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+{"text":"\\section{Introduction}\nGraphene's gapless, relativistic spectrum leads to many unusual transport properties~\\cite{castro2009electronic}, such as Klein tunneling~\\cite{katsnelson2006chiral} and the suppression of back scattering~\\cite{ando1998berry}. Graphene also exhibits strong optical transitions, with a universal absorption of $\\pi e^2\/ \\hbar c \\approx 2.3 \\%$ over a broad range of frequencies~\\cite{nair2008fine}. These defining features of graphene make electronic and optical control difficult to achieve. Indeed, for the realisation of optoelectronic devices which capitalize on the relative nature of graphene's dispersion, one must overcome certain undesirable relativistic features, without destroying all the attractive effects.\n\nFor optoelectronic devices applications, it would be highly  desirable to modify graphene's spectrum such that it possesses valence-like and conduction-like bands, separated by an externally tunable pseudo-bandgap (a true gap would require a drastic modification of the material by either functionalization, cutting or rolling), as well as reduce the freedom of quasiparticle  motion from two-dimensions down to one. In essence, we wish to form an analog of a narrow-gap carbon nanotube spectrum, without physically deforming the sheet by cutting it to make a ribbon, or rolling it to form a nanotube. In this paper, we propose a setup based on a bipolar potential, which does not open a true bandgap. However, at certain parameters of the potential, the electron dispersion becomes non-monotonous, and has pronounced extrema associated with avoided crossings between the states confined within a barrier and a well. We call the energy separation at a local dispersion minimum a pseudo-gap. Such a pseudo-gap would result in the giant enhancement of the probability of optical transitions~\\cite{hartmann2019interband}. However, unlike a nanotube or ribbon, whose bandgap is predefined by geometry, we seek to create a pseudo-gap which is fully tunable, without the need for huge magnetic fields~\\cite{portnoi2008terahertz,hartmann2015terahertz}.  \nRather than achieving the quantization of momentum through geometry like a nanoribbon, or nanotube does, one may instead quantize momentum via the application a quasi-one-dimensional (1D) electrostatically defined potential, i.e., by using electron waveguides~\\cite{pereira2006confined,cheianov2007focusing,tudorovskiy2007spatially,shytov2008klein,beenakker2009quantum,zhang2009guided,hartmann2010smooth,zhao2010proposal,sharma2011electron,hartmann2014quasi,he2014guided,hasegawa2014bound,xu2015guided,xu2016guided,mondal2019thz}. Unlike a physical tube whose radius cannot be changed, externally applied potentials can be easily varied. \nThere has been significant experimental progress since the pioneering work in the field of graphene electron waveguides~\\cite{huard2007transport, ozyilmaz2007electronic, gorbachev2008conductance,liu2008fabrication,williams2011gate,rickhaus2015guiding} and the recent breakthrough of utilizing a nanotube as a top gates~\\cite{cheng2019guiding} enabled the detection of individual guided modes within a single waveguide. However, apart from graphene waveguides possessing a threshold in the characteristic potential strength required to observe a fully bound mode~\\cite{hartmann2010smooth,hartmann2017two}, one could argue that a single graphene electron waveguides provides similar physics to that studied in quasi-1D channels within conventional semiconductor systems. Indeed, the absolute value of electron momentum along the direction of a waveguide formed by an attractive potential (quantum well) defines the electron's effective mass. Confined states of a deep well start with negative energy and for large values of momentum have a positive dispersion along the waveguide. Similarly, a potential barrier can also form a waveguide, where the confined electron states have negative dispersion, i.e., are hole-like. This raises the question what happens when branches of negative and positive dispersion meet? Answering this question is the focus of this paper. \nIn what follows we show that a bipolar electronic waveguide is a fully tunable quasi-1D system with a non-monotonous dispersion accompanied by pseudo-gaps, characterized by a giant enhancement of density of states and interband dipole transition probabilities in the energy range where graphene's own density of states is rather small. \nIn addition we present a general analytic formalism allowing us to find with a spectacular degree of accuracy the main features of a bipolar waveguide from the properties of a single quantum well.\n\n\n\n\nTransmission through single and multiple barriers in graphene has been a subject of extensive research~\\cite{katsnelson2006chiral,cheianov2006selective,williams2007quantum,cheianov2007focusing,williams2007quantum,pereira2007graphene,huard2007transport, ozyilmaz2007electronic,pereira2008resonant, gorbachev2008conductance,pereira2010klein,bahlouli2012tunneling,alhaidari2012relativistic,wei2013resonant} including periodic potentials~\\cite{brey2009emerging} and sinusoidal multiple-quantum-well systems~\\cite{xu2010resonant,pham2015tunneling}. \nDespite this significant body of research, the phenomenon of pseudo-gap formation in bipolar waveguides has been hitherto overlooked. \nIt should be emphasized that the idea of using bipolar waveguides stems directly from the essential feature of graphene, as a gapless material, that a potential barrier can contain guided electron modes, effectively acting as a potential well. For non-relativistic particles, the probability of tunneling between two wells results in the splitting of energy levels, to form a doublet state. In contrast, the probability of tunneling between a well and a barrier in graphene results in the bands forming an avoided crossing at finite $k_y$, where $\\hbar k_y$, is the momentum along the waveguide. As we demonstrate below, by modulating the applied voltage, bipolar waveguides have fully controllable pseudo-gaps, which exhibit extremely strong optical transitions. Not only this, but these transitions occur in the highly elusive and desirable THz frequency range. This part of the electromagnetic spectrum is notoriously difficult to generate and manipulate~\\cite{lee2007searching}. Therefore, by using a suitably chosen combination of guiding potentials, one can transform a graphene sheet into a narrow-gap nanotube without rolling, where the effective nanotube radius is controlled by the strength of the applied potential.\n\n\\begin{figure}[ht]\n    \\centering\n   \n    \\includegraphics[width=0.4\\linewidth]{figure_1_a}\n    \\includegraphics[width=0.4\\linewidth]{figure_1_b}\n    \\caption{\n    (a) The schematic of the proposed experimental setup and (b) a comparison between the potential created by two nanotubes, separated a distance $D=175$~nm,\n    with $h=40$~nm, and $t=20$~nm\n    (grey line) and the linear combination of a well and barrier defined by shifted $\\pm u_0 \/\\cosh(x\/L)$ functions (black dashed line) for the case of $\\phi_1=-\\phi_2=0.25$~V, matching both their peak values and second-derivative at their maxima.}\n    \\label{fig:set_up}\n\\end{figure}\nInspired by the most advanced experimentally attainable waveguides~\\cite{cheng2019guiding}, our proposed bipolar waveguide is defined by two nanotubes (top gates), of radius $r_{0}$, separated by a distance $D$, as shown in Fig.~\\ref{fig:set_up}~(a). Both nanotubes are placed at a height, $h$, above the metallic substrate. The potential profile in the graphene plane separated from the same substrate by another distance $t$ is given by the expression:\n\\begin{equation}\nU\\left(x\\right)=\\frac{e\\widetilde{\\phi}_{1}}{2}\\ln\\left[\\frac{\\left(x+\\frac{D}{2}\\right)^{2}+\\left(h-t\\right)^{2}}{\\left(x+\\frac{D}{2}\\right)^{2}+\\left(h+t\\right)^{2}}\\right]+\\frac{e\\widetilde{\\phi}_{2}}{2}\\ln\\left[\\frac{\\left(x-\\frac{D}{2}\\right)^{2}+\\left(h-t\\right)^{2}}{\\left(x-\\frac{D}{2}\\right)^{2}+\\left(h+t\\right)^{2}}\\right],\n\\end{equation}\nwhere $\\widetilde{\\phi}_{1,2}=\\phi_{1,2}\/\\ln\\left(\\frac{2h-r_{0}}{r_{0}}\\right)$, and $\\phi_{1}$ ($\\phi_{2}$) is the applied voltage between the left (right) nanotube and the back gate. The expression above can be easily modified when the top gates are fully embedded in a dielectric material. In the above-mentioned recent work~\\cite{cheng2019guiding}, a smooth electron waveguide was fabricated using a carbon nanotube as a top gate and the graphene sheet was sandwiched in between two layers of hexagonal boron nitride (h-BN). The top h-BN layer had a thickness, $h$, of between 4 and 100 nm, and the bottom layer had a thickness, $t$ of around 20 nm. It can be seen from Fig.~\\ref{fig:set_up}~(b) that the potential in the plane of the graphene sheet, derived using image charges in the substrate, can be very well approximated by a linear combination of two shifted hyperbolic secant functions. It is convenient to use this particular approximation because for a single well, the hyperbolic secant potential possesses quasi-exact solutions to the Dirac equation~\\cite{hartmann2010smooth,hartmann2011excitons,hartmann2014quasi}. This will enable us to treat both the size of the pseudo-gap and the optical transitions across it analytically.\n\n\\section{Negative dispersion and pseudo-gaps}\nThe low-energy quasiparticle behaviour in graphene is known to be described with spectacular accuracy by the 2D Dirac equation for massless fermions~\\cite{wallace1947band}. In the presence of a confining electrostatic potential, $U(x)$, the effective 1D matrix Hamiltonian for confined modes in a graphene waveguide can be written in the standard basis of graphene's two sub-lattices as\n\\begin{equation}\n\\hat{H}=\\hbar v_{\\mathrm{F}}\n\\left(\n\\hat{k}_{x}\\sigma_{x}+s_{\\mathrm{K}}k_{y}\\sigma_{y}\\right) +\\mathrm{I}U(x),\n\\label{eq:Ham_Dirac}\n\\end{equation}\nwhere $\\hat{k}_{x}=-i\\frac{\\partial}{\\partial x}$, $k_{y}$ is a wavenumber corresponding to the motion along the waveguide, $\\sigma_{x,\\,y,\\,z}$ are the Pauli spin matrices, $\\mathrm{I}$ is the 2 by 2 unit matrix, $v_{\\mathrm{F}}$ is the Fermi velocity, which is approximately $\\approx10^{6}$ m\/s, and $s_{\\mathrm{K}}$ is the valley quantum number, which has the value of $+1$ and $-1$ for the K and K' valley respectively. \n\nWe are interested in the situation when $U(x)$ is a combination of two fast decaying potentials separated by a distance $d$. For a better understanding of the underlying physics it is instructive to look both at the same and different sign constituent potentials. This allows a comparison with the familiar non-relativistic results for the double quantum well. The results are especially transparent for a quasi-one dimensional potential formed by a combination of either; a well and a barrier of the same strength, or, two equal wells:\n\\begin{equation}\nU\\left(x\\right)=\nu\\left(x+\\frac{d}{2}\\right) \\pm  \nu\\left(x-\\frac{d}{2}\\right),\n\\label{eq:pot_to_solve}\n\\end{equation}\nwhere, $u\\left(x\\right)$ is an individual symmetric potential well, for which the Hamiltonian, Eq.~(\\ref{eq:Ham_Dirac}), admits exact zero-energy eigenfunctions, which we denote as $\\Psi_{0}\\left(x\\right)$. The function $\\Psi_{0}\\left(x\\right)$ is normalized and assumed to rapidly decay outside of the well. The tunneling between wells, or between a well and a barrier, results in the energy level, $E=0$, splitting into two levels, $E_{1}$ and $E_{2}$. At $E=0$, the barrier wave function, denoted $\\Psi_{-}$, is the complex conjugate of the well wave function, $\\Psi_{-}=\\Psi_{0}^{\\star}\\left(x-\\frac{d}{2}\\right)$, reflecting the fact that the barrier for electrons is a well for holes; whereas, for the two-well case $\\Psi_{+}=\\Psi_{0}\\left(x-\\frac{d}{2}\\right)$.\nIn the weak wavefunction overlap approximation, resembling the tight-binding (H\\\"{u}ckel molecular orbital) methods widely used in solid-state and molecular physics, we may write the wave functions corresponding to eigenvalues $E_1$ and $E_2$ as:\n\\begin{equation}\n\\Psi_{1}=\\frac{1}{\\sqrt{2}}\\left[\\Psi_{\\pm}\\left(x-\\frac{d}{2}\\right)+\\Psi_{0}\\left(x+\\frac{d}{2}\\right)\\right],\n\\label{eq:Psi_E_1}\n\\end{equation}\n\\begin{equation}\n\\Psi_{2}=\\frac{1}{\\sqrt{2}}\\left[\\Psi_{\\pm}\\left(x-\\frac{d}{2}\\right)-\\Psi_{0}\\left(x+\\frac{d}{2}\\right)\\right].\n\\label{eq:Psi_E_2}\n\\end{equation}\nFollowing an approach similar to the non-relativistic case~\\cite{landau2013quantum}, but for a matrix Hamiltonian, the energy level splitting, $E_g=\\left|E_2-E_1\\right|$, can be shown to be\n\\begin{equation}\nE_{g}=2\\hbar v_{\\mathrm{F}}\\left|\\Psi_{\\pm}^{\\dagger}\\left(-\\frac{d}{2}\\right)\\sigma_{x}\\Psi_{0}\\left(\\frac{d}{2}\\right)\\right|.\n\\label{eq:gap_guess}\n\\end{equation}\nIt should be noted that there is a striking difference between the relativistic and non-relativistic case. For the non-relativistic case the splitting is proportional to the product of the single well function and its derivative~\\cite{landau2013quantum}; whereas, in the relativistic case, it depends only on the individual well and barrier functions (or the two shifted well functions for the double well). For simplicity, we considered above two potentials of equal strengths, and estimated the splitting of the $E=0$ state. These results can be easily generalized for non-equal potentials and non-zero values of energies as long as the energy levels in the individual potentials coincide~(see Appendix). This theorem demonstrates the utility of quasi-exact solutions to the Dirac equation~\\cite{hartmann2010smooth,hartmann2014quasi,hartmann2017two} for bipolar waveguides. Indeed, the exact solutions often correspond to the case where there is symmetry between the positive and negative energy solutions, allowing all pseudo-gaps to be treated within this formalism. Furthermore, knowledge of the exact wave-functions allows the matrix element of optical transitions across the pseudo-gaps to be calculated analytically. \n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.5\\linewidth]{Figure_2.jpg}\n    \\includegraphics[width=0.5\\linewidth]{Figure_3.jpg}\n    \\caption{\nThe energy spectrum in dimensionless units, $\\varepsilon=EL\/(\\hbar v_{\\mathrm{F}})$ \\textit{vs} $\\Delta=\\left|k_y\\right|L$, of confined states in a bipolar waveguide (shown as black dotted lines), defined by the potential $-u_{1}\/\\cosh\\left[\\left(x+\\frac{d}{2}\\right)\/L\\right]+u_{2}\/\\cosh\\left[\\left(x-\\frac{d}{2}\\right)\/L\\right]$, for (a) $u_1 L =u_2 L =0.75\\,\\hbar v_{\\mathrm{F}}$, $d\/L=8$, (b) $u_1 L=u_2 L =1.75\\,\\hbar v_{\\mathrm{F}}$, $d\/L=8$, (c) $u_{1} L =u_{2} L =0.75\\,\\hbar v_{\\mathrm{F}}$, $d\/L=12$ and (d) $u_{1} L =1.9\\, \\hbar v_{\\mathrm{F}}$, $u_{2} L =0.6\\, \\hbar v_{\\mathrm{F}}$, $d\/L=8$. The blue and red lines show the dispersion lines for an isolated well and barrier respectively, while the grey lines show the boundary at which the bound states merge with the continuum at $\\left|E\\right|=\\hbar v_{\\mathrm{F}} \\left|k_y\\right|$.\n}\n    \\label{fig:energy_fixed_dist}\n\\end{figure}\n\nIn what follows we shall model a bipolar waveguide as\n\\begin{equation}\nU(x)=-\\frac{u_{1}}{\\cosh\\left[\\left(x+\\frac{d}{2}\\right)\/L\\right]}+\\frac{u_{2}}{\\cosh\\left[\\left(x-\\frac{d}{2}\\right)\/L\\right]},\n\\label{eq:potential}\n\\end{equation}\nwhere $u_1$ and $u_2$ are the depth of the well and the height of the barrier, respectively, and $L$ is the effective width of the potential, which for now we assume to be the same for the well and the barrier. For $u_1=u_2=u_0>0$, this smooth potential indeed provides an excellent fit to the realistic potential generated by two oppositely charged nanotubes above the surface of graphene, see Fig.~\\ref{fig:set_up}~(b). Also each of the individual secant potentials supports exact analytic solutions at $E=0$~\\cite{hartmann2010smooth}, allowing the comparison of the numerical results with the approximate formula, Eq.~(\\ref{eq:gap_guess}). It is convenient to introduce similar dimensionless parameters as Ref.~\\cite{hartmann2010smooth}, namely $\\omega=u_{0} L \/ ( \\hbar v_{\\mathrm{F}})$ and $\\Delta=\\left|k_{y}\\right| L$. It should be noted that the number of bound states contained within a realistic confining potential is also defined by the product of the characteristic potential depth and its width \\cite{cheng2019guiding,hartmann2017two} rather than its exact form. Effects such as non-linear screening and the renormalization of the Fermi velocity~\\cite{elias2011dirac}, can be accommodated within the same dimensionless parameter, $u_0 L\/(\\hbar v_F)$. The particular choice of the potential we use does not influence the physical picture. The use of this dimensionless parameter also allows the application of our results to other 2D systems with linear dispersion beyond graphene, e.g., surface states of topological insulators.\n\nIn the absence of inter-potential tunneling, the dispersion lines of the well and barrier (indicated in Fig.~\\ref{fig:energy_fixed_dist} by blue and red lines respectively) cross when $\\Delta_n=\\omega - n -1\/2$, where $n$ is a positive integer~\\cite{hartmann2010smooth}. The corresponding exact wavefunctions when substituted into Eq.~(\\ref{eq:gap_guess}) yield the following approximate expression for the $n=0$ pseudo-gap (see Appendix)\n\\begin{equation}\nE_{g}\/\\left(\\hbar v_{\\mathrm{F}}\/L\\right)\\approx\\frac{2\\exp\\left(-\\Delta_{0}d\/L\\right)}{B\\left(1+\\Delta_{0},\\Delta_{0}\\right)},\n\\label{eq:approx_ana_gap}\n\\end{equation}\nwhere $B(m,n)$ is the Beta function. This formula gives an extremely good approximation to the numerical solution in the limit when the ratio between the wire separation and the effective width of the potential is large, i.e., $d\/L \\gg 1$. \n\n\nIn Fig.~\\ref{fig:energy_fixed_dist} we plot the numerically obtained energy dependence on $\\Delta$, i.e. the momentum along the barrier in dimensionless units, for the cases of $\\omega=0.75$ (panel~(a)) and $\\omega=1.75$ (panel~(b)). In both cases the two oppositely charged nanotubes are separated by a distance $d\/L=8$, which corresponds to approximately $175$~nm for the case of $h=40$~nm and $t=20 $~nm. \nAt this distance the energy-level splitting formula, Eq.~(\\ref{eq:gap_guess}), accurately predicts the value of the $n=0$ pseudo-gap within a few per cent error. This error becomes one order of magnitude smaller when $d\/L=12$. It is instructive to compare these electrostatically induced pseudo-gaps with curvature-induced gaps in carbon nanotubes. For a narrow-gap carbon nanotube, it is well established that the larger is its radius, the smaller is the curvature-induced gap~\\cite{kane1997size}. Therefore, the strength of the applied voltage, for a particular guided mode, can be mapped to the radius of the nanotube. Increasing the voltage results in more tightly confined guided modes, characterised by a higher value of $\\Delta_0$ entering Eq.~(\\ref{eq:approx_ana_gap}), therefore we arrive to a smaller value of the pseudo-gap, which moves to the right in Fig.~\\ref{fig:energy_fixed_dist}~(b).\nIt can also be seen from panel (b) of this figure that the deeper the well and higher the barrier, the more states are contained within each channel, increasing correspondingly the number of avoided crossings appearing in the dispersion. It should also be noted that although increasing the voltage leads to stronger confinement for lower order modes, it also results in the appearance of higher order modes which are more spread out, leading to additional wider pseudo-gaps. In Fig.~\\ref{fig:energy_fixed_dist}~(b), which corresponds to the case of a deeper well and higher barrier, we can see two pseudo-gaps at $E=0$ as well as additional pseudo-gaps at non-zero energy since there are more guided modes in the low-energy part of the spectrum.\n\n\n\n\n\n\nIt can be seen by comparing Fig.~\\ref{fig:energy_fixed_dist}~(c) to~(a) that increasing the distance between the nanotubes, decreases the size of the gap. This is a result of the decrease in the overlap between the well and barrier functions. Technologically it is quite difficult to have exact control over the precise tube separation. However, this is not so important since it is possible to control the value of the pseudo-gap by the applied voltage. Furthermore, it can be see from Fig.~\\ref{fig:energy_fixed_dist}~(d) that the effect of pseudo-gap opening is robust against asymmetry in the system. In Fig.~\\ref{fig:energy_fixed_dist}~(d), the dispersion is recalculated for two tubes separated at the same distance as in Fig.~\\ref{fig:energy_fixed_dist}~(a), but with the depth of the well increased to $\\omega=1.9$, while the size of the barrier decreased to $\\omega=0.6$. This demonstrates that top gates of mismatched radius, or dissimilar magnitudes of applied voltage will just shift the value in energy in which the avoided crossing occurs. Although Eqs.~(\\ref{eq:gap_guess},\\ref{eq:approx_ana_gap}) give for the double well and bipolar waveguide the same value of energy-level splitting at the value of $k_y$ corresponding to $E=0$ in a single well, for the double well there is no pseudo-gap and the dispersion remains monotonous, this case is considered in depth elsewhere~\\cite{Hartmann2020double}.\n\n\\section{Interband Transitions}\nIn what follows we shall demonstrate that much like in narrow-gap nanotubes~\\cite{hartmann2019interband}, the\nwavefunction intermixing leads to strongly allowed optical transitions across the pseudo-gaps.\nThe probability of optical transitions is proportional to the squared modulus of the matrix element of velocity operator between the relevant states. The velocity operator written in the same basis as the Hamiltonian given in Eq.~(\\ref{eq:Ham_Dirac}) is~\\cite{saroka2018momentum,hartmann2019interband} \n\\begin{equation}\n\\hat{\\boldsymbol{v}}=v_{\\mathrm{F}}\\left(\\sigma_{x}\\hat{\\boldsymbol{x}}+s_{\\mathrm{K}}\\sigma_{y}\\hat{\\boldsymbol{y}}\\right).\n\\label{eq:VME}\n\\end{equation}\nThe probability of a dipole transition is proportional to $\\left|\\left\\langle \\Psi_{f}\\left|\\hat{\\boldsymbol{v}}\\cdot\\textbf{e}\\right|\\Psi_{i}\\right\\rangle \\right|^{2}$,\nwhere $\\Psi_{i}$ and $\\Psi_{f}$ are the initial and final states, respectively, and $\\textbf{e}=(e_x,e_y)$ is the light polarization vector. For linearly polarized light, $\\textbf{e}=(\\cos\\left(\\varphi_{0}\\right),\\sin\\left(\\varphi_{0}\\right))$, while for right- and left-handed polarized light $\\textbf{e}=\\left(1,-i\\right)\/\\sqrt{2}$ and $\\textbf{e}=\\left(1,i\\right)\/\\sqrt{2}$. Within the small wavefunction overlap approximation, for a bipolar waveguide defined by the potential given by Eq.~(\\ref{eq:potential}) with $u_1=u_2=u_0>0$, the matrix element of velocity of the $n=0$ mode at $\\Delta=\\Delta_0$ is~(see Appendix)\n\\begin{equation}\n\\left|\\left\\langle \\Psi_{2}\\left|\\hat{\\boldsymbol{v}}\\right|\\Psi_{1}\\right\\rangle \\right|\/v_{\\mathrm{F}}\\approx\\left|\\frac{e_{x}B\\left(1-e^{-d\/L};\\frac{1}{2}+\\Delta_{0},\\,0\\right)e^{-\\Delta_{0}d\/L}}{B\\left(1+\\Delta_{0},\\,\\Delta_{0}\\right)}-i\\frac{k_{y}}{\\left|k_{y}\\right|}\\frac{e_{y}B\\left(\\frac{1}{2}+\\Delta_{0},\\,\\frac{1}{2}+\\Delta_{0}\\right)}{B\\left(1+\\Delta_{0},\\,\\Delta_{0}\\right)}\\right|,\n\\label{eq:n_0_vme}\n\\end{equation}\nwhere $B(x;m,n)$ is the incomplete Beta function. For the case of the double well, at the same value of momentum, the matrix element of velocity of the $n=0$ mode is the first term of Eq.~(\\ref{eq:n_0_vme}), whereas $v_y=0$. It reflects the fact that transitions in a double well are only caused by light polarized along the $x$-direction (normal to the waveguide), similar to the non-relativistic case and the 1D square-well potential in graphene~\\cite{avishai2020klein}.\n\nIn stark contrast, the transition across the pseudo-gap of a bipolar waveguide is strongly polarized along the $y$-axis (waveguide direction). The situation changes away from the pseudo-gap. For small values of $\\left|k_{y}\\right|$, the transitions are polarized normally to the $y$-direction, as expected from the momentum alignment phenomenon in graphene~\\cite{saroka2018momentum}. For large values of $\\left|k_{y}\\right|$, the overlap between the well and barrier wavefunctions becomes very small leading to vanishing transition probabilities for both polarizations. This effect very much resembles the situation in a narrow-gap carbon nanotube, where optical transitions polarized along the nanotube axis are allowed in the narrow energy interval around the curvature-induced gap~\\cite{hartmann2019interband}. The main difference from a nanotube can be clearly seen from Fig.~\\ref{fig:vme}, namely both transitions polarizations along and normal to the waveguide are allowed across the pseudo-gap. The contribution of the $x$-component exponentially decreases with an increase in top gate potential or the separation between the gates, as can be seen from Fig.~\\ref{fig:vme}. \n\n\\begin{figure}[th]\n    \\centering\n    \\includegraphics[width=0.55\\linewidth]{Figure_4}\n    \\caption{\nPolar plots, showing the dependence of the absolute value (the length of the green arrow) of the velocity matrix element (in units of $v_{\\mathrm{F}}$) across the $n=0$ pseudo-gaps for dipole transitions caused by normally-incident linearly polarized light on the angle between the polarization vector and the $x$-axis normal to the bipolar waveguide, defined by $\\omega=0.75$, for two different top gate separations: (a) $d\/L=8$ and (b) $d\/L=12$. The analytic approximation is depicted by the dashed grey line and the numerically obtained values are shown by the black solid line.\n\\label{fig:vme}\n}\n\\end{figure}\n\nThe presence of both polarizations for pseudo-gap transitions leads to an effect absent in both graphene and non-chiral nanotubes. Namely, right- and left-handed polarized light produces different populations of pseudo-valleys with opposite signs of $k_y$. This can be clearly seen from Eq.~(\\ref{eq:n_0_vme}).  This feature does not depend on the model describing the potential~(see Appendix).\n\nThe discussed transitions across the pseudo-gap, fully controlled by the top gate voltages, can be easily brought into the highly desirable THz frequency range, which is usually extremely difficult to control. The presence of the van Hove singularity at the pseudo-gap edge enhances the strength of these transitions. These effects opens the avenue for novel gate-controlled polarized sensitive THz detectors based on bipolar waveguides in graphene.\n\n\n\\section{Conclusions}\nTo conclude, we have shown that bipolar waveguides allows for the creation of a non-monotonous 1D dispersion along the electron waveguide. The repulsion of well and barrier states results in the appearance of pseudo-gaps in the spectrum, whose size and symmetry can be fully controlled by the top gate voltages. These gaps can be estimated analytically for exactly-solvable potentials. The opening of these pseudo-gaps results in strongly allowed THz transitions with non-trivial optical selection rules.\nThe predicted negative dispersion of the guided modes may lead to various other physical effects ranging from solitary waves~\\cite{gulevich2017exploring} to Gunn-diode type current oscillations~\\cite{gunn1963microwave}.\n\n\\section*{Acknowledgements}\nWe thank C.A. Downing for his critical reading of the manuscript. This work was supported by the EU H2020 RISE project TERASSE (H2020-823878). RRH acknowledges financial support from URCO (71 F U 3TAY18-3TAY19). The work of MEP was supported by the Ministry of Science and Higher Education of Russian Federation, Goszadanie no. 2019-1246. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\n\n\\section{INTRODUCTION}\n\nIn the context of Keplerian disks, while hydrodynamic simulations seem \nto confirm the absence of turbulence\nin the non-linear limit \\cite{hbsw}, the\nMagneto-Rotational Instability \\cite{chandra}, discovered by Balbus and\nHawley \\cite{bh} within ionized accretion flows, confirms the existence of\nMHD turbulence. This helps to understand the origin of shear stress or ``viscosity\".\nHowever the MRI driven instability fails to operate in\ndisks with small ionization fraction. Examples of such\nsystems are proto-planetary disks, outskirts of AGN accretion\ndisks \\cite{gam} etc.. As a result, the route to turbulence and subsequently\naccretion in neutral disks has remained one of the outstanding\npuzzles in modern astrophysics.\n\nLaboratory experiments of Taylor-Couette systems\n(which are similar to accretion systems) seem to indicate\nthat, the flow is unstable to turbulence for Reynolds numbers larger than a\nfew thousand \\cite{richard}, even for subcritical\nsystems. Based on this, Longaretti \\cite{long} concludes that \nthe absence of turbulence in previous simulations \\cite{hbsw} is due\nto their small effective Reynolds number.\n\nThe fundamental reason for subcritical turbulence may be the result of\n{\\it definite frequency} modes not being orthogonal in a shear flow. Therefore, a \nsuitably tuned linear combination of modes can\nshow an arbitrarily large transient energy growth\\footnote{Growth is defined as the\nratio of energy at a particular\ninstant to that at the beginning.}. This transient growth may possibly lead to\nsustained {\\it hydrodynamic turbulence} for large enough Reynolds numbers.\n\nHere we analyze this transient growth phenomenon in the framework of accretion disks.\nWe present the set of scaling relations for the transient\ngrowth as a function of the Reynolds number.\n\n\\section{BASIC EQUATIONS AND SCALING RELATIONS}\n\n\nLet us define the shear frequency $2A$ and the vorticity frequency\n$2B$ as, $2A = -q\\Omega, \\, 2B = (2-q)\\Omega$.\nThen the momentum balance equations give\n\\begin{eqnarray}\n\\hskip-1.2cm\n{du\\over dt}= 2\\Omega v - {\\partial \\hat{p}\\over \\partial x}\n+\\nu {\\nabla}^2 u,\\,\n{dv\\over dt} = -2B u - {\\partial \\hat{p}\\over \\partial y}\n+\\nu {\\nabla}^2 v,\\,\n{dw\\over dt}= - {\\partial \\hat{p}\\over \\partial z}\n+\\nu {\\nabla}^2 w,\n\\label{zmmtm}\n\\end{eqnarray}\nalong with the incompressibility relation ${\\partial u\/\\partial x} + {\\partial v\/\\partial y} + {\\partial\nw\/\\partial z} = 0$, where $\\hat{p}=P\/\\rho$.  Symbols $u,v,w$ and $\n\\Omega$ are three velocity components and angular frequency\nrespectively, $\\nu, P, \\rho$ are dynamical viscosity, pressure and density of the system respectively\nand $q$ is a parameter which controls the flow pattern (whether a Keplerian disk, a constant angular\nmomentum disk or plane Couette flow). \nThe Lagrangian time derivative $d\/dt$ is, \n${d\/dt} = \\partial\/{\\partial t} -q\\Omega x\\partial\/{\\partial y}$.