, only then we have the correct\nnumber of zero modes. Second, the $SU(2)_R$\nsymmetry in our problem comes from the $SO(4)$\nof $T^4$, while the $SU(2)_R$ of \\cite{Doug}\ndoes not act on the gauge field, since the gauge\nfield carries an index transverse to the D5-brane.\n\nFinally, the origin of $p-p$ strings is also easy\nto see. When $p=2$, the world-volume action\nis written down \\cite{BSS}. For $p=4$, one can\nconsider zero modes of the fundamental of\n$SU(2)\\times SU(2)\\in SU(4)$ in the background\ninstanton number 2 solution with a gauge group\n$SU(4)$. It is important to embed the instanton\nto $SU(4)$ rather than to a single $SU(2)$, in order\nto be able to higgs the off-diagonal strings. By\nan index theorem, there are $16$ real bosonic zero\nmodes. $8$ of them are W-bosons, and the\nother $8$ are massive Higgs. The 8-8 strings\nare discussed in \\cite{Hor}.\n\nWe have identified the stretched strings between\na $p$-brane and a $p'$-brane as  just the zero\nmodes of off-diagonal blocks; one would like\nto ask what about the massive modes of\n$p$-$p'$ strings in the M(atrix) theory.\nOn one hand, for short open strings these\nmodes, similar to the massive modes\nof short open 0-0 strings, are simply\nabsent in the M(atrix) model by postulate.\n(It would be interesting to examine the\nlong strings in M(atrix) theory ending\non $p$ $p'$-branes.)  On the other hand,\nit might be wise to leave the possibility\nopen that these massive modes on short strings\nand other massive modes, such as KK modes in\na higher dimensional super Yang-Mills theory\ncould be physically relevant so that their\ninclusion is necessary to make the high\ndimensional theory well-defined in the UV regime.\nWe leave investigation of this issue to the future.\n\nHow about the higher modes of the off-diagonal\nblocks? Could their effects approximate to those\nof the massive modes of $p$-$p'$ strings? We do\nnot think so, since  the latter is graded by $\\alpha'$,\nwhile the former is determined by the scale of the\nbackground field and the scale of the torus. The\nmodified M(atrix) model in the presence of\na longitudinal 5-brane proposed in Ref. \\cite{BD}\nshould be viewed as a low-energy effective theory\nof the fundamental M(atrix) model, in which the\nhigher modes of the off-diagonal block are ignored.\nIndeed, in this case the zero-modes of the\noff-diagonal block dominate the low-energy physics,\nsince surviving supersymmetry makes the\ncontributions of the higher modes cancel in the\nleading order at large distances.\n\nAlthough in this papers we have used\nthe \\mbox{\\II A\\hspace{.2em}} language for brane names, the above\ndiscussions are of M theory nature.\nIt may be amusing to consider an alternative \\mbox{\\II A\\hspace{.2em}} theory\nwhich is obtained by compactifying the ninth\ndirection and interchanging the role of the ninth\nand eleventh directions. What we called D0-branes\nabove become short strings, which are also understood\nas D0-branes by introducing unit electric flux to\nthe corresponding matrix element \\cite{DVV}.\nWe leave the complete analysis and related topics\nfor the future.\n\n\\section{Acknowledgment}\n\nThe work of P.M.H. and Y.S.W. is supported in part by\nU.S. NSF grant PHY-9601277. The work of M.L. is\nsupported by DOE grant DE-FG02-90ER-40560\nand NSF grant PHY-9123780.\n\n\\vskip .8cm\n\n\\baselineskip 22pt\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nLet $\\Omega$ be a lattice in $\\mathbb{C}$. We are interested in the Weierstrass $\\sigma$-function associated with $\\Omega$. This is defined as $$\\sigma_\\Omega(z)=z\\prod\\limits_{\\omega\\in \\Omega^*}\\left(1-\\dfrac{z}{\\omega}\\right)\\exp\\left(\\dfrac{z}{\\omega}+\\dfrac{z^2}{2\\omega^2}\\right)$$ where $\\Omega^*=\\Omega\\setminus\\{0\\}$. \n\nBy definition, $\\Omega=\\omega_1\\mathbb{Z}+\\omega_2\\mathbb{Z}$ where $\\omega_1$ and $\\omega_2$ are $\\mathbb{R}$-linearly independent. It is known that $\\omega_1$ and $\\omega_2$ may be chosen so that $\\frac{\\omega_2}{\\omega_1}$ lies in the upper half plane and has modulus at least 1 and real part in the interval $\\left[-\\frac{1}{2},\\frac{1}{2}\\right]$. Fix $\\omega_1$ and $\\omega_2$ with this property and set $\\tau=\\frac{\\omega_2}{\\omega_1}$. We prove two results concerning the number of algebraic points of bounded height and degree on the graph of $\\sigma_\\Omega$, under certain assumptions including that the imaginary part of $\\tau$ is at most $1.9$. This bound on $\\text{Im}(\\tau)$ ensures that a useful growth condition is satisfied. Given an algebraic number $z$, we write $H(z)$ for its multiplicative height. For a pair $(z,w)$, $H(z,w)=\\max\\{H(z),H(w)\\}$.