Papers by Dmitry Gourevitch
Proceedings of the American Mathematical Society
One of the key ingredients in the recent construction of the generalized doubling method is a new... more One of the key ingredients in the recent construction of the generalized doubling method is a new class of models, called ( k , c ) (k,c) models, for local components of generalized Speh representations. We construct a family of ( k , c ) (k,c) representations, in a purely local setting, and discuss their realizations using inductive formulas. Our main result is a uniqueness theorem which is essential for the proof that the generalized doubling integral is Eulerian.
arXiv (Cornell University), Sep 1, 2021
The classical Stone-von Neuman theorem relates the irreducible unitary representations of the Hei... more The classical Stone-von Neuman theorem relates the irreducible unitary representations of the Heisenberg group $H_n$ to non-trivial unitary characters of its center $Z$, and plays a crucial role in the construction of the oscillator representation for the metaplectic group. In this paper we extend these ideas to non-unitary and non-irreducible representations, thereby obtaining an equivalence of categories between certain representations of $Z$ and those of $H_n$. Our main result is a smooth equivalence, which involves the fundamental ideas of du Cloux on differentiable representations and smooth imprimitivity systems for Nash groups. We show how to extend the oscillator representation to the smooth setting and give an application to degenerate Whittaker models for representations of reductive groups. We also include an algebraic equivalence, which can be regarded as a generalization of Kashiwara's lemma from the theory of $D$-modules.
arXiv (Cornell University), Sep 23, 2021
One of the key ingredients in the recent construction of the generalized doubling method is a new... more One of the key ingredients in the recent construction of the generalized doubling method is a new class of models, called (k, c) models, for local components of generalized Speh representations. We construct a family of (k, c) representations, in a purely local setting, and discuss their realizations using inductive formulas. Our main result is a uniqueness theorem which is essential for the proof that the generalized doubling integral is Eulerian.
Annals of Mathematics, Aug 9, 2010
ABSTRACT
Forum of Mathematics, Sigma, 2021
We provide a microlocal necessary condition for distinction of admissible representations of real... more We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs.Let${\mathbf {G}}$be a complex algebraic reductive group and${\mathbf {H}}\subset {\mathbf {G}}$be a spherical algebraic subgroup. Let${\mathfrak {g}},{\mathfrak {h}}$denote the Lie algebras of${\mathbf {G}}$and${\mathbf {H}}$, and let${\mathfrak {h}}^{\bot }$denote the orthogonal complement to${\mathfrak {h}}$in${\mathfrak {g}}^*$. A${\mathfrak {g}}$-module is called${\mathfrak {h}}$-distinguished if it admits a nonzero${\mathfrak {h}}$-invariant functional. We show that the maximal${\mathbf {G}}$-orbit in the annihilator variety of any irreducible${\mathfrak {h}}$-distinguished${\mathfrak {g}}$-module intersects${\mathfrak {h}}^{\bot }$. This generalises a result of Vogan [Vog91].We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet mod...
Proceedings of Symposia in Pure Mathematics, 2019
Representations of Reductive Groups, 2019
The study of Whittaker models for representations of reductive groups over local and global field... more The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker models. In order to encompass other representations, one attaches a degenerate (or a generalized) Whittaker model $W_{\mathcal{O}}$, or a Fourier coefficient in the global case, to any nilpotent orbit $\mathcal{O}$. In this note we survey some classical and some recent work in this direction - for Archimedean, p-adic and global fields. The main results concern the existence of models. For a representation $\pi$, call the set of maximal orbits $\mathcal{O}$ with $W_{\mathcal{O}}$ that includes $\pi$ the Whittaker support of $\pi$. The two main questions discussed in this note are: (1) What kind of orbits can appear in the Whittaker support of a representation? (2) How does the Whittaker support of a given representation $\pi$ relate to other invariants of $\pi$, such as its wave-front set?
In this paper, we prove that any spherical character of any admissible representation of a real r... more In this paper, we prove that any spherical character of any admissible representation of a real reductive group G with respect to any pair of spherical subgroups is a holonomic distribution on G. This implies that the restriction of the spherical 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 the recent results of [KO13; KS] 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 BernsteinKashiwara 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, the space of G-equivariant distributions on any algebrai...
arXiv: Representation Theory, 2019
Let $F$ be a non-Archimedean local field. Let $G$ be an algebraic group over $F$. A $G$-variety $... more Let $F$ be a non-Archimedean local field. Let $G$ be an algebraic group over $F$. A $G$-variety $X$ defined over $F$ is said to be multiplicity-free if for any admissible irreducible representation $\pi$ of $G(F)$ the following takes place: $\dim Hom_{G(F)}(\mathcal{S}(X(F)), \pi) \le 1$ where $\mathcal{S}(X(F))$ is the space of Schwartz functions on $X(F)$. In this thesis we prove that $\mathbb{P}\mathfrak{gl}_{2}(F)$ is multiplicity-free as a $PGL_{2}(F)\times PGL_{2}(F)$-variety.