\nHere $x$ varies\nfrom $-L$ to $+L$, and $y$ and $z$ from $0$ to $2\\pi\/k_y$ and $2\\pi\/k_z$ respectively, where\n$k_x$, $k_y$, $k_z$, are the corresponding components of wave-vectors.\nThe boundary conditions are $u=v=w=0$ at $x=\\pm L$.\n\n\\subsection{Constant Angular Momentum Disks and Plane Couette Flow}\n\nIt is clear from (\\ref{zmmtm}) that the set of equations is very symmetrical for a constant\nangular momentum ($q=2$) and plane Couette ($\\Omega=0$) flow, except that $x$ and $y$ are interchanged.\nTherefore the growth is identical in the two cases. As\nthe previous analytical\/numerical studies (e.g. \\cite{bf,man}) showed that the growth is \nmaximum for vertical perturbations ($k_y=0$), here we consider such perturbation.\nNoting that the Reynolds number is, $R = |2A|L^2\/\\nu$,\nwe find that the maximum growth in energy for a given $k_z$ is\n\\begin{eqnarray}\nG_{max}(k_z,R) = \\left(\\frac{u^2(t)+v^2(t)+w^2(t)}{u^2(0)+v^2(0)+w^2(0)}\\right)_{\\rm optimum}\n={R^2k_z^2L^2\\,e^{-2}\\over \\left[{1\\over2}\\pi^2\n+k_z^2L^2\\right]^3}.\n\\end{eqnarray}\nMaximizing this over $k_z$, we find that the optimum wavevector is\n$k_zL = \\pi\/2 = 1.57$.\nThe maximum growth factor and the corresponding time are\n\\begin{eqnarray}\n\\nonumber\nG_{max}(R) = 0.82\\times10^{-3}R^2,\\,\\,\n|2A|t_{max}(R) = 0.13 R.\\\\\n\\end{eqnarray}\nClearly the growth scales as $R^2$ and therefore can be very large even for modest $R$.\nFor example, when $R=1200$, the above relations give \n$G_{max}(R)>1000$, which is probably large enough to induce non-linear feedback and turbulence.\n\n\\subsection{Keplerian Disk}\n\nHere we study eqns. (\\ref{zmmtm}) for $q=1.5$. As the earlier analysis \\cite{man} hinted that $k_z=0$\ngenerates the best growth, we consider such perturbations.\n\nWe consider a plane wave that is frozen into the fluid and is sheared\nalong with the background flow.  If the flow\nstarts at time $t=0$ with initial wave-vector $\\{k_{xi}, k_y\\}$, \nthe $k_x$ at later time is given by \n$k_x(t) = k_{xi} + q\\Omega k_yt$.\nThe maximum energy growth is then approximately \n\\begin{eqnarray}\nG_{max}(k_{xi},k_y,R) \\sim {k_{xi}^2L^2 \\over (1.7)^2+k_y^2L^2}\n\\exp\\left(-{2\\over 3R}{k_{xi}^3L^2\\over k_y}\\right).\n\\end{eqnarray}\nwhen $k_{x,min}L=1.7$.\nMaximizing this with respect to $k_{xi}$ and $k_y$, we obtain\n\\begin{eqnarray}\nG_{max}(R) \\sim 0.13R^{2\/3}, \\,\\,\\, |2A|t_{max}(R) \\sim 0.88 R^{1\/3}.\n\\end{eqnarray}\nAlso maximizing with respect to $k_{xi}$, keeping $k_y$ and $R$ fixed, we obtain\n\\begin{eqnarray}\n\\hskip-1.0cm\nG_{max}(k_y,R)= {(k_yL)^{2\/3} \\over (1.7)^2+(k_yL)^2}\ne^{-2\/3}R^{2\/3},\\,\\,|2A|t_{max}(k_y,R)= (k_yL)^{-2\/3} R^{1\/3}.\n\\end{eqnarray}\nWe see that the maximum growth scales as $R^{2\/3}$. Though this rate of increase is less than \nthat for a $q=2$ disk and plane Couette flow, still, at a large enough $R$, the growth can\nbecome large, and may cause turbulence in a Keplerian disk. For example, \nfor $R=10^6$, $G_{max}(R)> 1000$, which by analogy with a $q=2$ disk may be enough to cause\nturbulence. \n  \n\n\\section{CONCLUSION}\n\nWe have shown that significant transient growth of\nperturbations is possible in a Keplerian flow. Although the system does not \nhave any unstable eigenmodes, because of the non-normal nature of the eigenmodes a\nsignificant level of transient energy growth is possible at a large Reynolds number for\nappropriate choice of initial conditions. \nWe argue that a plausible critical Reynolds number for sustaining hydrodynamic turbulence\nin a Keplerian disk is\n$\\sim 10^4-10^6$ (for detailed discussions, see \\cite{man}).\nWe base this on the expectation that once the growth crosses the threshold, secondary instabilities of\nvarious kinds, such as the elliptical instability (see e.g. \\cite{ker}), might\nserve as a possible route to self-sustained turbulence.\nHowever, it remains to be verified that\nthese instabilities are present. Also, even if they are present, one will need to investigate whether\nthey lead to non-linear feedback and 3-dimensional turbulence.\n \nThis work was supported in part by NASA grant NAG5-10780 and NSF grant\nAST 0307433.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCosmic dust is widespread in the universe. It can be used to trace the evolutionary paths of planets, stars, and even black holes. Dust grains in the ISM of galaxies alter the appearance of the observable universe through absorption and scattering of photons. Thus, determining the effects of dust is needed for studying a wide range of astrophysical phenomena. Indeed a significant fraction of ISM is locked up in dust grains (e.g. \\citealt{Jenk09}). However, many aspects of cosmic dust remain poorly understood, such as: their chemical composition and origin; their physical properties and spectral signatures; their formation and evolution, and destruction mechanisms; and their role and impact on their environment. In AGN the properties of dust are particularly uncertain (see e.g. the review by \\citealt{Li07}). Also, dust in AGN can be associated to different possible origins, such as: dust lanes of recent galaxy merger remnants, or dusty winds from the AGN (see e.g. \\citealt{Komo97,Reyn97,Lee01,Cren01b}).\n\nDust is an essential player in the unification theory of AGN, where the supermassive black hole (SMBH) and the accretion disk are surrounded by an optically-thick dusty torus. The observational properties of AGN are hence strongly influenced by our viewing angle relative to the orientation of this obscuring torus (\\citealt{Anto85,Anto93,Urry95}). Yet, our knowledge of dust properties in the AGN torus and its environment is limited. \n\nAccretion onto SMBHs at the core of AGN is accompanied by winds of gas, which couple the SMBHs to their environment. The observed relations between SMBHs and their host galaxies, such as the M--$\\sigma$ relation \\citep{Ferr00}, indicate that SMBHs and their host galaxies are likely co-evolved through a feedback mechanism. The AGN winds can play a key role in this co-evolution as they can significantly impact star formation (e.g. \\citealt{Silk98}) and chemical enrichment of the surrounding intergalactic medium (e.g. \\citealt{Oppen06}). However, there are significant gaps in our understanding of the outflow phenomenon in AGN, which cause major uncertainties in determining their role and impact in galaxy evolution.\n\nThe origin and physical structure of AGN winds are generally poorly understood. Different mechanisms have been postulated for the launch and driving of winds from either the accretion disk or the AGN torus (e.g. \\citealt{Krol01,Prog04,Fuku10}). However, their association to the different kinds of winds found from observations is uncertain. AGN winds can originate as either thermally-driven \\citep{Krol01} or radiatively-driven \\citep{Doro08} winds from the dusty torus. Indeed the warm-absorber winds, which are commonly detected in the UV and X-ray spectra of bright AGN, are most consistent with being torus winds (e.g. \\citealt{Kaas12,Meh18}); thus dust could be mixed with such winds. Importantly, the infrared (IR) radiation pressure on dust grains can boost winds from the torus \\citep{Doro11}. Therefore, establishing the existence of dust in AGN winds is important for understanding the driving mechanism of AGN winds, and defining their impact on their environment. This is needed for assessing the contribution of such winds to AGN feedback. There are observational evidence, which indicate that AGN winds are likely carriers of dust into the ISM. For example, in the case of \\object{ESO~113-G010} \\citep{Meh12}, presence of dust embedded in the AGN wind was inferred based on the altering of the AGN emission, dust-to-gas ratio arguments, and the properties of the wind.\n\nIn order to advance our understanding of cosmic dust in the universe we need to utilise all available tools at our disposal. The X-ray energy band, which is invaluable for exploring both the cold and hot gas, is a relatively new scientific window for dust studies. The X-ray absorption fine structures (XAFS) at the K edge of O, Mg, Si, Fe, and the LII and LIII edges of Fe provide distinct and unblended signatures of dust grains in X-rays (e.g. \\citealt{Lee09,Cost12,Zeeg17,Roga18}). Therefore, in addition to the traditional low-energy domain observations, like in the IR, the X-ray band enables us to directly access the chemical composition of dust in the diffuse ISM of galaxies. Thus, high-resolution X-ray spectroscopy provides a powerful and sensitive diagnostic tool to probe the properties of both gas and dust. This is invaluable for understanding the formation and evolution history of galaxies, including the host galaxies of AGN. Indeed X-ray spectroscopy can help in studying the chemistry of dust in AGN (e.g. \\citealt{Lee13}), which is an important indicator of the evolutionary phase of AGN.\n\n\\object{IC~4329A} is bright nearby AGN at redshift ${z = 0.016054}$ \\citep{Will91}. It has been described as `an extreme Seyfert galaxy' \\citep{Disn73} and `the nearest quasar' \\citep{Wils79}, based on the spectroscopy of its broad optical emission lines. However, it is technically classified as a Seyfert 1.2 by \\citet{Vero06}. The host galaxy of the AGN is highly inclined (i.e. edge-on), with the observed ratio of minor to major axis ${b\/a = 0.28}$ \\citep{deVa91}. A prominent dust lane bisects the nucleus of {IC~4329A}\\xspace, which can be seen in the HST image of Fig. \\ref{dustlane_fig}. The dust lane is indicative of the past merger history of this galaxy. {IC~4329A}\\xspace is a member of a group of seven galaxies \\citep{Koll89}. It is displaced by a projected distance of 59 kpc from the giant lenticular galaxy \\object{IC 4329} \\citep{Wols95}. The relative orientations of the axes of the AGN and the disk of the host galaxy may have been influenced by interaction between {IC~4329A}\\xspace and its massive neighbour IC~4329 \\citep{Wols95}. \n\nThe mass of the supermassive black hole (\\ensuremath{{M_{\\rm BH}}}\\xspace) in {IC~4329A}\\xspace is not accurately determined. The \\ensuremath{{M_{\\rm BH}}}\\xspace from reverberation study is poorly constrained due to low quality lightcurves, and hence unreliable lag measurements, from which \\citet{Pet04} estimate an upper limit of $\\sim 3 \\times 10^{7}$~$M_{\\odot}$. However, measurements using other methods find higher \\ensuremath{{M_{\\rm BH}}}\\xspace. Using the empirical relation derived by \\citet{McHa06} between the bolometric luminosity $L_{\\rm bol}$, \\ensuremath{{M_{\\rm BH}}}\\xspace, and the break frequency in the X-ray power spectral density function (PSD), \\citet{Mark09} calculate $M_{\\rm BH} = 1.3_{-0.3}^{+1.0} \\times 10^{8}$~$M_{\\odot}$. Also, using relations between \\ensuremath{{M_{\\rm BH}}}\\xspace and the stellar velocity dispersion $\\sigma_*$, \\citet{Mark09} find $\\ensuremath{{M_{\\rm BH}}}\\xspace \\approx 2_{-1}^{+2} \\times 10^{8}$~$M_{\\odot}$. Furthermore, \\citet{deLa10} report $\\ensuremath{{M_{\\rm BH}}}\\xspace \\sim 1.2 \\times 10^{8}$~$M_{\\odot}$. From the AGN sample study of \\citet{Vasu10}, the black hole mass is estimated from the $K$-band luminosity of the host galaxy bulge, which in the case of {IC~4329A}\\xspace is found to be about $2 \\times 10^{8}$~$M_{\\odot}$. Therefore, in our calculations we assume \\ensuremath{{M_{\\rm BH}}}\\xspace is $\\sim$1--2~$\\times 10^{8}$~$M_{\\odot}$.\n\n\\citet{Stee05} studied the X-ray absorption in {IC~4329A}\\xspace with {\\it XMM-Newton}\\xspace RGS spectroscopy. They found absorption by neutral gas in the host galaxy of the AGN, as well as warm absorption by an AGN wind. Despite significant absorption, {IC~4329A}\\xspace is bright enough for high-resolution X-ray spectroscopy, which is often not the case for many reddened and absorbed AGN, making {IC~4329A}\\xspace a valuable target. The column density \\ensuremath{N_{\\mathrm{H}}}\\xspace of the neutral gas was measured to be about ${1.7 \\times 10^{21}}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace \\citep{Stee05}. The warm absorber was found to have four ionisation components, with a total \\ensuremath{N_{\\mathrm{H}}}\\xspace of about ${1.0 \\times 10^{22}}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace. The velocity of the warm absorber components ranges from ${-200 \\pm 100}$ to ${+20 \\pm 160}$~\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace. Possible X-ray spectral features of dust were not investigated in \\citet{Stee05}.\n\nDust reddened and absorbed AGN are often too faint for dust X-ray spectroscopy because of strong absorption. Therefore, finding a suitable target is key. For our X-ray spectroscopic investigation of dust in winds, the AGN target must meet these required selection criteria: (1) bright enough in X-rays; (2) displays significant intrinsic reddening; (3) exhibits evidence of dust features in both X-rays and IR; (4) shows the presence of an AGN wind; (5) minimum amount of reddening and absorption contamination by the Milky Way in our of sight. Among many candidates that we systematically searched, {IC~4329A}\\xspace is one of the most suitable AGN that meets all these criteria. Our newly acquired {\\it Chandra}\\xspace-HETGS observations of {IC~4329A}\\xspace are presented for the first time in this paper.\n\nThe structure of the paper is as follows. The observations and the reduction of data are described in Sect. \\ref{data_sect}. The modelling of the spectral energy distribution (SED) is presented in Sect. \\ref{sed_sect}. The modelling of the X-ray absorption by the ISM gas and the AGN wind are explained in Sect. \\ref{wind_sect}. The multi-wavelength analysis of dust in {IC~4329A}\\xspace based on reddening, IR emission features, and X-ray absorption features, is presented in Sect. \\ref{dust_sect}. We discuss all our findings in Sect. \\ref{discussion}, and give concluding remarks in Sect. \\ref{conclusions}.\n\nThe spectral analysis and modelling presented in this paper were done using the {\\tt SPEX} package \\citep{Kaa96} v3.04.00. The spectra shown in this paper are background-subtracted and are displayed in the observed frame. We use C-statistics for spectral fitting and give errors at the $1\\sigma$ confidence level. We adopt a luminosity distance of 69.61 Mpc in our calculations with the cosmological parameters ${H_{0}=70\\ \\mathrm{km\\ s^{-1}\\ Mpc^{-1}}}$, $\\Omega_{\\Lambda}=0.70$, and $\\Omega_{m}=0.30$. We assume proto-solar abundances of \\citet{Lod09} in all our computations in this paper.\n\n\\begin{figure}[!tbp]\n\\centering\n\\hspace{-0.8cm}\\resizebox{0.95\\hsize}{!}{\\includegraphics[angle=0]{IC4329A_hst_image_final.ps}\\hspace{-2.8cm}}\n\\caption{Nuclear region of {IC~4329A}\\xspace as observed by the HST, showing the presence of a dust lane covering the nucleus. The image is obtained from an observation taken with the ACS\/HRC F550M filter on 22 February 2006, and is displayed with a logarithmic intensity scale.}\n\\label{dustlane_fig}\n\\end{figure}\n\n\\section{Observations and data processing}\n\\label{data_sect}\n\nThe observation log of data used in our spectral analysis are provided in Table \\ref{log_table}. We describe the processing of data from different observatories in the following.\n\n\n\\subsection{X-ray data}\n\nAll {\\it Chandra}\\xspace observations of {IC~4329A}\\xspace have been taken with HETGS \\citep{Cani05}. In all the HETGS observations, the ACIS camera was operated in the timed exposure (TE) read mode and the faint data mode. The data were reduced using the Chandra Interactive Analysis of Observations ({\\tt CIAO}, \\citealt{Frus06}) v4.9 software and the calibration database (CALDB) v4.7.3. The {\\tt chandra\\_repro} script of {\\tt CIAO} and its associated tools were used for the reduction of the data and production of the final grating products (PHA2 spectra, RMF and ARF response matrices). The MEG and HEG +\/- first-order spectra and their response matrices were combined using the CIAO {\\tt combine\\_grating\\_spectra} script. The short-term X-ray variability, seen by the HETGS observations on days timescale, correspond to about 10\\% flux variation around the mean. This is a small variability, allowing us to stack the individual spectra in order to enhance the signal-to-noise ratio. The fitted spectral range is 2.5--26~\\AA\\ for MEG, and 1.55--14.5~\\AA\\ for HEG. Over these energy bands, the HEG\\,\/\\,MEG flux ratio is nearly constant at 0.952. We take into account this instrumental flux difference between HEG and MEG by re-scaling the normalisation of HEG relative to MEG in our spectral modelling. The low and high energy data outside of these ranges are ignored because of deviations in the HEG\\,\/\\,MEG flux ratio caused by the increasing calibration uncertainties of the instruments.\n\nThe {\\it XMM-Newton}\\xspace data were processed using the Science Analysis System (SAS v16.0.0). The RGS \\citep{denH01} instruments were operated in the standard Spectro+Q mode for the {\\it XMM-Newton}\\xspace observations of {IC~4329A}\\xspace (Table \\ref{log_table}). The data were processed through the {\\tt rgsproc} pipeline task; the source and background spectra were extracted and the response matrices were generated. We filtered out time intervals with background count rates $> 0.1\\ \\mathrm{count\\ s}^{-1}$ in CCD number 9. The {\\tt rgscombine} task was used to stack the RGS first-order spectra. Our fitted spectral range is 7--35~\\AA\\ for RGS. The EPIC-pn instrument \\citep{Stru01} was operated in the Full-Frame mode with the Medium Filter during the 2001 observation, and in the Small-Window mode with the Thin Filter during the 2003 observation. Periods of high flaring background for EPIC-pn (exceeding 0.4 $\\mathrm{count\\ s}^{-1}$) were filtered out by applying the {\\tt \\#XMMEA\\_EP} filtering. The EPIC-pn spectra were extracted from a circular region centred on the source with a radius of $40''$. The background was extracted from a nearby source-free region of radius $40''$ on the same CCD as the source. The pileup was evaluated to be small at about 2\\%. The single and double events were selected for the EPIC-pn ({\\tt PATTERN <= 4}). Response matrices were generated for the spectrum of each observation using the {\\tt rmfgen} and {\\tt arfgen} tasks. The {\\tt epicspeccombine} task was used for stacking the EPIC-pn spectra. Our fitted spectral range for EPIC-pn is from 1.38 keV (9 \\AA) to 10 keV, since the soft X-ray band (0.35--1.8 keV) is simultaneously modelled with RGS. The EPIC-pn\\,\/\\,RGS flux ratio at the overlapping energy band is found to be 0.91, which we take into account by re-scaling in our simultaneous fitting of the RGS and EPIC-pn spectra. In our spectral modelling we first derived a time-averaged model fitted to the stacked spectra from all available observations, and then applied this model to determine the variability in the X-ray continuum and the X-ray absorption between the 2003 and 2017 epochs, when the deepest X-ray observations were taken.\n\n\\begin{table}[!tbp]\n\\begin{minipage}[t]{\\hsize}\n\\setlength{\\extrarowheight}{3pt}\n\\caption{Observation log of the data used in our spectral modelling.}\n\\centering\n\\footnotesize\n\\renewcommand{\\footnoterule}{}\n\\begin{tabular}{l | c c c}\n\\hline \\hline\n & & Obs. date & Length  \\\\\nObservatory & Obs. ID & yyyy-mm-dd & (ks)  \\\\ \n\\hline\n{\\it Chandra}\\xspace\/HETGS & 2177 &  2001-08-26 & 59.1\t\\\\\n{\\it Chandra}\\xspace\/HETGS & 20070 & 2017-06-06 & 91.8\t\\\\\n{\\it Chandra}\\xspace\/HETGS & 19744 & 2017-06-12 & 12.4\t\\\\\n{\\it Chandra}\\xspace\/HETGS & 20095 & 2017-06-13 & 33.3\t\\\\\n{\\it Chandra}\\xspace\/HETGS & 20096 & 2017-06-14 & 19.8\t\\\\\n{\\it Chandra}\\xspace\/HETGS & 20097 & 2017-06-17 & 16.8\t\\\\\n\\hline\n{\\it XMM-Newton}\\xspace & 0101040401\t& 2001-01-31 & 13.9 \\\\\n{\\it XMM-Newton}\\xspace & 0147440101\t& 2003-08-06 & 136.0 \\\\\n\\hline\n{\\it Swift}\\xspace\/UVOT & 00033058001\t& 2013-12-21 & 1.9 \\\\\n\\hline\nHST\/WFPC2\/F814W & U5GU0401R\t& 2000-02-25 & 0.012 \\\\\nHST\/NICMOS\/F160W & N4JQ06010 & 1998-05-21 & 0.22 \\\\\nHST\/NICMOS\/F196N & N4JQ06070 & 1998-05-21 & 0.51 \\\\\nHST\/NICMOS\/F200N & N4JQ060A0 & 1998-05-21 & 0.90 \\\\\nHST\/NICMOS\/F222M & N4JQ06040 & 1998-05-21 & 0.26 \\\\\n\\hline\n{\\it Spitzer}\\xspace\/IRAC & 12472576 & 2005-07-18 & 0.047  \\\\\n{\\it Spitzer}\\xspace\/IRAC & 18038784 & 2006-08-13 & 1.3 \\\\\n{\\it Spitzer}\\xspace\/MIPS & 10642176 & 2006-02-15 & 0.53 \\\\\n{\\it Spitzer}\\xspace\/IRS & 4848640 & 2004-07-13 & 1.3 \\\\\n{\\it Spitzer}\\xspace\/IRS & 18506496 & 2007-07-29 & 0.96 \\\\\n\\hline\n{\\it Herschel}\\xspace\/PACS\/70-\\ensuremath{\\mu{\\mathrm{m}}}\\xspace & 1342236918 & 2012-01-07 & 0.052\\\\\n{\\it Herschel}\\xspace\/PACS\/160-\\ensuremath{\\mu{\\mathrm{m}}}\\xspace & 1342236919 & 2012-01-07 & 0.052 \\\\\n{\\it Herschel}\\xspace\/SPIRE & 1342236198 & 2012-01-02 & 0.169 \\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\tablefoot{\nThe dates correspond to the start time of the observations in UTC. In addition to the above data, we also use fluxes at 60 and 100 $\\mu$m from the IRAS survey observations.\n}\n\\label{log_table}\n\\end{table}\n\n\\subsection{Optical and UV data}\n\nIn order to determine the optical and UV part of the SED, we made use of photometric data from {\\it XMM-Newton}\\xspace OM \\citep{Mas01} and {\\it Swift}\\xspace UVOT \\citep{Romi05}. We use data taken with the V, B, U, UVW1, UVM2 and UVW2 filters. For a description of the reduction of OM and UVOT data, we refer to Appendix A in \\citet{Meh15a} and references therein, which applies to the data used here. The size of the circular aperture used for our photometry was set to a diameter of 12$\\arcsec$ for OM, and 10$\\arcsec$ for UVOT, which is the optimum aperture size based on the calibration of these instruments.\n\nThe continuum at the UV and optical, and lower energies, is approximated to be constant over time in our SED modelling. The optical and UV fluxes from the OM and UVOT are consistent with each other at the overlapping filters despite being taken years apart. Furthermore, from examining the long-term variability in the UVOT observations, we find that the flux variation in the V band is about 4\\% around the mean. Therefore, the continuum at the optical and lower energies is not too variable for the purpose of approximating the SED by simultaneously fitting the individual observations given in Table \\ref{log_table}.\n\n\\subsection{Infrared data}\n\nTo determine the SED spanning near-IR to far-IR energies, we have utilised data from {HST}\\xspace, {\\it Spitzer}\\xspace, IRAS, and {\\it Herschel}\\xspace. The {HST}\\xspace images from Wide Field and Planetary Camera 2 (WFPC2) F814W filter, and the Near Infrared Camera and Multi-Object Spectrometer (NICMOS) F160W, F196N, F200N, F222M filters, were retrieved from Mikulski Archive for Space Telescopes (MAST). We carried out aperture photometry with a diameter of 10\\arcsec on the central source. \n\nThe {\\it Spitzer}\\xspace Multi-band Imaging Photometer (MIPS) spectrum was retrieved from the Spitzer Heritage Archive (SHA). The MIPS instrument, operating in the SED mode, provides a low-resolution spectrum from 53 to 100 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace. Furthermore, the {\\it Spitzer}\\xspace Infrared Array Camera (IRAC) photometric measurements at 3.6, 4.5, 5.8, 8.0 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace were taken from the {\\it Spitzer}\\xspace study of a sample of AGN by \\citet{Gall10}. The {\\it Spitzer}\\xspace Infrared Spectrograph (IRS) observations, operating in the IRS Stare mode, were used to extract the spectra. The IRS spectra from the low-resolution modules, short-low (SL) and long-low (LL), were retrieved from the Combined Atlas of Sources with Spitzer IRS Spectra (CASSIS). The spectra from the high-resolution modules, short-high (SH) and long-high (LH), were processed through the c2d pipeline and optimal PSF extraction.   \n\nThe Infrared Astronomical Satellite (IRAS) flux measurement at 60 and 100 $\\mu$m were obtained from the IRAS Faint Source catalogue v2.0 \\citep{Mosh90}. We have also utilised far-infrared (FIR) data from {\\it Herschel}\\xspace PACS (70 and 160 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace filters) and SPIRE (250, 350, and 500 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace filters). The photometric flux measurements were obtained from the {\\it Herschel}\\xspace-{\\it Swift}\\xspace\/BAT AGN sample studies of \\citet{Mele14} and \\citet{Shim16}, which report on the PACS and SPIRE observations, respectively.\n\n\\section{Determination of the spectral energy distribution}\n\\label{sed_sect}\n\nHere we present our modelling of the components of the IR-optical-UV-X-ray continuum. We note that our continuum modelling is tied to the modelling of the X-ray absorption, reddening, and dust IR emission features, which are presented separately in later sections (Sects. \\ref{wind_sect} and \\ref{dust_sect}).\n\n\\subsection{Primary IR-optical-UV-X-ray continuum of the AGN}\n\\label{continuum_sect}\n\nWe started by applying a power-law component ({\\tt pow}) to fit the X-ray spectrum of {IC~4329A}\\xspace (Fig. \\ref{overview_fig}). The X-ray spectrum is strongly absorbed, and the modelling of this absorption is described in Sect. \\ref{wind_sect}. The X-ray power-law continuum model represents Compton up-scattering of the disk photons in an optically-thin and hot corona in {IC~4329A}\\xspace. The high-energy exponential cut-off of the power-law was set to 186~keV \\citep{Bren14} based on {\\it NuSTAR}\\xspace and {\\it Suzaku}\\xspace observations. A low-energy exponential cut-off was also applied to the power-law continuum to prevent exceeding the energy of the seed disk photons. The photon index $\\Gamma$ of the power-law is ${\\Gamma = 1.78 \\pm 0.01}$.\n\nIn addition to the power-law, the soft X-ray continuum of {IC~4329A}\\xspace includes a `soft X-ray excess' component \\citep{Capp96,Pero99,Stee05}. To model the soft excess in {IC~4329A}\\xspace, we use the broad-band model derived in \\citet{Meh15a} for NGC~5548, in which the soft excess is modelled by warm Comptonisation (see e.g. \\citealt{Mag98, Meh11, Done12, Petr13, Kubo18}). In this explanation of the soft excess, the seed disk photons are up-scattered in an optically thick and warm corona to produce the soft X-ray excess. The {\\tt comt} model in \\xspace{\\tt SPEX}\\xspace produces a thermal optical-UV disk component modified by warm Comptonisation, so that its high-energy tail fits the soft X-ray excess. In recent years, multi-wavelength studies have found warm Comptonisation to be a viable explanation for the soft excess in Seyfert-1 AGN (e.g. most recently in \\object{Ark~120}, \\citealt{Porq18}). The fitted parameters of the {\\tt comt} model are its normalisation, seed photons temperature $T_{\\rm seed}$, electron temperature $T_{\\rm e}$, and optical depth $\\tau$ of the up-scattering plasma. The primary continuum model components are shown in the SED of Fig. \\ref{SED_fig}. The best-fit parameters of the power-law component, and the warm Comptonisation component (disk + soft X-ray excess), are provided in Table \\ref{continuum_table}. The observed flux and intrinsic luminosity of {IC~4329A}\\xspace over different energy bands are given in Table \\ref{luminosity_table}. \n\n\\subsection{Reprocessed X-ray emission}\n\\label{reflection_sect}\n\nThe primary X-ray continuum undergoes reprocessing, which is evident by the presence of the Fe K\\ensuremath{\\alpha}\\xspace line in the HETGS and EPIC-pn spectra (Fig. \\ref{overview_fig}, top panel). We measure a rest line energy of ${6.39 \\pm 0.01}$ keV for this line, which is consistent with neutral Fe emission. The flux of the Fe K\\ensuremath{\\alpha}\\xspace line is $8\\pm 1 \\times 10^{-13}$~{\\ensuremath{\\rm{erg\\ cm}^{-2}\\ \\rm{s}^{-1}}}\\xspace. We applied an X-ray reflection component ({\\tt refl}), which reprocesses an incident power-law continuum to produce the Fe K\\ensuremath{\\alpha}\\xspace line and the Compton hump at hard X-rays. The {\\tt refl} model in \\xspace{\\tt SPEX}\\xspace computes the Fe K\\ensuremath{\\alpha}\\xspace line according to \\citet{Zyck94}, and the Compton-reflected continuum according to \\citet{Magd95}, as described in \\citet{Zyck99}. The photon index $\\Gamma$ of the incident power-law was set to that of the observed primary continuum ($\\Gamma = 1.78$). The exponential high-energy cut-off of this incident power-law is also set to that of the observed primary power-law component at 186~keV \\citep{Bren14}. In our modelling the normalisation of the incident power-law continuum is set to the average of the HETGS and {\\it XMM-Newton}\\xspace observations. The ionisation parameter of {\\tt refl} is set to zero to produce a cold reflection component with all abundances kept at their solar values. The reflection scale $s$ parameter of the {\\tt refl} model was fitted. The {\\tt refl} component was convolved with a Gaussian velocity broadening model to fit the width $\\sigma_{v}$ of the Fe K\\ensuremath{\\alpha}\\xspace line, which is about $3400 \\pm 500$~\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace. We do not require a relativistic line profile to fit the Fe K\\ensuremath{\\alpha}\\xspace line. \n\nApart from the Fe K\\ensuremath{\\alpha}\\xspace line, there is another emission line at rest energy ${6.95 \\pm 0.03}$ keV, which is consistent with a \\ion{Fe}{xxvi} Ly$\\alpha$ line (6.97 keV). We fitted this line with a simple Gaussian model. The flux of the \\ion{Fe}{xxvi} line is $3\\pm 1 \\times 10^{-13}$~{\\ensuremath{\\rm{erg\\ cm}^{-2}\\ \\rm{s}^{-1}}}\\xspace. The line profile is unresolved with $\\sigma_{v} < 2000$~\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace.\n\n\\begin{figure}[!tbp]\n\\centering\n\\resizebox{1.045\\hsize}{!}{\\hspace{-0.45cm}\\includegraphics[angle=0]{IC4329A_HETGS_RGS_EPIC_spec.ps}}\\vspace{-0.2cm}\n\\caption{Overview of the X-ray spectrum of {IC~4329A}\\xspace taken with {\\it XMM-Newton}\\xspace RGS and EPIC-pn in 2003, and {\\it Chandra}\\xspace HETGS in 2017. Our best-fit models to the data are also displayed (in red), with the power-law continuum being brighter in 2017. The underlying 2017 continuum model without any X-ray absorption is shown in dashed black line. For comparison, the continuum model with only absorption by the Milky Way is shown in dotted black line, which demonstrates the strong intrinsic absorption by {IC~4329A}\\xspace.}\n\\label{overview_fig}\n\\end{figure}\n\n\\begin{figure}[!tbp]\n\\centering\n\\resizebox{1.043\\hsize}{!}{\\hspace{-0.85cm}\\includegraphics[angle=0]{IC4329A_SED_absorbed.ps}}\\vspace{-0.3cm}\n\\resizebox{1.043\\hsize}{!