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We introduce a notion of distributions on $\mathbb{R}^n$, called distributions of C$^{{\mathrm{exp}}}$-class, based on wavelet transforms of distributions and the theory from [6] about C$^{{\mathrm{exp}}}$-class functions. We prove that the framework of C$^{{\mathrm{exp}}}$-class distributions is closed under natural operations, like push-forward, pull-back, derivation and antiderivation, and, in the tempered case, Fourier transforms. Our main result is the (real analytic) holonomicity of all distributions of C$^{{\mathrm{exp}}}$-class.

We prove that a quotient of a Nash manifold $X$ by a closed equivalence relation $R\subset X\times X$, which is submersive over $X$, yields a Nash manifold $X/R$., Comment: 18 pages, comments are welcome

We study equivariant morphisms from zero dimensional schemes to varieties and show that, under suitable assumptions, all such morphisms factor via a canonical one. We relate the above to Algebraic Representations of Ergodic Actions.

Let a complex algebraic reductive group $\bf G$ act on a complex algebraic manifold $\bf X$. For a $\bf G$-invariant subvariety $\Xi$ of the nilpotent cone $\mathcal{N}(\mathfrak{g}^*)\subset \mathfrak{g}^*$ we define a notion of $\Xi$-symplectic complexity of $\bf X$. This notion generalizes the notion of complexity defined in [Vin86]. We prove several properties of this notion, and relate it to the notion of $\Xi$-complexity defined in [AG] motivated by its relation with representation theory., Comment: 10 p

We show that the results of [BM97, DeB02b, Oka, Lus85, AA07, Tay16] imply a positive answer to the question of Moeglin-Waldspurger on wave-front sets in the case of depth zero cuspidal representations. Namely, we deduce that for large enough residue characteristic, the Zariski closure of the wave-front set of any depth zero irreducible cuspidal representation of any reductive group over a non-Archimedean local field is an irreducible variety. In more details, we use [BM97, DeB02b, Oka] to reduce the statement to an analogous statement for finite groups of Lie type, which is proven in [Lus85, AA07, Tay16]., Comment: 7p

Let $\Gamma$ be a finite group, let $\theta$ be an involution of $\Gamma$, and let $\rho$ be an irreducible complex representation of $\Gamma$. We bound $\dim \rho^{\Gamma^{\theta}}$ in terms of the smallest dimension of a faithful $\mathbb{F}_p$-representation of $\Gamma/Rad_p(\Gamma)$, where $p$ is any odd prime and $Rad_p(\Gamma)$ is the maximal normal $p$-subgroup of $\Gamma$. This implies, in particular, that if $\mathbf{G}$ is a group scheme over $\mathbb{Z}$ and $\theta$ is an involution of $\mathbf{G}$, then the multiplicity of any irreducible representation in $C^\infty \left( \mathbf{G}(\mathbb{Z}_p)/ \mathbf{G} ^{\theta}(\mathbb{Z}_p) \right)$ is bounded, uniformly in $p$., Comment: 44 pages; Version 2 - minor corrections

We prove the following result in relative representation theory of a reductive p-adic group $G$: Let $U$ be the unipotent radical of a minimal parabolic subgroup of $G$, and let $\psi$ be an arbitrary smooth character of $U$. Let $S \subset Irr(G)$ be a Zariski dense collection of irreducible representations of $G$. Then the span of the Bessel distributions $B_{\pi}$ attached to representations $\pi$ from $S$ is dense in the space $\mathcal S^*(G)^{U\times U,\psi \times \psi}$ of all $(U\times U,\psi \times \psi)$-equivariant distributions on $G.$ We base our proof on the following results: 1. The category of smooth representations $\mathcal M(G)$ is Cohen-Macaulay. 2. The module $ind_U^G(\psi)$ is a projective module., Comment: 17 pages

We explain and correct a mistake in Section 2.6 and Appendix C of the first and second author's paper "Representation Growth and Rational Singularities of the Moduli Space of Local Systems" arXiv:1307.0371., Comment: 3 pages, 2 figures, to appear in Inventiones Mathematicae

