Publications

Let Γ be a finite group, let θ be an involution of Γ and let p be an irreducible complex representation of Γ. We bound dimp^Γθ in terms of the smallest dimension of a faithful F_p-representation of Γ/Rad_p(Γ), where p is any odd prime and Rad_p(Γ) is the maximal normal p-subgroup of Γ. This implies, in particular, that if G is a group scheme over Z and θ is an involution of G, then the multiplicity of any irreducible representation in C^∞(G(Z_p)/G^θ(Z_p)) is bounded, uniformly in p.

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.

Let G be a real reductive algebraic group, and let H ⊂ G be an algebraic subgroup. It is known that the action of G on the space of functions on G/H is \u201ctame\u201d if this space is spherical. In particular, the multiplicities of the space 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 S(G/H) for small enough irreducible representations of G.

Let a complex algebraic reductive group G act on a complex algebraic manifold X. For a G-invariant subvariety Ξ of the nilpotent cone N(g^∗)⊂g^∗ we define a notion of Ξ-symplectic complexity of X. This notion generalizes the notion of complexity defined in Vinberg (1986). We prove several properties of this notion, and relate it to the notion of Ξ-complexity defined in Aizenbud and Gourevitch (2024) motivated by its relation with representation theory.

We show that recent results 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 results of Barbasch and Moy, DeBacker, and Okaka to reduce the statement to an analogous statement for finite groups of Lie type, which was proven by Lusztig, Achar and Aubert, and Taylor.

Given a finite group G and its representation ρ, the corresponding McKay graph is a graph Γ(G, ρ) whose vertices are the irreducible representations of G; the number of edges between two vertices π, τ of Γ(G, ρ) is dim Hom_G(π ⊗ ρ, τ). 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, ρ); 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.

We prove the local multiplicity at most one theorem underlying the definition and theory of local -, ∊- 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.

We explain and correct a mistake in Section 2.6 and Appendix C of the first and second authors paper \u201cRepresentation Growth and Rational Singularities of the Moduli Space of Local Systems\u201d [1].

Regrettably, the paper [1] had two incorrect lemmas. Nevertheless, the statements of the theorems in [1] are correct with no modification required. Specifically, [1, Lemma 6.4] does not hold for spherical spaces, and also the conclusion of [1, Lemma B.2] is not valid in the generality stated there. The effect of the error in [1, Lemma 6.4] on the proof of [1, Theorem 6.1] is significant. In § 1 one we provide a proof of [1, Theorem 6.1] that does not relay upon [1, Lemma 6.4]. The effect of the error in [1, Lemma B.2] is minor and we explain the nessecary modifications in §2.

We show that any stack X of finite type over a Noetherian scheme has a presentation X→X by a scheme of finite type such that X(F)→X(F) is onto, for every finite or real closed field F. Under some additional conditions on 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 Z/pn; the other is about the relation between real algebraic and Nash stacks.

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 C-exp-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 C-exp-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 C-exp-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 C-exp-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 C-exp-class. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.

For a spherical variety X of a reductive group G over a non-archimedean local field F, and for a smooth representation pi of G we study homological multiplicities dimExtG(S(X),pi). Based on Bernstein's decomposition of the category of smooth representations of G, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities are finite. The latter are known to be finite for symmetric varieties and for many other spherical varieties and conjectured to be finite for all spherical varieties. Furthermore, we show that the Euler-Poincare characteristic is constant in families parabolically induced from finite length representations of a Levi subgroup M

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 dimHom(rho, 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.Different aspects of this conjecture were studied in [3,11,6,7]. 

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 Q-rank is greater than 1, then 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 (in fact, any C > 40 suffices) such that r(n) (Gamma) = O(n(C)) for every such Gamma. This answers a question of Larsen and Lubotzky.

We prove that there is no non-trivial autoequvivalence of the category of schemes of finite type over Q.

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 GL(3)(C). Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently in [Sak16].

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 defined over , the space of -equivariant distributions on the manifold of real points of any algebraic -manifold is finite-dimensional if has finitely many orbits on .

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle E on X. Let h be the Lie algebra of H. Let S(X, E) be the space of Schwartz sections of E. We prove that hS(X, E) is a closed subspace of S(X, 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 π be a CasselmanWallach representation of G and V be the corresponding HarishChandra module. Then the natural morphism of coinvariants V _h → π _h is an isomorphism if and only if any linear h-invariant functional on V is continuous in the topology induced from π. 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.

The following changes to the main results of [1] are necessary: (1) In Theorem A and Corollary B the following assumption is required: the number of orbits of the complexification H_C on G_C/P_C is finite, where P is a minimal parabolic subgroup of G. (2) In Theorem C the following additional assumption is required: the number of orbits of H_C on X_C is finite.

Let G be a semisimple algebraic group defined over (Formula presented.) , and let (Formula presented.) be a compact open subgroup of (Formula presented.). We relate the asymptotic representation theory of (Formula presented.) and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both:(1)We prove that there is a constant C, independent of G, such that the number of n-dimensional representations of (Formula presented.) grows slower than (Formula presented.) , confirming a conjecture of Larsen and Lubotzky. In fact, we can take (Formula presented.). We also prove the same bounds for groups over local fields of large enough characteristic.(2)We prove that the coarse moduli space of G-local systems on a smooth projective curve of genus at least (Formula presented.) has rational singularities. 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.