}{\\hspace{-0.85cm}\\includegraphics[angle=0]{IC4329A_SED_unabsorbed.ps}}\n\\caption{SED of {IC~4329A}\\xspace from far-IR to hard X-rays. The SED in the {\\it top panel} includes the effects of reddening, X-ray absorption, host galaxy starlight emission, and the BLR and NLR emission. The SED in the {\\it bottom panel} is corrected for these processes, revealing the underlying continuum. The best-fit model to the data is shown in both panels in solid black line. The contribution of individual continuum components are displayed in the {\\it bottom panel}: a Comptonised disk component (solid magenta line), power-law continuum (dashed black line), X-ray reflection (dotted black line), thermal IR continuum (dotted magenta line). The model for the 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace silicate features are also plotted (dashed purple lines). In the 2017 epoch (HETGS observation), the X-ray power-law (dashed black line) is brighter than in the 2003 epoch ({\\it XMM-Newton}\\xspace observation).}\n\\label{SED_fig}\n\\end{figure}\n\n\\subsection{Thermal infrared continuum}\n\\label{thermal_sect}\n\nThe primary AGN continuum drops from optical-UV towards lower energies. However, the observed continuum rises again from near-IR towards the mid-IR energies, and then declines again towards the far-IR energies (see Fig. \\ref{SED_fig}). This is generally thought to be thermal emission from dust in AGN (e.g. \\citealt{Hern16}), which mainly comes from the AGN torus. To properly fit the broad-band continuum extending from near-IR to far-IR, we require three black body ({\\tt bb}) components, with different temperatures. The parameters of these components are given in Table \\ref{continuum_table}. We note that we report on the 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace dust features separately in Sect. \\ref{dust_ir_sect}.\n\n\\subsection{Emission from the BLR, the NLR, and the galactic bulge}\n\\label{NLR_sect}\n\nApart from the optical and UV continuum, the photometric filters of OM and UVOT contain emission from the broad-line region (BLR) and the narrow-line region (NLR) of the AGN. Therefore, in order to correct for this contamination, we applied the emission model derived in \\citet{Meh15a} for NGC~5548 as a template model for the optical and UV data of {IC~4329A}\\xspace. The model takes into account the Balmer continuum, the \\ion{Fe}{ii} feature, and the emission lines from the BLR and NLR. We normalised this model to the intrinsic luminosity of the H$\\beta$ line in {IC~4329A}\\xspace as derived by \\citet{Marz92}, which is $2.4 \\times 10^{42}$~{\\ensuremath{\\rm{erg\\ s}^{-1}}}\\xspace.\n\nTo take into account the host galaxy optical and UV stellar emission in the OM and UVOT filters, we used the galactic bulge model of \\citet{Kin96}, and normalised it to the {IC~4329A}\\xspace host galaxy flux derived from {HST}\\xspace images by \\citet{Ben13}. This is about ${3.57 \\times 10^{-15}}$ {\\ensuremath{\\rm{erg\\ cm}^{-2}\\ \\rm{s}^{-1}\\ {\\AA}^{-1}}}\\xspace at 5100 $\\AA$.\n\nApart from the BLR and NLR emission in the optical and UV band, there is also infrared emission from the NLR, which is present in the {\\it Spitzer}\\xspace\/IRS high-resolution spectrum (see Sect. \\ref{dust_ir_sect}). These narrow forbidden lines include [\\ion{S}{iv}] (10.511 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace), [\\ion{Ne}{ii}] (12.814 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace), [\\ion{Ne}{V}] (14.322 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace), [\\ion{Ne}{iii}] (15.555 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace), [\\ion{S}{iii}] (18.713 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace), [\\ion{Ne}{v}] (24.318 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace), [\\ion{O}{iv}] (25.913 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace). Therefore, in our SED modelling, we take into account these components by fitting narrow Gaussian emission lines to these lines.\n\n\\begin{table}[!tbp]\n\\begin{minipage}[t]{\\hsize}\n\\setlength{\\extrarowheight}{3pt}\n\\caption{Best-fit parameters of the broad-band continuum model components for {IC~4329A}\\xspace.}\n\\centering\n\\small\n\\renewcommand{\\footnoterule}{}\n\\begin{tabular}{l | c}\n\\hline \\hline\nParameter\t\t\t\t\t\t& Value\t\t\t\t\t\\\\\n\\hline\n\\multicolumn{2}{c}{Primary X-ray power-law component ({\\tt pow}):} \t\t\t\t\t\t\\\\\nNormalisation\t\t\t& ${1.77 \\pm 0.01}$~(2003) \\\\\n            \t\t\t& ${2.33 \\pm 0.01}$~(2017) \\\\\nPhoton index $\\Gamma$\t\t& ${1.78 \\pm 0.01}$\t\t\t\\\\\n\\hline\n\\multicolumn{2}{c}{Disk component: optical-UV and the soft X-ray excess ({\\tt comt}):} \t\t\t\t\t\t\\\\\nNormalisation\t&\t\t${1.4 \\pm 0.1}$ \\\\\n$T_{\\rm seed}$ (eV) &\t${0.8 \\pm 0.1}$ \\\\\n$T_{\\rm e}$ (eV) &\t\t${75 \\pm 3}$ \\\\\nOptical depth $\\tau$ &\t${33 \\pm 2}$ \\\\\n\\hline\n\\multicolumn{2}{c}{X-ray reflection ({\\tt refl}):} \t\t\t\t\t\t\\\\\nIncident power-law Norm.\t\t\t& ${2.05}$ (f) \\\\\nIncident power-law $\\Gamma$\t\t& $1.78$ (f)\t\t \t\t\t\\\\\nReflection scale\t$s$\t\t\t& $0.43 \\pm 0.02$\t\t\t\t\t\\\\\n$\\sigma_v$ (\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace)\t\t\t\t& $3400 \\pm 500$\t\t\t\t\t\\\\\n\\hline\n\\multicolumn{2}{c}{Thermal IR emission components ({\\tt bb}):}\t\t \t\t\t\t\t\t\\\\\nBB 1: $T$ (eV)\t\t\t\t\t& $0.0077 \\pm 0.0001$\t\t\t\\\\\nBB 1: Flux &  $1.59 \\pm 0.02$\t\t\t\\\\\nBB 2: $T$ (eV)\t\t\t\t\t& $0.042 \\pm 0.001$\t\t\t\\\\\nBB 2: Flux &  $2.3 \\pm 0.2$\t\t\t\\\\\nBB 3: $T$ (eV)\t\t\t\t\t& $0.121 \\pm 0.003$\t\t\t\\\\\nBB 3: Flux &  $3.4 \\pm 0.4$\t\t\t\\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\tablefoot{\nThe power-law normalisation of the {\\tt pow} and {\\tt refl} components is in units of $10^{52}$ photons~s$^{-1}$~keV$^{-1}$ at 1 keV. The normalisation of the Comptonised disk component ({\\tt comt}) is in units of $10^{57}$ photons~s$^{-1}$~keV$^{-1}$. The flux of the {\\tt bb} components are in $10^{-10}$ {\\ensuremath{\\rm{erg\\ cm}^{-2}\\ \\rm{s}^{-1}}}\\xspace. The high-energy exponential cut-off of the power-law for both {\\tt pow} and {\\tt refl} is fixed to 186~keV. The photon index $\\Gamma$ of the incident power-law for the reflection component ({\\tt refl}) is set to the $\\Gamma$ of the observed primary power-law continuum ({\\tt pow}).\n}\n\\label{continuum_table}\n\\end{table}\n\n\\begin{table}[!tbp]\n\\begin{minipage}[t]{\\hsize}\n\\setlength{\\extrarowheight}{3pt}\n\\caption{Observed flux ($F$) and intrinsic luminosity ($L$) of {IC~4329A}\\xspace over various energy bands. The values correspond to the SEDs shown in Fig. \\ref{SED_fig}. The continuum modelling is described in Sect. \\ref{sed_sect} (Table \\ref{continuum_table}).}\n\\centering\n\\small\n\\renewcommand{\\footnoterule}{}\n\\begin{tabular}{l | c c}\n\\hline \\hline\n              & $F$\t&  $L$\t\t\\\\\nEnergy range  & (${10^{-11}}$ {\\ensuremath{\\rm{erg\\ cm}^{-2}\\ \\rm{s}^{-1}}}\\xspace)\t&   (${10^{44}}$ {\\ensuremath{\\rm{erg\\ s}^{-1}}}\\xspace)\t\t\t\t\t              \\\\\n\\hline\nHard X-ray (2--10 keV) & 11.0 (2003) & 0.7 (2003) \\\\\n                       & 14.2 (2017) & 0.9 (2017) \\\\\nSoft X-ray (0.2--2 keV) & 3.0 (2003) & 1.4 (2003) \\\\\n                        & 3.7 (2017) & 1.6 (2017) \\\\\nEUV (100--1000 \\AA) & ${8 \\times 10^{-7}}$ & 5.8 \\\\\nUV (1000--4000 \\AA) & 0.2 & 4.5 \\\\\nOptical (4000--7000 \\AA) & 2.2 & 1.0 \\\\\nNear-IR (0.7--3 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace) & 23.5 & 1.5 \\\\\nMid-IR (3--40 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace) & 64.6 & 3.9 \\\\\nFar-IR (40--500 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace) & 10.8 & 0.7 \\\\\n\\hline\n1--1000 Ryd & 16.9 (2003) & 8.5 (2003) \\\\\n            & 21.6 (2017) & 9.1 (2017) \\\\\nBolometric (${10^{-6}}$--${10^{3}}$ keV) & 152.0 (2003) & 24.5 (2003) \\\\\n                                         & 164.9 (2017) & 26.4 (2017) \\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\tablefoot{The observed fluxes ($F$) include the effects of all the reddening and absorption components. The host galaxy optical and UV emission, and the AGN BLR and NLR emission, are excluded from the reported fluxes and luminosities. The IR fluxes and luminosities include all the dust emission components.}\n\\label{luminosity_table}\n\\end{table}\n\n\\section{Modelling of the ISM and the AGN wind}\n\\label{wind_sect}\n\n\\subsection{X-ray absorption by the diffuse interstellar gas}\n\\label{ism_sect}\n\nPrior to modelling the ISM gas absorption in {IC~4329A}\\xspace, we first determined our model for the Milky Way absorption. The X-ray continuum and line absorption by the Galaxy are taken into account by applying the {\\tt hot} model in \\xspace{\\tt SPEX}\\xspace. This model calculates the transmission of a plasma in collisional ionisation equilibrium at a given temperature, which for neutral ISM is set to 0.5~eV. The Milky Way column density \\ensuremath{N_{\\mathrm{H}}}\\xspace is fixed to ${4.61\\times 10^{20}\\ \\mathrm{cm}^{-2}}$\\citep{Kal05} in our line of sight to {IC~4329A}\\xspace. The Galactic molecular \\ensuremath{N_{\\mathrm{H}}}\\xspace in our line of sight is a small fraction (17\\%) of the total \\ensuremath{N_{\\mathrm{H}}}\\xspace according to \\citet{Will13}.\n\nApart from the Milky Way absorption, an additional neutral ISM component intrinsic to the source is required for modelling the X-ray spectrum. This component is essential to fit the strong suppression of the soft X-ray continuum in {IC~4329A}\\xspace (see Fig. \\ref{overview_fig}). Also, such a component is needed to fit the absorption features of neutral gas in the spectrum (e.g. seen in \\ion{O}{i}). We thus incorporate another {\\tt hot} component with its temperature fixed to 0.5~eV to fit the column density of neutral gas in {IC~4329A}\\xspace, which is found to be ${\\ensuremath{N_{\\mathrm{H}}}\\xspace = 3.1 \\pm 0.2} \\times 10^{21}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace. As this is a high column density of ISM in {IC~4329A}\\xspace, a mildly-ionised component is also detected through primarily \\ion{O}{iii}. Thus, we include another {\\tt hot} component, with its temperature and \\ensuremath{N_{\\mathrm{H}}}\\xspace fitted. This mildly-ionised ISM component is found to have a temperature ${T_{\\rm e} = 7.3 \\pm 0.8}$~eV with ${\\ensuremath{N_{\\mathrm{H}}}\\xspace = 8 \\pm 1 \\times 10^{20}}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace. In Sects. \\ref{outflow_sect} and \\ref{dust_abs_sect}, we model additional components of the absorption in {IC~4329A}\\xspace, produced by the AGN wind, and dust, respectively.\n\n\\subsection{X-ray absorption by the ionised AGN wind}\n\\label{outflow_sect}\n\nThe RGS and HETGS spectra of {IC~4329A}\\xspace exhibit a series of absorption lines belonging to outflowing ionised gas. We simultaneously modelled the RGS and HETGS spectra to derive a model for this ionised wind from the AGN. For photoionisation modelling and spectral fitting, we use the \\xspace{\\tt pion}\\xspace model in \\xspace{\\tt SPEX}\\xspace (see \\citealt{Meh16b}), which is a self-consistent model that calculates the thermal and ionisation balance together with the spectrum of a plasma in photoionisation equilibrium (PIE). The \\xspace{\\tt pion}\\xspace model uses the SED (Sect. \\ref{sed_sect}) from the continuum model components set in \\xspace{\\tt SPEX}\\xspace. During spectral fitting, as the continuum varies, the thermal and ionisation balance and the spectrum of the plasma are re-calculated at each stage. This means while using realistic broad-band continuum components to fit the data, the photoionisation is calculated accordingly by the \\xspace{\\tt pion}\\xspace model.\n\nTo properly fit all the absorption lines from various ionic species we require three photoionisation (\\xspace{\\tt pion}\\xspace) components with different values for the ionisation parameter $\\xi$ \\citep{Tar69,Kro81}. This parameter is defined as ${\\xi = L \/ n_{\\rm H}\\, r^2}$, where $L$ is the luminosity of the ionising source over the 1--1000 Ryd band (13.6 eV to 13.6 keV) in {\\ensuremath{\\rm{erg\\ s}^{-1}}}\\xspace, $n_{\\rm H}$ the hydrogen density in cm$^{-3}$, and $r$ the distance between the photoionised gas and the ionising source in cm. The \\xspace{\\tt pion}\\xspace components are added one at a time until all the observed features are modelled and thus our fit is no longer improved.\n\nAccording to our model, the lowest ionisation component (Comp. A) produces absorption from the Li-like ion \\ion{O}{vi}, the He-like ions \\ion{C}{v}, \\ion{N}{vi}, and \\ion{O}{vii}, and the H-like ion \\ion{C}{vi}. Moreover, this component is responsible for producing a shallow unresolved transition array (UTA, \\citealt{Beh01}) at about 16--17 $\\AA$ from the M-shell Fe ions. The inclusion of this component improves our fit by $\\Delta$\\,C-stat = 820. The next ionisation component (Comp. B) primarily produces lines from the H-like \\ion{N}{vii} and \\ion{O}{viii}, as well as the He-like \\ion{Ne}{ix} and \\ion{Mg}{xi}. The lines fitted by Comp. B provide a better fit by $\\Delta$\\,C-stat = 440. Finally, the highest ionisation component (Comp. C) produces the H-like ions \\ion{Ne}{x}, \\ion{Mg}{xii}, \\ion{Si}{xiv}, as well as the He-like \\ion{Si}{xiii}. This component also produces the high-ionisation Fe species in the form of \\ion{Fe}{xix}, \\ion{Fe}{xx}, and \\ion{Fe}{xxi}. The addition of the high-ionisation component further improves the fit by $\\Delta$\\,C-stat = 350. The column density \\ensuremath{N_{\\mathrm{H}}}\\xspace, the ionisation parameter $\\xi$, the outflow velocity $v_{\\rm out}$, and the turbulent velocity $\\sigma_{v}$ of each \\xspace{\\tt pion}\\xspace component are fitted. The covering fraction of all our absorption components in our modelling is fixed to unity. As the RGS and HETGS spectra are taken at different epochs, we allowed \\ensuremath{N_{\\mathrm{H}}}\\xspace and $\\xi$ of the components to be different for the two epochs. The model transmission spectrum of the three \\xspace{\\tt pion}\\xspace components (Comps. A to C) are displayed in Fig. \\ref{wind_fig}. The best-fit parameters of the AGN wind components, and the ISM components, are given in Table \\ref{wind_table}.\n\nWe fit all the X-ray absorption by the AGN wind in {IC~4329A}\\xspace with three \\xspace{\\tt pion}\\xspace photoionisation components. However, in the X-ray analysis of {IC~4329A}\\xspace by \\citet{Stee05}, four photoionisation components were incorporated. The three highest ionisation components in their study roughly correspond to our three \\xspace{\\tt pion}\\xspace components, but they include an extra photoionisation component, with a very low ionisation (${\\log \\xi \\sim - 1.37}$) and zero outflow velocity, which we do not require in our modelling. This discrepancy between the two works can be explained by differences in the modelling of the neutral X-ray absorption in {IC~4329A}\\xspace. The derived column density of the neutral ISM gas by \\citet{Stee05} (${\\ensuremath{N_{\\mathrm{H}}}\\xspace = 1.7 \\times 10^{21}}$~cm$^{-2}$) is significantly smaller than the one derived in our study (${\\ensuremath{N_{\\mathrm{H}}}\\xspace = 3.1 \\times 10^{21}}$~cm$^{-2}$). However, this difference in \\ensuremath{N_{\\mathrm{H}}}\\xspace is instead modelled by the cold photoionisation component in \\citet{Stee05}, whereas we associate it to the ISM absorption in {IC~4329A}\\xspace. Moreover, some differences in the parameterisation of the absorption components can be attributed to our inclusion of the dust model and the determination of the broad-band continuum, which result in a better fit to the spectra. Furthermore, there are extended updates and enhancements to the atomic database of \\xspace{\\tt SPEX}\\xspace v3 and the new \\xspace{\\tt pion}\\xspace photoionisation model, which were not available back in \\xspace{\\tt SPEX}\\xspace v2, used by \\citet{Stee05}.\n\n\\begin{figure}[!tbp]\n\\centering\n\\resizebox{1.041\\hsize}{!}{\\hspace{-0.35cm}\\includegraphics[angle=0]{IC4329A_wind_trans.ps}}\n\\caption{Model transmission spectrum of the three components of the AGN wind in {IC~4329A}\\xspace. Comp. A is the lowest-ionisation component, and Comp. C the highest. The model shown in blue is the one derived from the 2017 HETGS observations. The model derived from the archival 2003 {\\it XMM-Newton}\\xspace observations is also shown for comparison, which is plotted in red behind the blue one.}\n\\label{wind_fig}\n\\end{figure}\n\n\\section{Multi-wavelength analysis of dust in {IC~4329A}\\xspace}\n\\label{dust_sect}\n\n\\subsection{Dust reddening in {IC~4329A}\\xspace}\n\\label{dust_red_sect}\n\n{IC~4329A}\\xspace displays significant internal reddening. This is evident from both the Balmer decrement (i.e. the observed H\\ensuremath{\\alpha}\\xspace\/H\\ensuremath{\\beta}\\xspace flux ratio) and the steepness of the optical-UV continuum (see Fig. \\ref{SED_fig}, top panel). The Balmer decrement of {IC~4329A}\\xspace has been reported in different papers to be from about 9 to 12 \\citep{Wils79,Marz92,Wing96,Malk17}. This is significantly higher than the expected Balmer decrement, implying significant internal reddening. The theoretical calculations for `Case B' recombination, where gas is optically thick in Lyman lines, give H\\ensuremath{\\alpha}\\xspace\/H\\ensuremath{\\beta}\\xspace$\\approx 2.85$ \\citep{Bake38,Ost06}. Moreover, the observations of unreddened AGN show that the Balmer decrement of the BLR in AGN is H\\ensuremath{\\alpha}\\xspace\/H\\ensuremath{\\beta}\\xspace$\\approx 2.72$ \\citep{Gask17}. Importantly, the intrinsic reddening also affects the optical and UV continuum and changes the shape of the so-called `big-blue-bump' thermal emission from the accretion disk. Unlike SEDs of typical unreddened AGN (e.g. NGC~5548, \\citealt{Meh15a}), the flux of {IC~4329A}\\xspace drops rapidly towards higher UV energies (Fig. \\ref{SED_fig}, top panel), which is indicative of strong reddening.\n\n\\begin{table}[!tbp]\n\\begin{minipage}[t]{\\hsize}\n\\setlength{\\extrarowheight}{3pt}\n\\caption{Best-fit parameters of the final model for the AGN wind and the ISM in {IC~4329A}\\xspace. The parameters of the associated dust model in {IC~4329A}\\xspace are presented in Table \\ref{dust_table}.}\n\\centering\n\\small\n\\renewcommand{\\footnoterule}{}\n\\begin{tabular}{l | c}\n\\hline \\hline\nParameter\t\t\t\t\t\t              & Value\t\t\t\t\t\\\\\n\\hline\n\\multicolumn{2}{c}{AGN wind Comp. A ({\\tt pion}):} \\\\\n\\ensuremath{N_{\\mathrm{H}}}\\xspace ($10^{20}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace)\t              &\t${7 \\pm 1}$ (2003) \\\\\n                  \t              &\t${8 \\pm 1}$ (2017) \\\\\n$\\log~\\xi$ (erg~cm~s$^{-1}$)      &\t${0.8 \\pm 0.1}$ (2003) \\\\\n                                  &\t${1.0 \\pm 0.1}$ (2017) \\\\\n$v_{\\rm out}$ (\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace)              &\t${-410 \\pm 30}$ \\\\\n$\\sigma_v$ (\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace)                 &\t${90 \\pm 20}$ \\\\\n\\hline\n\\multicolumn{2}{c}{AGN wind Comp. B ({\\tt pion}):} \\\\\n\\ensuremath{N_{\\mathrm{H}}}\\xspace ($10^{20}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace)\t              &\t${6 \\pm 1}$ (2003) \\\\\n                  \t              &\t${6 \\pm 1}$ (2017) \\\\\n$\\log~\\xi$ (erg~cm~s$^{-1}$)      &\t${2.1 \\pm 0.1}$ (2003) \\\\\n                                  &\t${1.9 \\pm 0.1}$ (2017) \\\\\n$v_{\\rm out}$ (\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace)              &\t${0 \\pm 20}$ \\\\\n$\\sigma_v$ (\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace)                 &\t${60 \\pm 10}$ \\\\\n\\hline\n\\multicolumn{2}{c}{AGN wind Comp. C ({\\tt pion}):} \\\\\n\\ensuremath{N_{\\mathrm{H}}}\\xspace ($10^{20}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace)\t              &\t${11 \\pm 2}$ (2003) \\\\\n                  \t              &\t${22 \\pm 2}$ (2017) \\\\\n$\\log~\\xi$ (erg~cm~s$^{-1}$)      &\t${2.79 \\pm 0.07}$ (2003) \\\\\n                                  &\t${3.03 \\pm 0.03}$ (2017) \\\\\n$v_{\\rm out}$ (\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace)              &\t${-360 \\pm 20}$ \\\\\n$\\sigma_v$ (\\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace)                 &\t${210 \\pm 40}$ \\\\\n\\hline\n\\multicolumn{2}{c}{Neutral ISM component of {IC~4329A}\\xspace ({\\tt hot}):} \\\\\n\\ensuremath{N_{\\mathrm{H}}}\\xspace\t($10^{20}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace)               & ${31 \\pm 1}$ \\\\\n${T_{\\rm e}}$ (eV)                & ${0.5}$ (f)\t\\\\\n\\hline\n\\multicolumn{2}{c}{Mildly-ionised ISM component of {IC~4329A}\\xspace ({\\tt hot}):} \\\\\n\\ensuremath{N_{\\mathrm{H}}}\\xspace\t($10^{20}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace)               & ${8 \\pm 1}$ \\\\\n${T_{\\rm e}}$ (eV)                & ${7.3 \\pm 0.8}$\t\\\\\n\\hline\n\\multicolumn{2}{c}{C-stat \/ d.o.f. = 10069 \/ 9280} \\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\label{wind_table}\n\\end{table}\n\nBefore deriving the reddening in {IC~4329A}\\xspace, we first fixed our model for reddening in the ISM of the Milky Way, which in our line of sight has a colour excess ${E(B-V) = 0.052}$~mag \\citep{Schl11}. We applied the {\\tt ebv} model in \\xspace{\\tt SPEX}\\xspace to model this reddening, which incorporates the extinction curve of \\citet{Car89}, including the update for near-UV given by \\citet{ODo94}. The scalar specifying the ratio of total to selective extinction ${R_V = A_V\/E(B-V)}$ was fixed to 3.1. \n\nA general model for internal reddening in AGN is lacking. Studies of reddening in AGN have found different kinds of extinction laws. For example, \\citet{Hopk04} examined the SEDs of a large sample of quasars using broad-band photometry data from SDSS. They concluded that the reddening is best described by SMC-like extinction. On the other hand, the \\citet{Czer04} study of composite quasar spectra from SDSS finds a `grey' (flat) extinction curve, where the extinction curve is flatter than that of the diffuse Milky Way. They also find no trace of the `2175~$\\AA$ bump' that is seen in the Milky Way extinction curve. On the other hand, from a case study of a reddened AGN (\\object{NGC~3227}) using HST, and comparing its spectrum with that of an unreddened AGN (NGC 4151), \\citet{Cren01a} concluded that the extinction curve in the UV is even steeper than that of SMC. On the other hand, flat extinction curves have been found by \\citet{Maio01,Gask04,Gask07}. \n\nWe determined the colour excess \\ensuremath{{E(B-V)}}\\xspace in {IC~4329A}\\xspace by jointly modelling the reddening of the optical-UV continuum and the Balmer decrement. We tested different template extinction laws from \\citet{Car89} (Milky Way), \\citet{Czer04} (flat), and \\citet{Zafa15} (steep). The user-defined multiplicative model in \\xspace{\\tt SPEX}\\xspace ({\\tt musr}) was used to import these reddening models into \\xspace{\\tt SPEX}\\xspace. We find that a flat extinction curve \\citep{Czer04} with ${\\ensuremath{{E(B-V)}}\\xspace = 1.0 \\pm 0.1}$ is most appropriate for {IC~4329A}\\xspace. It gives the best-fit to the broad-band continuum with the most reasonable continuum parameters. The steeper extinction curves lead to unphysical UV-EUV continuum luminosity. This conclusion was also reached for the reddened AGN {ESO~113-G010}\\xspace \\citep{Meh12}. Therefore, in our modelling of {IC~4329A}\\xspace, we adopt the extinction curve of \\citet{Czer04}, with $R_V$ fixed to 3.1. In this case the relation between reddening and the Balmer decrement is ${\\ensuremath{{E(B-V)}}\\xspace = 1.475\\, \\log\\, (R_{\\rm obs} \/ R_{\\rm int})}$, where $R_{\\rm obs}$ and $R_{\\rm int}$ are the observed and intrinsic H\\ensuremath{\\alpha}\\xspace\/H\\ensuremath{\\beta}\\xspace, respectively. Thus, using the reported range of the Balmer decrement in {IC~4329A}\\xspace (${R_{\\rm obs} \\approx}$ 9 to 12), and ${R_{\\rm int} = 2.72}$, the reddening $\\ensuremath{{E(B-V)}}\\xspace$ ranges between about 0.8 and 1.0. This matches ${\\ensuremath{{E(B-V)}}\\xspace = 1.0 \\pm 0.1}$ derived from de-reddening of the continuum (see Fig. \\ref{SED_fig}). We later discuss the origin of the flat extinction curve in Sect. \\ref{discussion}.\n\n\\subsection{Dust IR emission features in {IC~4329A}\\xspace}\n\\label{dust_ir_sect}\n\nThe widely known 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace features in the mid-IR spectra of AGN belong to silicate dust (e.g. \\citealt{Stur05,Henn10}). The 9.7 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace feature is generally attributed to the stretching of the Si-O bonds in silicates, while the 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace feature is attributed to O-Si-O bending in the same material (e.g. \\citealt{Henn10}). These features have canonical wavelengths of 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace in the Milky Way diffuse ISM, however, in AGN they often show deviation to longer wavelengths (e.g. \\citealt{Stur05,Hatz15}). They also display much broader spectral width compared to that of the Milky Way \\citep{Li08}. These features are thought to originate from silicates in the AGN dusty torus (e.g. \\citealt{Pier92,Pier93,Sieb05,Maso15}). \n\nThe 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace silicate features are evident in emission in the {\\it Spitzer}\\xspace IRS spectrum of {IC~4329A}\\xspace (Fig. \\ref{spitzer_fig}). The presence of these emission features in {IC~4329A}\\xspace were also previously reported by the AGN sample studies with {\\it Spitzer}\\xspace (e.g. \\citealt{Gall10}). In our broad-band SED modelling, we fit these features by the means of Gaussian emission components ({\\tt gaus}). This parametrisation allows us to determine the peak wavelength, flux, and the spectral width of these features. Our best-fit model to the {\\it Spitzer}\\xspace IRS spectrum is shown in Fig. \\ref{spitzer_fig}. The results of our modelling are given in Table \\ref{dust_table}, and are discussed in Sect. \\ref{torus_prop_sect}.\n\n\\begin{figure}[!tbp]\n\\centering\n\\resizebox{1.037\\hsize}{!}{\\hspace{-1.4cm}\\includegraphics[angle=0]{IC4329A_Spitzer_spec.ps}}\n\\caption{{\\it Spitzer}\\xspace IRS spectrum of {IC~4329A}\\xspace. The 9.7- and 18-micron emission features are shown in dashed purple and green lines, respectively. The best-fit model to the data, which includes the narrow emission lines from the AGN NLR, is shown in dashed red line. The underlying thermal IR continuum model is shown in dotted black line for comparison.}\n\\label{spitzer_fig}\n\\end{figure}\n\n\\subsection{Dust X-ray absorption features in {IC~4329A}\\xspace}\n\\label{dust_abs_sect}\n\nA telltale sign of dust X-ray absorption is that despite atomic and ionic absorption, the photoelectric edges in the X-ray spectrum are not well fitted. This is because dust modifies the profile of the absorption edges. In the case of {IC~4329A}\\xspace, despite neutral and warm absorption by the ISM of the host galaxy, and the ionised gas absorption by the AGN wind, additional absorption is required at the K edge of O, the LII and LIII edges of Fe, and to lesser extent at the K edge of Si (see Fig. \\ref{edge_fig}). Such stronger than expected edges are indicative of absorption by dust. The wavelength, absorption profile, and strength of these features are not consistent with atomic or ionic gas. Furthermore, the Milky Way column density and reddening in our line of sight to {IC~4329A}\\xspace is too low to be responsible for these features. Therefore, they can only be attributed to dust grains in our line of sight in {IC~4329A}\\xspace, which contain O, Si, and Fe.\n\n\\begin{figure}[!tbp]\n\\centering\n\\resizebox{1.032\\hsize}{!}{\\hspace{-0.15cm}\\includegraphics[angle=0]{IC4329A_HETGS_Si_edge_v2.ps}}\\vspace{-0.33cm}\n\\resizebox{1.032\\hsize}{!}{\\hspace{-0.15cm}\\includegraphics[angle=0]{IC4329A_RGS_O_Fe_edges_v2.ps}}\n\\caption{Stacked HETGS ({\\it top panel}) and RGS ({\\it bottom panel}) spectra of {IC~4329A}\\xspace in the K-band of Si (HETGS) and O (RGS), and the L-band of Fe (RGS). The best-fit model shown in red includes dust, whereas the one in black does not include dust. The panels above each spectrum show the transmission model by the Si, O, and Fe elements in atomic (dotted line) and dust (solid line) form in {IC~4329A}\\xspace. The strongest absorption features in the spectrum are labelled, including the K-edge of O and Si, and the LII- and LIII- edges of Fe from dust in the AGN, as well as lines from the ionised AGN wind. The model with dust absorption (shown in red) fits the O K and Fe LIII edges significantly better than the one without dust (shown in black).}\n\\label{edge_fig}\n\\end{figure}\n\nWe use the {\\tt amol} model in \\xspace{\\tt SPEX}\\xspace to calculate the X-ray absorption by dust (see \\citealt{Pint10,Cost12}). In our modelling, we deplete the O, Si, and Fe elements from gas to dust form. For absorption in {IC~4329A}\\xspace the depletion factors are fitted, whereas for that of the Milky Way they are fixed. For the Milky Way contribution (albeit relatively tiny), the depletion factors are fixed to standard values taken from \\citet{Jenk09}: 22\\% for O, 68\\% for Si, and 94\\% for Fe. We apply a simple template model (hematite Fe$_2$O$_3$) to incorporate dust absorption by O and Fe by fitting the K edge of O and the LII and LIII edges of Fe. The inclusion of this dust model improves the fit to the edges significantly with {$\\Delta$C-stat ${= 292}$}. While the Fe edges were fitted well at this stage, there were some remaining residuals at the O edge, which we found to be fitted well with the inclusion of molecular oxygen. This improved the fit further by {$\\Delta$C-stat ${= 133}$}. Finally, we find that the fit to the Si K-edge is slightly improved by $\\Delta$C-stat ${= 30}$ with the inclusion of crystal Si. We note that using AGN spectra from the existing high-resolution X-ray spectrometers, we can only constrain the column density of these elements in dust form, rather than determining the exact chemical composition of dust. Therefore, the above model represents an ad-hoc template model. Visual comparison of the best-fit models to the spectra with and without the dust model are shown in Fig. \\ref{edge_fig}. The best-fit model with dust absorption (red line) fits the O K and Fe LIII edges significantly better than the one without dust (black line). The AGN wind parameters are not significantly affected by the inclusion of dust as their absorption lines are similarly fitted well regardless of the dust model. The obtained best-fit column densities of elements in dust form in {IC~4329A}\\xspace are given in Table \\ref{dust_table}. We discuss the results in Sect. \\ref{dust_comp_sect}.\n\n\\begin{table}[!tbp]\n\\begin{minipage}[t]{\\hsize}\n\\setlength{\\extrarowheight}{3pt}\n\\caption{Best-fit parameters of the dust model in {IC~4329A}\\xspace.