In this paper we extend the notions of Schwartz functions, tempered functions and generalized Sch... more In this paper we extend the notions of Schwartz functions, tempered functions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds and moreover, to the notions of Schwartz sections, tempered sections and generalized Schwartz sections of Nash bundles. We reprove for this case classically known properties of Schwartz functions on R n and prove some additional properties which are important in representation theory. Acknowledgements We would like to thank our teacher Prof. Joseph Bernstein for teaching us most of the mathematics we know and for his help in this work. We would also like to thank Prof. William Casselman who dealt with similar things in the early 90s and sent us his unpublished material. We thank Prof. Pierre Schapira for drawing our attention to the works [KS], [Mor] and [Pre] which are related to this paper. We would like to thank Prof. Ilya Tyomkin for helpful discussions. Also we would like to thank Dr. Ben-Zion Aizenbud, Shifra Reif, Ilya...
We study generalized and degenerate Whittaker models for reductive groups over local fields of ch... more We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker-Fourier coefficients of automorphic representations. For GLn(F) this implies that a smooth admissible representation π has a generalized Whittaker model WO(π) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wavefront set WF(π). Previously this was only known to hold for F nonarchimedean and O maximal in WF(π), see [MW87]. We also express WO(π) as an iteration of a version of ...
An important question of Representation Theory is "what happens to an irreducible representa... more An important question of Representation Theory is "what happens to an irreducible representation of a group when one restricts it to a subgroup?". Usually it stops being irreducible and it is interesting to know whether it decomposes to distinct irreducible representations. This report contains an exposition on this question and a short description of a recent proof that in the case of the pair (GLn+1, GLn) over local fields the answer is positive. The authors would like to mention that they were guided in this project by Eitan Sayag and Joseph Bernstein.
arXiv: Number Theory, 2019
In this work we prove the local multiplicity at most one theorem underlying the definition and th... more 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.
arXiv: Representation Theory, 2020
Let $G$ be a reductive group over a local field $F$ of characteristic zero, Archimedean or not. L... more Let $G$ be a reductive group over a local field $F$ of characteristic zero, Archimedean or not. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$, considered as a representation of $G$. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map, and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying our theorems to symmetric pairs $(G,H)$ we show that there exists an infinite-dimensional $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is non-compact. For quasi-split $G$ we also extend to the Archimedean case the theorem of Prasad stating that there exists a generic $H$-distinguished representation of $G$ if and only if the rea...
arXiv: Number Theory, 2018
In this paper we analyze Fourier coefficients of automorphic forms on adelic reductive groups $G(... more In this paper we analyze Fourier coefficients of automorphic forms on adelic reductive groups $G(\mathbb{A})$. Let $\pi$ be an automorphic representation of $G(\mathbb{A})$. It is well-known that Fourier coefficients of automorphic forms can be organized into nilpotent orbits $\mathcal{O}$ of $G$. We prove that any Fourier coefficient $\mathcal{F}_\mathcal{O}$ attached to $\pi$ is linearly determined by so-called 'Levi-distinguished' coefficients associated with orbits which are equal or larger than $\mathcal{O}$. When $G$ is split and simply-laced, and $\pi$ is a minimal or next-to-minimal automorphic representation of $G(\mathbb{A})$, we prove that any $\eta \in \pi$ is completely determined by its standard Whittaker coefficients with respect to the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $\mathrm{GL}_n$. In this setting we also derive explicit formulas expressing any maximal parabolic Fourier coe...
Canadian Journal of Mathematics, 2020
In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an ade... more In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$ . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.
Mathematische Zeitschrift, 2021
We consider a special class of unipotent periods for automorphic forms on a finite cover of a red... more We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal a...
Annales de l'Institut Fourier, 2021
Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be a finite c... more Let $F$ be either $\mathbb{R}$ or a finite extension of $\mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $\pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=\mathbb{R}$). For each nilpotent orbit $\mathcal{O}$ we consider a certain Whittaker quotient $\pi_{\mathcal{O}}$ of $\pi$. We define the Whittaker support WS$(\pi)$ to be the set of maximal $\mathcal{O}$ among those for which $\pi_{\mathcal{O}}\neq 0$. In this paper we prove that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $\pi$ is quasi-cuspidal then we show that all $\mathcal{O}\in\mathrm{WS}(\pi)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some conjectures from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17].
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Papers by Dmitry Gourevitch