Given a finite group $G$ and its representation $\rho$, the corresponding McKay graph is a graph $\Gamma(G,\rho)$ whose vertices are the irreducible representations of $G$; the number of edges between two vertices $\pi,\tau$ of $\Gamma(G,\rho)$ is $dim Hom_G(\pi \otimes \rho, \tau) $. The collection of all McKay graphs for a given group $G$ encodes, in a sense, its character table. Such graphs were also used by McKay to provide a bijection between the finite subgroups of $SU(2)$ and the affine Dynkin diagrams of types $A, D, E$, the bijection given by considering the appropriate McKay graphs. In this paper, we classify all (undirected) trees which are McKay graphs of finite groups and describe the corresponding pairs $(G,\rho)$; this classification turns out to be very concise. Moreover, we give a partial classification of McKay graphs which are forests, and construct some non-trivial examples of such forests., Comment: Ver2: Theorem A and its proof corrected, many examples added

Let $G$ be a real reductive algebraic group, and let $H\subset G$ be an algebraic subgroup. It is known that the action of $G$ on the space of functions on $G/H$ is "tame" if this space is spherical. In particular, the multiplicities of the space $\mathcal{S}(G/H)$ of Schwartz functions on $G/H$ are finite in this case. In this paper we formulate and analyze a generalization of sphericity that implies finite multiplicities in $\mathcal{S}(G/H)$ for small enough irreducible representations of $G$., Comment: 31p. Appendix B by Ido Karshon. v4: extended version - added results on the homogeneous case and branching problems - see sections 1.1 and 1.2

For a Nash submersion $\phi\colon X\to Y$, we study the complex $\mathcal{SDR}(\phi)$ of Schwartz sections of the relative de Rham complex of $\phi$. We define the notion of Schwartz sections of constructible sheaves on Nash manifolds and prove that $\mathcal{SDR}(\phi)$ is homotopy equivalent to the Schwartz sections of the proper push-forward $\phi_!\mathbb{R}_X$ of the constant sheaf $\mathbb{R}_X$. Using this equivalence, we show that $\mathcal{SDR}(\phi)$ depends (up to homotopy equivalence) only on the homology type of the map $\phi$. We also deduce that $\mathcal{SDR}(\phi)$ has Hausdorff homology spaces., Comment: 70 pages

We define a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. We study commutativity, cancellativity and associativity of this operation. We focus on rigid irreducible components and discuss inductive ways to construct them., Comment: final version

We show that under certain conditions the Gelfand-Kazhdan criterion for the Gelfand property is a necessary condition. We work in the generality of finite groups, however part of the argument carries over to p-adic and real groups.

We show that any stack $\mathfrak{X}$ of finite type over a Noetherian scheme has a presentation $X \rightarrow \mathfrak{X}$ by a scheme of finite type such that $X(F) \rightarrow \mathfrak{X}(F)$ is onto, for every finite or real closed field $F$. Under some additional conditions on $\mathfrak{X}$, we show the same for all perfect fields. We prove similar results for (some) Henselian rings. We give two applications of the main result. One is to counting isomorphism classes of stacks over the rings $\mathbb{Z}/p^n$; the other is about the relation between real algebraic and Nash stacks., Comment: 20 pages

In the context of geometry and analysis on non-archimedean local fields, we study two recent notions, $C^{\mathrm exp}$-class distributions from [11] and WF-holonomicity from [1], and we show that any distribution of $C^{\mathrm exp}$-class is WF-holonomic. Thus we answer a question from [1] by providing a framework of WF-holonomic distributions for non-archimedean local fields which is stable under taking Fourier transforms and which contains many natural distributions, in particular, the distributions studied in [1]. We further show that one can regularize distributions without leaving the $C^{\mathrm exp}$-class. Finally, we show a close link between zero loci and smooth loci for functions and distributions of $C^{\mathrm exp}$-class, by proving a converse to a result of [11]. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.

In this work we prove the local multiplicity at most one theorem underlying the definition and theory of local $\gamma$-, $\epsilon$- and $L$-factors, defined by virtue of the generalized doubling method, over any local field of characteristic 0. We also present two applications: one to the existence of local factors for genuine representations of covering groups, the other to the global unfolding argument of the doubling integral., Comment: To be published in the J. Eur. Math. Soc. The main body of the paper by Dmitry Gourevitch and Eyal Kaplan. Appendix A by Avraham Aizenbud and Dmitry Gourevitch. V2: Appendix A added

We study homological multiplicities of spherical varieties of reductive group $G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the category of smooth representations of a $p$-adic group, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities are finite, in particular this holds for symmetric varieties, conjectured for all spherical varieties and known for a large class of spherical varieties. Furthermore, we show that the Euler-Poincar\'e characteristic is constant in families induced from admissible representations of a Levi $M.$ In the case when $M=G$ we compute these multiplicities more explicitly., Comment: 14 pages