}\n\\centering\n\\small\n\\renewcommand{\\footnoterule}{}\n\\begin{tabular}{l | c}\n\\hline \\hline\nParameter\t\t\t\t\t\t                          & Value\t\t\t\t\t\\\\\n\\hline\n\\multicolumn{2}{c}{Reddening component of {IC~4329A}\\xspace ({\\tt musr}):} \\\\\n$E(B-V)$                                      &\t${1.0 \\pm 0.1}$  \\\\\n\\hline\n\\multicolumn{2}{c}{Dust IR emission features in {IC~4329A}\\xspace ({\\tt gaus}):} \\\\\n\\multicolumn{2}{c}{9.7 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace feature:}  \\\\\n$\\lambda_{\\rm peak}$ ($\\mu$m)                 &\t${13 \\pm 1}$ \\\\\nFWHM ($\\mu$m)                                 &\t${7 \\pm 3}$ \\\\\nFlux ($10^{-11}$ {\\ensuremath{\\rm{erg\\ cm}^{-2}\\ \\rm{s}^{-1}}}\\xspace)                    &\t${5 \\pm 1}$ \\\\\n\\multicolumn{2}{c}{18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace feature:}  \\\\\n$\\lambda_{\\rm peak}$ ($\\mu$m)                 &\t${22 \\pm 1}$ \\\\\nFWHM ($\\mu$m)                                 &\t${18 \\pm 1}$ \\\\\nFlux ($10^{-11}$ {\\ensuremath{\\rm{erg\\ cm}^{-2}\\ \\rm{s}^{-1}}}\\xspace)                    &\t${14 \\pm 1}$ \\\\\n\\hline\n\\multicolumn{2}{c}{Dust X-ray absorption in {IC~4329A}\\xspace ({\\tt amol}):} \\\\\n$N_{\\rm O-dust}$ ($10^{17}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace)              & ${6.2 \\pm 0.6}$ \\\\\n$N_{\\rm Si-dust}$ ($10^{17}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace)             & ${1.9 \\pm 0.4}$ \\\\\n$N_{\\rm Fe-dust}$ ($10^{17}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace)             & ${1.5 \\pm 0.1}$ \\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\tablefoot{\nThe AGN extinction curve of \\citet{Czer04} is used to derive the reddening $E(B-V)$ with $R_{V}$ fixed at 3.1. The $\\lambda_{\\rm peak}$ wavelengths correspond to the rest-frame of {IC~4329A}\\xspace.\n}\n\\label{dust_table}\n\\end{table}\n\n\\section{Discussion}\n\\label{discussion}\n\n\\subsection{Broad-band continuum of {IC~4329A}\\xspace}\n\\label{broadband_sect}\n\nThe broad-band continuum of {IC~4329A}\\xspace is strongly modified by internal reddening and X-ray absorption. By modelling these effects in this paper, the underlying intrinsic emission of the AGN is uncovered. While the observed SED of {IC~4329A}\\xspace appears significantly different from that of the archetypal Seyfert-1 galaxy NGC~5548, the underlying continuum from near-IR to hard X-rays is consistent with the global model derived for NGC~5548 in \\citet{Meh15a}. The continuum from near-IR to soft X-rays can be explained by a single component, which Compton up-scatters the disk photons in an optically-thick, warm corona. The high-energy tail of this component produces the soft X-ray excess. The hard X-ray spectrum of {IC~4329A}\\xspace is consistent with a typical power-law (${\\Gamma \\approx 1.78}$), produced in an optically-thin, hot corona, which is accompanied by a neutral X-ray reflection component. The accretion-powered radiation is reprocessed into lower energies by the dusty AGN torus, which its luminosity peaks at the mid-IR band. The broad-band modelling done in this paper enables us to derive the bolometric luminosity of the AGN, which is about ${2.45 \\times 10^{45}}$~{\\ensuremath{\\rm{erg\\ s}^{-1}}}\\xspace in 2003, and ${2.64 \\times 10^{45}}$~{\\ensuremath{\\rm{erg\\ s}^{-1}}}\\xspace in 2017 (Table \\ref{luminosity_table}). Taking into account the black hole mass of {IC~4329A}\\xspace (1--2~$\\times 10^{8}$~$M_{\\odot}$), these corresponds to bolometric luminosities at about 10--20\\% of the Eddington luminosity. The reprocessed emission by the AGN torus accounts for about 20\\% of the bolometric luminosity.\n\n\\subsection{Components of dust in {IC~4329A}\\xspace}\n\\label{dust_comp_sect}\n\nIn this paper we have carried out broad-band continuum modelling, together with X-ray and IR spectroscopy, to study dust in {IC~4329A}\\xspace. By examining our results from reddening, X-ray absorption and IR emission by dust, and comparing with previous polarisation studies, we can construct a physical picture of dust in {IC~4329A}\\xspace. We argue that the derived information from each of the above analyses points to the presence of two distinct components of dust in {IC~4329A}\\xspace: one in the ISM of the host galaxy and its associated dust lane, and the other a nuclear component in the AGN torus with likely association to the wind. In Fig. \\ref{cartoon_fig} we illustrate how our line of sight towards the central engine is intercepted by these dust regions in {IC~4329A}\\xspace.\n\nThe relationship between the hydrogen column density \\ensuremath{N_{\\mathrm{H}}}\\xspace and the reddening \\ensuremath{{E(B-V)}}\\xspace can be written as ${N_{\\rm{H}}\\ ({\\rm{cm}}^{-2}) = a \\times 10^{21}\\ \\ensuremath{{E(B-V)}}\\xspace\\ (\\rm{mag})}$, where $a$ is reported to have a value ranging from about 5.5 to 6.9 \\citep{Bohl78,Gor75,Pre95,Guv09}. Therefore, for our derived ${\\ensuremath{{E(B-V)}}\\xspace = 1.0 \\pm 0.1}$, the expected \\ensuremath{N_{\\mathrm{H}}}\\xspace ranges between 5.0 and ${7.6 \\times 10^{21}}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace. However, according to our modelling of the X-ray spectrum, the column density of neutral gas in {IC~4329A}\\xspace is ${\\ensuremath{N_{\\mathrm{H}}}\\xspace = 3.1 \\pm 0.1 \\times 10^{21}}$~cm$^{-2}$. Thus, in {IC~4329A}\\xspace the neutral gas alone is not sufficient to produce all the observed reddening. On the other hand, if one considers that part of the dust that causes the reddening is associated to the ionised gas in {IC~4329A}\\xspace (Table \\ref{wind_table}), in particular the less-ionised components of the wind (i.e. a dusty warm absorber), the sum of \\ensuremath{N_{\\mathrm{H}}}\\xspace matches the expected \\ensuremath{N_{\\mathrm{H}}}\\xspace inferred from the observed reddening. The AGN wind Comps. A and B, and the neutral+warm ISM give a total \\ensuremath{N_{\\mathrm{H}}}\\xspace ${5.2 \\pm 0.2 \\times 10^{21}}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace (from the 2003 {\\it XMM-Newton}\\xspace spectra) and \\ensuremath{N_{\\mathrm{H}}}\\xspace ${5.3 \\pm 0.2 \\times 10^{21}}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace (from the 2017 HETGS spectra). This excludes the highest ionisation component of the AGN wind. Therefore, the above results suggest that apart from the neutral ISM, the ionised gas (i.e. the warm ISM and the AGN wind) are likely dusty.\n\nApart from the above \\ensuremath{N_{\\mathrm{H}}}\\xspace-reddening relation, the measured column densities of dust (Table \\ref{dust_table}) also indicate that some of the dust is likely associated to the AGN wind. This is mainly because the column density of Fe in dust form is too high to be feasible with Fe depletion in the ISM alone (for both the neutral and mildly-ionised components). In other words, the measured number of Fe atoms in dust form exceeds the number of Fe atoms available in gas form in the ISM. However, by considering the total \\ensuremath{N_{\\mathrm{H}}}\\xspace, including that of the AGN wind, feasible depletion factors are obtained. Taking into account the possible range of the total \\ensuremath{N_{\\mathrm{H}}}\\xspace ($4.8$ to ${5.7 \\times 10^{21}}$~{\\ensuremath{\\rm{cm}^{-2}}}\\xspace), and the uncertainties on the column densities of O-dust, Si-dust, and Fe-dust, we derive the following depletion factors from gas to dust form in {IC~4329A}\\xspace: 17--23\\% for O, 70--100\\% for Si, and 77--98\\% for Fe. These depletion factors are for the total gas in {IC~4329A}\\xspace, and assume the proto-solar abundances of \\citet{Lod09}. The depletion factors are comparable to those in the diffuse Milky Way (e.g. \\citealt{Jenk09}). We further discuss the possibility of a dusty warm absorber from the AGN torus in Sect. \\ref{dust_torus_sect}.\n\n\\begin{figure}[!tbp]\n\\centering\n\\resizebox{1.015\\hsize}{!}{\\hspace{-0.3cm}\\includegraphics[angle=0]{IC4329A_cartoon.ps}}\n\\caption{Illustration of our line of sight through {IC~4329A}\\xspace. The host galaxy and its dust lane are viewed edge-on, whereas the AGN disk and torus are tilted towards us. Our line of sight goes through the nuclear wind from the torus. The Comps. A and B of the AGN wind are located outside of the torus and are likely dusty.}\n\\label{cartoon_fig}\n\\end{figure}\n\n\\subsection{Dusty AGN torus}\n\\label{torus_prop_sect}\n\n{IC~4329A}\\xspace displays significant optical polarisation \\citep{Mart82,Wols95}. From imaging polarimetry of {IC~4329A}\\xspace, \\citet{Wols95} find two components of polarisation. The first component is parallel to the galactic plane and the edge-on dust lane. The second component is a nuclear polarising component with a position angle approximately parallel to the galactic plane. The authors suggest that the first component arises from magnetic field in the plane of the galaxy, while the second component arises from dust scattering by an asymmetric geometry in the nucleus involving the AGN torus. Based on the polarisation of the nuclear component, \\citet{Wols95} argue that the torus must be significantly tilted in our line of sight, while we can still see the central engine. In this case, {IC~4329A}\\xspace is still perceived spectroscopically as a type-1 AGN, but the inner edge of the torus will be viewed as an elliptical ring which introduces an asymmetry into the scattering geometry.\n\nThe 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace silicate features are commonly seen in absorption in type-2 AGN, and in emission in type-1 AGN. This is because of strong silicate absorption by the AGN torus in type-2 AGN, which obscures the line of sight. However, in {IC~4329A}\\xspace, our view of the accretion disk is only partially obscured by the torus. Thus, silicate emission from the inner walls of the torus dominates over any weaker silicate absorption in our line of sight. Furthermore, highly-inclined (i.e. edge-on) galaxies often show the 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace features in absorption, rather than emission (e.g. \\citealt{Alon11}). This can be attributed to silicate absorption in the disk of the galaxy. However, in {IC~4329A}\\xspace the 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace features are seen in emission, rather than absorption \\citep{Deo09,Alon11}, which is also found by our analysis of the {\\it Spitzer}\\xspace\/IRS spectrum (Fig. \\ref{spitzer_fig}). As suggested by \\citet{Deo09}, this is likely because our line of sight does not intersect dense dust clouds in the ISM of the galaxy. \n\n\\citet{Tris09} have carried out a survey with MID-infrared Interferometric instrument (MIDI) at the VLTI on 13 bright AGN (including {IC~4329A}\\xspace). This is to resolve the nuclear dust emission in these AGN and study their spatial distribution. However, they find that {IC~4329A}\\xspace emission is unresolved, which implies that the bulk of the mid-IR emission (12~\\ensuremath{\\mu{\\mathrm{m}}}\\xspace) is concentrated at ${< 10.8}$~pc \\citep{Tris09}. This suggests that mid-IR dust emission from the nucleus (i.e. the dusty torus) dominates over non-nuclear emission (i.e. the dust lane).\n\nFrom our modelling of internal reddening in {IC~4329A}\\xspace (Sect. \\ref{dust_red_sect}), we derived ${E(B-V) \\approx 1.0}$. The reddening is best described with a grey extinction curve \\citep{Czer04}, which is relatively flat in the UV with no steep rise into the far-UV. This implies that dust in {IC~4329A}\\xspace is different from that of the Milky Way. From the {\\it Spitzer}\\xspace study of dust features in a sample of AGN, \\citet{Xie17} conclude that all sources require that the dust grains are micron-sized (typically ${\\sim 1.5}$ micron), which is much larger than the sub-micron sized Galactic interstellar grains. This would imply a grey (flat) extinction law for AGN \\citep{Xie17}, which is what we find for {IC~4329A}\\xspace from our SED modelling. Therefore, the torus dust grains in {IC~4329A}\\xspace are likely larger than those in the diffuse Milky Way.\n\nIn {IC~4329A}\\xspace, we find that the 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace silicate emission features peak at wavelengths higher than the standard ISM silicate dust (Table \\ref{dust_table}). A recent census of the silicate features in the mid-IR spectra of AGN by \\citet{Hatz15} shows that $\\lambda_{\\rm peak}$ of the 9.7 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace emission feature (in the rest frame) is often shifted to longer wavelengths relative to the nominal wavelength of 9.7 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace. This shift is much less present when the feature is in absorption. \\citet{Hatz15} report that the shift nearly always occurs in AGN dominated spectra, where the fractional contribution of the AGN component to the total luminosity between 5 and 15 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace is $> 0.7$. From our modelling we find this is indeed the case for {IC~4329A}\\xspace, as this fraction is $\\approx 0.80$. The peak wavelength, spectral width, and relative strength of the 9.7 and 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace features are thought to depend on the grain composition and size (e.g. \\citealt{Xie17}). For example, amorphous olivine gives the longest peak wavelength for the 9.7 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace feature and the highest ratio of the 18 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace feature to the 9.7 \\ensuremath{\\mu{\\mathrm{m}}}\\xspace feature \\citep{Xie17}, which is the case for {IC~4329A}\\xspace. Moreover, the shifting and broadening of the silicate features have been explained by \\citet{Li08} using porous composite dust, consisting of amorphous silicate. They suggest that such porous dust is expected in the dense circumnuclear region of the AGN, as a consequence of grain coagulation.\n\n\\subsection{Dusty wind from the AGN torus}\n\\label{dust_torus_sect}\n\nFrom our photoionisation modelling of the AGN wind in {IC~4329A}\\xspace (Sect. \\ref{wind_sect}), we find that it consists of three ionisation components. According to our modelling, the two lowest ionisation components (Comps. A and B) are consistent with having no significant variability between the 2003 and 2017 epochs. Their column density \\ensuremath{N_{\\mathrm{H}}}\\xspace and ionisation parameter $\\xi$ remain unchanged (Table \\ref{wind_table}). However, the highest ionisation component (Comp. C) displays significant variability in \\ensuremath{N_{\\mathrm{H}}}\\xspace and $\\xi$. This increase in $\\xi$ of Comp. C between the two epochs matches the increase in the luminosity of the X-ray continuum (Fig. \\ref{SED_fig}, Tables \\ref{continuum_table} and \\ref{luminosity_table}), suggesting that Comp. C has responded to the variability of the ionising SED. Therefore, the recombination timescale ($t_{\\rm rec}$) of Comp. C needs to be shorter than the spacing between the two epochs, while the lack of variability in Comps. A and B suggests that their $t_{\\rm rec}$ are longer than this limit. From our photoionisation modelling, the product $t_{\\rm rec} \\times n_{\\rm H}$ for each component is yielded. We find $t_{\\rm rec}$ is about ${60~n_{4}^{-1}}$ days for Comp. A, ${49~n_{4}^{-1}}$ days for Comp. B, and ${9~n_{4}^{-1}}$ days for Comp. C, where ${n_{4}}$ represents ${n_{\\rm H} = 10^{4}}$~cm$^{-3}$. Hence, from the limit on $t_{\\rm rec}$, based on the non-variability of Comps. A and B, and variability of Comp. C, we can place a limit on the density $n_{\\rm H}$ of each component. Consequently, from this $n_{\\rm H}$ limit and the definition of the ionisation parameter (${\\xi = L \/ n_{\\rm H}\\, r^{2}}$), limit on the location $r$ of each component from the ionising source is computed. We find Comp. A is at ${r > 350}$~pc, Comp. B at ${r > 83}$~pc, and Comp. C at ${r < 93}$~pc. For comparison, the inner radius of the AGN torus in {IC~4329A}\\xspace, derived by the torus modelling of \\citet{Alon11} is 2.7~pc. \n\nThe observational characteristics of the wind in {IC~4329A}\\xspace match the warm-absorber type winds in type-1 AGN. Such winds are most consistent with an origin from the AGN torus (e.g. \\citealt{Kaas12,Meh18}). The location of the wind components in {IC~4329A}\\xspace (Comps. A and B), and their relatively low outflow velocities, are consistent with a torus wind. As discussed in Sect. \\ref{dust_comp_sect}, a component of dust in {IC~4329A}\\xspace arises from the AGN torus. Thus, the reddening and X-ray absorption from this nuclear component in our line of sight is most likely associated to a wind from the torus. The two lowest ionisation components (Comps. A and B) of the AGN wind are most likely dusty. These components, which are located outside of the dusty torus, are beyond the dust sublimation radius, and therefore they can host dust. Interestingly, there are observational evidence of extended mid-IR emission from dust along the polar directions in type-2 AGN, such as the \\object{Circinus} galaxy \\citep{Tris14}, which may be produced by the dusty outflows from the AGN torus.\n\nCurrently using {\\it XMM-Newton}\\xspace and {\\it Chandra}\\xspace, the dust X-ray features are only detectable in a very few X-ray bright AGN. The proposed {\\it Arcus} mission \\citep{Smit16} with its unprecedented sensitivity and energy resolution in the soft X-ray band would provide a major breakthrough in the X-ray spectroscopy of dust. It would enable us to distinguish between different chemical compositions of dust in AGN, providing important insight into the evolutionary path of AGN. Furthermore, the upcoming {\\it Athena}\\xspace observatory \\citep{Nand13} extends the wind studies to a larger population of AGN, from different types and ages, which is key for understanding the role and impact of AGN winds in galaxy evolution.\n\n\\section{Conclusions}\n\\label{conclusions}\n\nIn this paper we have carried out broad-band continuum modelling spanning far-IR to hard X-rays, combined with high-resolution X-ray and IR spectroscopy, to investigate the nature and origin of dust in {IC~4329A}\\xspace. From the findings of our investigation we conclude the following. \n\n\\begin{enumerate}\n\\item There are two distinct components of dust in {IC~4329A}\\xspace: an ISM dust lane component, and a nuclear component. The nuclear dust component originates from the AGN torus and its associated wind. Dust in the AGN torus is seen through IR emission, while the dust in the torus wind is detected through reddening and X-ray absorption in our line of sight.  \n\\medskip\n\\item The AGN wind in {IC~4329A}\\xspace consists of three ionisation components. They have moderate outflow velocities: $-340$ to $-440$ \\ensuremath{\\mathrm{km\\ s^{-1}}}\\xspace for two of the components, while the other is consistent with zero outflow velocity.\n\\medskip\n\\item According to our variability analysis, the two lowest ionisation components of the AGN wind show no long-term changes between historical and new observations (${\\sim 14}$ years apart), while the highest component shows changes in \\ensuremath{N_{\\mathrm{H}}}\\xspace and ionisation parameter $\\xi$. From the recombination timescale analysis, we derive limits on the distance of the wind components from the central engine. The two lowest ionisation components are at ${r > 350}$~pc, and ${r > 83}$~pc, while the highest ionisation component is at ${r < 93}$~pc.\n\\medskip\n\\item The total internal reddening in {IC~4329A}\\xspace is $\\ensuremath{{E(B-V)}}\\xspace \\approx 1.0$. The reddening is most consistent with a grey (flat) extinction law. \n\\medskip\n\\item From high-resolution X-ray spectroscopy of dust in {IC~4329A}\\xspace we derive the total depletion factors from gas into dust for O, Si, and Fe. They correspond to 17--23\\% for O, 70--100\\% for Si, and 77--98\\% for Fe.\n\\medskip\n\\item The dust grains associated to the AGN torus and its wind in {IC~4329A}\\xspace are likely larger than the Milky Way ISM dust, and are in a porous composite form, containing amorphous silicate with iron and oxygen.\n\\medskip\n\\item From on our modelling of the continuum from far-infrared to X-rays, we derive a bolometric luminosity of about ${2.5 \\times 10^{45}}$~{\\ensuremath{\\rm{erg\\ s}^{-1}}}\\xspace for {IC~4329A}\\xspace, which corresponds to 10--20\\% of the Eddington luminosity. \n\\end{enumerate}\n\n\\begin{acknowledgements}\nM.M. and E.C. are supported by the Netherlands Organisation for Scientific Research (NWO) through The Innovational Research Incentives Scheme Vidi grant 639.042.525. This research has made use of data obtained from the Chandra Data Archive, and software provided by the Chandra X-ray Center (CXC). This work made use of data supplied by the UK {\\it Swift}\\xspace Science Data Centre at the University of Leicester. This work is based on observations made with the NASA\/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. This research has made use of the NASA\/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We thank Fred Lahuis for help with the reduction of the {\\it Spitzer}\\xspace IRS spectra, and Michiel Min for useful discussions. We thank the anonymous referee for the useful comments.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\n\n\\section{Introduction}\n\nIt is of great interest for solar as well as stellar astrophysicists \nto compare the activity of our Sun to those of a number of other \nsimilar solar-type stars, since it may provide us with an opportunity \nto infer the trend of solar activity on a very long astronomical \ntime scale (i.e., investigating the long-time behavior of a star \nmay be replaced by studying many similar stars at a given time).\n\nBaliunas and Jastrow (1990) argued based on the results of Mt. Wilson \nObservatory's HK survey project for 74 solar-type stars that the \ndistribution of $S$-index\n(nearly equivalent to \n$\\propto \\int_{\\rm line}{F_{\\lambda}}d\\lambda \/\\int_{\\rm cont}{F_{\\lambda}}d\\lambda$;\ni.e., the ratio of integrated core-flux of Ca~{\\sc ii} HK lines \nto the continuum flux; cf. Vaughan et al. 1978)\nis bimodal, with about 1\/3 showing appreciably smaller activities\nthan the Sun, which they interpreted as being in the ``Maunder-minimum''   \nstate of activity.  If this is true, it may mean that the spotless phase\nof considerably low-activity such as occurred in late 17th century \nin our Sun (Eddy 1976) may not necessarily be an unusual phenomenon \nin the long run.\n\nHowever, this result could not be confirmed by a similar analysis done by\nHall and Lockwood (2004), who reported based on many repeated observations\nof Ca~{\\sc ii} H and K lines for 57 Sun-like stars along with the Sun\nat Lowell Observatory that such a bimodal distribution of $S$-index \n(as suggesting the existence of a considerable fraction of appreciably \nlower activity stars than the current Sun) is not observed; actually, \neven the $S$-values of 10 ``flat-activity'' stars turned out to be \ncomparable with (or somewhat larger than) the typical solar-minimum value. \n\nFurthermore, Wright (2004) pointed out an important problem in \nBaliunas and Jastrow's (1990) sample selection. He concluded by examining\nthe absolute magnitudes of their sample based on Hipparcos parallaxes \nthat many of those ``Maunder-minimum stars'' with considerably low $S$ \nindices are old stars evolved-off the main sequence, which suggests \nthat their apparently low activity is nothing but due to the aging effect \nwithout any relevance to the cyclic or irregular change of activity\nin solar-type dwarfs.   \nThus, fairly speaking, the original claim by Baliunas and Jastrow \n(1990) appears to be rather premature and difficult to be justified \nfrom the viewpoint of these recent studies.\n\nYet, the issue of clarifying the status of solar activity\namong Sun-like stars does not seem to have been fully settled\nand further investigations still remain to be done:\\\\\n--- First, since understanding the activity of the Sun from a \ncomprehensive perspective is in question, comparison samples \nshould comprise stars as closer to the Sun as possible.\nAdmittedly, those authors surely paid attention to this point: \nBaliunas and Jastrow's (1990) HK project targets were in the \n$B-V$ range of 0.60--0.76, while Hall and Lockwood's (2004) \nSun-like stars sample were chosen from stars of \n$0.58 \\le B-V \\le 0.72$, both narrowly encompassing the solar \n$(B-V)_{\\odot}$ of 0.65 (Cox 2000). However, the homogeneity of \nthese samples are not yet satisfactory. Could they be made up \nof further more Sun-like stars or solar analogs? \\\\\n--- Second, it appears that the precision of detecting low-level \nactivity has been insufficient. Although Mt. Wilson $S$ index \nreflects the core-emission strength (equivalent width) of \nCa~{\\sc ii} HK lines, it would not be a sensitive activity indicator \nany more, when the emission becomes weak, as it stabilizes\nat a constant value determined by the photospheric absorption\nprofile. While an indicator for the pure-emission strength,\n$R'_{\\rm HK} (\\equiv R_{\\rm HK} - R_{\\rm phot}$; where\n$R_{\\rm HK} \\equiv F_{\\rm HK}\/F_{\\rm bol}$ and\n$R_{\\rm phot} \\equiv F_{\\rm phot}\/F_{\\rm bol}$), has also been \nintroduced to rectify this shortcoming and widely used,\nthe photospheric component $R_{\\rm phot}$ is in most cases \nonly roughly evaluated as a simple function of $B-V$ (e.g., \nNoyes et al. 1984) and thus its accuracy is rather questionable.\nWright (2004) also pointed out the importance of correctly\nsubtracting this component, in view of its possible dependence on\nother stellar parameters (i.e., not only on $B-V$ or $T_{\\rm eff}$, \nbut also on $\\log g$ and [Fe\/H]).\n\nThese requirements motivated us to conduct a new investigation\non this subject, since we have been engaged these years with \nthe extensive study of 118 Sun-like stars selected by the criteria\nof $0.62 \\ltsim B-V \\ltsim 0.67$ and $4.5 \\ltsim M_{V} \\ltsim 5.1$\n($M_{V,\\odot} = 4.82$; Cox 2000), a quantitatively as well as \nqualitatively ideal sample of solar-analog stars. \nThis project was originally started for the purpose of clarifying \nthe behavior of Li abundances ($A$(Li)), and revealed that the rotational \nvelocity ($v_{\\rm e}\\sin i$) is the most influential key parameter\n(Takeda et al. 2007; hereinafter referred to as Paper I).\nIn a successive study, we investigated the activities of these solar \nanalogs by using the residual flux at the core of the Ca~{\\sc ii} 8542 \nline ($r_{0}$(8542)), and confirmed a clear correlation between\n$A$(Li), $v_{\\rm e}\\sin i$, and $r_{0}$(8542), as expected\n(Takeda et al. 2010; hereinafter referred to as Paper II).\nHowever, it turned out hard to discriminate the differences \nin $r_{0}$(8542) when the activity is as low as that of the Sun, \nsince it tends to get settled at $\\sim 0.2$ and does not serve \nas a sensitive indicator any more. Actually, test calculations\nof non-LTE line formation suggested (cf. Appendix B in Paper II) \nthat the core flux of the Ca~{\\sc ii} 8498\/8542\/8662 triplet lines \nare rather inert to the chromospheric temperature rise in the upper \natmosphere when the activity is low, but that for the Ca~{\\sc ii} \n3934\/3968 doublet (H+K lines) is still sensitive and thus more \nadvantageous for detecting the low-level activity.\nThen, we recently studied the Be abundances of these sample stars \nby using the Be~{\\sc ii} 3131 line based on the near-UV spectra\nobtained with Subaru\/HDS (Takeda et al. 2011; hereinafter referred to \nas Paper III). Since these HDS spectra fortunately cover the Ca~{\\sc ii} H+K \nlines in the violet region, we decided to reinvestigate the activities\nof these 118 Sun-like stars by measuring the core-emission strength\nof the Ca~{\\sc ii} 3934 (K) line, in order to clarify the activity\nstatus of our Sun in comparison with similar solar analogs, where \nspecial attention was given to correctly removing the background \nline profile computed from the solar photospheric model, while \ntaking advantage of the fact that atmospheric parameters of all these \ntargets are well established. The purpose of this paper is to report\nthe outcome of this investigation.\n\n\\section{Basic Observational Data}\n\n\\subsection{Target Sample}\n\nWe use the same targets (118 solar analogs) as used in Papers I--III,\nwhich were selected by the criteria of having $B-V$ and $M_{V}$ values\nsufficiently similar to those of the Sun ($|\\Delta (B-V)| \\ltsim$~0.2--0.3\nand $|\\Delta M_{V}| \\ltsim 0.3$). See section 2 in Paper I for a \ndetailed description about the sample selection. We also determined\nthe atmospheric parameters ($T_{\\rm eff}$, $\\log g$, $v_{\\rm t}$,\nand [Fe\/H]) from the equivalent widths of Fe lines, and the stellar \nparameters ($M$ and $age$) by comparing the positions on the HR diagram \nwith the theoretical evolutionary tracks (cf. section 3 in Paper I).\nThese parameters for most of the targets were actually confirmed to \nproximally distribute around the solar values (i.e.,\n$|\\Delta T_{\\rm eff}| \\ltsim 100$~K, $|\\Delta \\log g| \\ltsim 0.1$~dex,\n$|\\Delta v_{\\rm t}| \\ltsim 0.2$~km~s$^{-1}$, \n$|\\Delta{\\rm [Fe\/H]}|\\ltsim 0.2$~dex,\n$|\\Delta M|\\ltsim 0.1\\;M_{\\odot}$, \nand $|\\Delta \\log age| \\ltsim 0.5$~dex; cf. figures 4 and 5 in Paper I).\n\nHowever, the following characteristics regarding the relations between \nthese parameters are to be noted, which we had better bear in mind \nin discussing the behavior of stellar activities.\\\\\n--- Since the effect of a decreased metallicity on $B-V$ is compensated \nby a lowering of $T_{\\rm eff}$, several outlier stars with appreciably \nlower $T_{\\rm eff}$ as well as [Fe\/H] \n($\\Delta T_{\\rm eff} \\ltsim -200$~K and [Fe\/H]~$\\ltsim -0.4$~dex)\nare included in our sample (cf. figure 4c in Paper I), \nsuch as HIP~26381, HIP~39506, HIP~40118, and HIP~113989; and they belong to \nthe oldest group ($age \\sim 10^{10}$~yr).\\\\\n--- These parameters are not independent from each other and some\ncorrelations appear to exist between specific combinations; such as \n$age$ vs. $T_{\\rm eff}$ (lower $T_{\\rm eff}$ stars tend to be older), \n$age$ vs. [Fe\/H] (lower [Fe\/H] stars tend be older), \n$M$ vs. $age$ (lower-mass stars tend to be older), and \n$v_{\\rm t}$ vs. $T_{\\rm eff}$ ($v_{\\rm t}$ tends to decrease with \na lowered $T_{\\rm eff}$), as recognized from figure 5 or figure 10 \nin Paper I, though the existence of outlier stars mentioned above \npartly plays a role in these tendencies.\n\nIt should also be remarked that the data for HIP~41484 given in Paper I\nwere incorrect, as reported in appendix A of Paper II, where the \ncorrect results derived from a reanalysis of this star are presented. \n\n\\subsection{Observational Material}\n\nThe spectroscopic observations of these 118 solar analogs and Vesta \n(substitute for the Sun) were carried out on 2009 August 6, \n2009 November 27, 2010 February 4, and 2010 May 24 (Hawaii Standard \nTime), with the High Dispersion Spectrograph (HDS; Noguchi et al. 2002) \nplaced at the Nasmyth platform of the 8.2-m Subaru Telescope\natop Mauna Kea, by which we obtained high-dispersion spectra covering \n$\\sim$~3000--4600~$\\rm\\AA$ with a resolving power of $R \\simeq 60000$. \nSee section 2 of Paper III and electronic table E1 therein for\ndetails of the observations and the data reduction, as well as \nthe basic data of the spectra (e.g., observing date, exposure time, \nS\/N ratio at $\\lambda \\sim 3131 \\rm\\AA$).