Multiplicity one theorem for $$({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))$$

Aizenbud, Avraham; Gourevitch, Dmitry

doi:10.1007/s00029-009-0544-7

Cited by 36 publications

(3 citation statements)

References 12 publications

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“…This corollary follows from Theorem E since Hom h (π, τ ) lies in the space of Δh-invariant functionals on the completed tensor product π ⊗τ ∈ Irr(G(R))×H (R))) (see [5,Corollary A.0.7 and Lemma A.0.8]). All symmetric pairs satisfying the conditions of the corollary were classified in [23].…”

Section: Applications To Representation Theorymentioning

confidence: 91%

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

Aizenbud

Gourevitch

Minchenko

2016

Sel. Math. New Ser.

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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 G defined over R, the space of G(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 . Mathematics Subject Classification

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Section: Applications To Representation Theorymentioning

confidence: 91%

“…For a good introduction to the algebraic theory of D-modules, we refer the reader to [9] and [10]. For a short overview, see [5,Appendix B]. By a D-module on a smooth algebraic variety X , we mean a quasi-coherent sheaf of right modules over the sheaf D X of algebras of algebraic differential operators.…”

Section: D-modulesmentioning

confidence: 99%

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

Aizenbud

Gourevitch

Minchenko

2016

Sel. Math. New Ser.

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“…When the initial manuscript of this paper was completed, the authors learned that A. Aizenbud and D. Gourevitch had proved the multiplicity one theorems for the pairs (GL n+1 (R), GL n (R)) and (GL n+1 (C), GL n (C)), independently and in a different approach. This has since appeared as [AG09b]. Now assume that G is one of the five Jacobi groups.…”

Section: ))mentioning

confidence: 99%

Multiplicity one theorems: the Archimedean case

Sun¹,

Zhu²

2012

Ann. Math.

134

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Shintani Functions, Real Spherical Manifolds, and Symmetry Breaking Operators

Kobayashi

2014

Developments and Retrospectives in Lie Theory

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For a pair of reductive groups G ⊃ G ′ , we prove a geometric criterion for the space Sh(λ, ν) of Shintani functions to be finite-dimensional in the Archimedean case. This criterion leads us to a complete classification of the symmetric pairs (G, G ′ ) having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of dim C Sh(λ, ν) is also obtained. Furthermore, we prove that symmetry breaking operators of the restriction of smooth admissible representations yield Shintani functions of moderate growth, of which the dimension is determined for (G, G ′ ) = (O(n + 1, 1), O(n, 1)).

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Multiplicity one theorem for $$({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))$$

Abstract: Let F be either R or C. Consider the standard embedding GLn(F )

Cited by 36 publications

References 12 publications

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

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

Multiplicity one theorems: the Archimedean case

Shintani Functions, Real Spherical Manifolds, and Symmetry Breaking Operators

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