Let $E/F$ be an unramified extension of non-archimedean local fields of residual characteristic different than $2$. We provide a simple geometric proof of a variation of a result of Y. Hironaka. Namely we prove that the module $\mathcal{S}(X)^{K_0}$ is free over the Hecke algebra $\mathcal{H}(SL_{n}(E),SL_{n}(O_E))$, where $X$ is the space of unimodular Hermitian forms on $E^n$ and $O_E$ is the ring of integers in $E$., Comment: 7 pages

We study the space of invariant generalized functions supported on an orbit of the action of a real algebraic group on a real algebraic manifold. This space is equipped with the Bruhat filtration. We study the generating function of the dimensions of the filtras, and give some methods to compute it. To illustrate our methods we compute those generating functions for the adjoint action of $\mathrm{GL}_3(\mathbb{C})$. Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently by Sakellaridis., Comment: 10 pages

We prove that there is no non-trivial aoutoequvivalence of the categoryof schemes of finite type over $\mathbb{Q}$.

Let $G$ be a reductive group scheme of type $A$ acting on a spherical scheme $X$. We prove that there exists a number $C$ such that the multiplicity $\dim Hom(\rho,\mathbb{C}[X(F)])$ is bounded by $C$, for any finite field $F$ and any irreducible representation $\rho$ of $G(F)$. We give an explicit bound for $C$. We conjecture that this result is true for any reductive group scheme and when $F$ ranges (in addition) over all local fields of characteristic $0$.

For a finite group $G$, Frobenius found a formula for the values of the function $\sum_{\mathrm{Irr} G} (\dim\, \pi)^{-s}$ for even integers $s$, where $\mathrm{Irr} G$ is the set of irreducible representations of $G$. We generalize this formula to the relative case: for a subgroup $H$, we find a formula for the values of the function $\sum_{\mathrm{Irr} G} (\dim\, \pi)^{-s} (\dim\, \pi ^H)^{-t}$. We apply our results to compute the E-polynomials of Fock--Goncharov spaces and to relate the Gelfand property to the geometry of generalized Fock--Goncharov spaces.

We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than one, let $r_n(\Gamma)$ be the number of irreducible $n$-dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(\Gamma)=O(n^C)$ for every such $\Gamma$. This answers a question of Larsen and Lubotzky., Comment: version 2: The prove of the main theorem was simplified and a result on arithmetic algebraic geometry was added. 29 pages

In this paper, we prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results from [KO13,KS16] on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. In order to deduce this application we prove relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group $G$ defined over $\mathbb{R}$, the space of $G(\mathbb{R})$-equivariant distributions on the manifold of real points of any algebraic $G$-manifold $X$ is finite-dimensional if $G$ has finitely many orbits on $X$., Comment: 16 pages. v2: Important corrections: confusion between algebraic varieties and their real points removed. The generality of Theorem E(i) and Corollary F(i) became narrower

For a real reductive group G, the center $\mathfrak{z}(\mathcal{U}(\mathfrak{g}))$ of the universal enveloping algebra of the Lie algebra $\mathfrak{g}$ of G acts on the space of distributions on G. This action proved to be very useful (see e.g. [HC63, HC65, Sha74, Bar03]). Over non-Archimedean local fields, one can replace this action by the action of the Bernstein center z of G, i.e. the center of the category of smooth representations. However, this action is not well studied. In this paper we provide some tools to work with this action and prove the following results. 1) The wave-front set of any z-finite distribution on G over any point $g\in G$ lies inside the nilpotent cone of $T_g^*G \cong \mathfrak{g}$. 2) Let $H_1,H_2 \subset G$ be symmetric subgroups. Consider the space J of $H_1\times H_2$-invariant distributions on G. We prove that the z-finite distributions in J form a dense subspace. In fact we prove this result in wider generality, where the groups $H_i$ are spherical groups of certain type and the invariance condition is replaced by equivariance. Further we apply those results to density and regularity of spherical characters. The first result can be viewed as a version of Howe's expansion of characters. The second result can be viewed as a spherical space analog of a classical theorem on density of characters of admissible representations. It can also be viewed as a spectral version of Bernstein's localization principle. In the Archimedean case, the first result is well-known and the second remains open., Comment: 27 pages. v2:version to appear in Advances in Mathematics