\n\nThe counts of raw echelle spectra (including the effect of blaze \nfunction) around $\\lambda \\sim 3950 \\rm\\AA$ (the broad maximum between \nthe two large depressions of Ca~{\\sc ii} 3934 (K) and 3968 (H) lines) \nare typically about $\\sim 15$ times as large as those at the UV region\nof $\\lambda \\sim 3131 \\rm\\AA$, while the counts at core of the Ca~{\\sc ii} \nK line at 3934~$\\rm\\AA$ is about $\\sim 10\\%$ of those at \n$\\lambda \\sim 3950 \\rm\\AA$. Therefore, the S\/N ratio at the deep \nabsorption core of the K line is not much different\nfrom the value at $\\lambda \\sim 3131 \\rm\\AA$ given in electronic \ntable E1 of Paper III; that is, on the order of S\/N~$\\sim 100$.\n\n\\section{Core Emission Measurement}\n\n\\subsection{K line of ionized calcium}\n\nMost activity studies of solar-type stars so far based on \nthe core emission strengths of Ca~{\\sc ii} resonance lines \nappear to utilize both K (3934~$\\rm\\AA$) and H (3968~$\\rm\\AA$) lines, \npresumably due to the intention of reducing the systematic \nerrors by averaging both two, since measurements tend to be done \nrather roughly by directly integrating raw spectra at the specified\nwavelength regions.\n\nIn this investigation, however, we focus only on the former K line \nat 3933.66~$\\rm\\AA$, since (1) it is by two times stronger than \nthe H line and thus comparatively more suitable as a probe of\nthe condition at the optically-thin chromospheric layer, and\n(2) the latter Ca~{\\sc ii} H line at 3968.47~$\\rm\\AA$ is blended \nwith the Balmer line (H$\\epsilon$ at 3970.07~$\\rm\\AA$) which\nwould make the situation more complicated in simulating\nthe photospheric profile to be subtracted.\n\nOur spectra in the selected wavelength region (3930--3937~$\\rm\\AA$)\nincluding the relevant Ca~{\\sc ii}~K line are shown in\nfigures 1 (Vesta\/Sun), 2 (all 118 stars including Vesta\/Sun),\nand 3 (all stars overplotted), where the continuum levels of\nthe observed spectra are so adjusted as to match the theoretical ones\nas explained below. We can see from these figures that the strengths\nof core emission considerably vary from star to star, while\nthat for the Sun is apparently weak.\n\n\\setcounter{figure}{0}\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(80mm,60mm){figure1.eps}\n   \n  \\end{center}\n\\caption{\nChromospheric emission feature at the core of the \nCa~{\\sc ii} 3933.663 line for the case of the Sun (Vesta),\nin comparison with the theoretical photospheric spectra\ncomputed from the classical model atmosphere. \nThree solid lines indicate the theoretical residual flux spectra \n(normalized by the continuum flux),\n$r_{\\lambda}^{\\rm th} (\\equiv F_{\\lambda}^{\\rm th}\/F_{\\rm cont}^{\\rm th})$,\nwhich are based on essentially the same Kurucz's (1993) \nATLAS9 solar atmospheric model but with different surface locations \n(i.e., the optical depth at the first mesh point), as depicted in the \n$T(\\tau_{5000})$ structures shown in the inset.\nRed line is the LTE line profile derived from the adopted model \natmosphere with its surface at $\\log\\tau_{5000} = -5$,\nwhile blue lines are the LTE and NLTE line profiles derived from \nthe specially-extrapolated model atmosphere with its surface at \n$\\log\\tau_{5000} = -7$.\nOpen symbols $\\cdots$ observed spectrum of Vesta\/Sun, \n$r_{\\lambda}^{\\rm obs} (\\equiv D_{\\lambda}^{\\rm obs}\/D_{\\rm cont}^{\\rm obs})$,\nwhere $D_{\\lambda}^{\\rm obs}$ is the actually recorded spectrum\n(ADU counts) while $D_{\\rm cont}^{\\rm obs}$ (regarded as an adjustable \nfree parameter) was adequately chosen by requiring a satisfactory match \nbetween $r_{\\lambda}^{\\rm th}$ and $r_{\\lambda}^{\\rm obs}$\nwithin $\\ltsim$~2--3~$\\rm\\AA$ from the line center (excepting the \ncore-emission region) as explained in subsection 3.2.\nThe specified line-core region (3932.8--3934.6~$\\rm\\AA$) is indicated\nby a horizontal bar, over which the integration was made for evaluating \nthe total chromospheric emission.\n}\n\\end{figure}\n\n\\setcounter{figure}{2}\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(80mm,60mm){figure3.eps}\n   \n  \\end{center}\n\\caption{\nIn blue lines are overplotted the observed spectra \n($r_{\\lambda}^{\\rm obs}$) of Ca~{\\sc ii} 3933.663 in the \n3930--3937~$\\rm\\AA$ region for all 118 stars (the same as those \nshown in figure 2), along with the Vesta\/Sun spectrum highlighted \nin the red thick line. The theoretical photospheric spectra\n($r_{\\lambda}^{\\rm th}$) for each of the stars are also overplotted \nin green lines (only the solar spectrum is in red) with a downward \noffset by $-0.05$. The wavelength scale of all spectra is adjusted \nto the laboratory frame.\n}\n\\end{figure}\n\n\n\\subsection{Photospheric profile matching}\n\nGiven that the very strong Ca~{\\sc ii} K and H lines with considerably \nextended damping wings are predominant at $\\sim$~3900--4000~$\\rm\\AA$,\nit is hopeless to empirically establish the continuum position\nfrom the observed spectrum $D_{\\lambda}^{\\rm obs}$ (where an echelle \norder covers only $\\sim 50$~$\\rm\\AA$).\nWe thus ``adjusted'' the continuum position ($D_{\\rm cont}^{\\rm obs}$) \nof the spectrum (judged by eye-inspection) in such a way that \n$r_{\\lambda}^{\\rm obs} (\\equiv D_{\\lambda}^{\\rm obs}\/D_{\\rm cont}^{\\rm obs})$\nsatisfactorily matches the theoretically calculated residual flux\n$r_{\\lambda}^{\\rm th} (\\equiv F_{\\lambda}^{\\rm th}\/F_{\\rm cont}^{\\rm th})$\\footnote{ \nWe here use the astrophysical flux ($F$) defined by $\\pi F_{\\lambda}^{\\rm th} \n\\equiv 2\\pi \\int_{0}^{1} \\mu I_{\\lambda}^{\\rm th}(0, \\mu) d\\mu$,\nwith which the effective temperature $T_{\\rm eff}$ is related as\n$\\pi F_{\\rm bol} = \\pi \\int_{0}^{\\infty} F_{\\lambda} d\\lambda = \\sigma T_{\\rm eff}^{4}$\n($\\sigma$: Stephan--Boltzmann constant).}\nin the inner wing of the line (within $|\\Delta\\lambda| \\ltsim$~2--3~$\\rm\\AA$ from \nthe line center, excepting the core-emission region).\nAn example of such an accomplished match is displayed in figure 1\nfor the case of the Sun (Vesta).\n\nRegarding the computation of theoretical spectra, we used \nKurucz's (1993) WIDTH9 program, which was modified by Y. Takeda \nto enable spectrum synthesis by including many lines.\nAs to the atomic line data, we invoked Kurucz and Bell's (1995)\ncompilation and included all available lines in the relevant region.\nIn particular, the data for the Ca~{\\sc ii} K line at 3933.663~$\\rm\\AA$\nof our primary concern are as follows: $\\chi_{\\rm low}$ = 0.00~eV, \n$\\log gf = +0.134$, $\\log \\Gamma_{\\rm R} = 8.20$ [radiation\ndamping width (s$^{-1}$)], \n$\\log \\Gamma_{e}\/N_{\\rm e} = -5.52$\n[Stark effect damping width (s$^{-1}$) per electron density \n(cm$^{-3}$) at $10^{4}$~K], and \n$\\log \\Gamma_{w}\/N_{\\rm H} = -7.80$\n[van der Waals damping width (s$^{-1}$) per hydrogen density \n(cm$^{-3}$) at $10^{4}$~K].\nSince the atmospheric parameters ($T_{\\rm eff}$, $\\log g$, [Fe\/H], and \n$v_{\\rm t}$) are already established (as summarized in table 1), \nthe model atmosphere for each star \nwas generated by 3-dimensionally interpolating Kurucz's (1993) ATLAS9 model grids \n(LTE, plane-parallel model) in terms of $T_{\\rm eff}$, $\\log g$, and [Fe\/H]. \nThen, the synthetic spectrum was computed by using the relevant atmospheric model\nalong with the metallicity-scaled abundances (for all elements including Ca; \ni.e., [X\/H] = [Fe\/H] for any X) as well as the microturbulence ($v_{\\rm t}$),\nand further broadened according to the macrobroadening parameter \ndetermined in Paper III. \n\n\\subsection{Evaluation of chromospheric emission}\n\nNow that the theoretical photospheric background profile ($r_{\\lambda}^{\\rm th}$) \nused for subtraction has been successfully fitted with $r_{\\lambda}^{\\rm obs}$ \nby appropriately adjusting the continuum position, we can calculate the absolute \nemission flux ($F'_{\\rm Kp}$) at the K-line core originating from the chromosphere \n(i.e., after subtraction of the photospheric component) as \n\\begin{equation}\nF'_{\\rm Kp} \\equiv \nF_{\\rm cont}^{\\rm th}\\int _{\\lambda_{1}}^{\\lambda_{2}}\n(r_{\\lambda}^{\\rm obs} - r_{\\lambda}^{\\rm th})d \\lambda\n\\end{equation}\nwhere $\\lambda_{1}$ and $\\lambda_{2}$ defining the integration range \nwere chosen to be 3932.8 and 3934.6~$\\rm\\AA$, respectively (cf. figure 1).\nIn order to demonstrate how this subtraction process works well, \nthe photospheric profile ($r_{\\lambda}^{\\rm th}$) \nat [$\\lambda_{1}$, $\\lambda_{2}$] is displayed by red dashed line \n(along with the observed spectrum shown by symbols) for each star in figure 2. \nFinally, we can obtain $R'_{\\rm Kp}$ (the ratio of the chromospheric \nemission flux at the K line to the total bolometric flux) as\n\\begin{equation}\nR'_{\\rm Kp} \\equiv F'_{\\rm Kp}\/ F_{\\rm bol} \n= \\pi F'_{\\rm Kp} \/ (\\sigma T_{\\rm eff}^{4}).\n\\end{equation}\nThe resulting $\\log R'_{\\rm Kp}$ values for each of the 118 solar analogs \n(+Sun) are presented in table 1, where other activity-related quantities \n($r_{0}$(8542) and $v_{\\rm e}\\sin i$) determined in Paper II are also \ngiven, along with the Li\/Be abundances and stellar parameters established \nin Papers I and III. \n\n\\subsection{Zero-Point Uncertainties in $R'_{Kp}$}\n\nWe would like to remark here that the ``absolute'' values of \n$R'_{\\rm Kp}$ are not very meaningful in the low-activity regime\nbecause of the uncertainties in its zero-point.\nThat is, the theoretical profile we have computed for subtraction of\nthe background photospheric component is by no means uniquely defined.\nActually, apparently different results may be obtained depending on \nhow it is calculated.\n\nThis situation is demonstrated in figure 1 for the solar case.\nThe solar atmospheric model (which was obtained by interpolating \nthe grids of Kurucz's ATLAS9 model atmospheres) we adopted for \ncalculating the LTE photospheric profile (red line) has its surface \nat $\\log \\tau_{5000}^{\\rm surf} = -5$ ($T^{\\rm surf} \\sim 4000$~K).\nHowever, we note that the same but simply extrapolated model up to \n$\\log \\tau_{5000}^{\\rm surf} = -7$ ($T^{\\rm surf} \\sim 3600$~K)\nyields an appreciably deeper core (upper blue line), reflecting \nthe fact that the residual flux at the center is determined by\n$\\sim B_{\\lambda}(T^{\\rm surf})\/B_{\\lambda}(T^{\\rm ph})$\n($T^{\\rm ph}$: photospheric temperature) and very sensitive to\n$T^{\\rm surf}$ in this violet region where Wien's approximation\nnearly holds. Moreover, the core of the non-LTE profile (simulated \nby using the departure coefficients computed in Paper II\nfor the case of Model E; cf. Appendix B therein) gets even more\ndeeper approaching a completely dark core (lower blue line).\nBesides, the core shape strongly depends on the turbulent\nvelocity field in the upper atmosphere; e.g., compare the non-LTE \nprofile in figure 1 (computed with a depth-independent microturbulence\nof 1~km~s$^{-1}$) with that for Model E depicted in figure B.1(b)\nof Paper II (where a variable microturbulent velocity field increasing\nwith height was adopted).\n\nAccordingly, $R'_{\\rm Kp}$ values [equation (2)] may \nbe uncertain by an arbitrary constant, because \n$\\int_{\\lambda_{1}}^{\\lambda_{2}}r_{\\lambda}^{\\rm th}d\\lambda$\nin equation (1) can have different values depending on\nhow $r_{\\lambda}^{\\rm th}$ is computed, though ``relative'' differences \nbetween $R'_{\\rm Kp}$ values of each star are surely meaningful \nas long as they are evaluated in the same system.\nTherefore, we should keep in mind that comparison of absolute \nvalues of our $R'_{\\rm Kp}$ with those of similar $R'$ parameters \nderived by other groups, which we will try in subsection 4.1,\nis not much meaningful when the low-activity region\n(e.g., $\\log R'_{\\rm Kp} \\ltsim -5$) is concerned. (On the other hand,\nsuch a zero-point problem should not be so serious for higher-activity \ncases, where $\\log R'_{\\rm Kp}$ tends to be dominated by the emission \ncomponent and the role of $r_{\\lambda}^{\\rm th}$ subtraction is \nless significant.) In any event, we consider that our choice of\nshallower $r_{\\lambda}^{\\rm th}$ is adequate, since it leads to\nsmaller values of $R'_{\\rm Kp}$ (as a result of larger subtraction), \nwhich eventually realizes a larger contrast in the $\\log R'_{\\rm Kp}$ \nvalues of low-activity stars. \n\n\\section{Discussion}\n\n\\subsection{Comparison of $R'_{\\rm Kp}$ with previous studies}\n\nThe $\\log R'_{\\rm Kp}$ values determined in subsection 3.3 are compared\nwith the equivalent activity indices\\footnote{\nSince Strassmeier et al. (2000) treated H and K lines separately\nand presented each data of $F'_{\\rm H}$ and $F'_{\\rm K}$, we could convert \ntheir $R'_{\\rm HK} [\\equiv (F'_{\\rm H} + F'_{\\rm K})\/F_{\\rm bol}]$ \n(as clearly defined by them)\ninto $R'_{\\rm K} (\\equiv F'_{\\rm K}\/F_{\\rm bol})$ which is directly \ncomparable with our $R'_{\\rm Kp}$. Meanwhile, the $R'_{\\rm HK}$ data \nderived by Wright et al.'s (2004) as well as Isaacson and Fischer (2010)\nappear to be essentially the {\\it average} \n(not the sum) of $R'_{\\rm H}$ and $R'_{\\rm K}$ as judged by their\nextents. Therefore, we should be cautious about different definitions \nin the meaning of $R'_{\\rm HK}$. This situation is manifestly displayed\nin figure 7 of Paper II, where we can see that $R'_{\\rm HK}$(Strassmeier)\nis systematically larger than $R'_{\\rm HK}$(Wright) by 0.3~dex.\nTherefore, we compared our $\\log R'_{\\rm Kp}$ with \n$R'_{\\rm HK}$(Wright, Isaacson) without applying any correction, \nsince the latter is practically \nequivalent to $R'_{\\rm K}$ in any case.} derived by Strassmeier \net al. (2000) [$\\log R'_{\\rm K}$], Wright et al. (2004) \n[$\\log R'_{\\rm HK}$], and Isaacson and Fischer (2010)\n[$\\log R'_{\\rm HK}$] in figures 4a, 4b, and 4c, respectively.\nThese figures reveal almost the same tendency of correlation \nbetween our $\\log R'$ and those of three studies: The agreement\nat $\\log R' \\gtsim -5$ is mostly good, though our $\\log R'$\ntends to be slightly larger by $\\sim 0.1$~dex for\nhigh-activity stars of $\\log R' \\gtsim -4.5$. Meanwhile,\na distinct difference is observed at the low-activity region\nwhere their $\\log R'$ values tend to settle down at $\\sim -5$, \nwhile ours are dispersed over $-5.5 \\ltsim \\log R' \\ltsim -5$.\nAdmittedly, there is not much meaning in comparing the absolute \n$R'$ values of different systems with each other, as remarked \nin subsection 3.4. The important point is, however, that we \ncould detect the subtle difference\nin the low-level activity by measuring the weak core-emission \nat Ca~{\\sc ii} K with a careful subtraction of the background \nphotospheric profile, while such a precision for distinguishing \nthe delicate difference in the weak emission strength could not \nbe accomplished by comparatively rough measurements in those \nprevious studies.\n\n\\setcounter{figure}{3}\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(70mm,140mm){figure4.eps}\n   \n  \\end{center}\n\\caption{\nCorrelation of the $\\log R'_{\\rm Kp}$ indices determined \nin this study with the literature values taken from three \nrepresentative papers:\n(a) Strassmeier et al.'s (2000) $\\log R'_{\\rm K}$ values,\nwhere 16 stars are in common. (b) Wright et al.'s (2004) \n$\\log R'_{\\rm HK}$ values, where 50 stars are in common.\n(c) Isaacson and Fischer's (2010) $\\log R'_{\\rm HK}$ values, \nwhere 66 stars are in common.\n(Since $\\log R'_{\\rm HK}$ of Wright et al. (2004) as well as \nIsaacson and Fischer (2010) should be equivalent to the mean \nof $\\log R'_{\\rm H}$ and $\\log R'_{\\rm K}$, it may be directly \ncompared with our $\\log R'_{\\rm Kp}$.) \n}\n\\end{figure}\n\n\\subsection{Connection with stellar parameters}\n\nFigures 5a, 5b, and 5c display how the three activity-related\nparameters ($r_{0}$(8542), $v_{\\rm e}\\sin i$, and $\\log age$;\ncf. Paper II) are correlated with $\\log R'_{\\rm Kp}$.\n\n\n\\setcounter{figure}{4}\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(70mm,140mm){figure5.eps}\n   \n  \\end{center}\n\\caption{\nDiagrams showing how the activity-related quantities\nderived in Paper I and Paper II ($r_{0}(8542)$, $v_{\\rm e}\\sin i$,\nand $\\log age$; also given in table 1) are correlated with \nthe activity index ($\\log R'_{\\rm Kp}$) derived in this study.\n(a) $r_{0}(8542)$ vs. $\\log R'_{\\rm Kp}$,\n(b) $v_{\\rm e}\\sin i$ vs. $\\log R'_{\\rm Kp}$, and \n(c) $\\log age$ vs. $\\log R'_{\\rm Kp}$.\nThe $\\log R'_{\\rm Kp,\\odot}$ value which we derived from the spectrum of\nVesta\/Sun (corresponding to the near-minimum phase of activity) is indicated \nby the bigger (red) circle, while the expected minimum--maximum\nrange of $\\log R'_{\\rm Kp,\\odot}$ (between $-5.35$ and $-5.15$;\ncf. subsection 4.3) is shown by a horizontal bar.\n}\n\\end{figure}\n\nWe can observe in figure 5a a marked sensitivity-difference\nbetween $\\log R'_{\\rm Kp}$ and $r_{0}$(8542) in the low-activity \nregion; i.e., the latter is inert to a variation of low-level \nactivity and stabilizes at $\\sim 0.2$, despite that the former \nstill shows an appreciable variability over \n$-5.5 \\ltsim \\log R'_{\\rm Kp} \\ltsim -5$.\nThis is just what we have expected (cf. Appendix B in Paper II),\nand demonstrates the superiority of the Ca~{\\sc ii} HK core emission\n(as long as correctly measured) to the line-center residual flux \nof Ca~{\\sc ii} 8542 when it comes to investigating the activities \nof solar-type stars as low as the Sun.\n\nFigure 5b shows a positive correlation between $R'_{\\rm Kp}$ \nand $v_{\\rm e}\\sin i$, suggesting that stellar activity depends\non the rotation rate, as we already confirmed in Paper II\n(cf. figure 5a therein). However, since $v_{\\rm e}\\sin i$ values\ncluster around $\\sim 2$~km~s$^{-1}$ at \n$-5.5 \\ltsim \\log R'_{\\rm Kp} \\ltsim -5$, we can not state much \nabout whether this activity--rotation connection persists down to\nsuch a low-activity region (also, uncertainties in the projection \nfactor prevent from a meaningful discussion).\n\nWhen we compare the $age$ vs. $R'_{\\rm Kp}$ relation depicted in \nfigure 5c with the similar $r_{0}$(8542) vs. $age$ plot \n(cf. figure 5c of Paper II), the anti-correlation is more clearly\n(or less unambiguously, to say the least) recognized in the present \ncase, thanks to the extended dynamic range of the activity indicator \nfor low-activity stars, though the dispersion is still considerably\nlarge. \n\nHow the abundances of Li and Be depend on $R'_{\\rm Kp}$ determined\nin this study is illustrated in figure 6, where their dependences\nupon $r_{0}$(8542) and $v_{\\rm e}\\sin i$ already discussed in Papers\nII and III are also shown for comparison. We can see from\nfigures 6a that the near-linear relation between $A$(Li) and \n$\\log R'_{\\rm Kp}$ ($A$(Li)~$\\simeq 7 + \\log R'_{\\rm Kp}$) holds \nwidely from highly active ($\\log R'_{\\rm Kp} \\sim -4$) to less \nactive ($\\log R'_{\\rm Kp} \\sim -5.5$) stars, \nin contrast to the case of $A$(Li) vs. $r_{0}$(8542) where\n$A$(Li) shows a considerable dispersion at $r_{0}$(8542)~$\\sim 0.2$\n(as if compressed) because of the less sensitivity of $r_{0}$(8542).\nThis substantiates the observational conclusion in Paper II (or \ncorroborates its validity even for less-active cases) that $A$(Li)\nclosely depends upon the stellar activity, which further lends \nsupport for our previous argument that the key parameter for controlling \nthe surface lithium in solar-analog stars is the stellar rotation.\n\nRegarding $A$(Be), we suspected in Paper III that the peculiar 4 stars\nshowing drastically depleted Be (by $\\gtsim 2$~dex) in comparison with \nthe solar abundance (while others have more or less near-solar Be) \nmight be very slowly-rotating and and thus less active star than \nthe Sun. However, since these 4 stars have appreciably different \n$\\log R'_{\\rm Kp}$ from each other\n($-5.4 \\ltsim \\log R'_{\\rm Kp} \\ltsim -4.9$; cf. figure 6a$'$) \nand are not necessarily less active than the Sun, this speculation\ndoes not seem likely. Some other explanation would have to be sought for.\n\n\\setcounter{figure}{5}\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(80mm,160mm){figure6.eps}\n   \n  \\end{center}\n\\caption{\nAbundances of Li (left panels) and Be (right panels) plotted \nagainst $\\log R'_{\\rm Kp}$ (top), $r_{0}$(8542)\n(middle), and $v_{\\rm e}\\sin i$ (bottom).\nThe solar values are indicated by the bigger (red) circle\nin each panel. The expected minimum--maximum range of\n$\\log R'_{\\rm Kp,\\odot}$ is shown by a horizontal bar.\nThe results for the determinable cases are shown by filled circles, \nwhile the open inverse triangles denote the upper limits for \nthe unmeasurable cases.\n}\n\\end{figure}\n\n\\subsection{Activities of solar analogs and the Sun}\n\nWe now discuss the activity trends of 118 solar analogs, \nwith special attention paid to the status of our Sun among \nits close associates, as the main subject of this study.\nWhile our discussion is based on the $\\log R'_{\\rm Kp}$ indices\ndetermined by ourselves, we should keep in mind  \nthat only one snapshot data is available for each star.\nThis means that uncertainties due to possible time-variations of stellar\nactivities (whichever cyclic or irregular) are inevitably involved.\nFor example, the Mt. Wilson $S$ value for the Sun varies over \nthe range of $0.16 \\ltsim S_{\\odot} \\ltsim 0.20$ (Baliunas et al. 1995),\nwhich may be translated into a variability in $\\log R'_{\\odot}$ by \n$\\sim 0.2$~dex ($-5.0 \\ltsim \\log R'_{\\odot} \\ltsim -4.8$)\naccording to the transformation formula given by Noyes et al. (1984). \nSince the observation time (2010 February 5) of our solar spectrum \n(Vesta) had better be regarded as corresponding to the near-minimum \nphase because of the appreciably retarded beginning of cycle 24 \nafter the minimum in 2008, our $\\log R'_{\\rm Kp,\\odot} (= -5.33)$\nmay as well be raised by $\\ltsim 0.2$~dex at the solar maximum phase. \nIt is thus reasonable to assume that our $\\log R'_{\\rm Kp,\\odot}$ \nranges from $-5.35$ (solar minimum) to $-5.15$ (solar maximum),\nas indicated by a short horizontal bar in figures 5, 6, and 7.\n\nThe distribution histogram of $\\log R'_{\\rm Kp}$ for all the 118 stars \n(and the Sun) and a similar histogram of $r_{0}$(8542) (for comparison; \ncf. Paper II) are shown in figures 7a and 7b, respectively.\nWe immediately notice in figure 7a the bimodal distribution of \n$\\log R'_{\\rm Kp}$ having two peaks at $\\sim -5.3$ and $\\sim -4.3$,\nconstituting a well-known Vaughan--Preston gap (Vaughan \\& Preston 1980),\nwhile this bimodal trend is not clear in the distribution of $r_{0}$(8542) \n(figure 7b) because of the densely peaked population around  \n$r_{0}$(8542)~$\\sim 0.2$, reflecting its insensitivity when the activity \nis low (cf. figure 5a).\n\nAs we can see from figure 7a, the Sun with $\\log R'_{\\rm Kp,\\odot}$\nof $-5.33$ manifestly belongs to the low-activity group (ranging \nfrom $\\sim -5.4$ to $\\sim -5.0$). As a matter of fact, only 11 \n($\\sim 10\\%$)\\footnote{\nThis ratio naturally depends on the reference solar activity,\nwhich may be subject to uncertainties due to cyclic variations\nas mentioned at the beginning of subsection 4.3. For example, if we assume \na somewhat higher value of $-5.2$ for $\\log R'_{\\rm Kp,\\odot}$ \ncorresponding to the active phase of the Sun, this fraction \nbecomes $\\sim 30\\%$.} \nout of 118 solar analogs have $\\log R'_{\\rm Kp}$ values smaller than $-5.33$. \nThis distinctly low-activity nature of the Sun is also recognized \nby eye-inspection of figure 2, revealing that the Ca~{\\sc ii} K line \nemission strength in our solar spectrum is near to the minimum level \namong other stars.\n\n\\setcounter{figure}{6}\n\\begin{figure}\n  \\begin{center}\n    \\FigureFile(70mm,120mm){figure7.eps}\n   \n  \\end{center}\n\\caption{\nHistograms showing the distributions of (a) $\\log R'_{\\rm Kp}$ \nand (b) $r_{0}(8542)$ for our sample of 118 solar analogs. \nThe position for the solar value is indicated by an downward arrow \nat each panel. The expected minimum--maximum range of\n$\\log R'_{\\rm Kp,\\odot}$ is shown by a horizontal bar.\n}\n\\end{figure}\n\nThus, we can rule out the possibility for the existence of a significant \nfraction of Maunder-minimum stars (i.e., solar-type stars with appreciably \nlower activity than the Sun, even showing an another peak well below \nthe current solar-minimum level), such as those once suggested by \nBaliunas and Jastrow (1990). \nThis result corroborates the arguments raised by recent studies \n(e.g., Hall \\& Lockwood 2004; Wright 2004), which cast doubts about \nthe reality of such a high frequency of Maunder-minimum stars.\nThus, our Sun belongs to the group of manifestly low activity \nlevel among solar analogs, the fraction of stars below which \nis essentially insignificant. \n\n\\subsection{Stars of Subsolar Activity}\n\nAlthough we have concluded that the Sun belongs to nearly the lowest\nactivity group, some stars do exist showing activities\nstill lower than that of the minimum-Sun, which are worth being \nexamined more in detail. \nSince we defined that solar $\\log R'_{\\rm Kp, \\odot}$ varies\nfrom $-5.35$ (minimum) to $-5.15$ (maximum) (cf. section 4.1), \nsuch stars may be sorted out by the criterion of $\\log R'_{\\rm Kp} < -5.35$, \nwhich resulted in the following 8 objects ($\\log R'_{\\rm Kp}$,\n$A$(Li), $\\log age$, [Fe\/H], remark):\nHIP~7918 ($-5.42$, 1.89,  9.56, +0.01),\nHIP~31965 ($-5.39$, 1.00,  9.87, +0.05),\nHIP~39506 ($-5.44$, $<0.9$, 10.36, $-0.62$) \nHIP~53721 ($-5.43$, 1.75,  9.89, $-0.02$, PHS),\nHIP~59610 ($-5.40$, 1.62,  9.63, $-0.06$, PHS),\nHIP~64150 ($-5.38$, $<1.0$,  9.63, +0.05, Be depleted),\nHIP~64747 ($-5.40$, $<1.1$,  9.80, $-0.18$), and\nHIP~96901 ($-5.38$, $<1.1$,  9.77, +0.08, PHS). \n\nWhile only one (HIP~39506) of these is an outlier of lower \n[Fe\/H] as well as lower $T_{\\rm eff}$ (belonging to rather old \npopulation) as mentioned in subsection 2.1, the remaining 7 stars \nare Sun-like stars with sufficiently similar parameters,\nwhich excludes the possibility of such a low-level activity being \ndue to stellar-evolution (i.e., evolved subgiants; cf. section 1). \n\nWe note here that (1) three (out of only five in our sample of \n118 stars) planet-host stars  are included,  (2) all these stars\nhave low-scale Li abundances ($A$(Li)~$\\ltsim 2$), and (3) their ages \nare similar to or older than that of the Sun. Combining these facts with\nthe consequences in Papers I and II, we consider that these stars have\nactually low activities even compared with the solar-minimum level,\nand this is presumably attributed to their intrinsically slow rotation \n(which is closely related with the Li abundance as well as with \nthe existence of giant planets). \n\nIt is, therefore, interesting to investigate the variabilities\nof these low-activity stars by long-term monitoring observations,\nin order to see how their activities behave with time (cyclic? flat? \nirregular?). Admittedly, activity observations for these stars \nhave been reported in several published studies so far, for example:\nHIP~7918, 53721, 64150, and 96901 by Duncan et al. (1991);\nHIP~53721, 59610, 64150, and 96901 by Wright et al. (2004);\nHIP~7918, 53721, and 96901 by Hall et al. (2007);\nHIP~31965, 59610, 64150, and 96901 by Isaacson and Fischer (2010).\nHowever, as they are still quantitatively insufficient for establishing\nthe long-term behavior of their activities\\footnote{As a comparatively \nwell studied case where a sufficient amount of data are available, we may \npresumably state that HIP~96901 = HD~186427 showed a Sun-like cyclic \nvariation in the 1995--2007 period; cf. figure 7 of Hall et al. (2007).},\nmuch more observations are evidently needed.\n\n\\section{Conclusion}\n\nThere have been several arguments regarding the status of solar \nactivity among similar Sun-like stars. which began with the\nimplication of Baliunas and Jastrow (1990) based on their Mt. Wilson\nHK survey project that a considerable portion ($\\sim 1\/3$) of \nsolar-type stars have activities significantly lower than\nthe present-day Sun, which they called ``Maunder-minimum stars.''\nHowever, their conclusion could not be confirmed by Hall and Lockwood's\n(2004) follow-up study, and Wright (2004) criticized the reality\nof such considerably low-active solar-type stars by pointing out \nthat most of them are not so much dwarfs as evolved subgiants.\n\nGiven this controversial situation, we decided to contend with \nthis problem by ourselves based on carefully selected sample of \n118 solar-analogs sufficiently similar to each other (which we \nalready investigated their stellar parameters as well as Li\/Be \nabundances in a series of our previous papers), with a special \nattention being paid to reliably evaluating their activities \ndown to a considerably low level. \n\nPractically, we measured the emission strength at the core of \nCa~{\\sc ii} 3933.663 line (K line) on the high-dispersion \nspectrogram obtained by Subaru\/HDS, where we gave effort\nto correctly evaluating the pure emission component by removing \nthe wing-fitted photospheric profile calculated from \nthe classical solar model atmosphere, which enabled us to detect \nlow-level activities down to $\\log R'_{\\rm Kp} \\sim -5.5$.\n\nA comparison of our $\\log R'_{\\rm Kp}$ results with the corresponding \n$\\log R'$ values of Strassmeier et al. (2000), Wright et al. (2004),\nand Isaacson and Fischer (2010) \nrevealed that low-active stars (for which they derived \n$\\log R' \\sim -5.1$ at the minimum limit) actually have a dispersion \nof $\\sim 0.4$~dex ($-5.5 \\ltsim \\log R'_{\\rm Kp} \\ltsim -5.0$) \nin our measurement,\nsuggesting that our $\\log R'_{\\rm Kp}$ has a higher sensitivity\nand thus advantageous. A similar situation holds regarding the \ncomparison with $r_{0}$(8542) we used in Paper II; i,e., this index \nstabilizes at $\\sim 0.2$ and becomes insensitive for low-active stars\nin contrast to $\\log R'_{\\rm Kp}$. \n\nAs another merit of using $\\log R'_{\\rm Kp}$, we can state\nthat the visibility of the $A$(Li)--activity relation as well as \nthe age--activity relation becomes comparatively clearer, because\nthis activity index turns out to have well diversified values\nfor low-activity stars thanks to its high sensitivity, which \ncan not be accomplished by using, e.g., $r_{0}$(8542).