\n\n\\begin{thm}\\label{alglattice}\nSuppose $\\text{Im}(\\tau)\\leq 1.9$. If $\\omega_1$ and $\\omega_2$ are both algebraic then there is an effective positive constant $c$ such that, for all $d\\geq e$ and $H\\geq e^e$, there are at most $$c d^6(\\log d)(\\log H)^{2}\\log\\log H$$ algebraic points $(z,\\sigma_\\Omega(z))$ such that $[\\mathbb{Q}(z,\\sigma_\\Omega(z)):\\mathbb{Q}]\\leq d$, $H(z,\\sigma_\\Omega(z))\\leq H$ and $z\\notin \\Omega$.\n\\end{thm}\n\nFor our second result, we consider the parameters $$g_2=60\\sum\\limits_{\\omega\\in\\Omega^*}\\omega^{-4}\\text{ and } g_3=140\\sum\\limits_{\\omega\\in\\Omega^*}\\omega^{-6}$$ associated with $\\Omega$.\n\n\\begin{thm}\\label{alggs}\nSuppose $\\text{Im}(\\tau) \\leq 1.9$. If $g_2$ and $g_3$ are both algebraic then there is an effective positive constant $c$ such that, for all $d\\geq e$ and $H\\geq e^e$, there are at most \n$$c d^{20}(\\log d)^5 (\\log H)^2 \\log \\log H$$\nalgebraic points $(z,\\sigma_\\Omega(z))$ such that $[\\mathbb{Q}(z,\\sigma_\\Omega(z)):\\mathbb{Q}]\\leq d$ and $H(z,\\sigma_\\Omega(z))\\leq H$. \n\\end{thm}\n\nThere have been many recent results on the general topic of bounding the number of rational points of bounded height lying on the graph of some transcendental function. These go back to fundamental work of Bombieri and Pila, and then Pila \\cite{BP,P}. In the latter of these papers, a bound of the form $cH^\\epsilon$ is established for the number of rational points of height at most $H$ lying on the graph of a transcendental analytic function $f:[0,1]\\to \\mathbb{R}$. A construction by Surroca \\cite{SurrocaShort} shows that this is best possible in the sense that there are examples for which no power of log bound will hold (see also \\cite{PilaSurface}). Bounds of the form $c(\\log H)^\\gamma$ were given for functions satisfying pfaffian differential equations by Pila \\cite{PilaPfaffCurve}, and for the Riemann zeta function, restricted to the interval $[2,3]$, by Masser \\cite{Masser}. Pila's result is of a real nature, and wouldn't apply for instance to a pfaffian entire function if instead of rational points one considered points of bounded degree. Masser proves his result for points of bounded degree on a compact set containing $[2,3]$. \n\nOf the many other papers on this topic the most relevant to us are those by Besson \\cite{Besson} and by the first and last authors \\cite{BJ2}. In these, results similar to Theorems \\ref{alglattice} and \\ref{alggs} were proved, with no condition on $\\tau$, but only for points lying on the graph of the restriction of $\\sigma_\\Omega$ to a disk. The result in \\cite{Besson} is obtained with the aid of a zero estimate for $\\sigma_\\Omega$, also proved in \\cite{Besson}, and this leads to a slightly better bound than is obtained \\cite{BJ2}. The results in \\cite{BJ2} apply generally to entire functions of finite order and positive lower order, with the exponent $\\gamma$ in the bound $c(\\log H)^\\gamma$ depending only on the order and lower order. The constant $c$ depends the function, the disk and an upper bound $d$ for the degrees of the points considered. In the results in \\cite{Masser} and \\cite{Besson}, the dependence on $d$ was shown to be polynomial. For the interest in obtaining polynomial dependence on $d$ see \\cite{HarryGaloisBounds,BJS}. \n\nWhat is striking about Theorem \\ref{alggs} is that we do not have to restrict the function at all. We count algebraic points of bounded height and degree on the entire graph. For Theorem \\ref{alglattice}, where the lattice points are algebraic, it is clear that the condition $z\\notin\\Omega$ could not be omitted. In earlier work of the authors, some results were obtained for restrictions of entire functions to unbounded sets, but generally a lot of points had to be omitted. In \\cite{BJ1} the first and third authors obtained, for example, a bound for points on the graph of the Riemann zeta function restricted to a sector $-\\frac{\\pi}{2}<\\theta\\leq \\arg{z}\\leq \\phi<\\frac{\\pi}{2}$. In recent work of the second author \\cite{Chalebgwa}, bounds were given for restrictions of certain entire functions, defined by infinite products, to domains of the form $\\mathbb{C}\\setminus S$, where $S$ is an arbitrarily small sector whose interior contains the positive real line. \n\nTo prove Theorems \\ref{alglattice} and \\ref{alggs}, we draw on the reasoning in \\cite{Besson}. We combine Besson's zero estimate with a proposition of Masser from \\cite{Masser}. Besson does this too, but we must work on increasingly large disks and our challenge is to show that, with perhaps a manageable number of exceptions, for any given $d$ and $H$ the $z$'s that we are counting all lie in a disk of sufficiently small radius. It is possible to avoid reliance on the zero estimate from \\cite{Besson} and instead use techniques from earlier papers of the first and third authors, though at the cost of larger exponents. We expect it would also be possible to use the zero estimate of Coman and Poletsky, given as Corollary 7.2 in \\cite{CP}. \n\nThe current project began in the PhD research of the second author. A version of Theorem \\ref{alglattice}, with $\\Omega=\\mathbb{Z}+\\mathbb{Z}i$, was given in his thesis. \n\nIn \\S 2 we discuss a growth property of $\\sigma_\\Omega$ in the case where $\\text{Im}(\\tau)\\le  1.9$. In \\S 3 we prove Theorem \\ref{alglattice}. In \\S 4 we prove Theorem \\ref{alggs}. Then in \\S 5 we briefly outline a proof of a weaker version of Theorem \\ref{alggs}, which avoids use of the zero estimate from \\cite{Besson}.