We prove vanishing of distribution on p-adic spherical spaces that are equivariant with respect to a generic character of the nilradical of a Borel subgroup and satisfy a certain condition on the wave-front set. We deduce from this non-vanishing of spherical Bessel functions for Galois symmetric pairs., Comment: 9 pages. v2: extended version, to appear in IMRN

Let $H$ be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold $X$ and a real algebraic bundle $\mathcal{E}$ on $X$. Let $\mathfrak{h}$ be the Lie algebra of $H$. Let $\mathcal{S}(X,\mathcal{E})$ be the space of Schwartz sections of $\mathcal{E}$. We prove that $\mathfrak{h}\mathcal{S}(X,\mathcal{E})$ is a closed subspace of $\mathcal{S}(X,\mathcal{E})$ of finite codimension. We give an application of this result in the case when $H$ is a real spherical subgroup of a real reductive group $G$. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let $\pi$ be a Casselman-Wallach representation of $G$ and $V$ be the corresponding Harish-Chandra module. Then the natural morphism of coinvariants $V_{\mathfrak{h}}\to \pi_{\mathfrak{h}}$ is an isomorphism if and only if any linear $\mathfrak{h}$-invariant functional on $V$ is continuous in the topology induced from $\pi$. The latter statement is known to hold in two important special cases: if $H$ includes a symmetric subgroup, and if $H$ includes the nilradical of a minimal parabolic subgroup of $G$., Comment: v4: version appearing in Math. Z + erratum added in the end

We prove vanishing of distributions on a split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup, and eigen with respect to the center of the universal enveloping algebra of the Lie algebra of G. This is a generalization of a result by Shalika, that concerned the group case. Shalika's result was crucial in the proof of his multiplicity one theorem. We view our result as a step in the study of multiplicities of quasi-regular representations on spherical varieties. As an application we prove non-vanishing of spherical Bessel functions., Comment: 7 pages. v2: extended version

We relate the asymptotic representation theory of $SL(d,\mathbb{Z}_p)$ and the singularities of the moduli space of $SL(d)$-local systems on a smooth projective curve, proving new theorems about both. Regarding the former, we prove that, for every d, the number of n-dimensional representations of $SL(d,\mathbb{Z}_p)$ grows slower than $n^{22}$, confirming a conjecture of Larsen and Lubotzky. Regarding the latter, we prove that the moduli space of $SL(d)$-local systems on a smooth projective curve of genus at least 12 has rational singularities. Most of our results apply more generally to semi-simple algebraic groups. For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities., Comment: preliminary version, comments are welcome. v2. Revised version, now covering all semi simple groups

We study the Fourier transform of the absolute value of a polynomial on a finite-dimensional vector space over a local field of characteristic 0. We prove that this transform is smooth on an open dense set. We prove this result for the Archimedean and the non-Archimedean case in a uniform way. The Archimedean case was proved in [Ber]. The non-Archimedean case was proved in [HK] and [CL]. Our method is different from those described in [Ber,HK,CL]. It is based on Hironaka's desingularization theorem, unlike [Ber] which is based on the theory of D-modules and [HK,CL] which is based on model theory. Our method also gives bounds on the open dense set where the Fourier transform is smooth. These bounds are explicit in terms of resolution of singularities. We also prove the same result on the Fourier transform of other measures of algebraic origins., Comment: 40 pages

The notion of derivatives for smooth representations of $GL(n,\mathbb{Q}_p)$ was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the "adduced" representation. In [AGS] derivatives of all orders were defined for smooth admissible Frechet representations (of moderate growth). A key ingredient of this definition is the functor of twisted coinvariants with respect to the nilradical of the mirabolic subgroup. In this paper we prove exactness of this functor and compute it on a certain class of representations. This implies exactness of the highest derivative functor, and allows to compute highest derivatives of all monomial representations. In [AGS] these results are applied to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations., Comment: 28 pages. v2: minor corrections, version to appear in the Israel Journal of Mathematics

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12]., Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematics

A fundamental result in representation theory is Kostant's theorem which describes the algebra of polynomials on a reductive Lie algebra as a module over its invariants. We prove a quantum analogue of this theorem for the general linear group, and from this deduce the analogous result for reflection equation algebras., Comment: 13 pages