\n\nFrom the distribution histogram of $\\log R'_{\\rm Kp}$, we could\nrecognize a clear Vaughan--Preston gap between two peaks\nat $\\sim -5.3$ and $\\sim -4.3$.\nOur result of $\\log R'_{\\rm Kp,\\odot} = -5.33$ manifestly suggests \nthat the Sun belongs to the group of the former peak and has a \ndistinctly low-active nature among solar analogs. \nActually, a fraction of stars with \n$\\log R'_{\\rm Kp} \\le \\log R'_{\\rm Kp,\\odot}$ is only $\\sim 10\\%$.\nThis consequence exclude the possibility for the existence\nof a considerable fraction (e.g., $\\sim 1\/3$) of \n``Maunder-minimum stars'' such that having activities \nsignificantly lower than the current solar-minimum level as once \nsuggested by Baliunas and Jastrow (1990).\n\nYet, some stars (only a minor fraction of the sample) do exist \nshowing activities still lower than that of the solar-minimum level. \nHaving examined such 8 low-activity stars, we found that they tend to\ninclude planet-host stars and have low Li abundances, from which \nwe suspect that their activities are actually low as a result of \nintrinsically slow rotation. It would be an important task to \nclarify the behavior of their activity variations by long-term \nmonitoring observations.  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tLet $G$ be a group, the (complex) representation theory of $G$  has been largely studied because it encodes many properties associated to $G$.  Unfortunately, representation theory of $G$ could be difficult to describe in elemental terms directly. \n\t Presently, there is an interesting approach in order to describe the irreducible representations of a locally compact $G$ in terms of the projective representations of a normal subgroup $A$: the beautiful theory known as \\textit{Mackey machine} \\cite{Mac}.\n\n\t\n\tIn this paper we  generalize the Mackey machine's ideas to the context of finite dimensional $G$-vector bundles over proper $G$-spaces, with $G$ a Lie group and where the normal subgroup $A$ is compact and acts trivially on the base space obtaining a natural decomposition in equivariant K-theory. \n\t\n\tMore specifically, consider an extension $$1\\to A\\to G\\to Q\\to 1,$$ and let $\\Irr(A)$ the space of isomorphism classes of irreducible representations of $A$ (with the discrete topology), there is an action of $G$ (and $Q$) over $\\Irr(A)$ given by conjugation (see Section \\ref{sectiona}). Let $[\\rho]\\in \\Irr(A)$ and $G_{[\\rho]}$ (respectively $Q_{[\\rho]}$) be the isotropy group of $[\\rho]$ under the mentioned action, we construct $S^1$-central extensions \n\t(see Prop. \\ref{notwist})\n\t$$1\\to S^1\\to \\widetilde{Q}_{[\\rho]}\\to Q_{[\\rho]},$$and \t$$1\\to S^1\\to \\widetilde{G}_{[\\rho]}\\to G_{[\\rho]}.$$That codifies the obstruction to $\\rho$ can be lifted to an irreducible representation of $G_{[\\rho]}$. \n\t\n\tUsing both extensions we can decompose a $G$-vector bundle $E$ over a proper $G$-space $X$ where $A\\subseteq G$ is a compact normal subgroup acting trivially over $X$ as follows:\n\t\n\tThe bundle $E$ viewed as a $A$-vector bundle can be decomposed in isotypical parts in the following way (see \\cite{segal})\n\t\n\t$$E\\cong\\bigoplus_{[\\rho]\\in\\Irr(A)}\\rho\\otimes \\Hom_A(\\rho,E).$$\n\tThis is an isomorphism of $A$-vector bundles. Now the idea is to recover the $G$-action on the left hand side from an \\textit{ad hoc} $G$-action defined on the right hand side, this $G$ action is defined in two steps:\n\t\n\tFirst, we associate terms in the direct sum on the right hand side that are related by the $G$-action over $\\Irr(A)$, then the above decomposition can be expressed as\n\t\n\t$$E\\cong\\bigoplus_{[\\rho]\\in G\\setminus \\Irr(A)}\\left(\\bigoplus_{g\\in G_{[\\rho]}\/A}g\\cdot\\rho\\otimes \\Hom_A(g\\cdot\\rho,E)\\right).$$\n\t\n\tSecond, each of the terms $$\\bigoplus_{g\\in G_{[\\rho]}\/A}g\\cdot\\rho\\otimes\\Hom_{A}(g\\cdot \\rho,E)$$ can be completely recovered (as an induction) if we know only one of the terms in the sum (with its $G_{[\\rho]}$-action). This is the content of Thm. \\ref{vectordecomp}. Now if we choose one specific $\\rho\\otimes\\Hom_A(\\rho, E)$ it can be recovered (with its $G_{[\\rho]}$-action) from $\\Hom_A(\\rho, E)$ (with an action of some $S^1$-central extension of  $Q_{[\\rho]}$). It is the content of Thm. \\ref{toe}.\n\t\n\tFor a Lie group $G$ and a proper $G$-space $X$ where $A$ acts trivially such that proper equivariant K-theory (denoted by $K_G(X)$) can be generated by finite dimensional $G$-vector bundles (assumption (K)), the above procedure produces a decomposition of $K_G(X)$ in terms of equivariant (respect to the isotropy groups $Q_{[\\rho]}$, see Def. \\ref{twisteddef}) twisted K-theory groups of $X$. More specifically we have the following result \n\t\n\t\\begin{thm}\\label{decomp}\n\t\tLet $G$ be a Lie group satisfying (K), and $X$ be a proper $G$-space  on which the normal subgroup $A$ acts trivially. There is a natural isomorphism \n\t\t\\begin{align*}\\Psi_X:K^*_G(X)&\\to\\bigoplus_{[\\rho]\\in G\\setminus\\Irr(A)}{}^{\\widetilde{Q}_{[\\rho]}}K^*_{Q_{[\\rho]}}(X)\\\\ [E]&\\mapsto\\bigoplus_{[\\rho]\\in G\\setminus\\Irr(A)}[\\Hom_A(\\rho,E)].\\end{align*} This isomorphism is functorial on $G$-maps $X\\to Y$ of proper $G$- spaces on which $A$ acts trivially. \n\t\\end{thm}\n\t\n\t\n\tIt is the content of Thm. \\ref{decomp}. The main purpose of this paper is prove that decomposition in this generality. \n\t\n\n\t\n\tOur results generalize the decomposition obtained in \\cite{GU} for finite groups and in \\cite{AGU} for compact Lie groups. For finite groups, the same  decomposition can be found in \\cite{dual} using different methods.\n\t\nOn the other hand we also present some examples and applications of this decomposition, firstly, consider the antipodal action of $\\mathbb{Z}_2$ on $\\mathbb{S}^2$, we compute the $Q_8$-equivariant K-theory groups of $\\mathbb{S}^2\\times\\mathbb{S}^2$ where the action is given by the canonical projection $Q_8\\to \\mathbb{Z}_2\\times\\mathbb{Z}_2$, in this case there is a copy of $\\mathbb{Z}_2\\subseteq Q_8$ acting trivially on $\\mathbb{S}\n^2\\times\\mathbb{S}^2$, then we can apply Thm. \\ref{decomp}. In this case we obtain $$K_{Q_8}^*(\\mathbb{S}^2\\times\\mathbb{S}^2)\\cong K^*(\\mathbb{RP}^2\\times\\mathbb{RP}^2)\\oplus{}^\\alpha K^*(\\mathbb{RP}^2\\times \\mathbb{RP}^2),$$where $\\alpha$ is the unique (up to isomorphism) non trivial twisting over $\\mathbb{R} P^2\\times\\mathbb{R} P^2$. The first term of the direct sum on the left hand side is easily computed using the Kunneth formula for K-theory, the computation of the second term is non trivial (There is no a Kunneth formula for twisted K-theory useful for this case), for this we use a universal coefficient theorem for twisted K-theory and the Atiyah-Hirzebruch spectral sequence for twisted K-theory (with $\\mathbb{Z}_2$-coefficients). We also use some computations in \\cite{BaVe2016} to compute the equivariant K-theory of the classifying space for proper actions of the Steinberg group $\\operatorname{St}(3,\\mathbb{Z})$, this computations correspond the right hand side of the Baum-Connes conjecture (this conjecture is not know for $\\operatorname{St}(3,\\mathbb{Z})$). \n\nWe can use the above decomposition to study proper actions of  Lie group with only one isotropy type and obtain another natural decomposition, this is the content of Thm. \\ref{oneiso}, we use that decomposition to study the equivariant K-theory groups of  actions of $SU(2)$ on simply connected 5-dimensional manifolds with only one isotropy type and orbit space $S^2$.\n\nFinally, we use similar ideas to obtain a decomposition in equivariant connective K-homology for actions of compact Lie groups, for this we use the configuration space model as in \\cite{ve2019}, this is the content of Thm. \\ref{k-homdecomp}.\n\t\nThis paper is organized as follows: The Section \\ref{sectiona} is devoted to adapt the Mackey machine \\cite{Mac} to equivariant vector bundles. Theorem \\ref{decomp}  describes how any $G$-equivariant vector bundle over a topological space $X$ can be obtained as the sum of inductions of twisted equivariant (with respect to certain groups) vector bundles over $X$. In the Subsection \\ref{sectionaa}, we apply the induced decomposition of equivariant $K$-theory to examples coming from the quaternions and the Steinberg group. \n\nIn Section \\ref{sectionb} we study proper actions of Lie groups with only one isotropy type and apply the decomposition of equivariant $K$-theory of Section \\ref{sectiona}. We apply this decomposition to compute the equivariant $K$-theory of $5$-dimensional simply connected manifolds with actions of $SU(2)$ and only one isotropy type.\n\nIn Section \\ref{sectionc}, we prove an analogue decomposition as the presented in Section 2, for equivariant connective $K$-homology and actions of compact Lie groups on a topological space $X$. \n\nFinally, in the Appendix A, we make some additional proofs and remarks about the property $(K)$ in order to explain how this property is satisfied for a very general class of topological groups.\n\t\nThe second author thanks  the first and third author for being helpful in the development of the initial idea of this paper.   \n\\section{Mackey machine for equivariant vector bundles and compact normal subgroup} \\label{sectiona}\n\t\nIn \\cite{Mac}, G. W. Mackey presented a technique to describe the unitary representations of a locally compact group $G$ in terms of the  representations of a closed normal subgroup $A$. This technique --called \\emph{Mackey machine}-- is a powerful tool when the structure of the unitary representations of $A$ is known. In this paper we restrict ourselves  \nto the case when $A$ is also a compact group because the Mackey machine works particularly well and the representation theory of $A$ is suitable to be applied into the setup of equivariant $K$-theory. Following this approach, we proceed to explain how the natural action of $G$ on the discrete set of irreducible representations of $A$ implies the existence of twisted-vector bundles parametrized by a family of subgroups of $G\/A$. In order to obtain useful applications to $K$-theory, we present here an exposition of how the Mackey machine can be generalized for equivariant vector bundles if the compact normal subgroup $A$ acts trivially on the space $X$. Throughout this section we suppose that $G$ is a Lie group and $A$ is a compact normal subgroup of $G$.\n\t\n\n\t\n\t\n\tLet $$1 \\to A\\to G\\to Q\\to 1$$ be the corresponding extension with $Q:=G\/A$. We set $\\Irr(A)$ to be the set of unitary equivalence classes of irreducible representations of $A$ endowed with the Fell topology (\\cite{Kaniuth}, Section 1). By compactness of $A$, $\\Irr(A)$ is a discrete set, see \\cite{Kaniuth}, Proposition 1.70. Let \n\t$$\n\t\\rho:A\\to U(V_\\rho),\n\t$$\n\tbe an irreducible representation of $A$ on the vector space $V_{\\rho}$ which is supposed to have an $A$-invariant metric. Let $g\\in G$. We define another irreducible representation of $A$ by\\begin{align*}g\\cdot\\rho:A&\\to U(V_\\rho)\\\\a&\\mapsto\\rho(g^{-1}ag).  \\end{align*}resulting in a left action of $G$ over $\\Irr(A)$. \n\t\n\tLet $G_{[\\rho]}$ be the isotropy group of the isomorphism class of $[\\rho] \\in \\Irr(A)$ under the action of $G$. This means that for each $g \\in G_{[\\rho]}$ there is a unitary matrix $U_g$ such that, for all $a\\in A$\n\t$$\n\t\\rho(g^{-1}ag) = U_g^{-1}\\rho(a)U_g.\n\t$$\n\tFor a fixed $[\\rho] \\in \\Irr(A)$ and $x\\in A$, $a\\cdot\\rho(x)=\\rho(x^{-1}ax)=\\rho(x^{-1})\\rho(a)\\rho(x)=\\rho(a)$ for all $a\\in A$. Take $U_x=\\rho(x)$. It follows that $A\\subseteq G_{[\\rho]}$, and it is possible to define $Q_{[\\rho]}=G_{[\\rho]}\/A$. Therefore, there is an extension:\n\t$$\n\t1 \\to A \\stackrel{\\iota  }{\\rightarrow} G_{[\\rho]} \\stackrel{}{\\rightarrow} Q_{[\\rho]} \\rightarrow  1.\n\t$$\n\t\n\t\\begin{remark}\n\t\tIf $A$ is central in $G$, then the action of $G$ on $\\Irr(A)$ is trivial, $G_{[\\rho]}=G$ and $Q_{[\\rho]}=Q$.\n\t\\end{remark}\n\t\n\t\n\t\n\tFor any pair $g,h\\in G_{[\\rho]}$, and for all $a\\in A$:$$U_{(gh)^{-1}}\\rho(a)U_{gh}=U_{h^{-1}}U_{g^{-1}}\\rho(a)U_{g}U_{h}.$$By straightforward calculations, the equation above can be rewritten as:$$U_{gh}U_{h^{-1}}U_{g^{-1}}\\rho(a)=\\rho(a)U_{gh}U_{h^{-1}}U_{g^{-1}}.$$\n\tAs $\\rho$ is supposed to be irreducible, by Schur's lemma $U_{gh}$ must be a multiple of $U_{g}U_{h}$ and therefore there is a well defined homomorphism  $\\Upsilon:G_{[\\rho]}\\to PU(V_\\rho)$ that extends $\\rho: A \\rightarrow U(V_\\rho)$. Summarizing, we have the following result.\n\t\\begin{lemma}\\label{lema41}\n\t\tThere is a unique homomorphism $\\Upsilon:G_{[\\rho]}\\to PU(V_\\rho)$ such that the following diagram commutes\n\t\t$$\\xymatrix{A\\ar[r]^\\iota \\ar[d]^\\rho&G_{[\\rho]}\\ar[d]^{\\Upsilon}\\{\\mathcal U}(V_\\rho)\\ar[r]^p&PU(V_\\rho).}$$\n\t\\end{lemma}\n\t\n\t\n\tRecall the canonical central extension\n\t\\[\n\t1  \\to S^{1} \\stackrel{i}{\\rightarrow} U(V_{\\rho})\\stackrel{p}{\\rightarrow} PU(V_{\\rho})\\to 1\n\t\\]\n\twhere $PU(V_\\rho) = \\Inn(U(V_\\rho))$ and $p(u)$ is the conjugation by $u$.\n\t\n\t\n\tDefine the Lie group  $\\widetilde{G}_{[\\rho]} : = G_{[\\rho]} \\times_{PU(V_\\rho)} U(V_\\rho) = \\{ (g,u) \\mid {\\Upsilon}(g) = p(u)\\}$. There is a central extension\n\t\\[\n\t1 \\to S^{1} \\stackrel{i}{\\rightarrow} \\widetilde{G}_{[\\rho]} \\stackrel{\\pi_1}{\\rightarrow}  G_{[\\rho]} \\to 1, \n\t\\]\n\twhere $i : S^1 \\rightarrow \\widetilde{G}_{[\\rho]}$ is given by the inclusion $1 \\times S^1 \\subseteq G_{[\\rho]} \\times U(V_\\rho)$, which actually lands in $\\widetilde{G}_{[\\rho]}$, because $p(S^1)=1$. \n\t\n\t\n\t\n\tThere is a commutative diagram of $S^1$-central extensions of Lie groups\n\t$$\\xymatrix{1\\ar[r]&S^1\\ar[r]\\ar[d]^{id}&\\widetilde{G}_{[\\rho]}\\ar[r]^{\\pi_1} \\ar[d]^{\\widetilde{\\Upsilon}}&G_{[\\rho]}\\ar[r]\\ar[d]^{\\Upsilon}&1\\\\1\\ar[r]&S^1\\ar[r]&U(V)\\ar[r]&PU(V)\\ar[r]&1}$$\n\t\n\t\n\tThe homomorphism $ \\iota \\times \\rho :  A \\rightarrow G_{[\\rho]} \\times U(V_\\rho)$ has the image contained in $\\widetilde{G}_{[\\rho]}$. Lets see that $(\\iota \\times \\rho) (A) $ is normal in $\\widetilde{G}_{[\\rho]}$. To proof this we need to check that if $a \\in A$ and $(g,u)  \\in G_{[\\rho]} \\times_{PU(V_\\rho)} U(V_\\rho) $ then \n\t$$\n\t(g,u) (\\iota \\times \\rho ) (a)  (g,u)^{-1} \\in (\\iota \\times \\rho) (A). \n\t$$\n\tWe have,\n\t$$\n\t(g,u) (\\iota \\times \\rho ) (a)  (g,u)^{-1} = ( gag^{-1},u \\rho(a) u^{-1})\n\t$$\n\tsince $(g,u) \\in G_{[\\rho]} \\times_{PU(V_\\rho)} U(V_\\rho)$, we have that $\\Upsilon(g) = p(u)$, i.e \n\t$$\n\t\\rho(gag^{-1} ) = u \\rho(a) u^{-1}\n\t$$\n\tand therefore\n\t$$\n\t(g,u) (\\iota \\times \\rho ) (a)  (g,u)^{-1} =  (\\iota \\times \\rho ) ( gag^{-1})\n\t.$$This prove that $(\\iota\\times\\rho)(A)$ is normal in $\\widetilde{G}_\\rho$.\n\t\n\t\n\tThus the quotient $\\widetilde{G}_{[\\rho]}\/ (\\iota \\times \\rho) (A)$\n\tis a Lie group and denoted by $\\widetilde{Q}_{[\\rho]}$ and there is a sequence\n\t$$\n\t1 \\to A \\stackrel{\\iota \\times \\rho }{\\rightarrow} \\widetilde{G}_{[\\rho]} \\stackrel{}{\\rightarrow} \\widetilde{Q}_{[\\rho]} \\rightarrow  1.\n\t$$\n\t\n\tWe want to see now that there is a central extension\n\t\\begin{equation} \\label{centralexquot}\n\t1 \\to S^{1} \\stackrel{\\widetilde{i}}{\\rightarrow} \\widetilde{Q}_{[\\rho]} \\rightarrow Q_{[\\rho]} \\to 1\n\t\\end{equation}\n\t\n\twhere $\\widetilde{\\iota} : S^1 \\rightarrow \\widetilde{G}_{[\\rho]}\/ (\\iota \\times \\rho) (A)$ is induced by  $1 \\times S^1 \\subseteq G_{[\\rho]} \\times_{PU(V_\\rho)} U(V_\\rho)$.\n\t\n\t\n\tNote that $ (\\{e \\} \\times S^1) \\cap  (\\iota \\times \\rho) (A) = \\{e \\}$ and therefore \n\t$$\n\tS^{1} \\stackrel{i}{\\rightarrow} \\widetilde{G}_{[\\rho]} \\rightarrow \\widetilde{G}_{[\\rho]}\/ (\\iota \\times \\rho) (A)\n\t$$\n\tis injective and gives a map $\\widetilde{\\iota} : S^1 \\rightarrow \\widetilde{G}_{[\\rho]}\/ (\\iota \\times \\rho) (A)=\\widetilde{Q}_{[\\rho]} $. \n\t\n\tThe image of $\\widetilde{\\iota}$ can be written as $(\\{e \\} \\times S^1)(\\iota \\times \\rho) (A)\/ (\\iota \\times \\rho) (A)$ (second isomorphism theorem), and the quotient $\\widetilde{Q}_{[\\rho]} \/ S^1$ is:\n\t\n\t$$\n\t\\frac{\\frac{\\widetilde{G}_{[\\rho]}}{ (\\iota \\times \\rho) (A)}}{ \\frac{ (\\{e \\} \\times S^1)(\\iota \\times \\rho) (A)}{(\\iota \\times \\rho) (A)}  }  \\cong \\frac{ \\widetilde{G}_{[\\rho]} } { (\\{e \\} \\times S^1)(\\iota \\times \\rho) (A)} \\cong  \\frac{ \\frac{ \\widetilde{G}_{[\\rho]}} { \\{e \\} \\times S^1} }  { \t\\frac{(\\{e \\} \\times S^1)(\\iota \\times \\rho) (A)}{ \\{e \\} \\times S^1}} \\cong \\frac{G_{[\\rho]}}{\\iota(A)} \\cong Q_{[\\rho]}.   \n\t$$\n\t\n\t\n\tThere is a commutative diagram of extensions of Lie groups\n\t$$\\xymatrix{1 \\ar[r]&A\\ar[r]\\ar[d]^{id}& G_{[\\rho]}\\ar[r] \\ar[d]^{}& Q_{[\\rho]} \\ar[r]\\ar[d]& 1 \\\\ 1 \n\t\t\\ar[r]&A\\ar[r] &\\widetilde{G}_{[\\rho]}\\ar[r]^{\\pi_1} &\\widetilde{Q}_{[\\rho]} \\ar[r] & 1 \n\t}$$\n\tand also a commutative diagram of $S^1$-central extensions of Lie groups\n\t$$\\xymatrix{1 \\ar[r]&S^1\\ar[r]\\ar[d]^{id}&\\widetilde{G}_{[\\rho]}\\ar[r]^{\\pi_1} \\ar[d]^{}&G_{[\\rho]}\\ar[r]\\ar[d]& 1 \\\\\n\t\t1\\ar[r] &  S^{1}  \\ar[r]^{\\widetilde{i}} &  \\widetilde{Q}_{[\\rho]} \\ar[r] & Q_{[\\rho]} \\ar[r] & 1.\n\t}$$\n\t\n\tNow we will study when these extensions are trivial.\n\t\n\t\\begin{remark}\n\t\tIf $\\rho$ is a $1$-dimensional irreducible representation of $A$, then the $S^1$-extension \n\t\t\\[\n\t\t1 \\to S^{1} \\stackrel{i}{\\rightarrow} \\widetilde{G}_{[\\rho]} \\stackrel{\\pi_1}{\\rightarrow}  G_{[\\rho]} \\to 1, \\]\n\t\tis trivial. This is because in this case $PU(V_\\rho)=1$.\n\t\t\n\t\t\n\t\t\n\t\\end{remark}\n\t\n\tExactly as in \\cite{AGU}, we have:\n\t\n\t\\begin{prop} \\label{notwist}\n\t\tThe irreducible representation $\\rho$ can be extended to $G_{[\\rho]}$ if and only if the $S^1$-central extension of Lie groups \n\t\t$$1 \\to S^1\\to \\widetilde{Q}_{[\\rho]}\\to Q_{[\\rho]} \\to 1$$\n\t\tis trivial.\n\t\\end{prop}\\begin{proof}\n\t\tSuppose that the above central extension is trivial, then $$1\\to S^1\\to\\widetilde{G}_{[\\rho]}\\to G_{[\\rho]}\\to1$$ is also trivial.  Since $\\widetilde{G}_{[\\rho]}\\cong G_{[\\rho]}\\times S^1$ as Lie groups, it means that there is a Lie group homomorphism $\\sigma:G_{[\\rho]}\\to\\widetilde{G}_{[\\rho]}$ that is a right inverse of the quotient map. Let $\\widetilde{\\Upsilon}:\\widetilde{G}_{[\\rho]}\\to U(V_\\rho)$ be the associated representation to ${\\Upsilon}:G_{[\\rho]}\\to PU(V_\\rho)$. $\\widetilde{\\Upsilon}\\circ\\sigma:G_{[\\rho]}\\to U(V_\\rho)$ is an extension of $\\rho$, and for $a\\in A$\\begin{align*}\n\t\t\\widetilde{\\Upsilon}(\\sigma(i(a)))&=\\widetilde{\\Upsilon}(\\widetilde{i}(a))\\\\ &=\\rho(a),\n\t\t\\end{align*}\n\t\twhere the last equality follows by the definition of $\\widetilde{i}$.\n\t\t\n\t\tOn the other hand, if $\\widetilde{\\rho}:G_{[\\rho]}\\to U(V_\\rho)$ is an extension of $\\rho$ we can take $\\pi\\circ\\widetilde{\\rho}:G_{[\\rho]}\\to PU(V_\\rho)$ as a definition of ${\\Upsilon}$ in Lemma \\ref{lema41}. Recall that $$\\widetilde{G}_{[\\rho]}=\\{(g,\\chi)\\in G_{[\\rho]}\\times U(V_\\rho)\\mid \\pi(\\widetilde{\\rho}(g))=\\pi(\\chi)\\}.$$ Define \\begin{align*}\\sigma:G_{[\\rho]}&\\to\\widetilde{G}_{[\\rho]}\\{\\mathfrak{genus}}_G&\\mapsto(g,\\widetilde{\\rho}(g)).\\end{align*}\n\t\tThen $\\widetilde{G}_{[\\rho]}\\cong G_{[\\rho]}\\times S^1$ as Lie groups. It implies that $\\widetilde{Q}_{[\\rho]}\\cong Q_{[\\rho]}\\times S^1$ as Lie groups and thus the extension is trivial\n\t\\end{proof}\n\tWhen the exact sequence in Proposition \\ref{notwist} does not represent a trivial $S^1$-central extension of Lie groups, we refer to it as being the twisting associated with the irreducible representation.\n\t\n\t\n\t\\subsubsection{Semidirect products with abelian normal subgroup}\n\t\n\tNow we will study the case of a semidirect product. As it is usual in representation theory, for a  group $G$, we denote by $$\\widehat{G}=\\{f:G\\to \\mathbb{C}\\mid f\\text{ is constant on conjugacy classes}\\},$$ the space of characters of $G$ (with the compact open topology).\n\t\n\tLet $A \\mathrel{\\unlhd} G$ be a normal subgroup of $G$. Then the group $G$ acts on $\\widehat{A}$ by taking a character $\\chi$ of $A$, and defining $(g \u00b7 \\chi)(a) = \\chi(gag^{-1})$ where $g\\in G$. For a character $\\gamma \\in \\widehat{G}$, consider its restriction to the subgroup $A\\subset G$ as an element $\\gamma|_A\\in \\widehat{A}$. If $\\chi=\\gamma|_A$ for some $\\gamma\\in\\widehat{G}$, then for every $g\\in G$, $g \\cdot \\chi =\\chi$. Thus the restriction homomorphism $\\widehat{G} \\rightarrow \\widehat{A}$ has its image in the fixed point group $\\widehat{A}^G$. \n\t\\begin{prop}\n\t\tIf $G = A \\rtimes Q$, then there is a group isomorphism\n\t\t$$\n\t\t\\Gamma: \\widehat{G} \\rightarrow {\\widehat{A}}^Q \\times \\widehat{Q},\n\t\t$$given by restriction.\n\t\t\\begin{proof}\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t$\\Gamma$ is injective because it is the restriction to both $A$ and $Q$, and because $G=AQ$. To see that it is surjective, take $\\alpha \\in {\\widehat{A}}^Q $ and $\\beta \\in \\widehat{Q}$ and consider the product $\\alpha \\rtimes \\beta$ defined for $g\\in G$ as follows: there is a unique pair $(a,q)\\in A\\times Q$ such that $g=aq$; define  $\\alpha\\rtimes\\beta(g)=\\alpha(a)\\beta(q)$. Let us see that $\\alpha\\rtimes\\beta$ is a character.\n\t\t\t$$\n\t\t\t\\begin{aligned} \n\t\t\t\\alpha \\rtimes \\beta \\left( h_1 q_1 h_2 q_2 \\right) & = \\alpha \\rtimes \\beta \\left( h_1 q_1 h_2 q_1 ^ { - 1 } q_1 q_2 \\right) \\\\\n\t\t\t& = \\alpha \\left( h_1 q_1 h_2 q_1^ { - 1 } \\right) \\beta \\left( q_1 q_2 \\right) \\\\\n\t\t\t& = \\alpha ( h_1 ) \\alpha \\left( q_1 h_2 q_1 ^ { - 1 } \\right) \\beta ( q_1 ) \\beta \\left( q_2 \\right) \\\\ \n\t\t\t& = \\alpha ( h_1 ) \\alpha \\left( h_2 \\right) \\beta ( q_1 )  \\beta \\left( q_2 \\right) \\\\\n\t\t\t& = \\alpha ( h_1 ) \\beta ( q_1 ) \\alpha \\left( h_2 \\right) \\beta \\left( q_2 \\right) \\\\\n\t\t\t& = \\left (\\alpha \\rtimes \\beta \\right) ( h_1 q_1 )  \\left (\\alpha \\rtimes \\beta \\right) \\left( h_2 q_2 \\right). \n\t\t\t\\end{aligned}\n\t\t\t$$Clearly $\\Gamma(\\alpha\\rtimes\\beta)=(\\alpha,\\beta)$.\n\t\t\\end{proof}\n\t\\end{prop}\n\t\n\t\\begin{prop}\n\t\tLet $A$ be an abelian group and $G = A \\rtimes Q$. Then for every irreducible representation $\\rho$ of $A$, the $S^1$-central extension of Lie groups \n\t\t$$1 \\to S^1\\to \\widetilde{Q}_{[\\rho]}\\to Q_{[\\rho]} \\to 1$$\n\t\tis trivial.\n\t\t\\begin{proof}\n\t\t\tSuppose $A$ is abelian, and take $\\rho \\in \\widehat{A}$. For $g\\in G_{[\\rho]}$ \n\t\t\t$$\n\t\t\t\\rho(g^{-1}ag) =U_g^{-1}\\rho(a)U_g = \\rho(a)\n\t\t\t$$\n\t\t\tsince $U_g$, $\\rho(a)$ and $U_{g}^{-1}$ are elements of  $S^1$; therefore every character of $A$ is invariant under $G_{\\rho}$. Also, $G_{[\\rho]} = A \\rtimes Q_{[\\rho]}$ since clearly $G_{[\\rho]} \\supseteq A \\rtimes Q_{[\\rho]}$ and for $(\\widetilde{a},q) \\in G_{[\\rho]}$\n\t\t\t$$\n\t\t\t\\rho(a) = \\rho((\\widetilde{a}q)^{-1}a (\\widetilde{a}q)) = \\rho(q^{-1}\\widetilde{a}^{-1}a \\widetilde{a}q) = \\rho(q^{-1}aq)\n\t\t\t$$\n\t\t\tand therefore $q \\in Q_{[\\rho]}$.\n\t\t\t\n\t\t\tTaking the trivial character of $Q_{[\\rho]}$ results in a character of $G_{[\\rho]}$ that extends $\\rho$. By  Proposition \\ref{notwist} we are done.\n\t\t\t\n\t\t\t\n\t\t\\end{proof}\n\t\\end{prop}\n\t\n\tNext, we describe the Mackey machine for equivariant vector bundles when the normal subgroup is compact.\n\t\n\t\n\t\\subsection{Decomposition of $G$-vector bundles}\n\tLet $E$ be a $G$-vector bundle over a $G$-space $X$, and let $A$ be a compact normal subgroup of $G$ acting trivially over $X$. Since $A$ is compact we can assume that $E$ is endowed with an $A$-invariant Hermitian metric. Each fiber is therefore a unitary representation of $A$. By a result in \\cite{segal} we know that the restriction $E\\mid_A$ considered as an $A$-vector bundle can be decomposed into isotypical parts  $$E\\mid_A \\quad \\cong\\bigoplus_{[\\rho]\\in \\Irr(A)}  \\rho \\otimes \\Hom_A(\\rho,E),$$where $\\rho$ is considered as the trivial $A$-bundle $X \\times V_\\rho \\rightarrow X$ and $\\Hom_A(E_1,E_2)$ is the bundle of $A$-equivariant maps between two $A$-bundles $E_1,E_2$ endowed with the trivial $A$-action. Let $E_\\rho=\\rho\\otimes\\Hom_A(\\rho,E)$ and call it the \\emph{$\\rho$-isotypical part of $E$}.\n\t\\begin{defn}\n\t\tLet $X$ be a $G$-space, where the action of $A$ is trivial and $\\rho:A\\to U(V)$ is an irreducible $A$-representation. A $G_{[\\rho]}$-vector bundle $p:E\\to X$\n\t\tis $(G_{[\\rho]},\\rho)$-isotypical if each fiber $E_x$ is an $A$-representation isomorphic to a multiple of $\\rho$.\n\t\\end{defn}\n\t\n\n\t\n\tAnother way to say that each fiber $E_x$ is an $A$-representation isomorphic to a multiple of $\\rho$ is that the map\n\t\\begin{align*}\n\t\\beta:E_\\rho&\\to E\\\\\n\tv\\otimes f&\\mapsto f(v)\n\t\\end{align*}\n\tis an isomorphism of $A$-vector bundles.\\\\\n\t\n\tAs in \\cite{AGU}, $(G_{[\\rho]},\\rho)$-isotypical vector bundles are in correspondence with $\\widetilde{Q}_{[\\rho]}$-vector bundles on which $S^1$-acts by multiplication by inverses. The  proof of \\cite{AGU} did not depend on the compactness of $G$. We include it below for completeness.\n\t\n\t\\begin{thm}\\label{toe}\n\t\tLet $X$ be a $G$-space where $A$ acts trivially on $X$.  There is a natural one to one correspondence between isomorphism classes of $(G_{[\\rho]},\\rho)$-isotypical vector bundles over $X$ and isomorphism classes of $\\widetilde{Q}_{[\\rho]}$-equivariant vector bundles over $X$ on which $S^1$-acts by multiplication by inverses. \n\t\t\n\t\\end{thm}\n\t\\begin{proof}\n\t\tSuppose that $p:E\\to X$ is a $(G_{[\\rho]},\\rho)$-isotypical vector bundle over $X$. Then $\\Hom_A(\\rho,E)$ is a complex vector bundle. In order to get a more readable proof we proceed to divide it in steps.\n\t\t\n\t\tStep 1: We keep the notation in Lemma \\ref{lema41} and endow $\\Hom_A(\\rho,E)$  with a linear $\\widetilde{G}_{[\\rho]}$-action in which ${\\Upsilon}^*(A)$ acts trivially.\n\t\t\n\t\tLet $\\phi\\in\\Hom_A(\\rho,E)$ and $\\widetilde{g}\\in\\widetilde{G}_{[\\rho]}$. Define\n\t\t$$(\\widetilde{g}\\bullet\\phi)(v)=\\tau(\\widetilde{g})\\phi\\left(\\left(\\widetilde{\\Upsilon}\\left(\\widetilde{g}\\right)\\right)^{-1}v\\right)$$ where $\\tau:\\widetilde{G}_{[\\rho]}\\to G_{[\\rho]}$ is the natural projection. This action is continuous. \n\t\t\n\t\tLet $\\widetilde{a}\\in \\tau^{-1}(A)$ with $\\tau(\\widetilde{a})=a$. Then\\begin{align*}\\left(\\widetilde{a}\\bullet\\phi\\right)(v)&=a\\left(\\left(\\phi(\\widetilde{\\Upsilon}\\left(\\widetilde{a}\\right)\\right)^{-1}v\\right) \\\\&=a\\phi\\left(\\rho(a)^{-1}v\\right)\\\\&=\\phi(v).\\end{align*}Thus the action of $\\tau^{-1}(A)$ is trivial, resulting in an action of $\\widetilde{Q}_{[\\rho]}.$\n\t\t\n\t\tStep 2: Show that the action of $S^1=\\ker(\\tau)$ acts by multiplication by  inverses.\n\t\t\n\t\tFor $\\lambda\\in S^1=\\ker(\\tau)$,\n\t\t$$(\\lambda\\bullet\\phi)(v)=\\phi\\left(\\widetilde{\\Upsilon}(\\lambda)^{-1}v\\right)=\\phi\\left(\\lambda^{-1}v\\right)=\\lambda^{-1}\\phi(v).$$\n\t\t\n\t\tDefine a transformation$$[E]\\mapsto[\\Hom_A(\\rho,E)]$$between isomorphism classes of $(G_{[\\rho]},\\rho)$-isotypical vector bundles over $X$ and isomorphism classes of $\\widetilde{Q}_{[\\rho]}$-equivariant vector bundles over $X$  for which $S^1$-acts by multiplication by inverses. We will see that this transformation is a bijective correspondence.\n\t\t\n\t\tLet $p:F\\to X$ be a $\\widetilde{Q}_{[\\rho]}$-vector bundle on which $S^1$-acts by multiplication by its inverses. Consider the vector bundle $\\rho\\otimes F$.\n\t\t\n\t\tStep 3: Endow  $\\rho\\otimes F$ with a linear $G_{[\\rho]}$-action.\n\t\t\n\t\tDefine a $\\widetilde{G}_{[\\rho]}$-action, let $\\widetilde{g}\\in \\widetilde{G}_{[\\rho]}$ and $v\\otimes f\\in\\rho\\otimes F$\n\t\t\n\t\t$$\\widetilde{g}\\cdot(v\\otimes f)=\\left(\\widetilde{\\Upsilon}\\left(\\widetilde{g}\\right)v\\right)\\otimes(\\widetilde{g}\\tau^{-1}(A)\\cdot f)$$ where $\\widetilde{g}\\tau^{-1}(A)$ is the coset corresponding to $\\widetilde{g}$ in $\\widetilde{Q}_{[\\rho]}.$ This action is continuous. Moreover, if $\\lambda\\in S^1=\\ker(\\tau)$,\n\t\t$$\\lambda\\cdot(v\\otimes f)=\\left(\\widetilde{\\Upsilon}(\\lambda)v\\right)\\otimes\\left(\\left(\\lambda\\tau^{-1}(A)\\right)\\cdot f\\right)=\\lambda v\\otimes\\lambda^{-1}f=v\\otimes f.$$\n\t\tThus $\\ker(\\tau)$ acts trivially, and there is an action of $G_{[\\rho]}=\\widetilde{G}_{[\\rho]}\/S^1$.\n\t\t\n\t\tStep 4: $\\rho\\otimes F$ is a $(G_{[\\rho]},\\rho)$-isotypical vector bundle. \n\t\t\n\t\tLet $a\\in A$, and let $\\widetilde{a}\\in \\tau^{-1}(a)$,\n\t\t$$a\\cdot(v\\otimes f)=\\widetilde{a}(v\\otimes f)=\\left(\\widetilde{\\Upsilon}\\left(\\widetilde{a}\\right)v\\right)\\otimes\\left(\\widetilde{a}\\tau^{-1}(A)\\cdot f\\right)=\\rho(a)v\\otimes f.$$\n\t\tThis implies that $\\rho\\otimes F$ is a $(G_{[\\rho]},\\rho)$-isotypical vector bundle.\n\t\t\n\t\tDefine a transformation\n\t\t\n\t\t$$[F]\\mapsto[\\rho\\otimes F]$$\n\t\tbetween isomorphism classes of $\\widetilde{Q}_{[\\rho]}$-equivariant vector bundles over $X$  for which $S^1$ acts by multiplication by inverses and isomorphism classes of  $(G_{[\\rho]},\\rho)$-isotypical vector bundles over $X$. We will see that this transformation is the inverse of the transformation defined in Step 2.\n\t\t\n\t\tStep 5: $\\rho\\otimes\\Hom_A(\\rho,E)\\cong E$ as $(G_{[\\rho]},\\rho)$-isotypical vector bundles. \n\t\t\n\t\tNote that \\begin{align*}\n\t\tev:\\rho\\otimes\\Hom_A(\\rho,E)&\\to E\\\\v\\otimes\\phi&\\mapsto\\phi(v)\n\t\t\\end{align*}\n\t\tis an isomorphism of vector bundles. We will see that $ev$ is $G_{[\\rho]}$-equivariant.\n\t\t\n\t\tLet $g\\in G_{[\\rho]}$ and $\\widetilde{g}\\in\\widetilde{G}_{[\\rho]}$ with $\\tau(\\widetilde{g})=g$.\n\t\t\\begin{align*}\n\t\tev(g\\cdot(v\\otimes\\phi))&=ev\\left(\\widetilde{g}\\cdot(v\\otimes\\phi\\right))\\\\&=ev\\left(\\left(\\widetilde{\\Upsilon}(\\widetilde{g})v\\right)\\otimes\\left(\\widetilde{g}\\bullet\\phi\\right)\\right)\\\\&=\\left(\\widetilde{g}\\bullet\\phi\\right)\\left(\\widetilde{\\Upsilon}\\left(\\widetilde{g}\\right)v\\right)\\\\&=g\\phi\\left(\\widetilde{\\Upsilon}\\left(\\widetilde{g}\\right)^{-1}\\widetilde{\\Upsilon}\\left(\\widetilde{g}\\right)v\\right)\\\\&=g\\phi(v).\n\t\t\\end{align*}\n\t\tThen $\\rho\\otimes\\Hom_A(\\rho,E)\\cong E$ as $(G_{[\\rho]},\\rho)$-isotypical vector bundles.\n\t\t\n\t\tStep 6: $F\\cong\\Hom_A(\\rho,\\rho\\otimes F)$ as $\\widetilde{Q}_{[\\rho]}$-equivariant vector bundles over $X$  for which $S^1$-acts by multiplication by inverses.