\n\n\n\\section{Growth}\n\nRecall that $\\Omega=\\omega_1\\mathbb{Z}+\\omega_2\\mathbb{Z}$, where $\\tau=\\frac{\\omega_2}{\\omega_1}$ lies in the upper half plane and has modulus at least 1 and real part in the interval $\\left[-\\frac{1}{2},\\frac{1}{2}\\right]$. Let $P$ be the closed region in $\\mathbb{C}$ enclosed by the parallelogram with vertices $\\frac{\\pm\\omega_1\\pm \\omega_2}{2}$. \n\n\\begin{prop}\\label{growth}\nSuppose $\\text{Im}(\\tau)\\leq 1.9$. There are effective positive constants $r$ and $c$ such that, for all $z\\in\\mathbb{C}$ with $|z|\\geq r$, there exists $z_0\\in P$ such that $z-z_0\\in\\Omega$ and $|\\sigma_{\\Omega}(z)|\\geq |\\sigma_{\\Omega}(z_0)|e^{c|z|^2}$.\n\\end{prop}\n\nThe proof is based on reasoning given in \\cite{Besson}.\n\n\\proof Let $\\Omega'=\\mathbb{Z}+\\mathbb{Z}\\tau$. Since $\\Omega=\\omega_1\\Omega'$, we have $\\sigma_{\\Omega}(\\omega_1 z)=\\omega_1\\sigma_{\\Omega'}(z)$ and so it is sufficient to prove Proposition \\ref{growth} with $\\Omega'$, $\\sigma_{\\Omega'}$ and $P'=\\frac{1}{\\omega_1}P$ in place of $\\Omega$, $\\sigma_{\\Omega}$ and $P$.\n\nLet $z_0\\in P'$. For any $n,m\\in\\mathbb{Z}$, $$\\sigma_{\\Omega'}(z_0+m+n\\tau)=(-1)^{m+n+mn}\\sigma_{\\Omega'}(z_0)e^{(m\\eta_1+n\\eta_2)(z_0+\\frac{m}{2} + \\frac{n}{2}\\tau)}$$ where $\\eta_1$ and $\\eta_2$ depend on $\\tau$ (and we follow the convention which assigns them a value twice that in \\cite{Besson}). The above equation is standard (see, for example, (20.21) on page 255 of \\cite{MasserBook}). We are interested in how $$\\text{Re}\\left[\\left(m\\eta_1+n\\eta_2\\right)\\left(z_0+\\frac{m}{2} + \\frac{n}{2}\\tau\\right)\\right]$$ $$=\\text{Re}\\left(\\frac{\\eta_1}{2}\\right)m^2+\\text{Re}\\left(\\frac{\\eta_1\\tau+\\eta_2}{2}\\right)mn+\\text{Re}\\left(\\frac{\\eta_2\\tau}{2}\\right)n^2+\\text{Re}\\left(\\eta_1z_0\\right)m+\\text{Re}\\left(\\eta_2z_0\\right)n$$ behaves as $\\max\\{|m|,|n|\\}$ increases. Following Besson, we consider the discriminant of $\\text{Re}\\left(\\frac{\\eta_1}{2}\\right)m^2+\\text{Re}\\left(\\frac{\\eta_1\\tau+\\eta_2}{2}\\right)mn+\\text{Re}\\left(\\frac{\\eta_2\\tau}{2}\\right)n^2$, $$\\Delta=\\left(\\text{Re}\\left(\\frac{\\eta_1\\tau+\\eta_2}{2}\\right)\\right)^2-4\\left(\\text{Re}\\left(\\frac{\\eta_1}{2}\\right)\\right)\\left(\\text{Re}\\left(\\frac{\\eta_2\\tau}{2}\\right)\\right).$$ If $\\Delta<0$ then $\\left(\\text{Re}\\left(\\frac{\\eta_1}{2}\\right)\\right)\\left(\\text{Re}\\left(\\frac{\\eta_2\\tau}{2}\\right)\\right)\\neq 0$ and $$\\left|\\text{Re}\\left(\\frac{\\eta_1}{2}\\right)m^2+\\text{Re}\\left(\\frac{\\eta_1\\tau+\\eta_2}{2}\\right)mn+\\text{Re}\\left(\\frac{\\eta_2\\tau}{2}\\right)n^2\\right|$$ $$\\geq \\max\\left\\{\\left|\\frac{\\Delta}{4\\text{Re}\\left(\\frac{\\eta_2\\tau}{2}\\right)}\\right|m^2,\\left|\\frac{\\Delta}{4\\text{Re}\\left(\\frac{\\eta_1}{2}\\right)}\\right|n^2\\right\\},$$ by the general fact that if $a,b,c\\in\\mathbb{R}$ and $b^2-4ac<0$ then the minimum value of $\\left|ax^2+bx+c\\right|$ is $\\left|\\frac{4ac-b^2}{4a}\\right|$. \n\nSince $\\left|z_0\\right|\\leq 1+\\left|\\tau\\right|$, $$\\left|\\text{Re}(\\eta_1z_0)m+\\text{Re}(\\eta_2z_0)n\\right|\\leq (1+\\left|\\tau\\right|)\\max\\{\\left|\\eta_1\\right|,\\left|\\eta_2\\right|\\}\\max\\{\\left|m\\right|,\\left|n\\right|\\}.$$ Let $c_1=\\min\\left\\{\\left|\\dfrac{\\Delta}{4\\text{Re}\\left(\\frac{\\eta_2\\tau}{2}\\right)}\\right|,\\left|\\dfrac{\\Delta}{4\\text{Re}\\left(\\frac{\\eta_1}{2}\\right)}\\right|\\right\\}$ and $c_2=(1+\\left|\\tau\\right|)\\max\\{\\left|\\eta_1\\right|,\\left|\\eta_2\\right|\\}$. Suppose $\\Delta<0$. For each $z\\in \\mathbb{C}$, there exist $z_0\\in P'$ and $m,n\\in \\mathbb{Z}$ such that $z=z_0+m+n\\tau$. Then $$\\text{Re}\\left[(m\\eta_1+n\\eta_2)\\left(z_0+\\frac{m}{2} + \\frac{n}{2}\\tau\\right)\\right]\\geq c_1\\left(\\max\\{|m|,|n|\\}\\right)^2-c_2\\max\\{|m|,|n|\\}.$$ Since $\\max\\{\\left|m\\right|,\\left|n\\right|\\}\\geq \\dfrac{|z|-|z_0|}{1+\\left|\\tau\\right|}$, we obtain positive constants $r$ and $c$, depending effectively on $\\omega_1$ and $\\omega_2$, such that $$\\text{Re}\\left[(m\\eta_1+n\\eta_2)\\left(z_0+\\frac{m}{2} + \\frac{n}{2}\\tau\\right)\\right]\\geq c\\left|z\\right|^2$$ whenever $\\left|z\\right|\\geq r$. Then, for all $z\\in \\mathbb{C}$ with $\\left|z\\right|\\geq r$, there exists $z_0\\in P'$ such that $z-z_0\\in \\Omega'$ and $$\\left|\\sigma_{\\Omega'}(z)\\right|\\geq \\left|\\sigma_{\\Omega'}(z_0)\\right|e^{c|z|^2}.