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Dmitry Gourevitch

Publications and source records attributed to Dmitry Gourevitch.

At least 37 records · Page 2Linked to original sources
arXiv2021

Eulerianity of Fourier coefficients of automorphic forms

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a `hidden' invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier-Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.

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arXiv2020

Whittaker supports for representations of reductive groups

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 $π$ 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 $π_{\mathcal{O}}$ of $π$. We define the Whittaker support WS$(π)$ to be the set of maximal $\mathcal{O}$ among those for which $π_{\mathcal{O}}\neq 0$. In this paper we prove that all $\mathcal{O}\in\mathrm{WS}(π)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $π$ is quasi-cuspidal then we show that all $\mathcal{O}\in\mathrm{WS}(π)$ 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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arXiv2019

$\mathbb{P}\mathfrak{gl}_{2}$ is Multiplicity-Free as a $PGL_{2} \times PGL_{2}$-Variety

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 $π$ of $G(F)$ the following takes place: $\dim Hom_{G(F)}(\mathcal{S}(X(F)), π) \le 1$ 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.

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arXiv2019

Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

In this paper we analyze Fourier coefficients of automorphic forms on a finite cover $G$ of an adelic split simply-laced group. Let $π$ be a minimal or next-to-minimal automorphic representation of $G$. We prove that any $η\in π$ 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 $GL_n$. 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 non-existence 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 towards applications to scattering amplitudes in string theory.

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arXiv2019

Generalized and degenerate Whittaker quotients and Fourier coefficients

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 $π$, call the set of maximal orbits $\mathcal{O}$ with $W_{\mathcal{O}}$ that includes $π$ the Whittaker support of $π$. 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 $π$ relate to other invariants of $π$, such as its wave-front set?

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arXiv2017

Analytic continuation of equivariant distributions

We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein's theory of analytic continuation of holonomic distributions. We use this to construct $H$-equivariant functionals on principal series representations of $G$, where $G$ is a real reductive group and $H$ is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations, and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the $p$-adic case using a recent result of Hong and Sun.

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arXiv2017

Composition series for degenerate principal series of GL(n)

In this note we consider representations of the group GL(n,F), where F is the field of real or complex numbers or, more generally, an arbitrary local field, in the space of equivariant line bundles over Grassmannians over the same field F. We study reducibility and composition series of such representations. Similar results were obtained already in [HL99,Al12,Zel80], but we give a short uniform proof in the general case, using the tools from [AGS15a]. We also indicate some applications to cosine transforms in integral geometry.

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arXiv2017

Generalized and degenerate Whittaker models

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 $\mathrm{GL}_n(F)$ this implies that a smooth admissible representation $π$ has a generalized Whittaker model $\mathcal{W}_{\mathcal{O}}(π)$ corresponding to a nilpotent coadjoint orbit $\mathcal{O}$ if and only if $\mathcal{O}$ lies in the (closure of) the wave-front set $\mathrm{WF}(π)$. Previously this was only known to hold for $F$ non-archimedean and $\mathcal{O}$ maximal in $\mathrm{WF}(π)$, see [MW87]. We also express $\mathcal{W}_{\mathcal{O}}(π)$ as an iteration of a version of the Bernstein-Zelevinsky derivatives [BZ77,AGS15a]. This enables us to extend to $\mathrm{GL_n}(\mathbb{R})$ and $\mathrm{GL_n}(\mathbb{C})$ several further results from [MW87] on the dimension of $\mathcal{W}_{\mathcal{O}}(π)$ and on the exactness of the generalized Whittaker functor.

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arXiv2017

Invariant generalized functions supported on an orbit

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 $\mathrm{GL}_3(\mathbb{C})$. Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently by Sakellaridis.

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arXiv2016

Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let $H$ be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold $X$ and a real algebraic bundle $\mathcal{E}$ on $X$. Let $\mathfrak{h}$ be the Lie algebra of $H$. Let $\mathcal{S}(X,\mathcal{E})$ be the space of Schwartz sections of $\mathcal{E}$. We prove that $\mathfrak{h}\mathcal{S}(X,\mathcal{E})$ is a closed subspace of $\mathcal{S}(X,\mathcal{E})$ 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 Casselman-Wallach representation of $G$ and $V$ be the corresponding Harish-Chandra module. Then the natural morphism of coinvariants $V_{\mathfrak{h}}\to π_{\mathfrak{h}}$ is an isomorphism if and only if any linear $\mathfrak{h}$-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 $G$.

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arXiv2016

Holonomicity of relative characters and applications to multiplicity bounds for spherical pairs

In this paper, 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 from [KO13,KS16] 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 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 $X$.

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arXiv2015

Invariant Functionals on the Speh representation

We study Sp(2n,R)-invariant functionals on the spaces of smooth vectors in Speh representations of GL(2n,R). For even n we give expressions for such invariant functionals using an explicit realization of the space of smooth vectors in the Speh representations. Furthermore, we show that the functional we construct is, up to a constant, the unique functional on the Speh representation which is invariant under the Siegel parabolic subgroup of Sp(2n,R). For odd n we show that the Speh representations do not admit an invariant functional with respect to the subgroup U(n) of Sp(2n,R) consisting of unitary matrices. Our construction, combined with the argument in [GOSS12], gives a purely local and explicit construction of Klyachko models for all unitary representations of GL(2n,R).

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arXiv2015

z-Finite distributions on p-adic groups

For a real reductive group G, the center $\mathfrak{z}(\mathcal{U}(\mathfrak{g}))$ of the universal enveloping algebra of the Lie algebra $\mathfrak{g}$ of G acts on the space of distributions on G. This action proved to be very useful (see e.g. [HC63, HC65, Sha74, Bar03]). Over non-Archimedean local fields, one can replace this action 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. 1) The wave-front set of any z-finite distribution on G over any point $g\in G$ lies inside the nilpotent cone of $T_g^*G \cong \mathfrak{g}$. 2) Let $H_1,H_2 \subset G$ be symmetric subgroups. Consider the space J of $H_1\times H_2$-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 groups of certain type and the invariance condition is replaced by equivariance. Further we apply those results to density and regularity of spherical 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 admissible 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.

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arXiv2015

On an analytic description of the $α$-cosine transform on real Grassmannians

The goal of this paper is to describe the $α$-cosine transform on functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector space. in analytic terms as explicitly as possible. We show that for all but finitely many complex $α$ the $α$-cosine transform is a composition of the $(α+2)$-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of $α$ except one we interpret the $α$-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value $α$, which is $-(min\{i,n-i\}+1)$, is still an open problem.

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arXiv2014

Degenerate Whittaker functionals for real reductive groups

In this paper we establish a connection between the associated variety of a representation and the existence of certain degenerate Whittaker functionals, for both smooth and K-finite vectors, for all quasi-split real reductive groups, thereby generalizing results of Kostant, Matumoto and others.

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arXiv2014

Vanishing of certain equivariant distributions on spherical spaces

We prove vanishing of 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, and eigen with respect to the center of the universal enveloping algebra of the Lie algebra of G. This is a generalization of a result by Shalika, that concerned the group case. Shalika's 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.

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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.

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