\n\t\t\n\t\tAs $A$ acts trivially on $F$ it is the case that \\begin{align*}\n\t\tF&\\to\\Hom_A(\\rho,\\rho\\otimes F)\\\\f&\\mapsto (v\\mapsto v\\otimes f)\n\t\t\\end{align*}\n\t\tis an isomorphism of $\\widetilde{Q}_{[\\rho]}$-vector bundles. Therefore the correspondence defined in Step 2 is one to one.\n\t\\end{proof}\n\t\n\tNow using the above theorem we will decompose every $G$-vector bundle over an $A$-trivial $G$-space in terms of equivariant $A$-vector bundles, which turn out to be twisted $Q_{[\\rho]}$-equivariant vector bundles (see Theorem \\ref{decomp} below). To do that, we need to recall the construction of the induced vector bundle.\n\t\n\t\\subsection{Induction}\n\t\n\tLet $H\\subseteq G$ be a subgroup of $G$ of finite index and $E\\xrightarrow{\\pi} X$ an $H$-vector bundle over a $G$-space $X$. If we choose an element from each left coset of $H$, we obtain a subset $R$ of $G$ called a \\textit{system of representatives of $G\/H$}; each $g\\in G$ can be written uniquely as $g=rs$, with $r\\in R=\\{r_1,\\ldots,r_n\\}$ and $s\\in H$, $G=\\coprod_{i=1}^nr_iH$. We assume that $r_1$ is the identity of the group $G$. Consider the vector bundle $F=\\bigoplus_{i=1}^n(r_i^{-1})^*E$, with projection $\\pi_F:F\\to X$, and consider the following $G$-action defined over $F$.\n\t\n\n\tLet $f\\in F$. Then $$f=f_{r_1}\\oplus\\cdots\\oplus f_{r_n},$$ where $f_{r_i}\\in(r_i^{-1})^*E$. If $\\pi_F(f)=x$ then $f_{r_i}=(x,e)$, with $e\\in E_{r_i^{-1}x}$.\n\n\tLet  $g\\in G$; note that $gr_i$ is in the same right coset as some $r_j$, i.e. $gr_i=r_js$ for some $s\\in H$. Define $$g(f_{r_i})=(gx,se)\\in (r_j^{-1})^*E,$$ and extend the action of $g$ on $f$ by linearity.\n\t\n\tWe will denote the $G$-vector bundle $F$ defined above by $\\operatorname{Ind}_H^G(E)$. The properties of $\\operatorname{Ind}_H^G(E)$ can be summarized in the following theorem.\n\t\n\t\\begin{thm}[Frobenius reciprocity] \\label{inductionprop}\n\t\n\t\\end{thm}\n\t\\begin{proof} The argument in Thm. 2.9 in \\cite{CRV} applies to any Lie group. \n\t\\end{proof}\n\n\tIn particular, note that each $(r_i^{-1})^*E$ has a natural structure of $r_iHr_i^{-1}$-vector bundle because if $g\\in r_iHr_i^{-1}$, then $g=r_ihr_i^{-1}$ with $h\\in H$. If $(x,e)\\in (r_i^{-1})^*E$, then\n\t$$g\\cdot(x,e)=(gx,he),$$hence $g\\cdot(x,e)\\in (r_i^{-1})^*E$.\n\t\n\t\n\tAs the vector bundle $E$ is finite-dimensional there is only a finite number of isomorphism classes $[\\rho]$ such that $$\\Hom_A(\\rho,E)\\neq0.$$ Moreover if $\\Hom_A(\\rho,E)\\neq0$, for every $g\\in G$ we have $\\Hom_A(g\\cdot\\rho,E)\\neq0.$ Let $\\phi:(V,\\rho)\\to E$ be a non trivial $A$-equivariant homomorphism. Then \\begin{align*}\n\t\\widetilde{\\phi}:(V,g\\cdot\\rho)&\\to E\\\\\n\tv&\\mapsto g\\phi(v)\n\t\\end{align*}\n\tis a nontrivial element in $\\Hom_A(g\\cdot\\rho,E)$. Hence, if $[\\rho]$ appears in $E$, then every element in its orbit appears in $E$. This implies that $G_{[\\rho]}$ has finite index in $G$, since $G\/G_{[\\rho]}\\cong G\\cdot[\\rho]$, the orbit of $[\\rho]$. Therefore we can apply the induction defined previously.\n\t\n\tLet $R=\\{r_1,\\ldots,r_n\\}$ be a system of representatives of $G\/G_{[\\rho]}$. There is a canonical isomorphism of $G_{[r_i\\cdot\\rho]}$-vector bundles $$(r_i^{-1})^*E_\\rho\\cong E_{r_i\\cdot\\rho}$$\n\tdefined as follows. Let $\\xi=(x,\\phi)\\in(r_i^{-1})^*E_\\rho$, with $\\phi\\in\\Hom_A(\\rho,E_{r_i^{-1}x})$, and define\n\t\\begin{align*}\n\t\\widetilde{\\phi}:(V,r_i\\cdot\\rho)&\\to E_x\\\\v&\\mapsto r_i\\phi(v).\n\t\\end{align*}\n\t$\\widetilde{\\phi}$ is $A$-equivariant,\\begin{align*}\n\t\\widetilde{\\phi}(a\\cdot v)&=\\widetilde{\\phi}(r_i^{-1}ar_iv)\\\\&=r_i\\phi(r_i^{-1}ar_iv)\\\\&=ar_i\\phi(v)=a\\widetilde{\\phi}(v).\n\t\\end{align*}\n\tFinally, note that the bundle map\n\t\\begin{align*}\n\t\\operatorname{Ind}_{G_{[\\rho]}}^G(E_\\rho)&\\to E\\\\(x,v\\otimes\\phi)\\in (r_i^{-1})^*(E_\\rho)&\\mapsto r_i\\phi(v)\\end{align*}is $G$-equivariant. Thus we have proved the following theorem:\n\t\\begin{thm}\\label{vectordecomp}Let $G$ be a Lie group and $X$ be a $G$-space. Suppose that $A$ is a compact normal subgroup of $G$ acting trivially over $X$. Let $p:E\\to X$ be a finite-dimensional $G$-vector bundle,. Then we have the isomorphism of $G$-vector bundles\n\t\t$$\\bigoplus_{[\\rho]\\in G\\setminus \\Irr(A)}\\operatorname{Ind}_{G_{[\\rho]}}^{G}\\left(\\rho\\otimes\\Hom_A(\\rho,E)\\right)\\to E.$$where $[\\rho]$ runs over representatives of the orbits of the $G$-action on the set of isomorphism classes of representations of $A$.\n\t\\end{thm}\n\t\n\tThe previous theorems applied to $X=*$ are the usual Mackey machine which describes how to get the finite-dimensional unitary representations of $G$ in terms of irreducible representations of $A$ and $\\widetilde{Q}_{[\\rho]}$-twisted representations of $Q_{[\\rho]}$: \n\t\\begin{enumerate}\n\t\t\\item Every finite-dimensional  representation of $G$ is the sum of induced representations of $(G_{[\\rho]},\\rho)$-isotypical ones (Theorem \\ref{vectordecomp}).\n\t\t\\item $(G_{[\\rho]},\\rho)$-isotypical representations can be obtained by tensoring the extension of $\\rho$ to $G_{[\\rho]}$ with  $\\widetilde{Q}_{[\\rho]}$-twisted representations of $Q_{[\\rho]}$ (Theorem \\ref{toe}).\n\t\\end{enumerate}\n\t\n\tThe following examples show how useful (and sometimes not so useful) the Mackey machine is.\n\t\n\t\\begin{exam}[$G=D_8,A=\\mathbb{Z}_4$]\n\t\t$$\n\t\tD_8= \\mathbb{Z}_4 \\rtimes \\mathbb{Z}_2 = \\left\\langle a , b \\mid a ^ { 4 } = b ^ { 2 } = e , b a b ^ { - 1 } = a ^ { - 1 } \\right\\rangle\n\t\t$$\n\t\t\n\t\t\n\t\tSince $D_8=\\mathbb{Z}_4 \\rtimes \\mathbb{Z}_2$ is a semidirect product, we know that the characters of $D_8$ are given by $\\widehat{\\mathbb{Z}_4}^{\\mathbb{Z}_2} \\times \\widehat{\\mathbb{Z}_2}$,  $\\widehat{\\mathbb{Z}_2} = \\{1,sign\\}$.\n\t\t\n\t\t$\\mathbb{Z}_2$ acts on\n\t\t$$\n\t\t\\widehat{\\mathbb { Z }_4 } = \\{ 1 , \\rho , \\rho ^ { 2 } , \\rho ^ { 3 } \\}$$\n\t\twith orbits\n\t\t$$\\mathbb{Z}_2 \\backslash \\widehat{\\mathbb{Z}_4}  = \\{ \\{ 1 \\} , \\left\\{ \\rho , \\rho ^ { 3 } \\right\\} , \\left\\{ \\rho ^ { 2 } \\right\\} \\} \n\t\t$$\n\t\tand therefore $\\widehat{\\mathbb{Z}_4}^{\\mathbb{Z}_2} = \\{1,\\rho^2\\}$ and there are four characters of $D_8:$\n\t\t$$\n\t\t1 \\rtimes 1, 1\\rtimes sign, \\rho^2 \\rtimes 1, \\rho^2 \\rtimes sign.\n\t\t$$\n\t\tFrom\n\t\t$$\n\t\t\\mid G\\mid = \\Sigma_{\\sigma \\in \\operatorname{Irr}(G)} \\dim(\\sigma)^2\n\t\t$$\n\t\twe know that there should be a $2$-dimensional irreducible representation of $D_8$.\n\t\t\n\t\tTake $A=\\mathbb{Z}_4$. Then $D_8$ acts on\n\t\t$$\n\t\t\\operatorname { Irr } ( \\mathbb { Z }_4 ) = \\{ 1 , \\rho , \\rho ^ { 2 } , \\rho ^ { 3 } \\}$$\n\t\twith orbits\n\t\t$$D _ { 8 } \\backslash \\operatorname { Irr } ( \\mathbb { Z }_4 ) = \\{ \\{ 1 \\} , \\left\\{ \\rho , \\rho ^ { 3 } \\right\\} , \\left\\{ \\rho ^ { 2 } \\right\\} \\}.\n\t\t$$\n\t\tWe have $G_{[1]} =G_{[\\rho^2]}= D_8$, $Q_{[1]}=Q_{[\\rho^2]} = D_8 \/\\mathbb{Z}_4 =\\mathbb{Z}_2$, $G _ { [ \\rho ] } = \\mathbb { Z } \/ 4$, $Q_{ [ \\rho ] }= 1$, and the twistings are trivial. \n\t\t\n\t\tThe two-dimensional irreducible representation of $D_8$ should be an induction from an irreducible representation of $G_{[\\rho]}$ that restricts to $A$ as a multiple of $\\rho$. In this case $G_{[\\rho]}=A$ and therefore we take the induction from $\\rho$, $\\operatorname{Ind}_{\\mathbb{Z}_4}^{D_8}(\\rho)$, as the $2$-dimensional irreducible representation of $D_8$,\n\t\t$$\n\t\tR(D_8) \\cong R(\\mathbb{Z}_2) \\oplus R(e)  \\oplus R(\\mathbb{Z}_2)\n\t\t$$\n\t\twhich is explicitly\n\t\t$$\n\t\tR(D_8) \\cong \\mathbb{Z} \\langle 1 \\rtimes 1, 1 \\rtimes sign \\rangle \\oplus \\mathbb{Z}\\langle \\operatorname{Ind}_{\\mathbb{Z}_4}^{D_8}(\\rho)\\rangle \\oplus \\mathbb{Z}\\langle\\rho^2 \\rtimes 1,\\rho^2 \\rtimes sign\\rangle\n\t\t.$$\n\t\\end{exam}\n\t\n\t\n\t\n\t\\begin{exam}[$G=D_8,A=\\mathbb{Z}_2$]\n\t\t$$\n\t\tD_8= \\mathbb{Z}_4 \\rtimes \\mathbb{Z}_2 = \\left\\langle a , b \\mid a ^ { 4 } = b ^ { 2 } = e , b a b ^ { - 1 } = a ^ { - 1 } \\right\\rangle\n\t\t$$\n\t\t\n\t\t\n\t\tLets take $A=\\mathbb{Z}_2=\\{e,a^2\\}$, the center of $D_8$. Since $A$ is central,  $D_8$ acts trivially on $\\operatorname { Irr } ( \\mathbb { Z } \/ 2 ) = \\{1,sign\\}$ with $G_{[1]} =G_{[sign]}= D_8$ and $Q_{[1]}=Q_{[sign]} = D_8 \/\\mathbb{Z}_2 =\\mathbb{Z}_2 \\times \\mathbb{Z}_2$.\n\t\t\n\t\tThe trivial representation extends to all $D_8$, but the $sign$ representation does not extend to all $D_8 = G_{[sign]}$, since the center of $D_8$ is the commutator subgroup and all $1$-dimensional representations factor through abelianization. Therefore there is a twisting, which is the obstruction to the $S^1$-extension\n\t\t\\[\n\t\t1 \\to S^{1} \\stackrel{\\widetilde{i}}{\\rightarrow} \\widetilde{Q}_{[\\rho]} \\rightarrow Q_{[\\rho]} \\to 1\n\t\t\\]\n\t\tto be trivial. It is the nontrivial element of\n\t\t$H^3(\\mathbb{Z}_2 \\times \\mathbb{Z}_2 ; \\mathbb{Z}) = \\mathbb{Z}_2 $, giving the extension\n\t\t\\[\n\t\t1 \\to S^{1} \\stackrel{\\widetilde{i}}{\\rightarrow} D_8 \\times_{\\mathbb{Z}_2} S^1 \\rightarrow \\mathbb{Z}_2\\times\\mathbb{Z}_2 \\to 1.\n\t\t\\]\n\t\tNevertheless, since the $sign$ representation is $1$-dimensional, the $S^1$-extension \n\t\t\\[\n\t\t1 \\to S^{1} \\stackrel{i}{\\rightarrow} \\widetilde{G}_{[sign]} \\stackrel{\\pi_1}{\\rightarrow}  G_{[sign]} \\to 1, \\]\n\t\tis the trivial extension\n\t\t\\[\n\t\t1 \\to S^{1} \\stackrel{i}{\\rightarrow} D_8 \\times S^1 \\stackrel{\\pi_1}{\\rightarrow}  D_8 \\to 1. \\]\n\t\t\n\t\t\n\t\tTherefore\n\t\t$$\n\t\tR(D_8) \\cong R(\\mathbb{Z}_2 \\times \\mathbb{Z}_2 ) \\oplus {}^{\\widetilde{Q}_{[sign]}} R(\\mathbb{Z}_2 \\times \\mathbb{Z}_2 ).\n\t\t$$\n\t\t\n\t\t\n\t\tLet's see what the Mackey machine says about the term ${}^{\\widetilde{Q}_{[sign]}} R(\\mathbb{Z}_2 \\times \\mathbb{Z}_2 )$.\n\t\t\n\t\tSince the $sign$ representation is $1$-dimensional, $\\widetilde{G}_{[sign]}=D_8 \\times S^1$. Also, $\\widetilde{\\Upsilon} : D_8 \\times S^1  \\stackrel{\\pi_2}{\\rightarrow} S^1$ is the extension of the sign representation to $ \\widetilde{G}_{[sign]}$. Additionally, $\\widetilde{Q}_{[sign]} = D_8 \\times_{\\mathbb{Z}_2} S^1$.\n\t\t\n\t\tThe second part of the Mackey machine says that to find irreducible representations of $G_{[sign]}=D_8$ that restrict $A=\\mathbb{Z}_2$ to copies of the $sign$ representation, we need to take a representation $\\sigma$ of $\\widetilde{Q}_{[\\rho]} \\cong D_8 \\times_{\\mathbb{Z}_2} S^1$ such that $S^1$ acts by multiplication by inverses, and tensor it with $\\widetilde{\\Upsilon}$.\n\t\t\n\t\tBut if $\\sigma : \\widetilde{Q}_{[sign]} = D_8 \\times_{\\mathbb{Z}_2} S^1 \\rightarrow U(n)$ is a representation such that $S^1$ acts by multiplication, then\n\t\t$\\sigma([g,\\lambda])=\\sigma([g,1])\\,\\sigma([e,\\lambda])=\\lambda^{-1}\\sigma([g,1])$, so it is determined by $\\sigma([g,1])$, which is a representation of $D_8$ that restricts $\\mathbb{Z}_2$ to multiples of the sign representation. We do not gain anything new with the second part of the Mackey machine in this case.\n\t\t\n\t\tComparing the two decompositions in the examples, we see that we must have \n\t\t$\n\t\t\\operatorname{Ind}_{\\mathbb{Z}_4}^{D_8}(\\rho)\n\t\t$ as the generator of  ${}^{\\widetilde{Q}_{[sign]}} R(\\mathbb{Z}_2 \\times \\mathbb{Z}_2)$. But this should be understood in the following sense: $\\operatorname{Ind}_{\\mathbb{Z}_4}^{D_8}(\\rho)$ is a $D_8 \\times_{\\mathbb{Z}\/2} S^1 $-representation given by\n\t\t$$\n\t\t[g,\\lambda] \\cdot \\vec{v} = \\lambda^{-1} \\left ( g \\cdot \\vec{v} \\right ).\n\t\t$$\n\t\t\n\t\t\n\t\\end{exam}\n\t\n\t\n\t\n\t\n\t\n\t\\subsection{Decomposition in equivariant $K$-theory (proper case)}\\label{proper$K$-theory}\n\tFor a general Lie  group $G$, we extend the definition in \\cite{lo} of  equivariant $K$-theory for proper actions, to twisted equivariant $K$-theory for the twisting relevant to our decompositions. See also \\cite{cantat}.\n\t\n\tFor general Lie groups we work on the category of locally compact Hausdorff second countable spaces.  If the group is discrete or compact we can also work on the category of proper $G$-CW complexes. \n\t\n\t\n\t\\begin{defn}\\label{twisteddef}\n\t\tLet $G$ be a Lie group acting properly on a  $G$-space $X$ and \n\t\t\\begin{equation}\\label{centralexten}\n\t\t1\\to S^1\\to\\widetilde{G}\\to G\\to 1\n\t\t\\end{equation}\n\t\tbe an $S^1$-central extension. Define  ${}^{\\widetilde{G}}\\mathbb{K}_G^0(X)$ as the Grothendieck group of the monoid of isomorphism classes of $\\widetilde{G}$-vector bundles over $X$ where $S^1$ acts by multiplication by inverses. Define for $n\\geq0$ $${}^{\\widetilde{G}}\\mathbb{K}_G^{-n}(X)=\\ker\\left({}^{\\widetilde{G}}\\mathbb{K}_G(X\\times S^n)\\xrightarrow{i^*}{}^{\\widetilde{G}}\\mathbb{K}_G(X)\\right),$$ where $G$ acts trivially on $S^n$.\n\t\tFor any proper $G$-pair $(X,A)$,  set$${}^{\\widetilde{G}}\\mathbb{K}_G^{-n}(X,A)=\\ker\\left({}^{\\widetilde{G}}\\mathbb{K}_G^{-n}(X\\cup_AX )\\xrightarrow{i_2^*}{}^{\\widetilde{G}}\\mathbb{K}_G^{-n}(X)\\right).$$ Note that, when $\\widetilde{G}\\cong S^1\\times G$, a $\\widetilde{G}$-vector bundle over $X$ where $S^1$ acts by multiplication by inverses is the same as a $G$-vector bundle, and we use the notation \n\t\t$\\mathbb{K}_G^{*}(X) = {}^{\\widetilde{G}}\\mathbb{K}_G^{*}(X)  \\text{ and  }\\mathbb{K}_G^{*}(X,A) = {}^{\\widetilde{G}}\\mathbb{K}_G^{*}(X,A)$.\n\t\\end{defn}\n\t\n\t\n\tSuppose that the Lie group $G$ satisfies the following property:\n\t\\begin{equation}\n\t\\tag{K}\\label{eq:A}\n\t\\parbox{\\dimexpr\\linewidth-4em}{%\n\t\t\\strut%\n\t\t\\emph{For every compact normal subgroup $A$ and  irreducible representation $\\rho \\in \\Irr(A)$, \n\t\t\tthe functors ${}^{\\widetilde{Q}_{[\\rho]}}\\mathbb{K}_{Q_{[\\rho]}}^*(-)$ define  $\\mathbb{Z}\/2$-graded $Q_{[\\rho]}$-cohomology  theories. This means that the twisted equivariant $K$-theories satisfy the following axioms:\n\t\t\t\\begin{enumerate}\n\t\t\t\\item $Q_{[\\rho]}$-homotopy invariance.\n\t\t\t\\item Long exact sequence of a pair.\n\t\t\t\\item Excision.\n\t\t\t\\item Disjoint union axiom.\n\t\t\t\\end{enumerate}  \n\t\t}\\strut\n\t}\\end{equation}For more details on the axioms see \\cite{luckcoh}.\n\tNote that by taking $A$ the trivial group, we have $Q_{[\\rho]}=G_{[\\rho]}=G$ and $\\widetilde{Q}_{[\\rho]} = S^1 \\times G $ and therefore we are requiring in particular that $\\mathbb{K}_G^{*}(-)$ is a $G$-cohomology theory.\n\t\n\t\n\t\n\tWhen $G$ satisfies the above assumption we write ${}^{\\widetilde{G}}K_G^{-n}(X,A)$ for ${}^{\\widetilde{G}}\\mathbb{K}_G^{-n}(X,A).$\n\t\\begin{remark}\\label{nota1}\n\t\tFrom \\cite{lo} and \\cite{dwyer}, a discrete group $G$ satisfies (K) (in the category of finite, proper $G$-CW complexes). Any compact Lie group also satisfies (K) (in the category of finite $G$-CW complexes and in the category of locally compact Hausdorff second countable $G$-spaces).\n\t\tOn the other hand, in the appendix  we will build upon the work of \\cite{Phil2}  to prove that  almost connected and linear Lie groups satisfy $(K)$ (in the category of locally compact Hausdorff second countable $G$-spaces). There are Lie groups where (K) fails to be true; see   \\cite{Phil} and   \\cite{lo}. \n\t\\end{remark}\n\n\t\n\tFor a general Lie group there is an induction scheme: if  $H$ is a subgroup of $G$ and $Y$ is a proper $H$-space, then\n\t$$\n\t\\mathbb{K}_G^*(G \\times_H Y) \\cong \\mathbb{K}_H^*(Y)\n\t$$\n\t(see lemma 3.4 of \\cite{lo}). There is a corresponding result for free actions, analogue to Lemma 3.5 of \\cite{lo}.\n\t\n\tLet $G$ be a Lie group acting properly on a  $G$-space $X$ and \n\t\\begin{equation\n\t1\\to S^1\\to\\widetilde{G}\\to G\\to 1\n\t\\end{equation}\n\tbe a $S^1$-central extension. For $G$ acting freely on $X$,\n\t$$\n\t{}^{\\widetilde{G}} \\mathbb{K}^*_{G}(X) \\cong {}^{\\alpha} K^*(X\/G)\n\t$$\n\twhere the right hand side denotes the \\emph{classical} twisted $K$-theory (defined for example in \\cite{AS}), and  $\\alpha$ is a cohomology class induced from the central extension (\\ref{centralexten}). This is done for finite groups in Proposition 3.8 of \\cite{GU}, and for proper actions of Lie groups on manifolds using $C^*$-algebras in Proposition 3.6 of \\cite{twistedstacks}.\n\tThe twisting $\\alpha$ can be described as follows: for the central extension\n\t$$\n\t1 \\to S^{1} \\stackrel{\\widetilde{\\iota}}{\\rightarrow} \\widetilde{G} \\rightarrow G \\to 1\n\t$$\n\tconsider the cohomology class $\\alpha_{\\widetilde{G}} \\in H^2(BG;S^1)$  that classifies it (see Thm. 10 in \\cite{moore}). We have the fibration\n\t$\n\tp : EG \\times_G X \\rightarrow BG,\n\t$\n\tand since the action is free, there exists a homotopy equivalence\n\t$\n\tEG \\times_G X \\rightarrow X\/G\n\t$\n\tgiving a commutative diagram\n\t$$\n\t\\xymatrix{\n\t\tEG \\times_G X \\ar[r]^{p} \\ar[d]^{\\simeq} & BG \\\\\n\t\tX\/G \\ar[ur]\n\t}\n\t$$\n\tConsider the cohomology class $\\alpha \\in H^2(X\/G;S^1) \\cong H^2_G(X;S^1)$ given by $p^*(\\alpha_{\\widetilde{G}})$. \n\t\n\t\n\t\\begin{thm}\\label{decomp}\n\t\tLet $G$ be a Lie group satisfying (K), and $X$ be a proper $G$-space  on which the normal subgroup $A$ acts trivially. There is a natural isomorphism \n\t\t\\begin{align*}\\Psi_X:K^*_G(X)&\\to\\bigoplus_{[\\rho]\\in G\\setminus\\Irr(A)}{}^{\\widetilde{Q}_{[\\rho]}}K^*_{Q_{[\\rho]}}(X)\\\\ [E]&\\mapsto\\bigoplus_{[\\rho]\\in G\\setminus\\Irr(A)}\\Hom_A(\\rho,E).\\end{align*} This isomorphism is functorial on $G$-maps $X\\to Y$ of proper $G$- spaces on which $A$ acts trivially. \n\t\\end{thm}\n\t\\begin{proof}\n\t\tIt is a straightforward application of Theorems \\ref{toe} and \\ref{vectordecomp}.     \n\t\\end{proof}\n\t\\subsection{Examples}\\label{sectionaa}\n\t\n\t\\begin{exam}\n\t\tConsider the group\n\t\t$$\n\t\tD_8= \\mathbb{Z}_4 \\rtimes \\mathbb{Z}_2 = \\left\\langle a , b \\mid a ^ { 4 } = b ^ { 2 } = e , b a b ^ { - 1 } = a ^ { - 1 } \\right\\rangle\n\t\t$$\n\t\t\n\t\tTake $A=\\mathbb{Z}_4$. $D_8$ acts on\n\t\t$$\n\t\t\\operatorname { Irr } ( \\mathbb { Z }_4 ) = \\{ 1 , \\rho , \\rho ^ { 2 } , \\rho ^ { 3 } \\}$$\n\t\twith orbits\n\t\t$$D _ { 8 } \\backslash \\operatorname { Irr } ( \\mathbb { Z }_4 ) = \\{ \\{ 1 \\} , \\left\\{ \\rho , \\rho ^ { 3 } \\right\\} , \\left\\{ \\rho ^ { 2 } \\right\\} \\}. \n\t\t$$\n\t\tWe have $G_{[1]} =G_{[\\rho^2]}= D_8$, $Q_{[1]}=Q_{[\\rho^2]} = D_8 \/\\mathbb{Z}_4 =\\mathbb{Z}_2$, $G _ { [ \\rho ] } = \\mathbb { Z } \/ 4$, $Q_{ [ \\rho ] }= 1$, and the twistings are trivial. \n\t\t\n\t\t\n\t\tIf $X$ is a space with an action of $D_8$ that restricts to the  trivial action of $ \\mathbb{Z}_4$, then\n\t\t$$\n\t\tK_{D_8}^*(X) \\cong K_{\\mathbb{Z}_2}^* (X) \\oplus K^*(X) \\oplus K_{\\mathbb{Z}_2}^* (X).\n\t\t$$\n\t\t\n\t\tCompare with example 6.1 of \\cite{GU}.\n\t\t\n\t\\end{exam}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\begin{exam}\n\t\tConsider the quaternions $Q_8= \\{ \\pm 1, \\pm i, \\pm j, \\pm k\\}$. It is a central extension of $\\mathbb{Z}_2 \\times \\mathbb{Z}_2$ by $\\mathbb{Z}_2$\n\t\t\\begin{equation*}\n\t\t0\\to\\mathbb{Z}\/2 \\to Q_8\\to \\mathbb{Z}_2 \\times \\mathbb{Z}_2\\to 0.\\end{equation*}\n\t\t\n\t\t\n\t\tTake  $A=\\mathbb{Z}_2=\\{1,-1\\}$. Since $A$ is central, $Q_8$ acts trivially on $\\operatorname { Irr } ( \\mathbb{Z}_2 ) = \\{1,sign\\}$ with $G_{[1]} =G_{[sign]}= Q_8$ and $Q_{[1]}=Q_{[sign]} = Q_8 \/\\mathbb{Z}_2 =\\mathbb{Z}_2 \\times \\mathbb{Z}_2$.\n\t\t\n\t\tThe trivial representation extends to $Q_8$, but the $sign$ representation does not extend to $Q_8 = G_{[sign]}$. Therefore there is a nontrivial twisting. If $X$ is a space with an action of $Q_8$ with trivial action of $ \\mathbb{Z}_2$, then\n\t\t$$\n\t\tK_{Q_8}^*(X) \\cong K_{\\mathbb{Z}_2 \\times \\mathbb{Z}_2 }^*(X) \\oplus {}^{\\widetilde{Q}_{[sign]}} K_{\\mathbb{Z}_2 \\times \\mathbb{Z}_2 }^*(X)\n\t\t$$\n\t\tand the twisting is represented by the nontrivial element of\n\t\t$H^3(\\mathbb{Z}_2 \\times \\mathbb{Z}_2 ; \\mathbb{Z}) = \\mathbb{Z}_2 $.\n\t\t\n\t\tConsider the projection $Q_8 \\rightarrow \\mathbb{Z}_2 \\times \\mathbb{Z}_2$, and consider the action of $\\mathbb{Z}_2 \\times \\mathbb{Z}_2$ on $\\mathbb{S}^2 \\times \\mathbb{S}^2$. Consider the decomposition \n\t\t$$\n\t\tK_{Q_8}^*(\\mathbb{S}^2 \\times \\mathbb{S}^2) \\cong K_{\\mathbb{Z}_2 \\times \\mathbb{Z}_2 }^*(\\mathbb{S}^2 \\times \\mathbb{S}^2) \\oplus {}^{\\widetilde{Q}_[sign]} K_{\\mathbb{Z}_2 \\times \\mathbb{Z}_2 }^*(\\mathbb{S}^2 \\times \\mathbb{S}^2).\n\t\t$$\n\t\tSince the action is free, \n\t\t$$\n\t\tK_{\\mathbb{Z}_2 \\times \\mathbb{Z}_2 }^*(\\mathbb{S}^2 \\times \\mathbb{S}^2) \\cong K^*(\\mathbb{RP}^2 \\times \\mathbb{RP}^2 ) \n\t\t$$\n\t\tand\n\t\t$$\n\t\t{}^{\\widetilde{Q}_[sign]} K_{\\mathbb{Z}_2 \\times \\mathbb{Z}_2}^*(\\mathbb{S}^2 \\times \\mathbb{S}^2) \\cong {}^\\alpha K^*(\\mathbb{RP}^2 \\times \\mathbb{RP}^2 ) \n\t\t$$\n\t\twhere $\\alpha$ is the nontrivial class of $H^3(\\mathbb{RP}^2 \\times \\mathbb{RP}^2;\\mathbb{Z})=\\mathbb{Z}_2$.\n\t\t\n\t\t\n\t\t\n\t\tFrom the Kunneth's theorem for $K$-theory \\cite{AtiyKunneth}, there is a short exact sequence: \n\t\t$$\n\t\t0 \\rightarrow \\bigoplus_{i+j=*} K^i(\\mathbb{RP}^{2}) \\otimes K^j(\\mathbb{RP}^{2}) \\rightarrow K^*(\\mathbb{RP}^{2} \\times \\mathbb{RP}^{2}) \\rightarrow \\operatornamewithlimits{Tor}\\limits_{i+j=*+1}(K^i(\\mathbb{RP}^{2}), K^j(\\mathbb{RP}^{2})) \\rightarrow 0.\n\t\t$$\n\t\t\n\t\tFor $*=0$, there is no $\\Tor$ term and therefore the $K$-theory is just the tensor product, which is $(\\mathbb { Z } \\oplus \\mathbb{Z}_2) \\otimes (\\mathbb { Z } \\oplus \\mathbb{Z}_2) = \\mathbb { Z } \\oplus (\\mathbb{Z}_2)^3 $.\n\t\t\n\t\tFor $*=1$ there is no tensor term and the $K$-theory is the $\\Tor$ part, which is $Tor ((\\mathbb { Z } \\oplus \\mathbb{Z}_2),(\\mathbb { Z } \\oplus \\mathbb{Z}_2)) = \\mathbb{Z}_2 $\n\t\t\n\t\t$$\n\t\tK^*(\\mathbb{RP}^{2} \\times \\mathbb{RP}^{2}\n\t\t) \\cong \\left\\{ \\begin{array} { c l } { \\mathbb { Z } \\oplus \\left ( \\mathbb{Z}_2 \\right ) ^3 , } & { * = 0 } \\\\ { \\mathbb{Z}_2 , } & { *=1 } \\end{array} \\right..\n\t\t$$\n\t\t\n\t\t\n\t\tLet us compute the twisted $K$-theory ${}^\\alpha K^*(\\mathbb{RP}^2 \\times \\mathbb{RP}^2 )$ with the twisted Atiyah-Hirzebruch spectral sequence \\cite{AtiySegAHSS}.\n\t\tWith coefficients $\\mathbb{Z}_2$ we have\n\t\t$$\n\t\tH^*(\\mathbb{RP}^2 \\times \\mathbb{RP}^2;\\mathbb{Z}_2) \\cong \\mathbb{Z}_2[x,y]\/(x^3,y^3).\n\t\t$$\n\t\t\n\t\t\n\t\tOn the other hand, for the cohomology  with integer coefficients,\n\t\t$$\n\t\tH^*(\\mathbb{RP}^2 \\times \\mathbb{RP}^2;\\mathbb{Z}) \\cong \\mathbb Z[x_2,y_2,z_3]\/ (2x_2,2y_2,2z_3,x_2^2,y_2^2,z_3^2,x_2z,y_2z_3)\n\t\t$$\n\t\twith $x_2$ and $y_2$ having degree $2$ and $x_2 \\Mod{2} =x^2$, $y_2 \\Mod{2} = y^2$,  $z_3$ having degree $3$ and $z_3 \\Mod{2} = x^2y+xy^2$. We have the following components on each degree\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{ |c|c|c|c|c|c|c| }\n\t\t\t\t\\hline\n\t\t\t\tdim 0 & dim 1 & dim 2 & dim 3 & dim 4 & dim 5 & dim 6  \\\\ \n\t\t\t\t\\hline\n\t\t\t\t$\\mathbb{Z}$ &  $0$ & $(\\mathbb{Z}_2)^2$ & $\\mathbb{Z}_2$  & $\\mathbb{Z}_2$ & $0$ & $0$ \\\\ \\hline\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t\tThen the third page of the Atiyah-Hirzebruch spectral sequence for twisted K-theory looks like\n\t\t\n\t\t$$\n\t\t\\begin{array}{c|ccccccc}\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t2 &  \\mathbb{Z} &  0 & \\mathbb{Z}_2 \\oplus \\mathbb{Z}_2   & \\mathbb{Z}_2 & \\mathbb{Z}_2 & 0 & 0  \\\\\n\t\t1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 & \\mathbb{Z} &  0 & \\mathbb{Z}_2 \\oplus \\mathbb{Z}_2   & \\mathbb{Z}_2 & \\mathbb{Z}_2 & 0 & 0\\\\\n\t\t-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t-2 & \\mathbb{Z} &  0 & \\mathbb{Z}_2 \\oplus \\mathbb{Z}_2   & \\mathbb{Z}_2 & \\mathbb{Z}_2 & 0 & 0\\\\\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t\\hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \n\t\t\\end{array}\n\t\t$$\n\t\twhere\n\t\t$d^3 :  H^0 \\to H^3$ sends 1 to $\\alpha$\n\t\tand\n\t\t$d^3 : H^1 \\to H^4$ is zero because $Sq^2$ on $0$ and $1$-dimensional classes is zero. The $E_4$-page is the $E_\\infty$-page and gives\n\t\t$$\n\t\tE_\\infty^{0,2k} = 2 \\mathbb{Z}, E_\\infty^{2,2k} = \\mathbb{Z}_2\\oplus \\mathbb{Z}_2, E_\\infty^{3,2k} = 0, E_\\infty^{4,2k} = \\mathbb{Z}_2\n\t\t$$ \n\t\tLet's see what the result is for ${}^\\alpha K^*(X)$. Working backwards, $F^4 ({}^\\alpha K^{0}(\\mathbb{RP}^2 \\times \\mathbb{RP}^2)) = \\mathbb{Z}_2$ which is also $F^3$, and then $F^2 \/ F^3 \\cong \\mathbb{Z}_2 \\oplus \\mathbb{Z}_2$, so we need to find possible extensions\n\t\t$$\n\t\t\\{0\\}\\rightarrow \\mathbb{Z}_2 \\rightarrow F^2 ({}^\\alpha K^{0}(\\mathbb{RP}^2 \\times \\mathbb{RP}^2)) \\rightarrow \\mathbb{Z}_2 \\oplus \\mathbb{Z}_2  \\rightarrow \\{0\\}\n\t\t$$\n\t\twhich could be $\\mathbb{Z}_4 \\oplus \\mathbb{Z}_2$ or $\\mathbb{Z}_2 \\oplus \\mathbb{Z}_4$ or $(\\mathbb{Z}_2)^3$.\n\t\t\n\t\t\n\t\tSo $F^2=F^1$, and thus we have the reduced $K$-theory equal to $(\\mathbb{Z}_2)^3$ or $\\mathbb{Z}_2 \\oplus \\mathbb{Z}_4$\n\t\t\n\t\tAlso note that in the extension\n\t\t$$\n\t\t\\{0\\}\\rightarrow F^1 \\rightarrow F^0 ({}^\\alpha K^{0}(\\mathbb{RP}^2 \\times \\mathbb{RP}^2)) \\rightarrow \\mathbb{Z}  \\rightarrow \\{0\\}\n\t\t$$\n\t\tsince $\\mathbb{Z}$ es free, the extension is trivial. Therefore,\n\t\t\n\t\t$$\n\t\t{}^\\alpha K^0(\\mathbb{RP}^2 \\times \\mathbb{RP}^2) \\cong \\mathbb{Z} \\oplus (\\mathbb{Z}_2)^3\n\t\t$$\n\t\tor\n\t\t$$\n\t\t{}^\\alpha K^0(\\mathbb{RP}^2 \\times \\mathbb{RP}^2) \\cong \\mathbb{Z} \\oplus \\mathbb{Z}_4 \\oplus \\mathbb{Z}_2.\n\t\t$$\n\t\t\n\t\tFor $*=1$, the only $E_\\infty$-page that contributes is $E_\\infty^{3,-2}=0$\n\t\twhich gives\n\t\t$$\n\t\t{}^\\alpha K^1(\\mathbb{RP}^2 \\times \\mathbb{RP}^2) = 0.\n\t\t$$\n\t\t\n\t\tTo be able to resolve the extension problems, we will use twisted $K$-theory with $\\mathbb{Z}\/2$-coefficients, i.e. twisted Morava $K$-theory at the prime $p=2$. Twisted Morava $K$-theory was introduced in \\cite{sati} where the corresponding twisted Atiyah Hirzebruch spectral sequence was studied.\n\t\t\n\t\tThe following Universal Coefficient Theorem was used by Aliaksandra Yarosh in \\cite{Yarosh}: for $H _ { 3 } \\in H ^ { 3 } ( X ; \\mathbb { Z } )$, there is an exact sequence:\n\t\t$$\n\t\t0 \\rightarrow K _ { n } ^ { H _ { 3 } } ( X ) \\otimes \\mathbb{Z}_2 \\rightarrow K ( 1 ) _ { n } \\left( X ; H _ { 3 } \\right) \\rightarrow \\operatorname { Tor } _ { 1 } \\left( K _ { n - 1 } ^ { H _ { 3 } } ( X ) , \\mathbb{Z}_2 \\right)\n\t\t$$ where $K(1)_n(X, H_3)$ denotes the twisted Morava $K$-homology theory which becomes the twisted $K$-theory with coefficients in $\\mathbb{Z}_2$. For details on the relationship between twisted Morava $K$-theory and twisted $K$-theory with $\\mathbb{Z}_2$ coefficients  see \\cite{Yarosh}, Theorem 6.12. \n\t\t\n\t\tUsing the theory of $C^*$-algebras there is a short exact sequence\n\t\t$$\n\t\t0 \\rightarrow {}^ { H _ { 3 } }K ^{ n }  ( X ) \\otimes \\mathbb{Z}_2 \\rightarrow K ( 1 )^{ n } \\left( X ; H _ { 3 } \\right) \\rightarrow \\operatorname { Tor } _ { 1 } \\left( {}^ { H _ { 3 }}K ^ { n + 1 }   ( X ) , \\mathbb{Z}_2 \\right).\n\t\t$$\n\t\t\n\t\t\n\t\t\n\t\t\n\t\tFor $n=0$, it translates to\n\t\t$$ 0 \\rightarrow {}^ { \\alpha }K ^{ 0 }  ( \\mathbb{RP}^2 \\times \\mathbb{RP}^2) \\otimes \\mathbb{Z}_2 \\rightarrow K ( 1 )^{ 0 } \\left( \\mathbb{RP}^2 \\times \\mathbb{RP}^2 ; \\alpha \\right) \\rightarrow \\operatorname { Tor } _ { 1 } \\left( {}^ { \\alpha}K ^ {   1 }(\\mathbb{RP}^2 \\times \\mathbb{RP}^2 )    , \\mathbb{Z}_2 \\right).$$\n\t\tSince \n\t\t$$\n\t\t{}^\\alpha K^1(\\mathbb{RP}^2 \\times \\mathbb{RP}^2) = 0\n\t\t$$\n\t\tthis gives $ {}^ { \\alpha }K ^{ 0 }  ( \\mathbb{RP}^2 \\times \\mathbb{RP}^2) \\otimes \\mathbb{Z}_2 \\cong  K ( 1 )^{ 0 } \\left( \\mathbb{RP}^2 \\times \\mathbb{RP}^2 ; \\alpha \\right )$.\n\t\t\n\t\tLet us compute $K(1)^*( \\mathbb{RP}^2 \\times \\mathbb{RP}^2;\\alpha)$ using the twisted Atiyah Hirzebruch spectral sequence. \n\t\t\n\t\t$$\n\t\tH^*(\\mathbb{RP}^2 \\times \\mathbb{RP}^2;\\mathbb{Z}_2) \\cong \\mathbb{Z}_2[x,y]\/(x^3,y^3).\n\t\t$$\n\t\tOn each degree we have\n\t\t\n\t\t\\begin{center}\n\t\t\t\\begin{tabular}{ |c|c|c|c|c|c|c| }\n\t\t\t\t\\hline\n\t\t\t\tdim 0 & dim 1 & dim 2 & dim 3 & dim 4  \\\\ \n\t\t\t\t\\hline\n\t\t\t\t$\\mathbb{Z}_2$ &  $(\\mathbb{Z}_2)^2$ & $(\\mathbb{Z}_2)^3$ & $(\\mathbb{Z}_2)^2 $ & $\\mathbb{Z}_2$  \\\\ \\hline\n\t\t\t\t1 & $x,y$& $x^2,xy,y^2$ & $x^2y,y^2x$ & $x^2y^2$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{center}\n\t\tAnd the third page of the spectral sequence looks like\n\t\t\n\t\t$$\n\t\t\\begin{array}{c|ccccccc}\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t2 &  \\mathbb{Z}_2 &  (\\mathbb{Z}_2)^2 & (\\mathbb{Z}_2)^3  & (\\mathbb{Z}_2)^2 & \\mathbb{Z}_2 & 0 & 0  \\\\\n\t\t1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 &  \\mathbb{Z}_2 &  (\\mathbb{Z}_2)^2 & (\\mathbb{Z}_2)^3  & (\\mathbb{Z}_2)^2 & \\mathbb{Z}_2 & 0 & 0 \\\\\n\t\t-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t-2 &  \\mathbb{Z}_2 &  (\\mathbb{Z}_2)^2 & (\\mathbb{Z}_2)^3  & (\\mathbb{Z}_2)^2 & \\mathbb{Z}_2 & 0 & 0 \\\\\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t\\hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \n\t\t\\end{array}\n\t\t$$\n\t\tThe differentials are given by\n\t\t\\[ d _ { 2 ^ { n + 1 } - 1 } \\left( x v _ { n } ^ { k } \\right) = \\left( Q _ { n } ( x ) + ( - 1 ) ^ { | x | } x \\cup \\left( Q _ { n - 1 } \\cdots Q _ { 1 } ( H ) \\right) \\right) v _ { n } ^ { k - 1 } \\]\n\t\t(from \\cite{sati}) and for $n=1$,\n\t\t$$ d _ { 3 } \\left( x v _ { 1 } ^ { k } \\right) = \\left( Q _ { 1} ( x ) +  x \\cup   H \\Mod{2}  \\right) v _ { 1 } ^ { k - 1 }\n\t\t$$\n\t\t\n\t\tThe $Q_i$ are the Milnor primitives at prime $2$, defined inductively by $Q_0=Sq^1$ and $ Q _ { j + 1 } = S q ^ { 2 ^ { j+1 } } Q _ { j } - Q _ { j } S q ^ { 2 ^ { j+1 } } $. Therefore $Q_1= Sq^2Sq^1-Sq^1Sq^2$. On the other hand, $Sq^1Sq^2 = Sq^3$ (see for example Mosher-Tangora \\cite{mosher} page 23), making $Q_1 = Sq^2Sq^1+Sq^3$.\\\\\n\t\tThus we conclude:\n\t\t$$\n\t\td^3(x) = Q_1(x) + x \\cup  H \\Mod{2}.\n\t\t$$\n\t\tIn our example, $Q_1(1)=0$ and\\\\\n\t\t$$Q_1(x)=Sq^2Sq^1(x)=Sq^2(x^2)=x^4=0,$$ $$Q_1(y)=Sq^2Sq^1(y)=Sq^2(y^2)=y^4=0,$$\n\t\ttherefore $d^3 :  H^0 \\to H^3$ sends 1 to $\\alpha \\Mod{2} = x^2y+yx^2$;\n\t\t$d^3 : H^1 \\to H^4$ sends $x \\to x (x^2y+y^2x) = y^2x^2 $ and $y \\to y (x^2y+y^2x) = y^2x^2 $; thus $x+y \\in \\ker(d^3)$.\n\t\t\n\t\t\n\t\tThe $E_4$-page is the $E_\\infty$-page and gives\n\t\t$$\n\t\t\\begin{array}{c|ccccccc}\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t2 &  0 &  \\mathbb{Z}_2 & (\\mathbb{Z}_2)^3  & \\mathbb{Z}_2 & 0& 0 & 0  \\\\\n\t\t1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 &  0 &  \\mathbb{Z}_2 & (\\mathbb{Z}_2)^3  & \\mathbb{Z}_2 & 0 & 0 & 0 \\\\\n\t\t-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t-2 &  0 &  \\mathbb{Z}_2 & (\\mathbb{Z}_2)^3  & \\mathbb{Z}_2 & 0 & 0 & 0 \\\\\n\t\t\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\t\t\\hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \n\t\t\\end{array}\n\t\t$$\n\t\t\n\t\t\n\t\t$$\n\t\tE_\\infty^{1,2k} = \\mathbb{Z}_2, E_\\infty^{2,2k} =  (\\mathbb{Z}_2)^3, E_\\infty^{3,2k} = \\mathbb{Z}_2.