$$\n\nIt remains to show that $\\Delta<0$ if $\\text{Im}(\\tau)\\leq 1.9$. This is discussed in \\cite{Besson}, but we provide further details. A simple calculation gives $$\\Delta=\\text{Im}(\\tau)(\\left|\\eta_1\\right|^2\\text{Im}(\\tau)-2\\pi\\text{Re}(\\eta_1)).$$ This is done on page 5 of \\cite{Besson}, but recall that the $\\eta_1$ and $\\eta_2$ there are half ours. We have $\\eta_1=\\frac{\\pi^2}{3}E_2(\\tau)$, where $$E_2(\\tau)=1-24\\sum\\limits_{n\\geq 1}\\frac{nq^n}{1-q^n}$$ and $q=e^{2\\pi i \\tau}$ (see \\cite{Lang}). Letting $y=\\text{Im}(\\tau)$ and $\\phi(y)=\\dfrac{24e^{-2\\pi y}}{(1-e^{-2\\pi y})^3}$, we have $$\\text{Re}(E_2(\\tau))\\geq 1-\\phi(y)$$ and $$\\left|E_2(\\tau)\\right|\\leq 1+\\phi(y)$$ (see page 128 of \\cite{IJT}). Therefore $\\Delta<0$ provided $$y<\\frac{6(1-\\phi(y))}{\\pi(1+\\phi(y))^2}.$$ We compute a sequence $y_0,y_1,...$ with $y_0=\\frac{\\sqrt{3}}{2}$ and $$y_{n+1}=\\frac{6(1-\\phi(y_n))}{\\pi(1+\\phi(y_n))^2}.$$ We obtain $y_0<y_1<...<y_4\\approx 1.909$.  Since $\\phi(y)$ is positive and decreasing for $y\\geq \\frac{\\sqrt{3}}{2}$, this ensures that $\\Delta<0$ when $\\frac{\\sqrt{3}}{2}\\leq y\\leq 1.9$. The inequality $\\frac{\\sqrt{3}}{2}\\leq y$ is implied by the fact that $\\left|\\tau\\right|\\geq 1$ and $-\\frac{1}{2}\\leq\\text{Re}(\\tau)\\leq \\frac{1}{2}$. \\endproof\n\n\n\n\n\\section{When the lattice points are algebraic}\n\nIn this section we prove Theorem \\ref{alglattice}. The following zero estimate was obtained in \\cite{Besson}, for any lattice $\\Omega=\\omega_1\\mathbb{Z}+\\omega_2\\mathbb{Z}$ (see Th\\'eor\\`eme 1.2 in \\cite{Besson}).\n\n\\begin{thm}\\label{Besson_zero} There is an effective positive constant $c$, depending only on $\\omega_1$ and $\\omega_2$, such that, for every integer $L\\ge 1$, real number $R\\ge 2$ and nonzero polynomial $P(X,Y)$, with complex coefficients and degree at most $L$ in each variable, the function $P(z,\\sigma_\\Omega(z))$ has at most $cL (R+\\sqrt{L})^2 \\log (R+L)$ zeroes in the disk $|z|\\le R$.\n\\end{thm}\n\nBesson uses this, in combination with a result of Masser, to prove the following (see Th\\'eor\\`eme 1.1 in \\cite{Besson}).\n\n\\begin{thm}\\label{Besson}\nThere is an effective positive constant $c$ such that, for all $d\\geq 1$, $H\\geq 3$ and $R\\geq 2$, there are at most $$cR^{10}(\\log R)\\frac{d^4(\\log H)^2}{\\log(d\\log H)}$$ algebraic points $(z,\\sigma_\\Omega(z))$ such that $[\\mathbb{Q}(z,\\sigma_\\Omega(z)):\\mathbb{Q}]\\leq d$, $H(z,\\sigma_\\Omega(z))\\leq H$ and $\\left|z\\right|\\leq R$.\n\\end{thm}\n\nThis result and Theorem \\ref{Besson_zero} are actually stated in \\cite{Besson} under the assumption that $\\omega_1=1$. However, it is clear from the argument that this is not needed. In particular, the identity $\\sigma_{\\Omega}(\\omega_1 z)=\\omega_1\\sigma_{\\Omega'}(z)$, when $\\Omega'=\\mathbb{Z}+\\mathbb{Z}\\tau$ and $\\Omega=\\omega_1\\Omega'$, ensures that Theorem \\ref{Besson_zero} holds for any lattice. \n\nIn order to apply Besson's work to prove Theorem \\ref{alglattice}, we show that, for each $d$ and $H$, all the relevant $z$'s lie in a disk whose radius is polynomial in $d$ and $\\log H$. For values of $z$ not close to a lattice point, we use Proposition \\ref{growth}. We do not consider the lattice points themselves. That leaves only those values of $z$ which are close but not equal to lattice points. We use basic facts about height and degree to eliminate the possibility of any of these. \n\nWe could then simply apply Theorem \\ref{Besson} to obtain a result along the lines of Theorem \\ref{alglattice}. However, this would lead to a weaker estimate. So instead we use Besson's method of proof. This involves combining his zero estimate with Proposition 2 of \\cite{Masser}, which is as follows.\n\n\n\n\n\\begin{prop}\\label{propn2} Given integers $d\\ge 1, T\\ge \\sqrt{8d}$ and positive real numbers $A,Z,M$ and $H$, with $H\\geq 1$, let $f_1$ and $f_2$ be analytic functions on a neighbourhood of the disk $|z|\\le 2Z$. Suppose $|f_1(z)|\\le M$ and $|f_2(z)|\\le M$ for all $\\left|z\\right|\\leq 2Z$. Let $\\mathcal Z\\subseteq\\mathbb{C}$ be finite such that, for all $z,z'\\in \\mathcal Z$, \n\\begin{itemize}\n\\item[(a)] $|z|\\le Z$,\n\\item[(b)] $ |z-z'| \\le 1\/A$,\n\\item[(c)] $[\\mathbb Q (f_1(z),f_2(z)) :\\mathbb{Q}]\\le d$ and\n\\item[(d)] $H(f_1(z),f_2(z)) \\le H$.\n\\end{itemize}\nIf \n\\begin{equation}\\label{propn2condition}\n(AZ)^T > (4T)^{96d^2\/T} (M+1)^{16 d} H^{48d^2}\n\\end{equation}\nthen there is a non-zero polynomial $P \\in \\mathbb{C}[X,Y]$, of total degree at most $T$, such that $$P(f_1(z),f_2(z))=0$$ for all $z \\in \\mathcal Z$.