arXiv subjects

Avraham Aizenbud

Publications and source records attributed to Avraham Aizenbud.

At least 55 records · Page 3Linked to original sources

arXiv2014

Twisted homology for the mirabolic nilradical

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.

math.RT↗

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↗

arXiv2014

Representation Growth and Rational Singularities of the Moduli Space of Local Systems

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.

math.AG↗

arXiv2014

The wave front set of the Fourier transform of algebraic measures

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.

math.AG↗

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

A quantum analogue of Kostant's theorem for the general linear group

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.

math.QA↗

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

Functoriality in Morse theory on closed manifolds

We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We give sufficient conditions under which composition is preserved. As an application we provide a new proof that the cup product as defined in Morse theory on the chain level agrees with the cup product in singular cohomology. In appendices we present a proof (due to Paul Biran) that the unstable manifolds of a Morse-Smale system are the open cells of a CW structure on the underlying manifold, and also we show that the Morse complex of the triple is canonically isomorphic to the cellular complex of the CW structure. This gives a new proof that the Morse complex is actually a complex and that it computes the homology of the manifold.

math.DG↗

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

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↗

arXiv2008

A partial analog of integrability theorem for distributions on p-adic spaces and applications

Let X be a smooth real algebraic variety. Let $ξ$ be a distribution on it. One can define the singular support of $ξ$ to be the singular support of the $D_X$-module generated by $ξ$ (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.

math.RT↗

arXiv2008

Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R))

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 $(π,E)$ 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].

math.RT↗

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↗