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For a finite group G, Frobenius found a formula for the values of the function ∑IrrG(dim π)−s for even integers s, where IrrG 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 ∑IrrG(dim π)−s(dim πH)−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. [ABSTRACT FROM AUTHOR]

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 |$G$|⁠. [ABSTRACT FROM AUTHOR]

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" [1]. [ABSTRACT FROM AUTHOR]

Many phenomena in geometry and analysis can be explained via the theory of D-modules, but this theory explains close to nothing in the non-archimedean case, by the absence of integration by parts. Hence there is a need to look for alternatives. A central example of a notion based on the theory of D-modules is the notion of holonomic distributions. We study two recent alternatives of this notion in the context of distributions on non-archimedean local fields, namely Cexp-class distributions from Cluckers et al. ['Distributions and wave front sets in the uniform nonarchimedean setting', Trans. Lond. Math. Soc.5(1) (2018), 97–131] and WF-holonomicity from Aizenbud and Drinfeld ['The wave front set of the Fourier transform of algebraic measures', Israel J. Math.207(2) (2015), 527–580 (English)]. We answer a question from Aizenbud and Drinfeld ['The wave front set of the Fourier transform of algebraic measures', Israel J. Math.207(2) (2015), 527–580 (English)] by showing that each distribution of the Cexp-class is WF-holonomic and thus provides a framework of WF-holonomic distributions, which is stable under taking Fourier transforms. This is interesting because the Cexp-class contains many natural distributions, in particular, the distributions studied by Aizenbud and Drinfeld ['The wave front set of the Fourier transform of algebraic measures', Israel J. Math.207(2) (2015), 527–580 (English)]. We show also another stability result of this class, namely, one can regularize distributions without leaving the Cexp-class. We strengthen a link from Cluckers et al. ['Distributions and wave front sets in the uniform nonarchimedean setting', Trans. Lond. Math. Soc.5(1) (2018), 97–131] between zero loci and smooth loci for functions and distributions of the Cexp-class. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory. [ABSTRACT FROM AUTHOR]

Abstract 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 (ρ , C [ X (F) ]) is bounded by C , for any finite field F and any irreducible representation ρ 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. Different aspects of this conjecture were studied in [3,11,6,7]. [ABSTRACT FROM AUTHOR]

We prove that there is no non-trivial autoequvivalence of the category of schemes of finite type over Q . [ABSTRACT FROM AUTHOR]

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 of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the 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 $${{X}}$$ . [ABSTRACT FROM AUTHOR]

Let G be a semisimple algebraic group defined over $$\mathbb {Q}_p$$ , and let $$\Gamma $$ be a compact open subgroup of $$G(\mathbb {Q}_p)$$ . We relate the asymptotic representation theory of $$\Gamma $$ and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both: 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. [ABSTRACT FROM AUTHOR]

For a real reductive group G , the center z ( U ( g ) ) of the universal enveloping algebra of the Lie algebra g of G acts on the space of distributions on G . This action proved to be very useful (see e.g. [17,18,32,8] ). Over non-Archimedean local fields, one can replace the action of z ( U ( g ) ) 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. • The wavefront set of any z -finite distribution ξ on G over any point g ∈ G lies inside the nilpotent cone of T g ⁎ G ≅ g . • Let H 1 , H 2 ⊂ G be symmetric subgroups. Consider the space J of H 1 × 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 subgroups of certain type and the invariance condition is replaced by equivariance. Further we apply those results to density and regularity of relative 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 finite length 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. [ABSTRACT FROM AUTHOR]

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 nonvanishing of spherical Bessel functions for Galois symmetric pairs. [ABSTRACT FROM AUTHOR]

We establish the existence of a transfer, which is compatible with Kloosterman integrals, between Schwartz functions on GLn(R) and Schwartz functions on the variety of non-degenerate Hermitian forms. Namely, we consider an integral of a Schwartz function on GLn (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 GLn(R) is replaced with the variety of non-degenerate hermitian forms. We also obtain similar results for gln(R). The non-Archimedean case was done by H. Jacquet (Duke Math. J., 2003) and our proof is based on the ideas of this work. However we have to face additional difficulties that appear only in the Archimedean case. Those results are crucial for the comparison of the Kuznetsov trace formula and the relative trace formula of GLn with respect to the maximal unipotent subgroup and the unitary group, as done by H. Jacquet, and by B. Feigon, E. Lapid, and O. Offen. [ABSTRACT FROM AUTHOR]

Abstract: A fundamental result in representation theory is Kostantʼs 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. [Copyright &y& Elsevier]

Abstract: We show the disjointness property of Klyachko for and . [Copyright &y& Elsevier]

The goal of this paper we extend the notions of Schwartz functions, tempered functions, and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case the classically known properties of Schwartz functions on and build some additional tools that are important in representation theory. [ABSTRACT FROM PUBLISHER]