\n\n\\end{prop}\n\nWe now start working towards our bound on the radius. As in the previous section, let $P$ be the parallelogram with vertices $\\frac{\\pm \\omega_1 \\pm \\omega_2}{2}$. Since $\\lim\\limits_{w\\rightarrow 0}\\frac{\\sigma_\\Omega(w)}{w}=1$ and $w=0$ is the only zero of $\\sigma_\\Omega$ in $P$, there is some $\\delta>0$ such that $\\left|\\log\\left|\\sigma_\\Omega(w)\\right|-\\log\\left|w\\right|\\right|\\leq 1$ whenever $\\log|\\sigma_\\Omega(w)|\\leq -\\delta$. We fix some such $\\delta$. Recall that $\\tau=\\frac{\\omega_2}{\\omega_1}$. \n\n\\begin{lem}\\label{common}\nAssume $\\text{Im}(\\tau)\\leq 1.9$. Let $d\\geq 1$ and $H\\geq e$. Suppose $z\\in \\mathbb{C}\\setminus\\Omega$ is such that $\\sigma_{\\Omega}(z)$ is algebraic with $H(\\sigma_\\Omega(z))\\leq H$ and $[\\mathbb{Q}(\\sigma_{\\Omega}(z)):\\mathbb{Q}]\\leq d$. Let $z_0\\in P$ be such that $z-z_0\\in \\Omega$. Assume $\\left|z\\right|\\geq r$, where $r$ is as in Proposition \\ref{growth}. For all $B>0$ there exists $A>0$, depending only on $B, \\omega_1$ and $\\omega_2$, such that, for all $N\\geq \\sqrt{d\\log H}$, if $\\left|z\\right|\\geq AN$ then $\\log\\left|z_0\\right|\\leq -BN^2$. \n\\end{lem}\n\n\\proof Let $B>0$, $A>0$ and $N\\geq \\sqrt{d\\log H}$. Let $c$ be as in Proposition \\ref{growth}. Then $$|\\sigma_\\Omega(z)|\\geq |\\sigma_\\Omega(z_0)|e^{c|z|^2}.$$ Since $\\sigma_\\Omega(z)$ is algebraic with degree at most $d$ and height at most $H$, $|\\sigma_\\Omega(z)|\\leq H^d$. So $$N^2\\geq d\\log H\\geq\\log\\left|\\sigma_\\Omega(z)\\right|\\geq \\log\\left|\\sigma_\\Omega(z_0)\\right| + c\\left|z\\right|^2.$$ If $\\left|z\\right|\\geq AN$ then $$N^2\\geq \\log\\left|\\sigma_\\Omega(z_0)\\right| + cA^2N^2$$ and so $$\\log\\left|\\sigma_\\Omega(z_0)\\right|\\leq (1-cA^2)N^2$$ which implies $$\\log\\left|z_0\\right|\\leq (1-cA^2)N^2+1\\leq (2-cA^2)N^2$$ if $A$ has been chosen so that $$1-cA^2\\leq -\\delta.$$ We obtain the result with $A=\\max\\left\\{\\sqrt{\\frac{1+\\delta}{c}},\\sqrt{\\frac{2+B}{c}}\\right\\}$. \\endproof\n\nThe following lemma must be well known and is easily proved using the cosine rule. \n\n\\begin{lem}\\label{cosine}\nThere is a positive constant $c$, depending only on $\\omega_1$ and $\\omega_2$, such that for all $\\omega\\in \\Omega$ there are integers $k, l$ such that $\\left|k\\right|\\leq c\\left|\\omega\\right|$, $\\left|l\\right|\\leq c\\left|\\omega\\right|$ and $\\omega=k\\omega_1+l\\omega_2$. \n\\end{lem}\n\n\\begin{prop}\\label{applgrowth}\nAssume $\\text{Im}(\\tau)\\leq 1.9$ and both $\\omega_1$ and $\\omega_2$ are algebraic. Let $d\\geq 1$ and $H\\geq e$. Suppose $z\\in \\mathbb{C}\\setminus\\Omega$ is such that both $z$ and $\\sigma_{\\Omega}(z)$ are algebraic with $H(z,\\sigma_\\Omega(z))\\leq H$ and $[\\mathbb{Q}(z,\\sigma_{\\Omega}(z)):\\mathbb{Q}]\\leq d$. There exists $A>0$, depending only on $\\omega_1$ and $\\omega_2$, such that $|z|\\leq A d\\sqrt{\\log H}$. \n\\end{prop}\n\n\\proof The constants $c_1,c_2,c_3$ in this proof will only depend on $\\omega_1$ and $\\omega_2$. Choose $z_0\\in P$ such that $z-z_0\\in\\Omega$ and let $\\omega=z-z_0$. Then $\\left|\\omega\\right|\\leq \\left|z\\right|+\\left|\\omega_1\\right|+\\left|\\omega_2\\right|$. Combining this with Lemma \\ref{cosine}, we obtain $c_1>0$ and integers $k$ and $l$ such that $\\omega=k\\omega_1+l\\omega_2$ and $\\left|k\\right|,\\left|l\\right|\\leq c_1\\left|z\\right|$.  \n\nSince $z_0=z-\\omega$, we have $$H(z_0)\\leq 2H(z)H(\\omega)\\leq 4H(z)H(k)H(\\omega_1)H(l)H(\\omega_2).$$ Since $H(z)\\leq H$, $[\\mathbb{Q}(z):\\mathbb{Q}]\\leq d$, $H(k)=\\left|k\\right|$ and $H(l)=\\left|l\\right|$, we have $\\left|z\\right|\\leq H^d$ and then $H(k),H(l)\\leq c_1H^d$. So there exists $c_2>0$ such that $$\\log (H(z_0))\\leq (2d+1)\\log H+c_2.$$ The degree of $\\omega$ is at most $[\\mathbb{Q}(\\omega_1,\\omega_2):\\mathbb{Q}]$ and so $$[\\mathbb{Q}(z_0):\\mathbb{Q}]=[\\mathbb{Q}(z-\\omega):\\mathbb{Q}]\\leq [\\mathbb{Q}(\\omega_1,\\omega_2):\\mathbb{Q}]d.$$ Letting $c_3=[\\mathbb{Q}(\\omega_1,\\omega_2):\\mathbb{Q}]$, it follows that $$\\log\\left|z_0\\right|\\geq -[\\mathbb{Q}(z_0):\\mathbb{Q}]\\log(H(z_0))\\geq -c_3d((2d+1)\\log H+c_2)> -3c_3(1+c_2)d^2\\log H.$$ Letting $B=3c_3(1+c_2)$ and $N=d\\sqrt{\\log H}$, the result follows by Lemma \\ref{common} (assuming, as we may, that $\\left|z\\right|$ is at least the $r$ of Proposition \\ref{growth}). \\endproof\n\nWe can now complete the proof of Theorem \\ref{alglattice}. We apply Proposition \\ref{propn2} with $f_1(z)=z$ and $f_2(z)=\\sigma_\\Omega(z)$. We set $Z= 4 A d\\sqrt{\\log H}$, with the $A$ from Proposition \\ref{applgrowth}, and put\n\\[\n\\mathcal Z = \\{ z \\in \\mathbb C\\setminus\\Omega : [\\mathbb Q(z,\\sigma_\\Omega(z)) : \\mathbb Q]\\le d , H(z,\\sigma_\\Omega(z) ) \\le H\\}.