We establish the existence of a transfer, which is compatible with Kloosterman integrals, between Schwartz functions on GL(n,R) and Schwartz functions on the variety of non-degenerate Hermitian forms. Namely, we consider an integral of a Schwartz function on GL(n,R) along the orbits of the two sided action of the groups of upper and lower unipotent matrices twisted by a non-degenerate character. This gives a smooth function on the torus. We prove that the space of all functions obtained in such a way coincides with the space that is constructed analogously when GL(n,R) is replaced with the variety of non-degenerate hermitian forms. We also obtain similar results for gl(n,R). The non-Archimedean case is done by Jacquet and our proof follows the same lines. However we have to face additional essential difficulties that appear only in the Archimedean case., Comment: 24 pages, no figures

Let F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G:=GL(n+k,F) and let M:=GL(n,F) x GL(k,F)

Extending results of Kazhdan to the relative case, we relate harmonic analysis over some spherical spaces G(F)/H(F), where F is a field of positive characteristic, to harmonic analysis over the spherical spaces G(E)/H(E), where E is a suitably chosen field of characteristic 0. One of the Ingredients of the proof is a condition for finite generation of some modules over the Hecke algebra. We apply our results to show that the pair (GL_{n+1},GL_n) is a strong Gelfand pair for all local fields, and that the pair (GL_{n+k},GL_n x GL_k) is a Gelfand pair for all local fields of odd characteristic., Comment: v2. Section 2 changed. v3. A change in the definition of uniform spherical pair, which also caused some changes in the formulations of the theorems. To appear in Commentarii Mathematici Helvetici

Let F be either R or C. Let $(\pi,V)$ be an irreducible admissible smooth \Fre representation of GL(2n,F). A Shalika functional $\phi:V \to \C$ is a continuous linear functional such that for any $g\in GL_n(F), A \in \Mat_{n \times n}(F)$ and $v\in V$ we have $$ \phi[\pi g & A 0 & g)v] = \exp(2\pi i \re(\tr (g^{-1}A))) \phi(v).$$ In this paper we prove that the space of Shalika functionals on V is at most one dimensional. For non-Archimedean F (of characteristic zero) this theorem was proven in [JR]., Comment: 9 pages. v2:corrected version, to appear in Pacific Journal of Mathematics

In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna Slice Theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pairs (GL(n+k,F), GL(n,F) x GL(k,F)) and (GL(n,E), GL(n,F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proven earlier by Jacquet and Rallis and the second by Flicker. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-Archimedean F the latter is a classical theorem of Gelfand and Kazhdan., Comment: A merge of arXiv:0803.3395 and arXiv:0803.3397 (with no additional material). Appendix D by Avraham Aizenbud, Dmitry Gourevitch and Eitan Sayag. v2,v3: minor changes. v4: correction in the speciality criterion (7.3.7). v5: minor correction in the proof of Proposition 7.2.1

Let X be a smooth real algebraic variety. Let $\xi$ be a distribution on it. One can define the singular support of $\xi$ to be the singular support of the $D_X$-module generated by $\xi$ (some times it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of $T^*X$. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see e.g. [AG_AMOT, AS, Say]). The aim of this paper is to give an analog of this theorem to the non-Archimedean case. The theory of D-modules is not available to us so we need a different definition of the singular support. We use the notion wave front set from [Hef] and define the singular support to be its Zariski closure. Then we prove that the singular support satisfies some property that we call weakly coisotropic, which is weaker than being coisotropic but is enough for some applications. We also prove some other properties of the singular support that were trivial in the Archimedean case (using the algebraic definition) but not obvious in the non-Archimedean case., Comment: 18 pages

We study invariant distributions on the tangent space to a symmetric space. We prove that an invariant distribution with the property that both its support and the support of its Fourier transform are contained in the set of non-distinguished nilpotent orbits, must vanish. We deduce, using recent developments in the theory of invariant distributions on symmetric spaces that the symmetric pair (GL(2n,R),Sp(2n,R)) is a Gelfand pair. More precisely, we show that for any irreducible smooth admissible Frechet representation $(\pi,E)$ of GL(2n,R) the space of continuous functionals $Hom_{Sp_{2n}(R)}(E,C)$ is at most one dimensional. Such a result was previously proven for p-adic fields in [HR] for the field of complex numbers in [S]., Comment: 11 pages

Let F be either R or C. Consider the standard embedding GL(n,F)