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Dmitry Gourevitch

Publications and source records attributed to Dmitry Gourevitch.

At least 55 records · Page 3Linked to original sources
arXiv2014

Derivatives for smooth representations of GL(n,R) and GL(n,C)

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].

math.RT↗
arXiv2013

Intertwining operators between line bundles on Grassmannians

Let G=GL(n,F) where F is a local field of arbitrary characteristic, and let $π_1,π_2$ be representations induced from characters of two maximal parabolic subgroups $P_1,P_2$. We explicitly determine the space $Hom_G(π_1,π_2)$ of intertwining operators and prove that it has dimension at most 1 in all cases.

math.RT↗
arXiv2012

Irreducible representations of product of real reductive groups

Let $G_1,G_2$ be real reductive groups and $(π,V)$ a smooth, irreducible, admissible representation of $G_1 \times G_2$. We prove that $(π,V)$ is the completed tensor product of $(π_i,V_i)$, $i=1,2$, where $(π_i,V_i)$ is a smooth,irreducible,admissible representation of $G_i$, $i=1,2$. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proven in [AG] and the other direction we prove here. As a corollary, we deduce that strong Gelfand property for a pair $H \subset G$ of real reductive groups is equivalent to the usual Gelfand property of the pair $ΔH \subset G \times H$.

math.RT↗
arXiv2012

Annihilator varieties, highest derivatives, Whittaker functionals, and rank for unitary representations of GL(n,R)

In this paper we study irreducible unitary representations of GL(n,R) and prove a number of results. Our first result establishes a precise connection between the annihilator of a representation and the existence of degenerate Whittaker functionals, for both smooth and K-finite vectors, thereby generalizing results of Kostant, Matumoto and others. Our second result relates the annihilator to the sequence of highest derivatives, as defined in this setting by one of the authors. Based on those results, we suggest a new notion of rank of a smooth admissible representation of GL(n,R), which for unitarizable representations refines Howe's notion of rank. Our third result computes the highest derivatives for (almost) all unitary representations in terms of the Vogan classification. We also indicate briefly the analogous results over complex and p-adic fields.

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arXiv2011

Spherical Pairs Over Close Local Fields

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.

math.RT↗
arXiv2010

Smooth Transfer (the Archimedean case)

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.

math.RT↗
arXiv2009

Some regular symmetric pairs

In [AG2] 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, and in particular that any connected symmetric pair over C is a Gelfand pair. In this paper we show that the pairs $$(GL(V),O(V)), (GL(V),U(V)), (U(V),O(V)), (O(V \oplus W),O(V) \times O(W)), (U(V \oplus W),U(V) \times U(W))$$ are regular where V and W are quadratic or hermitian spaces over arbitrary local field of characteristic zero. We deduce from that that the pairs $(GL_n(\C),O_n(\C))$ and $(O_{n+m}(\C),O_n(\C) \times O_m(\C))$ are Gelfand pairs.

math.RT↗
arXiv2009

Multiplicity free Jacquet modules

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)<G be a maximal Levi subgroup. Let U< G be the corresponding unipotent subgroup and let P=MU be the corresponding parabolic subgroup. Let J denote the Jacquet functor from representations of G to representations of M (i.e. the functor of coinvariants w.r.t. U). In this paper we prove that J is a multiplicity free functor, i.e. dim Hom(J(π),ρ)<= 1, for any irreducible representations πof G and ρof M. To do that we adapt the classical method of Gelfand and Kazhdan that proves "multiplicity free" property of certain representations to prove "multiplicity free" property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

math.RT↗
arXiv2009

Uniqueness of Shalika functionals (the Archimedean case)

Let F be either R or C. Let $(π,V)$ be an irreducible admissible smooth \Fre representation of GL(2n,F). A Shalika functional $ϕ:V \to \C$ 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 $$ ϕ[πg & A 0 & g)v] = \exp(2πi \re(\tr (g^{-1}A))) ϕ(v).$$ 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].