\n\\] \nBy Proposition \\ref{applgrowth}, we have $|z|\\le Z$ for all $z\\in \\mathcal Z$. We take $A$ in Proposition \\ref{propn2} to be $2\/Z$. From the identity $$\\sigma_{\\Omega}(z_0+m\\omega_1+n\\omega_2)=(-1)^{m+n+mn}\\sigma_{\\Omega}(z_0)e^{(m\\eta_1+n\\eta_2)(z_0+\\frac{m}{2}\\omega_1 + \\frac{n}{2}\\omega_2)}$$ (see (20.21) on page 255 of \\cite{MasserBook}) and Lemma \\ref{cosine}, there exist positive constants $c_1$ and $c_2$ such that $\\left|\\sigma_\\Omega(z)\\right|\\leq c_1e^{c_2\\left|z\\right|^2}$ for all $z\\in\\mathbb{C}$. We set $M = c_1e^{c_2 Z^2}$. With these choices, (a) - (d) of Proposition \\ref{propn2} hold. If we take $T=c_3 d^3 \\log H$, for a suitable $c_3>0$, then we will have \\eqref{propn2condition}. We then apply Besson's zero estimate, Theorem \\ref{Besson_zero}, to conclude.\n\n\n\n\n\n\\section{When $g_2$ and $g_3$ are algebraic}\n\n\n\nIn this section we prove Theorem \\ref{alggs}. Our approach is similar to that followed in \\S 3 and we again use Theorem \\ref{Besson_zero}. We suppose throughout that $g_2$ and $g_3$ are algebraic. First, we require a transcendence measure, a very special case of a theorem of David and Hirata-Kohno.\n\\begin{lem}\\label{measure}There is a positive $c$, effectively computable from $g_2$ and $g_3$, with the following property. Let $H\\ge 3, d\\ge 1$ and suppose $\\alpha$ is algebraic, with $H(\\alpha)\\le H$ and $[\\Q(\\alpha):\\Q]\\le d$. If $\\omega\\in\\Omega$ is nonzero then \n\\[\n \\log |\\alpha-\\omega| \\ge -c d^4 (\\log d)^2 (\\log H) |\\omega|^2 (1+\\max\\{0,\\log |\\omega|\\})^3.\n \\]\n \\end{lem}\n \\begin{proof}  This is immediate from Theorem 1.6 in \\cite{DavidHirataKohno}.\n\\end{proof}\n\n\n\nWe now obtain a result similar to Proposition \\ref{applgrowth}. This time we allow for some exceptions but can ensure that the set of exceptional $z\\in\\mathbb{C}$ has $\\mathbb{Q}$-linear dimension at most one. \n\n\n\\begin{prop}\\label{applgrowth2}\nAssume $\\text{Im}(\\tau)\\leq 1.9$. There are positive constants $A$ and $B$ such that, for all $d,H\\geq e$ and $z,z'\\in \\mathbb{C}$, if $z,z',\\sigma_\\Omega(z)$ and $\\sigma_\\Omega(z')$ are algebraic, with $H(z,\\sigma_\\Omega(Z)), H(z',\\sigma_\\Omega(z'))\\leq H$ and $[\\mathbb{Q}(z,\\sigma_\\Omega(z)):\\mathbb{Q}], [\\mathbb{Q}(z',\\sigma_\\Omega(z')):\\mathbb{Q}]\\leq d$, then $$\\min\\{|z|,|z'|\\}\\leq A\\sqrt{d^9(\\log d)^2\\log H}$$ or there exist $\\omega,\\omega'\\in\\Omega$ such that $$\\max\\{\\log|z-\\omega|,\\log|z'-\\omega'|\\}<-Bd^9(\\log d)^2\\log H\\text{ and }\\frac{z'}{z}=\\frac{\\omega'}{\\omega}\\in\\mathbb{Q}.$$\n\\end{prop}\n\n\\proof Let $d, H\\geq e$ and let $z,z'\\in\\mathbb{C}$ be such that $H(z,\\sigma_\\Omega(Z)), H(z',\\sigma_\\Omega(z'))\\leq H$ and $[\\mathbb{Q}(z,\\sigma_\\Omega(z)):\\mathbb{Q}], [\\mathbb{Q}(z',\\sigma_\\Omega(z')):\\mathbb{Q}]\\leq d$. Choose $z_0, z_0'\\in P$ so that $z-z_0,z'-z_0'\\in\\Omega$. With $r$ as in Proposition \\ref{growth}, we may assume $\\left|z\\right|,\\left|z'\\right|\\geq r$. Applying Lemma \\ref{common}, with any $B>0$ and with $N=\\sqrt{d^9(\\log d)^2\\log H}$, we obtain $A>0$ such that $$\\min\\{\\left|z\\right|,\\left|z'\\right|\\}\\leq A\\sqrt{d^9(\\log d)^2\\log H}$$ or $$\\max\\{\\log\\left|z_0\\right|,\\log\\left|z_0'\\right|\\}\\leq-Bd^9(\\log d)^2\\log H.$$ At some points in the argument we shall want $B$ to be larger than certain values computable from $g_1$ and $g_2$, and may assume that this is the case.\n\nIf $\\min\\{|z|,|z'|\\}> A\\sqrt{d^9(\\log d)^2\\log H}$ then we have $\\omega,\\omega'\\in\\Omega$ such that $$\\log|z-\\omega|=\\log|z_0|\\leq-Bd^9(\\log d)^2\\log H$$ and $$\\log|z'-\\omega'|=\\log\\left|z_0\\right|\\leq-Bd^9(\\log d)^2\\log H.$$ We may assume $\\omega\\omega'\\neq 0$. We show that $\\frac{\\omega'}{\\omega}\\in\\mathbb{Q}$. Suppose not. Then $\\frac{\\omega'}{\\omega}\\notin\\mathbb{R}$ and we have some $a,b\\in \\mathbb{Q}$ such that $a\\omega+b\\omega'=\\omega_1$. We want to bound the height of $a$ and $b$ and for this we consider how they are obtained. By Lemma \\ref{cosine}, we have $k\\omega_1+l\\omega_2=\\omega$ and $k'\\omega_1+l'\\omega_2=\\omega'$ for some integers $k,l,k',l'$ such that such that $\\left|k\\right|,\\left|l\\right|\\leq c\\left|\\omega\\right|$ and $\\left|k'\\right|,\\left|l'\\right|\\leq c\\left|\\omega'\\right|$, where $c$ is positive and depends only on $g_1$ and $g_2$. We may assume $c\\geq 1$. Since $\\frac{\\omega'}{\\omega}\\notin\\mathbb{R}$, $kl'-lk'\\neq 0$. \n\n\nWe then have $a=\\frac{l'}{kl'-lk'}$ and $b=\\frac{-l}{kl'-lk'}$. It follows that $$H(a,b)\\leq 2c^2\\max\\{1,\\left|\\omega\\right|\\}\\max\\{1,\\left|\\omega'\\right|\\}.$$ Since $z$ and $z'$ have height at most $H$ and degree at most $d$, we have $\\left|z\\right|,\\left|z'\\right|\\leq H^d$. We also have $\\left|z-\\omega\\right|<1$ and $\\left|z'-\\omega'\\right|<1$ and so $\\left|\\omega\\right|\\leq \\left|z\\right|+1$ and $\\left|\\omega'\\right|\\leq \\left|z'\\right|+1$. Then $$H(a,b)\\leq 2c^2(H^d+1)^2\\leq 8c^2H^{2d}$$ and $$H(\\left|a\\right|+\\left|b\\right|)\\leq 16c^2H^{2d}.