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 xi on G over any point g is an element of G lies inside the nilpotent cone of T-g*G congruent to g. Let H-1, H-2 subset of G be symmetric subgroups. Consider the space J of H-1 x 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. (C) 2015 Elsevier Inc. All rights reserved.

We prove vanishing of (Formula presented.)-eigen 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. This is a generalization of a result by Shalika, that concerned the group case. Shalikas 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.

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 [Ber1]. The non-Archimedean case was proved in [HK] and [CL1], [CL2]. Our method is different from those described in [Ber1], [HK], [CL1], [CL2]. It is based on Hironakas desingularization theorem, unlike [Ber1] which is based on the theory of D-modules and [HK], [CL1], [CL2] which is based on model theory. Our method also gives bounds on the open dense set where the Fourier transform is smooth and, moreover, on the wave front set of the Fourier transform. These bounds are explicit in terms of resolution of singularities and field-independent. We also prove the same results on the Fourier transform of more general measures of algebraic origins.

The notion of derivatives for smooth representations of GL(n, 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 \u201cadduced\u201d representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real 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 the companion paper [AGS] 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, GS13a].

The notion of derivatives for smooth representations of GL(n, 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 \u201cadduced\u201d representation. In [AGS] derivatives of all orders were defined for smooth admissible Fréchet 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.

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.

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 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 GL_n 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.

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) × GL_k(F) *lt; G be a maximal Levi subgroup. LetU G _M : M(G) → M(M) be the Jacquet functor, i.e., the functor of coinvariants with respect toU. In this paper we prove that J is a multiplicity free functor, i.e., dimHom_M( J(π), ρ) ≤ 1, for any irreducible representations π of G and ρ of M. We adapt the classical method of Gelfand and Kazhdan, which proves the "multiplicity free" property of certain representations to prove the "multiplicity free" property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

Extending results of [Kaz86] 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. We apply our results to show that the pair .(GLnC1.F ); GL _n.(F)) is a strong Gelfand pair for all local fields of arbitrary characteristic, and that the pair .GL _nCk.(F); GL _n.(F)̃GLk.(F)) is a Gelfand pair for local fields of any characteristic different from 2. We also give a criterion for finite generation of the space of K-invariant compactly supported functions on G.E/=H.E/ as a module over the Hecke algebra.

In an earlier paper we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, including all connected symmetric pairs over ℂ. In this paper we show that the pairs (GL(V ),O(V )), (GL(V ), U(V )), (U(V ),O(V )), (O(V ⊕ W),O(V ) × O(W)), (U(V ⊕ W), U(V ) × U(W)) are regular, where V and W are quadratic or Hermitian spaces over an arbitrary local field of characteristic zero. We deduce from this that the pairs (GLⁿ (ℂ),O_n (ℂ)) and (O_n+m (ℂ),O_n (ℂ) × Om(ℂ)) are Gelfand pairs.

In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i. e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of the de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute the cohomologies of the Lie algebra g of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as the Shapiro lemma.

In the local, characteristic 0, non-Archimedean case, we consider distributions on GL.(n+1) which are invariant under the adjoint action of GL.(n). We prove that such distributions are invariant by transposition. This implies multiplicity at most one for restrictions from GL.(n+1) to GL.(n). Similar theorems are obtained for orthogonal or unitary groups.

Let F be either R{double-struck} or C{double-struck}. Let (π, V) be an irreducible admissible smooth Fréchet representation of GL_2n(F). A Shalika functional φ: V → C{double-struck} is a continuous linear functional such that for any g ∈ GL_n(F), A ∈ Mat_n×n(F) and v ∈ V we have. In this paper we prove that the space of Shalika functionals on V is at most one-dimensional. For nonarchimedean F (of characteristic zero) this theorem was proved by Jacquet and Rallis.

In the first part of this article, 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, we apply this technique to symmetric pairs. In particular, we prove that the pairs (GL_n+k(F), GL_n(F) × 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 proved earlier by Jacquet and Rallis [JR] and the second result was proved by Flicker [F]. 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.

Let F be either ℝ or ℂ. Consider the standard embedding GL_n(F) {right arrow, hooked} GL_n+1(F) and the action of GL_n(F) on GL_n+1(F) by conjugation. We show that any GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL_n+1(F) and τ of GL_n(F), dim Hom_GLn(F)(π, τ) ≤ 1. For p-adic fields those results were proven in [AGRS].

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let W=V Fe with the form Q extending q with Q(e) = 1. Consider the standard embedding O(V) → O (W) and the two-sided action of O(V)× O(V) on O(W) . In this note we show that any O(V) × O(V) -invariant distribution on O(W) is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581-593, 1986) for F=ℝ , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for F=ℂ and in Bosman and van Dijk (Geometriae Dedicata 50:261-282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), we show that this result on invariant distributions implies that the pair (O(V), O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Fréchet representation (π, E) of O(W) we have dim Hom_O(V)(E,ℂ) ≤ 1. A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).

Let F be an arbitrary local field. Consider the standard embedding GL _n(F) → GL_n+1(F) and the two-sided action of GL _n(F)× GL_n(F) on GL_n+1(F). In this paper we show that any GL_n(F) × GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We show that this implies that the pair (GL_n+1(F), GL_n(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GL_n+1(F), Hom_GLn(F)(E,ℂ) ≤ 1. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

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 ⁿ and build some additional tools that are important in representation theory.