\n\t\t$$ \n\t\tIn this manner we obtain:\n\t\t$$\n\t\tK ( 1 )^{ 0 } \\left( \\mathbb{RP}^2 \\times \\mathbb{RP}^2 ; \\alpha \\right ) \\cong (\\mathbb{Z}_2)^3\n\t\t$$\n\t\tand\n\t\t$$\n\t\tK ( 1 )^{ 1 } \\left( \\mathbb{RP}^2 \\times \\mathbb{RP}^2 ; H _ { 3 }\\right ) \\cong (\\mathbb{Z}_2)^2.\n\t\t$$\n\t\tWe have either\n\t\t$$\n\t\t{}^\\alpha K^0(\\mathbb{RP}^2 \\times \\mathbb{RP}^2) \\cong \\mathbb{Z} \\oplus (\\mathbb{Z}_2)^3\n\t\t$$\n\t\tor\n\t\t$$\n\t\t{}^\\alpha K^0(\\mathbb{RP}^2 \\times \\mathbb{RP}^2) \\cong \\mathbb{Z} \\oplus \\mathbb{Z}_4 \\oplus \\mathbb{Z}_2\n\t\t$$\n\t\tbut tensoring with $\\mathbb{Z}_2$ we see that the only possibility, consistent with universal coefficients, is\n\t\t$$\n\t\t{}^\\alpha K^0(\\mathbb{RP}^2 \\times \\mathbb{RP}^2) \\cong \\mathbb{Z} \\oplus \\mathbb{Z}_4 \\oplus \\mathbb{Z}_2.\n\t\t$$\n\t\tSummarizing, \n\t\t$$\n\t\t{}^\\alpha K^0(\\mathbb{RP}^2 \\times \\mathbb{RP}^2) \\cong \\left\\{ \\begin{array} { c l } { \\mathbb{Z} \\oplus \\mathbb{Z}_4 \\oplus \\mathbb{Z}_2,} & { * = 0 } \\\\ { 0 , } & { *=1 } \\end{array} \\right.,\n\t\t$$\n\t\tand all of these observations put together result in\n\t\t$$\n\t\tK^*_{Q_8}(\\mathbb{S}^{2} \\times \\mathbb{S}^{2}\n\t\t) \\cong \\left\\{ \\begin{array} { c l } { \\left (\\mathbb { Z } \\right)^2 \\oplus \\left ( \\mathbb{Z}_2 \\right ) ^4 \\oplus \\mathbb{Z}_4 , } & { * = 0 } \\\\ { \\mathbb{Z}_2 , } & { *=1 } \\end{array} \\right..\n\t\t$$\n\t\tCompare with example 6.2 of \\cite{AGU}, where $K^*_{D_8}(\\mathbb{S}^{\\infty} \\times \\mathbb{S}^{\\infty})$ is computed with the help of the Atiyah-Segal completion theorem.  \n\t\t\n\t\\end{exam}\n\t\\begin{exam}\n\t\tLet $\\operatorname{St}_n(R)$ be the Steinberg group \\cite{steinberg}  associated to a commutative ring $R$, and consider the central extension\n\t\t\\begin{equation}\\label{ext}0\\to\\mathbb{Z}_2\\to\\operatorname{St}_n(\\mathbb{Z})\\to\\operatorname{Sl}_n(\\mathbb{Z})\\to 0.\\end{equation}\n\t\tNote that if we take a finite $\\operatorname{Sl}_n(\\mathbb{Z})$-CW-complex model of $\\underbar{E}\\operatorname{Sl}_n(\\mathbb{Z})$, this space with the induced action of $\\operatorname{St}_n(\\mathbb{Z})$ is a finite $\\operatorname{St}_n(\\mathbb{Z})$-CW-complex model for $\\underbar{E}\\operatorname{St}_n(\\mathbb{Z})$. \n\t\t\n\t\tMoreover, as the extension is central, the action of $\\,\\operatorname{Sl}_n(\\mathbb{Z})$ over $\\Irr(\\mathbb{Z}_2)=\\{[triv],[sgn]\\}$ is trivial. Thus $Q_{[triv]}=Q_{[sgn]}=\\operatorname{Sl}_n(\\mathbb{Z})$. Now we will compute the $S^1$-central extensions corresponding to each irreducible representation.\n\t\t\n\t\tFor $[triv]$, the $S^1$-central extension is trivial and therefore there is no twisting.\n\t\t\n\t\tFor $[sgn]$, the associated $S^1$-central extension is constructed as follows:\n\t\t\n\t\tNote that $V_{[sgn]}$ is 1-dimensional; thus $PU(V_{[sgn]})=\\{0\\}$ and therefore $\\widetilde{G}_{[sgn]}=\\operatorname{St}_n(\\mathbb{Z})\\times S^1$ and $\\widetilde{Q}_{[sgn]}=(\\operatorname{St}_n(\\mathbb{Z})\\times S^1)\/\\Delta(\\mathbb{Z}_2)$. It follows that there is a pullback of central extensions\n\t\t$$\\xymatrix{1 \\ar[r]&\\mathbb{Z}_2\\ar[r]\\ar[d]^{i}& \\operatorname{St}_n(\\mathbb{Z})\\ar[r] \\ar[d]^{}& \\operatorname{Sl}_n(\\mathbb{Z}) \\ar[r]\\ar[d]^{id}& 1 \\\\ 1 \n\t\t\t\\ar[r]&S^1\\ar[r] &\\widetilde{Q}_{[sgn]}\\ar[r] &\\operatorname{Sl}_n(\\mathbb{Z}) \\ar[r] & 1. \n\t\t}$$ This means that the extension presented in (\\ref{ext}) is the pullback of the $S^1$-central extension associated to $[sgn]$.\n\t\n\t\n\t\tIn \\cite{dwyer} it was proved that for a discrete group $Q$ and a proper $Q$-CW-complex, if a $\\mathbb{Z}_n$-central extension $\\widetilde{Q'}$ of $Q$ is the pullback of an $S^1$-central extension $\\widetilde{Q}$ of $Q$  then $${}^{\\widetilde{Q}}K_Q(X)\\cong {}^{\\alpha_{\\widetilde{Q'}}}K_Q(X)$$ with the twisting $\\alpha_{\\widetilde{Q}'}$ on the right side defined in terms of cocycles in group cohomology (for details consult \\cite{dwyer} or \\cite{BaVe2016}). In this terminology, in \\cite{BaVe2016}, calculations of the equivariant twisted $K$-theory groups for $\\underbar{E}\\operatorname{Sl}_3(\\mathbb{Z})$ were obtained. Thus we have to determine the cocycle  $\\alpha_{\\widetilde{Q'}_{[sgn]}}$ in terms of the generators $u_1$ and $u_2$ of $H^3(\\operatorname{Sl}_3(\\mathbb{Z});\\mathbb{Z})\\cong \\mathbb{Z}_2\\oplus\\mathbb{Z}_2$. From the discussion on page 59 in \\cite{BaVe2016}, we know that there is a unique element $u_1+u_2\\in H^3(\\operatorname{Sl}_3(\\mathbb{Z});\\mathbb{Z})$ restricting to nontrivial extensions of the three different copies of $S_3$ contained in $\\operatorname{Sl}_3(\\mathbb{Z})$; on the other hand, Lemma 9 (i) in \\cite{soule} implies that $\\operatorname{St}_3(\\mathbb{Z})$ has the same property. Hence $u_1+u_2$ represents extension (\\ref{ext}) . Then using Theorem \\ref{decomp}\n\t\t\\begin{align*}K_{\\operatorname{St}_n(\\mathbb{Z})}^*(\\underbar{E}\\operatorname{St}_3(\\mathbb{Z}))\\cong & K_{\\operatorname{Sl}_3(\\mathbb{Z})}^*(\\underbar{E}\\operatorname{Sl}_3(\\mathbb{Z}))\\oplus{}^{u_1+u_2} K_{\\operatorname{Sl}_3(\\mathbb{Z})}^*(\\underbar{E}\\operatorname{Sl}_3(\\mathbb{Z}))\\\\&\\cong \\begin{cases}\n\t\t\\mathbb{Z}^{13}&\\text{ if $*$ is even}\\\\\n\t\t\\mathbb{Z}_2\\oplus\\mathbb{Z}_2&\\text{ if $*$ is odd.}\n\t\t\\end{cases}\\end{align*}\n\t\\end{exam}\n\t\n\t\n\t\n\t\\section{Structure of actions with only one isotropy type}\\label{sectionb}\n\t\n\t\n\t\n\tWhen a compact Lie group acts freely on a space $X$, the projection $X \\rightarrow X\/G$ is a fiber bundle with fiber $G$. More generally, when the action is not free but has only one isotropy type $(G\/H)$, then $X \\rightarrow X\/G$ is a fiber bundle with fiber $G\/H$ and structure group $NH\/H$.\n\t\n\tBorel observed that there is another fibration over the orbit space of a $G$-space with only one isotropy with fiber $NH\/H$.\n\t\n\t\n\t\\begin{thm}[Corollary 5.10 \\cite{bredon}, page 89] \\label{bredon1} \n\t\tLet $G$ be a compact Lie group acting continuously on a completely regular Hausdorff space $X$. Suppose $H$ is a closed subgroup of $G$ (not necessarily normal). If the action of $G$ on $X$ has  only one orbit type $G\/H$ (all isotropy groups are conjugate to $H$), then the action of $NH\/H$ on $X^H$ is free and $X^H\/NH$ is homeomorphic to $X\/G$. Also, there is a principal $NH\/H$-bundle\n\t\t$$\n\t\tNH\/H \\rightarrow X^H \\rightarrow X\/G.\n\t\t$$\n\t\\end{thm}\n\t\n\tIn fact, a $G$-space with only one orbit type  is  the associated fiber bundle over $X\/G$ with fiber $G\/H$ and structure group $NH\/H$.\n\t\n\t\\begin{thm}[Corollary 5.11 \\cite{bredon} page 89] \\label{bredon2}\n\t\tLet $G$ be a compact Lie group acting continuously on a completely regular Hausdorff space $X$. Suppose $H$ is a closed subgroup of $G$ (not necessarily normal). If the action of $G$ on $X$ has  only one orbit type $G\/H$ (all isotropy groups are conjugate to $H$), then the map \t\\begin{align*}\n\t\t\\Phi:G\/H \\times_{N(H)\/H} X^H &\\rightarrow X\\\\ [gH,x]&\\mapsto gx\n\t\t\\end{align*}\n\t\tis a $G$-homeomorphism. \n\t\\end{thm}\n\t\n\t\n\t\n\tThis theorem gives a correspondence between principal $NH\/H$-bundles over $B$, and actions of $G$ with only one isotropy type $(G\/H)$ and orbit space $B$. This correspondence takes a principal $NH\/H$-bundle $P \\rightarrow B$ and assigns the $G$ space  $G\/H \\times_{NH\/H} P$. For example, the $G$-actions with only one isotropy type $(G\/H)$ and quotient space $S^n$ are classified by $\\pi_{n-1}(NH\/H)$.\n\t\n\t\n\tThese results follow from the next theorem:\n\t\n\t\n\t\\begin{thm}[Theorem 5.9 \\cite{bredon} page 89] \\label{bredon3}\n\t\tLet $G$ be a compact Lie group acting continuously on a completely regular Hausdorff space  $X$. Suppose $H$ is a closed subgroup of $G$ (not necessarily normal). If the action of $G$ on $X$ has  only one orbit type $G\/H$ (all isotropy groups are conjugate to $H$), then the map $\\Phi([g,x])= gx$\n\t\t$$\n\t\t\\Phi: G \\times_{NH} X^H \\rightarrow X\n\t\t$$\n\t\tis a $G$-homeomorphism. \n\t\\end{thm}\n\t\n\tFor $G$ a compact Lie group, many $G$-spaces are of the form $G \\times_H X$ for $X$ an $H$-space.  Consider a $G$-space $Y$ with a $G$-equivariant continuous map $f: Y \\rightarrow G\/H$.\n\tIf we take $X=f^{-1}(eH)$, the natural map\n\t$$\n\tF : G \\times_H X \\rightarrow Y\n\t$$\n\tis a $G$-equivariant homeomorphism (for $G$ a compact Lie group, $H$ closed).\n\t\n\tWe want to extend these theorems to more general groups. This is condition (S) of \\cite{uribe} and it is satisfied, for example, if the group  is locally compact, second countable and has finite covering dimension. In particular, Lie groups satisfy condition (S).\n\t\n\t\n\t\\begin{thm}[\\cite{uribe} Lemma 4.2] \\label{conditionS}Let $ f : Y \\rightarrow G \/ H$  be a $ G$-equivariant  map for some subgroup $H \\subseteq G$.  Suppose that condition (S) is satisfied . Then the $G$-equivariant map  $ F: G \\times_H f^{-1}(eH) \\rightarrow Y$ given by $[g, e] \\rightarrow  g \\cdot e$  is a $G$-homeomorphism.\n\t\\end{thm}\n\t\n\t\n\t\n\tWe will use the previous theorem to study proper actions of Lie groups with only one isotropy type. When the Lie group acts smoothly and properly on a manifold, the analogs of Theorems \\ref{bredon1} and \\ref{bredon2} are obtained.\n\t\n\t\\begin{thm}\\label{Th4.1}\n\t\tLet $G$ be a Lie group acting continuously and properly on a manifold $M$. Suppose $H$ is a closed subgroup of $G$ (not necessarily normal). If the action of $G$ on $M$ has  only one orbit type $G\/H$ (all isotropy groups are conjugate to $H$), then the action of $NH\/H$ on $M^H$ is free and $M^H\/NH$ is homeomorphic to $M\/G$. Hence, there is a principal $NH\/H$-bundle\n\t\t$$\n\t\tNH\/H \\rightarrow M^H \\rightarrow M\/G\n\t\t$$\n\t\\end{thm}\n\t\n\t\\begin{thm}\n\t\tKeeping the same assumptions as in Theorem \\ref{Th4.1}, the map \n\t\\begin{align*}\n\t\t\\Phi:G\/H \\times_{N(H)\/H} M^H &\\rightarrow M\\\\ [gH,x]&\\mapsto gx\n\t\t\\end{align*}\n\t\tis a diffeomorphism of  $G$-spaces. \n\t\\end{thm}\n\t\n\tThese results follow for example from \\cite{Kolk} Theorem 2.6.7, Part iii)  and iv) respectively.\n\t\n\tThe following theorems extend the previous results to proper actions of Lie groups. The key fact is the existence of tubes and slices for proper actions of groups. In the case of completely regular Hausdorff spaces, see Theorem 2.3.3 of \\cite{palais}; for $G$-CW complexes, see Theorem 7.1 of  \\cite{uribe}.\n\t\n\t\\begin{thm} \\label{oneisoproper}\n\t\tLet $G$ be a Lie group acting properly  on a completely regular Hausdorff space $X$. Suppose $H$ is a compact subgroup of $G$ (not necessarily normal). If the action of $G$ on $X$ has  only one orbit type $G\/H$ (all isotropy groups are conjugate to $H$), then the map $\\Phi([g,x])=gx$\n\t\t$$\n\t\t\\Phi : G \\times_{NH} X^H \\rightarrow X\n\t\t$$\n\t\tis a $G$-homeomorphism. \n\t\t\\begin{proof}\n\t\t\tConsider the function $\\phi : X \\rightarrow G\/NH$ defined for $x \\in X$ as follows: the stabilizer at $x$ is $gHg^{-1}$ for some $g \\in G$, and therefore the stabilizer of  $g^{-1}x$ is $H$. $\\phi$ is defined sending $x$ to the coset $gNH$.\n\t\t\t\n\t\t\tThis function is $G$-equivariant because $\\overline{g}x$ has stabilizer $\\overline{g}gHg^{-1}\\overline{g}^{-1}$ and therefore $\\phi(\\overline{g}x) = \\overline{g}gNH=\\overline{g}\\phi(x)$.\n\t\t\t\n\t\t\tIf $\\phi(x) =gNH$, then $G_x = gHg^{-1}$ so that $x \\in X^{gHg^{-1}}$. Also if $x \\in X^{gHg^{-1}}$, then $gHg^{-1} \\subseteq G_x$, and since we are assuming only one isotropy type, then $G_x = \\overline{g} H \\overline{g}^{-1}$ for some $\\overline{g} \\in G$. Now by \\cite{Kolk}, Corollary 1.11.9,\n\t\t\t$$\n\t\t\tgHg^{-1} \\subseteq \\overline{g} H \\overline{g}^{-1} \\text { implies } gHg^{-1} =  \\overline{g} H \\overline{g}^{-1}.\n\t\t\t$$\n\t\t\tSince $H$ is compact, $gH g^{-1}$ and $\\overline{g}H \\overline{g}^{-1}$ have a finite number of components, and both are Lie groups. Therefore \n\t\t\t$$\n\t\t\t\\phi(x) = gNH  \\iff x \\in X^{gHg^{-1}}.\n\t\t\t$$\n\t\t\t\n\t\t\tSince the action of $G$ on $X$ is proper and $X$ is a completely regular Hausdorff space, by \\cite{palais}, Theorem 2.3.3, there is a slice and tube around $x$. Take a slice at $x$; this is a subset $S \\subseteq X$ that is $G_x$-equivariant and such that the natural map $G \\times_{G_{x}} S \\rightarrow X$ is a $G$-homeomorphism onto an open subset of $X$. \n\t\t\t\n\t\t\tSince  the stabilizer at $x$ is $gHg^{-1}$, the tube $U \\cong G \\times_{G_{x}} S$ has isotropy at $[\\overline{g},s]$ given by $G_{[\\overline{g},s]} = \\overline{g}{{G_{x}}_s}\\overline{g}^{-1}  \\subseteq \\overline{g}gHg^{-1} \\overline{g}^{-1}$. But since the isotropy type of $\\overline{g}s$ is also $H$, there exists $\\widetilde{g}$ such that\n\t\t\t$$\n\t\t\tG_{[\\overline{g},s]} = \\widetilde{g} H \\widetilde{g}^{-1} \\subseteq \\overline{g}gHg^{-1} \\overline{g}^{-1}.\n\t\t\t$$\n\t\t\t\n\t\t\tBut since $H$ is compact, $\\widetilde{g} H \\widetilde{g}^{-1}$ and $\\overline{g}gHg^{-1} \\overline{g}^{-1}$ have a finite number of components, and since they are both Lie groups, once again by Corollary 1.11.9 in \\cite{Kolk}, $\\widetilde{g} H \\widetilde{g}^{-1} \\subseteq \\overline{g}gHg^{-1} \\overline{g}^{-1}$ \n\t\t\timplies $\\widetilde{g} H \\widetilde{g}^{-1} = \\overline{g}gHg^{-1} \\overline{g}^{-1}$, and therefore $\\widetilde{g}NH = \\overline{g}g  NH$. \n\t\t\tOn the tube $G \\times_{G_x} S$ we have \n\t\t\t$$\n\t\t\t\\phi([\\overline{g},s]) = \\overline{g} g NH\n\t\t\t$$\n\t\t\twhich means $\\phi$ is constant on the slice and is continuous on the tube at $x$. \\\n\t\t\t\n\t\t\tAs $\\phi^{-1}(eNH) =X^H$, using Theorem \\ref{conditionS} the proof is finished.\n\t\t\\end{proof}\n\t\\end{thm}\n\t\n\t\n\tSimilarly for $G$-CW complexes, there exist slices by Theorem 7.1 of \\cite{uribe}. Therefore the following theorem holds.\n\t\n\t\\begin{thm} \\label{oneisoproperCW}\n\t\tLet $G$ be a Lie group acting properly  on a $G$-CW complex $X$. Suppose $H$ is a compact subgroup of $G$ (not necessarily normal). If the action of $G$ on $X$ has  only one orbit type $G\/H$ (all isotropy groups are conjugate to $H$), then the map $\\Phi([g,x])=gx$\n\t\t$$\n\t\t\\Phi : G \\times_{NH} X^H \\rightarrow X\n\t\t$$\n\t\tis a $G$-homeomorphism. \n\t\\end{thm}\n\t\n\t\n\t\\subsection{$K$-theory of actions with only one isotropy type}\n\t\n\tIn this section we study $G$-equivariant $K$-theory for actions with only one isotropy type. The are two easy types of actions with one isotropy type.\n\t\\begin{enumerate}\n\t\t\\item  Free actions\n\t\t\\item Trivial actions\n\t\\end{enumerate}\n\t\n\tFor free actions of a compact Lie group $G$, we know that the equivariant $K$-theory is the $K$-theory of the quotient:\n\t\n\t$$\n\tK^*_G(X) \\cong K^*(X\/G).\n\t$$\n\t\n\tFor trivial actions, the equivariant  $K$-theory is the tensor product of the group ring with the $K$-theory of the space, which can be written as a sum over the irreducible representations of $G$ of the $K$-theory of the space:\n\t$$\n\tK^*_G(X) \\cong R(G) \\otimes K^*(X) \\cong \\bigoplus_{[\\rho] \\in   \\Irr(G)} K^*(X).\n\t$$More generally, the $G$-action on $G\/H \\times X$, acting only on the first component, has one isotropy type, and the equivariant $K$-theory is:\n\t$$\n\tK^*_G(G\/H \\times X) \\cong K^*_H(X) \\cong R(H) \\otimes K^*(X) \\cong \\bigoplus_{[\\rho] \\in  \\Irr(H)} K^*(X).\n\t$$\n\t\n\t\n\tWe will prove a decomposition formula for actions with only one isotropy type that generalizes the previous examples.\n\t\n\tIf $G$ is a compact Lie group acting continuously on the space $X$\n\twith only one orbit type $(G\/H)$, we know that there is a $G$-homeomorphism\n\t$$\n\t\\Phi: G \\times_{NH} X^H \\rightarrow X.\n\t$$\n\t\n\tSince $H$ is closed, the normalizer $NH$ is a closed subgroup, and since $G$ is compact we have that $NH$ is a compact group.\n\t\n\tFrom the $G$-homeomorphism and induction structure of $K$-theory:\n\t$$\n\tK_G^*(X) \\cong  K^*_G(G\\times_{NH} X^H) \\cong K_{NH}^*(X^H).\n\t$$\n\t\n\t$H$ is a closed normal subgroup of $NH$ that acts trivially on $X^H$. From \\cite{AGU} there is a decomposition\n\t$$\n\tK_{NH}^*(X^H) \\cong  \\bigoplus_{[\\rho] \\in NH \\backslash \\Irr(H)}\n\t{}^{\\widetilde{W}_{[\\rho]}} K^*_{W_{[\\rho]}}(X^H),$$\n\twhere the sum is over  representatives of the orbits of the \n\t$NH$-action on the set of isomorphism classes of irreducible $H$-representations, and $W_{[\\rho]}=NH_{[\\rho]}\/H$. \n\t\n\t\n\tWe have an extension of compact Lie groups\n\t$$\n\t1  \\to H\\stackrel{\\iota}\\rightarrow NH\\stackrel{\\pi}\\rightarrow NH\/H \\to  1\n\t$$\n\tand for each $\\rho : H \\to U(V_{\\rho})$  complex, finite-dimensional, \n\tirreducible representation of $H$, there are central extensions\n\t\\[\n\t1 \\to S^{1} \\stackrel{\\widetilde{\\iota}}{\\rightarrow} \\widetilde{W}_{[\\rho]} \\rightarrow W_{[\\rho]} \\to 1.\n\t\\]\n\t\n\tNow, under the conditions of only one isotropy type, $NH\/H$ acts freely on $X^H$ and therefore $W_{[\\rho]}$ acts freely on $X^H$. Thus\n\t$$\n\t{}^{\\widetilde{W}_{\\rho}} K^*_{W_\\rho}(X^H) \\cong {}^{\\alpha_{[\\rho]}} K^*(X^H\/W_{[\\rho]})\n\t$$\n\twhere $\\alpha_{[\\rho]}$ is the twisting coming from the central extension as in Section \\ref{proper$K$-theory}.\n\t\n\t\n\t\n\t$NH_{[\\rho]}$ is the stabilizer of the action of $NH$ on $\\Irr(H)$, which is a discrete space. Therefore $NH\/NH_{[\\rho]}$ is a discrete space, and in fact a finite set of size $[NH_{[\\rho]}:NH]$. Thus,  there is a covering space:\n\t$$\n\t\\xymatrix{\n\t\tX^H \/ NH_{[\\rho]} \\ar[d] \\\\\n\t\tX^H\/NH\n\t}\n\t$$\n\tof degree $[NH_{[\\rho]}:NH]$.\n\t\n\tBut $G \\times_{NH} X^H$ is $G$-homeomorphic to $X$ and therefore \n\t$$\n\tX\/G \\cong X^H \/ NH\n\t$$\n\tand there is a covering space $X^H\/NH_{[\\rho]} \\rightarrow X\/G$.\n\t\n\t\n\tSince $H$ acts trivially on $X^H$, \n\t$$\n\tX^H\/(NH\/H) = X^H \/NH \\cong X\/G\n\t$$\n\tand $X^H\/NH_{[\\rho]} = X^H\/W_{[\\rho]}$. These observations put together result in\n\t$$\n\tK_G^*(X) \\cong K_{NH}^*(X^H) \\cong \\bigoplus_{[\\rho] \\in NH \\backslash \\Irr(H)}\n\t{}^{\\widetilde{W}_{[\\rho]}} K^*_{W_{[\\rho]}}(X^H)  \\cong  \\bigoplus_{[\\rho] \\in NH \\backslash \\Irr(H)}   {}^{\\alpha_{[\\rho]}} K^*(X^H\/{W}_{[\\rho]})\n\t$$\n\tand $X^H\/ {W}_{[\\rho]} = X^H\/NH_{[\\rho]} \\rightarrow X^H\/NH = X\/G$ is a covering space with fiber $NH\/NH_{[\\rho]}$. \n\t\n\t\\begin{thm} \\label{oneiso}\n\t\tSuppose a compact Lie group $G$ acts continuously on the topological space $X$ with only one orbit type $(G\/H)$. Then \n\t\t$$\n\t\tK_G^*(X) \\cong   \\bigoplus_{[\\rho] \\in NH \\backslash \\Irr(H)}   {}^{\\alpha_{[\\rho]}} K^*(X^H\/NH_{[\\rho]})\n\t\t$$\n\t\tand for every irreducible representation $\\rho$ of $H$ there is a covering space of degree $[NH_{[\\rho]}:NH]$\n\t\t$$\n\t\tX^H\/NH_{[\\rho]} \\rightarrow X\/G\n\t\t$$\n\t\tand a central extension \n\t\t\\[\n\t\t1 \\to S^{1} \\stackrel{\\widetilde{\\iota}}{\\rightarrow} \\widetilde{W}_{[\\rho]} \\rightarrow W_{[\\rho]} \\to 1\n\t\t\\]\n\t\tclassified by a cohomology class $\\alpha_{\\widetilde{W_{[\\rho]}}} \\in H^2(BW_{[\\rho]};S^1)$, and\n\t\t$\\alpha_{[\\rho]} \\in H^2(X^H\/NH_{[\\rho]};S^1)$ is the pullback $p^*(\\alpha_{\\widetilde{W_{[\\rho]}}})$ with \n\t\t$$\n\t\t\\xymatrix{\n\t\t\tEW_{[\\rho]} \\times_{W_{[\\rho]}} X^H \\ar[rr]^{p} \\ar[d]^{\\simeq} & & BW_{[\\rho]} \\\\\n\t\t\tX^H\/NH_{[\\rho]} \\ar[urr]\n\t\t}\n\t\t$$\n\t\\end{thm}\n\t\n\tTheorem \\ref{oneiso} generalizes Theorem 3.9 of \\cite{GU}.\n\t\n\t\\begin{exam}\n\t\tLets look at actions of $SU(2)$ on simply connected $5$-dimensional manifolds with only one isotropy type and orbit space $S^2$. These manifolds have been classified up to equivariant diffeomorphism in  \\cite{Simas}. There can only be three orbit types: $(e)$, $(SU(2)\/(\\mathbb{Z}_2))$ and $(SU(2)\/(\\mathbb{Z}_m))$. We will describe now the possible equivariant diffeomorphic types:\n\t\t\n\t\t\n\t\tConsider for natural numbers $l,m$ with $(l,m)=1$,\n\t\t$$\n\t\t\\mathcal { N } _ { m , m } ^ { l } = \\mathrm { SU } ( 2 ) \\times _ {{ S } ^ { 1 }}   S  ^ { 3 } \n\t\t$$\n\t\twith $S^1$ acting on $SU(2) \\times S^3$ by \n\t\t$$\n\t\tz \\cdot (g,(w_1,w_2)) = (g z^{-l}, (z^m w_1, z^m w_2))\n\t\t$$\n\t\tand $\\mathrm{SU}(2)$ acting on $\\mathcal { N } _ { m , m } ^ { l }$ by\n\t\t$$\n\t\t\\overline{g} \\cdot [g, (w_1,w_2)] = [\\overline{g}g,(w_1,w_2)]\n\t\t$$\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\\item $\\mathcal { N } _ { 1 , 1 } ^ { 0}=SU(2) \\times S^2$ is the free $SU(2)$-manifold. In this case $$K_{SU(2)}^*(\\mathcal{N}_{1,1}^0)\\cong K^*(S^2)\\cong \\mathbb{Z}[H]\/(H-1)^2$$ concentrated in degree 0.\n\t\t\n\t\t\t\\item  $SO(3) \\times S^2$ and $\\mathcal { N } _ { 2 , 2 } ^ { 1}$ are the $SU(2)$-manifolds with isotropy  type $(SU(2)\/\\mathbb{Z}_2)$. $SO(3) \\times S^2$ is not simply connected, but $\\mathcal { N } _ { 2 , 2 } ^ { 1}$ is in fact simply connected.\n\t\t\\end{itemize}\n\t\t\n\t\tFor  isotropy type $(SU(2)\/(\\mathbb{Z}_m))$, with $m\\geq 3$, we have the manifolds\n\t\t$$\n\t\t\\mathcal { N } _ { m , m } ^ { l }.\n\t\t$$ \n\t\t\n\t\tActually, by the theorem of Barden-Smale about  $5$-dimensional simply connected manifolds \n\t\t$$\n\t\t\\mathcal { N } _ { m , m } ^ { l } \\cong S^3 \\times S^2. \n\t\t$$\n\t\t\n\t\tTo use theorem \\ref{oneiso} we need to know the normalizers and the Weyl groups:\n\t\t\n\t\t\n\t\t$$\n\t\t\\begin{array}{l|lll} \n\t\tH & 1 & \\mathbb{Z}_2 & \\mathbb{Z}_m \\\\\n\t\tNH & SU(2) & SU(2) & Pin(2)  \\\\\n\t\tNH\/H & SU(2) & SO(3) & Pin(2) \n\t\t\\end{array}\n\t\t$$\n\t\t\n\t\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t%\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\tLets consider the case $m \\geq 3$. The normalizer in this case is $N\\mathbb{Z}_m=Pin(2)$. \n\t\tBy theorem \\ref{oneiso}\n\t\t\n\t\t$$\n\t\tK_{SU(2)}^*(\\mathcal { N } _ { m , m } ^ { l }  ) \\cong   \\bigoplus_{[\\rho] \\in Pin(2) \\backslash \\Irr(\\mathbb{Z}_m)}   {}^{\\alpha_{[\\rho]}} K^*(\\left (\\mathcal { N } _ { m , m } ^ { l } \\right )^{\\mathbb{Z}_m}\/Pin(2)_{[\\rho]})\n\t\t$$\n\t\twhere for every irreducible representation $\\rho$ of $\\mathbb{Z}_m$ there is a covering space  \n\t\t$$\n\t\t\\left ( \\mathcal { N } _ { m , m } ^ { l }\\right ) ^{\\mathbb{Z}_m}\/Pin(2)_{[\\rho]} \\rightarrow \\mathcal { N } _ { m , m } ^ { l }\/SU(2)\n\t\t$$\n\t\tof degree $[Pin(2)_{[\\rho]}:Pin(2)]$. But $\\mathcal{ N } _ { m , m } ^ { l }\/SU(2) \\cong S^2$, which is simply connected, and therefore the nontrivial coverings are disconnected,\n\t\t$$\n\t\t\\left ( \\mathcal { N } _ { m , m } ^ { l }\\right ) ^{\\mathbb{Z}_m}\/Pin(2)_{[\\rho]} \\cong S^2 \\times [Pin(2)_{[\\rho]}:Pin(2)].\n\t\t$$\n\t\tSince $S^2$ is $2$-dimensional, there are no twistings.\n\t\t\n\t\t\n\t\tTherefore\n\t\t$$\n\t\tK_{SU(2)}^*(\\mathcal { N } _ { m , m } ^ { l }  )  \\cong \\bigoplus_{[\\rho] \\in Pin(2) \\backslash \\Irr(\\mathbb{Z}_m)}    K^*(S^2 \\times [Pin(2)_{[\\rho]}:Pin(2)])   \n\t\t$$\n\t\twhich is just\n\t\t$$\n\t\t\\bigoplus_{[\\rho] \\in Pin(2) \\backslash \\Irr(\\mathbb{Z}_m)} \\bigoplus_{[Pin(2)_{[\\rho]}:Pin(2)]}  K^*(S^2) \\cong  \\bigoplus_{[\\rho] \\in  \\Irr(\\mathbb{Z}_m)}  K^*(S^2)\n\t\t$$\n\t\tThis can be written as\n\t\t\n\t\t$$\n\t\tK_{SU(2)}^*(\\mathcal { N } _ { m , m } ^ { l }  )  \\cong R(\\mathbb{Z}_m) \\otimes K^*(S^2) \\cong \\mathbb{Z}[\\sigma ] \/ (\\sigma^m-1) \\otimes \\mathbb{Z}[\\gamma]\/(\\gamma-1)^2\n\t\t$$\n\t\tconcentrated in degree zero (the isomorphisms are just additive).  The vector bundle $\\gamma$ denotes the tautological\n\t\tline bundle over $S^2$. \n\t\t\n\t\tThe case of isotropy $\\mathbb{Z}_2$ is completely analogue, resulting in a similar answer. We finish by saying that for a simply connected  5-manifold $M$, with action of $SU(2)$ with one isotropy type $(SU(2)\/H)$ and orbit space $S^2$, there is an isomorphism of abelian groups\n\t\t$$K_{SU(2)}^*(M)\\cong R(H)\\otimes \\mathbb{Z}[\\gamma]\/(\\gamma-1)^2.$$\n\t\t\n\t\tNote that this is the equivariant $K$-theory of $SU(2)\/H \\times S^2$, which is an example of a $5$-dimensional $SU(2)$-manifold with only one isotropy type, but nevertheless is not simply connected.\n\t\t\n\t\\end{exam}\n\t\n\tWe can generalize Theorem \\ref{oneiso} to the case of proper actions of Lie groups by using Theorems \\ref{decomp} and \\ref{oneisoproper}.\n\t\n\t\\begin{thm}\n\t\tSupppose the Lie groups  $G$ and $NH$ satisfy (K) and  act properly  on a locally compact Hausdorff second countable space $X$  with only one orbit type $(G\/H)$. Then \n\t\t$$\n\t\tK_G^*(X) \\cong   \\bigoplus_{[\\rho] \\in NH \\backslash \\Irr(H)}   {}^{\\alpha_{[\\rho]}} K^*(X^H\/NH_{[\\rho]})\n\t\t$$\n\t\twhere for every irreducible representation $\\rho$ of $H$ there is a covering space of degree $[NH_{[\\rho]}:NH]$\n\t\t$$\n\t\tX^H\/NH_{[\\rho]} \\rightarrow X\/G\n\t\t$$\n\t\tand a central extension \n\t\t\\[\n\t\t1 \\to S^{1} \\stackrel{\\widetilde{\\iota}}{\\rightarrow} \\widetilde{W}_{[\\rho]} \\rightarrow W_{[\\rho]} \\to 1\n\t\t\\]\n\t\tclassified by a cohomology class $\\alpha_{\\widetilde{W_{[\\rho]}}} \\in H^2(BW_{[\\rho]};S^1)$, and\n\t\t$\\alpha_{[\\rho]} \\in H^2(X^H\/NH_{[\\rho]};S^1)$ is the pullback $p^*(\\alpha_{\\widetilde{W_{[\\rho]}}})$ with \n\t\t$$\n\t\t\\xymatrix{\n\t\t\tEW_{[\\rho]} \\times_{W_{[\\rho]}} X^H \\ar[rr]^{p} \\ar[d]^{\\simeq} & & BW_{[\\rho]} \\\\\n\t\t\tX^H\/NH_{[\\rho]} \\ar[urr]\n\t\t}\n\t\t$$\n\t\t\n\t\\end{thm}\n\t\n\t\n\t\n\t\n\t\n\t\\section{Decomposition in equivariant connective K-homology (compact case)}\\label{sectionc}\n\tIn this section we will provide a decomposition for equivariant connective K-homology in terms of twisted equivariant connective K-homology groups, in a similar manner as in Section \\ref{proper$K$-theory}. A configuration space model as constructed in \\cite{ve2015} and \\cite{ve2019} will be used. It will be assumed throughout this section that $G$ is a compact Lie group.\n\t\n\t\\begin{defn}Let $(X,x_0)$ be a finite, based $G$-CW-complex that is $G$-connected. Consider an $S^1$-central extension\n\t\t$$0\\to S^1\\to\\widetilde{G}\\to G\\to0.$$Define the space $${}^{\\widetilde{G}}\\mathcal{C}(X,x_0,G)=\\bigcup_{n\\geq0}\\Hom^*(C(X,x_0),M_n(L^2(\\widetilde{G},S^1)))$$ with the compact open topology, where $C(X,x_0)$ denotes the $C^*$-algebra of complex valued continuous maps from $X$ vanishing at $x_0$, $$L^2(\\widetilde{G},S^1)=\\{f\\in L^2(\\widetilde{G})\\mid \\text{for }\\lambda\\in S^1 \\text{ and } \\widetilde{g}\\in\\widetilde{G}, f(\\lambda\\widetilde{g})=\\lambda^{-1}f(\\widetilde{g})  \\},$$ and $M_n(-)$ denotes the ring of matrices of size $n\\times n$. \n\t\\end{defn}\n\tResults in \\cite{ve2015} imply that $\\mathcal{C}(X,x_0,G)$  can be described as a configuration space whose elements are finite formal sums\n\t$$\\sum_{i=1}^k(x_i,V_i),$$where  $x_i \\in X-\\{x_0\\}$ with $i\\neq j$ implies $x_i\\neq x_j$, and each $V_i$ is an element of  $L^2(G)^\\infty$ (this is the space of finite-dimensional subspaces of $L^2(G)$) such that if $i\\neq j$ then $V_i$ and $V_j$ are  orthogonal.\n\t\n\tWe have a natural action of $G$ over $\\mathcal{C}(X,x_0,G)$ that we can define in the configuration space description as follows:\n\t$$g\\bullet\\left(\\sum_{i=1}^k(x_i,V_i)\\right)=\\sum_{i=1}^k(gx_i,gV_i).$$\n\t\n\t\\begin{thm}For a compact Lie group $G$ and a finite, based, $G$-CW-complex $(X,x_0)$ that is $G$-connected, there is a natural isomorphism$$\\pi_i\\left(\\mathcal{C}(X,x_0,G)\/G,{\\bf0}\\right)\\cong k^G_i(X,x_0),$$ where $k_*^G(-)$ stands for the $G$-equivariant connective K-homology of the pair $(X,x_0)$. \n\t\\end{thm}\n\t\\begin{remark}\n\t\tWhen $G$ is a finite group, in \\cite{ve2015} it is proved that the homotopy groups of the $G$-invariant part of the configuration space correspond with equivariant  connective K-homology groups. In fact, if $G$ is finite there is a based homotopy equivalence$$\\mathcal{C}(X,x_0,G)\/G\\simeq \\mathcal{C}(X,x_0,G)^G.$$ In the case of compact Lie groups of positive dimension, if $G$ acts on $X$ with only infinite orbits, the $G$-action on $\\mathcal{C}(X,x_0,G)$ is free outside of $x_0$. That is the reason to consider the orbit space.\n\t\\end{remark}\n\tThe elements in ${}^{\\widetilde{G}}\\mathcal{C}(X,X_0,G)\/\\widetilde{G}$ have a description as finite formal sums $$\\sum_{i=1}^k(Gx_i,E_i),$$where $Gx_i$ denotes the $G$-orbit of $x_i$ and $E_i$ is a $\\widetilde{G}$-vector bundle over $Gx_i$, with $S^1$ acting by multiplication by inverses.\n\t\\begin{remark}\n\t\tWhen the based $G$-CW-complex $(X,x_0)$ is not supposed to be $G$-connected, we define the configuration space $$^{\\widetilde{G}}\\mathcal{C}(X,x_0,G)=\\Omega_0\\left({}^{\\widetilde{G}}\\mathcal{C}(\\Sigma X,x_0,G)\\right),$$ where $\\Omega_0$ denotes the based loop space and $\\Sigma$ denotes the reduced suspension.\n\t\\end{remark}\n\t\n\t\\begin{defn}Let $X$ be a finite, $G$-CW-complex. Consider an $S^1$-central extension\n\t\t$$0\\to S^1\\to\\widetilde{G}\\to G\\to0.$$ The $\\widetilde{G}$-twisted, $G$-equivariant connective K-homology groups of $X$are defined by $${}^{\\widetilde{G}}k^G_n(X,x_0)=\\pi_i\\left({}^{\\widetilde{G}}\\mathcal{C}(X_+,+,G)\/\\widetilde{G},{\\bf0}\\right)$$ where $X_+=X\\coprod\\{+\\}$ and $+$ is an extra point with a trivial $G$-action.\n\t\\end{defn}\n\tIn this context, a decomposition is obtained after applying Theorem \\ref{vectordecomp}.\n\t\\begin{thm}\\label{k-homdecomp}\n\t\tLet $G$ be a compact Lie group, with $X$ being a finite $G$-CW-complex on which a normal subgroup $A$ acts trivially. There is a natural isomorphism  \n\t\t\\begin{align*}\\Psi_X:k_*^G(X)&\\to\\bigoplus_{[\\rho]\\in G\\setminus\\Irr(A)}{}^{\\widetilde{Q}_{[\\rho]}}k_*^{Q_{[\\rho]}}(X)\\end{align*} induced by the following map on configuration spaces: \\begin{align*}\\mathcal{C}(\\Sigma X_+,+,G)\/G&\\to\\prod_{[\\rho]\\in G\\setminus\\Irr(A)}{}^{\\widetilde{Q}_{[\\rho]}}\\mathcal{C}(\\Sigma X_+,+,Q_{[\\rho]})\/\\widetilde{Q}_{[\\rho]},\\\\\\sum_{i=1}^k(Gx_i,E_i)&\\mapsto\\bigoplus_{[\\rho]\\in G\\setminus\\Irr(A)}\\sum_{i=1}^k\\left(Gx_i,\\Hom_A(\\rho,E_i)\\right).\\end{align*} This isomorphism is functorial on $G$-maps $X\\to Y$ of $G$-CW-complexes spaces on which $A$ acts trivially. \n\t\\end{thm}\n\t\\begin{proof}\n\t\tJust apply Theorem \\ref{vectordecomp} to each $G$-vector bundle $E_i$ in the sum.\n\t\\end{proof}\n\t\\begin{remark}\n\t\tA decomposition for nonconnective K-homology may require the use of $C^*$-algebras. In a future work we will address this problem.\n\t\\end{remark}\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}