$$\n\n\n\nNow $$\\log|(az+bz')-\\omega_1|=\\log\\left|a(z-\\omega)+b(z'-\\omega')\\right|$$ $$\\leq\\log(|a|+|b|)-Bd^9(\\log d)^2\\log H$$ $$\\leq \\log(16c^2)+2d\\log H-Bd^9(\\log d)^2\\log H.$$ \n\nWe have $H(az+bz')\\leq 2(8c^2)^2H^{4d+2}$ and $[\\mathbb{Q}(az+bz'):\\mathbb{Q}]\\leq d^2$. By Lemma \\ref{measure} there is a positive constant $c'$, depending effectively on $g_1$ and $g_2$, such that $$\\log |(az+bz')-\\omega_1|\\ge -c' d^9(\\log d)^2 \\log H |\\omega_1|^2 (1+\\max\\{0,\\log |\\omega_1|\\})^3.$$ We have a contradiction given that $B$ is large enough. So $\\frac{\\omega'}{\\omega}\\in\\mathbb{Q}$. \n\nWe complete the proof by showing that $\\frac{z'}{z}=\\frac{\\omega'}{\\omega}$. Since $\\omega=k\\omega_1+l\\omega_2$, $\\omega'=k'\\omega_1+l'\\omega_2$ and $\\omega_1$ and $\\omega_2$ are $\\mathbb{Q}$-linearly independent, the rationality of $\\frac{\\omega'}{\\omega}$ implies that there exists $q\\in \\mathbb{Q}$ such that $qk=k'$ and $ql=l'$. In fact, $q=\\frac{\\omega'}{\\omega}$. We then have $$H\\left(\\frac{\\omega'}{\\omega}\\right)\\leq\\max\\{\\left|k\\right|,\\left|k'\\right|\\}\\leq c(\\max\\{\\left|z\\right|,\\left|z'\\right|\\}+1)\\leq c(H^d+1).$$\n\nAssuming without loss of generality that $|\\omega|\\geq|\\omega'|$, note that $$\\left|\\frac{\\omega' z}{\\omega}-z'\\right|= \\left|\\frac{\\omega' z}{\\omega}-\\omega'+\\omega'-z'\\right|\\leq\\frac{\\left|\\omega'\\right|}{\\left|\\omega\\right|}\\left|z-\\omega\\right|+\\left|\\omega'-z'\\right|\\leq \\left|z-\\omega\\right|+\\left|z'-\\omega'\\right|$$ and so $$\\log\\left|\\frac{\\omega' z}{\\omega}-z'\\right|\\leq \\log{2}-Bd^9(\\log d)^2\\log H.$$ But $\\frac{\\omega' z}{\\omega}-z'$ is algebraic with $$H\\left(\\frac{\\omega' z}{\\omega}-z'\\right)\\leq 2H\\left(\\frac{\\omega'}{\\omega}\\right)H(z)H(z')\\leq 2c(H^d+1)H^2$$ and $$\\left[\\mathbb{Q}\\left(\\frac{\\omega' z}{\\omega}-z'\\right):\\mathbb{Q}\\right]\\leq d^2.$$ Given that $B$ is large enough, we must have $\\frac{\\omega'z}{\\omega}-z'=0$ and so $\\frac{z'}{z}=\\frac{\\omega'}{\\omega}$. \\endproof\n\nProposition \\ref{applgrowth2} is almost enough for our purposes. To finish the proof of Theorem \\ref{alggs}, we bound the cardinality of the exceptional set\n\\[\n\\mathcal{E}=\\{ z \\in \\mathbb{C} : H(z,\\sigma_\\Omega(z) )\\leq H, [\\mathbb Q(z,\\sigma_\\Omega(z)) :\\mathbb Q] \\le d, |z| > \\max\\{r,A \\sqrt{ d^9 (\\log d)^2 \\log H}\\}\\}\n\\]\nwhere $d,H\\geq e$, $A$ is as in Proposition \\ref{applgrowth2} and $r$ is as in Proposition \\ref{growth}. Suppose that $z,z'\\in\\mathcal{E}$ and let $\\omega,\\omega'\\in \\Omega$ be such that $\\max\\{\\log|z-\\omega|,\\log|z'-\\omega'|\\}<-Bd^9(\\log d)^2\\log H$ and $\\frac{z'}{z}=\\frac{\\omega'}{\\omega}\\in\\mathbb{Q}$, with $B$ as in Proposition \\ref{applgrowth2}. We may assume $\\omega'\\neq 0$. Let $\\omega^*\\in\\Omega$ be non-zero and of minimum modulus lying on the line through $0$ and $\\omega$. So $\\omega = p \\omega^*$ for some non-zero integer $p$. Let $z^*=z\/p$. Then $z'$ is also an integer multiple of $z^*$, and so it suffices to show that there are few integers $p$ for which $pz^*\\in\\mathcal{E}$. We shall use $c_1,...,c_{11}$ to denote various positive constants which only depend on $g_1$ and $g_2$. \n\nWe can write $\\omega= k \\omega_1+l\\omega_2$,  with $k,l\\in \\mathbb{Z}$. By Lemma \\ref{cosine} we have $\\left|k\\right|,\\left|l\\right|\\leq c_1\\left|\\omega\\right|$. Since $p$ divides both $k$ and $l$, we have $|p| \\le c_1|\\omega| \\le c_1 (|z|+1) \\le c_2 H^d$. And so  $\\log H(z^*)\\leq c_3 d\\log H$. \n\nBy Proposition \\ref{growth}, $$\\log\\left|\\sigma_\\Omega(z)\\right|\\geq \\log\\left|\\sigma_\\Omega(z_0)\\right|+c_4|z|^2\\geq \\log\\left|z_0\\right|-c_5+c_4|z|^2$$ $$=\\log\\left|z-\\omega\\right|-c_5+c_4|z|^2\\geq \\log\\left|z^*-\\omega^*\\right|-c_5+c_4p^2|z^*|^2$$ where $z_0\\in P$ is such that $z-z_0\\in\\Omega$. By Lemma \\ref{measure}, $$\\log\\left|z^*-\\omega^*\\right|\\geq-c_6d^4(\\log d)^2d(\\log H)\\left|\\omega^*\\right|^2(1+\\max\\{0,\\log\\left|\\omega^*\\right|\\})^3$$ $$\\geq-c_7d^5(\\log d)^2(\\log H)\\left|z^*\\right|^2(1+\\max\\{0,\\log\\left|z^*\\right|\\})^3.$$\n\n\nWe then have $$d\\log H\\geq \\log\\left|\\sigma_\\Omega(z)\\right|\\geq -c_7d^5(\\log d)^2(\\log H)\\left|z^*\\right|^2(1+\\max\\{0,\\log\\left|z^*\\right|\\})^3-c_5+c_4p^2\\left|z^*\\right|^2.$$\n\nIt follows that $$p^2\\leq c_8d^5(\\log d)^2(\\log H)(1+\\max\\{0,\\log|z^*|\\})^3\\leq  c_9d^5(\\log d)^2(\\log H)(1+d\\log H)^3,$$ and $\\mathcal{E}$ has size at most \n$$\nc_{10}\\sqrt{d^5(\\log d)^2(\\log H)(1+d\\log H)^3}.