math.RT↗
arXiv2009

Multiplicity One Theorems and Invariant Distributions

This is my PhD thesis submitted to the Weizmann Institute of Science. It is based on the papers [AG08c], [AG08d], [AGRS07], [AGS08], [AGS09], [Aiz08] and [SZ08]. This thesis includes an introduction to Gelfand pairs and invariant distributions, a list of tools to work with invariant distributions oriented towards proving Gelfand property and a proof that the pair (GL(n+1,F),GL(n,F)) is a strong Gelfand pair. Namely, we prove that if $π$ is an irreducible admissible smooth representation of GL(n+1,F) and $ρ$ is an irreducible admissible smooth representation of GL(n,F) then $$dim Hom_{GL(n,F)}(π,ρ)\leq 1.$$

math.RT↗
arXiv2009

Generalized Harish-Chandra descent and applications to Gelfand pairs

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 arbitrary local field F of characteristic zero. Our main tool is Luna slice theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pair (GL(n,C),GL(n,R)) is a Gelfand pair. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-archimedean F the later is a classical theorem of Gelfand and Kazhdan. We use the techniques developed here in our subsequent work [AG3] where we prove an archimedean analog of the theorem on uniqueness of linear periods by H. Jacquet and S. Rallis.

math.RT↗
arXiv2009

(O(V+F), O(V)) is a Gelfand pair for any quadratic space V over a local field F

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let $W=V \oplus Fe$ with the form Q extending q with Q(e)=1. Consider the standard embedding of O(V) into O(W) and the two-sided action of $O(V)\times O(V)$ on $O(W)$. In this note we show that any $O(V)\times O(V)$-invariant distribution on O(W) is invariant with respect to transposition. This result was earlier proven in a bit different form in [vD] for F=R, in [AvD] for F=C and in [BvD] for p-adic fields. Here we give a different proof. Using results from [AGS], 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 Frechet representation E of O(W) we have $dim Hom_{O(V)}(E,C) \leq 1.$ A stronger result for p-adic fields is obtained in [AGRS].

math.RT↗
arXiv2009

(GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F

Let F be an arbitrary local field. Consider the standard embedding of GL(n,F) into GL(n+1,F) and the two-sided action of GL(n,F) \times GL(n,F) on GL(n+1,F). In this paper we show that any GL(n,F) \times 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), $$dimHom_{GL(n,F)}(E,\cc) \leq 1.$$ For the proof in the archimedean case we develop several new tools to study invariant distributions on smooth manifolds.

math.RT↗
arXiv2009

Geometry of obstructed equisingular families of projective hypersurfaces

We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructness, we give a detailed description of such families corresponding to quasihomogeneous singularities. Next we study the behavior of these properties with respect to stable equivalence of singularities. We show that under certain conditions, stabilization of singularities ensures the existence of a reduced component of expected dimension. For minimally obstructed families the whole family becomes irreducible. As an application we show that if the equisingular family of a projective hypersurface H has a reduced component of expected dimension then the deformation of H induced by the linear system |H| is complete with respect to one-parameter deformations.

math.AG↗
arXiv2008

Multiplicity one theorem for (GL(n+1,R),GL(n,R))

Let F be either R or C. Consider the standard embedding GL(n,F)<GL(n+1,F) and the action of GL(n,F) on GL(n+1,F) by conjugation. In this paper we show that any GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. We show that this implies that for any irreducible admissible smooth Frechet representations $π$ of GL(n+1,F) and $τ$ of GL(n,F), $$dim Hom_{GL(n,F)}(π,τ) \leq 1.$$ For p-adic fields those results were proven in [AGRS].

math.RT↗
arXiv2008

An archimedean analog of Jacquet - Rallis theorem

In this paper we prove that the symmetric pair $(GL_{n+k}(F),GL_n(F) \times GL_k(F))$ is a Gelfand pair for any local field F of characteristic 0. For non-archimedean F it has been proven in [JR]. We use techniques developed in [AG2] to generalize their proof to general local fields.

math.RT↗
arXiv2008

De-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

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 de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute cohomologies of the Lie algebra 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 Shapiro lemma.

math.AG↗