\n$$\n\nSo when proving Theorem \\ref{alggs} it suffices to consider those $z$ with $|z|\\le A\\sqrt{d^9(\\log d)^2\\log H}$. We then follow the proof of Theorem \\ref{alglattice}, with this larger radius. Set $Z= 4A \\sqrt{d^9(\\log d)^2\\log H}$. Making appropriate modifications in the choice of parameters for Proposition \\ref{propn2}, there is a non-zero polynomial $P$ of degree at most $c_{11} d^{10}(\\log d)^2 \\log H$ such that it suffices to bound the number of zeroes of $P(z,\\sigma_\\Omega(z))$ with $|z| \\le  A\\sqrt{d^9(\\log d)^2\\log H}$. We apply Theorem \\ref{Besson_zero} to conclude.\n\n\n\n\n\n\\section{Alternative reasoning}\n\nWe outline an alternative argument that uses reasoning from \\cite{BJ2}, instead of Theorem \\ref{Besson_zero}, to obtain a conclusion similar to that of Theorem \\ref{alggs}, but with a slightly weaker bound. As in the previous section, we obtain constants $c_{10}, c_{11}$ and $A$ and a polynomial $P$, with degree at most $T=c_{11} d^{10}(\\log d)^2 \\log H$, such that, with at most $c_{10}\\sqrt{d^5(\\log d)^2\\log H(1+d\\log H)^3}$ exceptions, all $z\\in \\mathbb{C}$ with $\\left[\\mathbb{Q}(z,\\sigma_\\Omega(z)):\\mathbb{Q}\\right]\\leq d$ and $H(z,\\sigma_\\Omega(z))\\leq H$ lie in the closed disk with centre zero and radius $A\\sqrt{d^9(\\log d)^2\\log H}$, and satisfy the equation $P(z,\\sigma_\\Omega(z))=0$. In the proof of Proposition 2 of \\cite{Masser}, Masser provides further information about the coefficients of $P$ assuming, as we may, that there are enough points $(z,\\sigma_\\Omega(z))$ such that $H(z,\\sigma_\\Omega(z))\\leq H$, $[\\mathbb{Q}(z,\\sigma_\\Omega(z)):\\mathbb{Q}]\\leq d$ and $\\left|z\\right|\\leq A\\sqrt{d^9(\\log d)^2\\log H}$. Specifically, $P$ may be chosen to have integer coefficients, each with absolute value at most $2^{1\/d}(T+1)^2H^T$.\n\nWe use reasoning from the proof of Theorem 2.5 in \\cite{BJ2} to bound the number of zeroes of $P(z,\\sigma_\\Omega(z))$ in the closed disk with centre zero and radius $A\\sqrt{d^9(\\log d)^2\\log H}$. Let $k$ be the degree of $Y$ in $P(X,Y)$. We get a good bound if $k=0$. Assume $k\\geq 1$. We set $\\tilde{P}(X,Y)=Y^kP(X,\\frac{1}{Y})$, $R(X)=\\tilde{P}(X,0)$, $Q(X,Y)=\\tilde{P}(X,Y)-R(X)$ and $\\tilde{Q}(X,Y)=\\frac{1}{Y}Q(X,Y)$. These are all polynomials with integer coefficients and we have bounds for their degrees and the absolute values of their coefficients. From now on we shall use $c$ to represent various positive constants, possibly different on each occasion, depending effectively on $g_2$ and $g_3$. \n\nWe seek some $w\\in\\mathbb{C}$ such that $\\left|w\\right|$ is not too large and $\\left|P(w,\\sigma_\\Omega(w))\\right|$ is not too small, restricting our attention to those $w$ for which $\\left|\\sigma_\\Omega(w)\\right|=M_{\\sigma_\\Omega}(\\left|w\\right|)$, where $M_{\\sigma_\\Omega}(s)$ denotes $\\max\\{\\left|\\sigma_\\Omega(z)\\right|:\\left|z\\right|\\leq s\\}$. We first find a disk around zero outside of which, for any such $w$, $\\left|Q\\left(w,\\frac{1}{\\sigma_\\Omega(w)}\\right)\\right|\\leq \\frac{1}{2}$. A disk with radius $cT$ will work as, outside of that, we shall have $\\left|w\\right|>cT$, $$\\left|\\tilde{Q}\\left(w,\\frac{1}{\\sigma_\\Omega(w)}\\right)\\right|\\leq 2^{1\/d}(T+1)^4H^T\\left|w\\right|^T\\max\\{1,\\left|\\sigma_\\Omega(w)\\right|^{-T}\\},$$ $$\\left|\\sigma_\\Omega(w)\\right|=M_{\\sigma_\\Omega}(\\left|w\\right|)\\geq ce^{c\\left|w\\right|^2}$$ and $$\\left|Q\\left(w,\\frac{1}{\\sigma_\\Omega(w)}\\right)\\right|=\\left|\\sigma_\\Omega(w)\\right|^{-1}\\left|\\tilde{Q}\\left(w,\\frac{1}{\\sigma_\\Omega(w)}\\right)\\right|.$$ \n\nWe then choose a $w$ outside but close to this disk, with $\\left|\\sigma_\\Omega(w)\\right|=M_{\\sigma_\\Omega}(\\left|\\omega\\right|)$ and $\\left|R(w)\\right|\\geq 1$. By the Boutroux-Cartan lemma (stated as Fact 2.3 in \\cite{BJ2}), there is such a $w$ in the disk with centre zero and radius $cT+T+14.$ We then have $$\\left|\\tilde{P}\\left(w,\\frac{1}{\\sigma_\\Omega(w)}\\right)\\right|\\geq \\frac{1}{2}$$ and so $$\\left|P(w,\\sigma_\\Omega(w))\\right|=\\left|\\sigma_\\Omega(w)\\right|^k\\left|\\tilde{P}\\left(w,\\frac{1}{\\sigma_\\Omega(w)}\\right)\\right|\\geq \\frac{1}{2}.$$ We are concerned with points on the disk $D_1=\\{z\\in\\mathbb{C}:\\left|z\\right|\\leq A\\sqrt{d^9(\\log d)^2\\log H}\\}$. We work with the disk $D_2=\\{z\\in\\mathbb{C}:\\left|z-w\\right|\\leq cT\\}$, where this $c$ is chosen large enough to ensure that $2D_1\\subseteq D_2$. We have $\\max\\{\\left|P(z,\\sigma_\\Omega(z))\\right|:z\\in D_2\\}\\leq (T+1)^4H^T(cT)^Te^{cT^3}$.\n\n\nBy a well-known consequence of Jensen's formula (stated as Fact 2.4 in \\cite{BJ2}), $P(z,\\sigma_\\Omega(z))$ has at most $c\\log\\left((T+1)^4H^T(cT)^Te^{cT^3}\\right)$ zeroes in $D_2$ and hence also in $D_1$. So we obtain the conclusion of Theorem \\ref{alggs} but with the weaker bound of $cd^{30}(\\log d)^6(\\log H)^3$. \n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}