The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions
We prove the nonvanishing hypothesis at infinity for Rankin-Selberg convolutions for $\GL(n)\times \GL(n-1)$.
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arXiv subjects
Binyong Sun
Publications and source records attributed to Binyong Sun.
The Howe duality conjecture: quaternionic case
We complete the proof of the Howe duality conjecture in the theory of local theta correspondence by treating the remaining case of quaternionic dual pairs in arbitrary residual characteristic.
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Chevalley's theorem for affine Nash groups
We formulate and prove Chevalley's theorem in the setting of affine Nash groups. As a consequence, we show that the semi-direct product of two almost linear Nash groups is still an almost linear Nash group.
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Conservation relations for local theta correspondence
We prove Kudla-Rallis conjecture on first occurrences of local theta correspondence, for all type I irreducible dual pairs and all local fields of characteristic zero.
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Almost linear Nash groups
A Nash group is said to be almost linear if it has a Nash representation with finite kernel. Structures and basic properties of these groups are studied.
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Uniqueness of Rankin-Selberg periods
Let $k$ be a local field of characteristic zero. Rankin-Selberg's local zeta integrals produce linear functionals on generic irreducible admissible smooth representations of $GL_n(k)\times GL_r(k)$, with certain invariance properties. We show that up to scalar multiplication, these linear functionals are determined by the invariance properties.
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Vanishing of quasi-invariant generalized functions
Determination of quasi-invariant generalized functions is important for a variety of problems in representation theory, notably character theory and restriction problems. In this note, we review some new and easy-to-use techniques to show vanishing of quasi-invariant generalized functions, developed in the recent work of the authors (Uniqueness of Ginzburg-Rallis models: the Archimedean case, Trans. Amer. Math. Soc. 363, (2011), 2763-2802). The first two techniques involve geometric notions attached to submanifolds, which we call metrical properness and unipotent $χ$-incompatibility. The third one is analytic in nature, and it arises from the first occurrence phenomenon in Howe correspondence. We also highlight how these techniques quickly lead to two well-known uniqueness results, on trilinear forms and Whittaker models.
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Fourier transform and rigidity of certain distributions
Let $E$ be a finite dimensional vector space over a local field, and $F$ be its dual. For a closed subset $X$ of $E$, and $Y$ of $F$, consider the space $D^{-ξ}(E;X,Y)$ of tempered distributions on $E$ whose support are contained in $X$ and support of whose Fourier transform are contained in $Y$. We show that $D^{-ξ}(E;X,Y)$ possesses a certain rigidity property, for $X$, $Y$ which are some finite unions of affine subspaces.
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Uniqueness of Fourier-Jacobi models: the Archimedean case
We prove uniqueness of Fourier-Jacobi models for general linear groups, unitary groups, symplectic groups and metaplectic groups, over an archimedean local field.
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Multiplicity one theorems for Fourier-Jacobi models
For every genuine irreducible admissible smooth representation $π$ of the metaplectic group $\widetilde{\Sp}(2n)$ over a p-adic field, and every smooth oscillator representation $ω_ψ$ of $\widetilde{\Sp}(2n)$, we prove that the tensor product $π\otimes ω_ψ$ is multiplicity free as a smooth representation of the symplectic group $\Sp(2n)$. Similar results are proved for general linear groups and unitary groups.
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MVW-extensions of real quaternionic classical groups
Let $G$ be a real quaternionic classical group $\GL_n(\bH)$, $\Sp(p,q)$ or $\oO^*(2n)$. We define an extension $\breve G$ of $G$ with the following property: it contains $G$ as a subgroup of index two, and for every $x\in G$, there is an element $\breve g\in \breve G\setminus G$ such that $\breve g x\breve{g}^{-1}=x^{-1}$. This is similar to Moeglin-Vigneras-Waldspurger's extensions of non-quaternionic classical groups.
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A note on special orthogonal groups following Waldspurger
The purpose of this note is to verify that the archimedean multiplicity one theorems shown for orthogonal groups (as well as general linear and unitary groups) in a previous paper of the authors remain valid for special orthogonal groups. The necessary ingredients to establish this variant are due to Waldspurger.
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Multiplicity one theorems: the Archimedean case
Let $G$ be one of the classical Lie groups $\GL_{n+1}(\R)$, $\GL_{n+1}(\C)$, $\oU(p,q+1)$, $\oO(p,q+1)$, $\oO_{n+1}(\C)$, $\SO(p,q+1)$, $\SO_{n+1}(\C)$, and let $G'$ be respectively the subgroup $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\oO(p,q)$, $\oO_n(\C)$, $\SO(p,q)$, $\SO_n(\C)$, embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\GL_{n}(\R)\ltimes \oH_{2n+1}(\R)$, $\GL_{n}(\C)\ltimes \oH_{2n+1}(\C)$, $\oU(p,q)\ltimes \oH_{2p+2q+1}(\R)$, $\Sp_{2n}(\R)\ltimes \oH_{2n+1}(\R)$, $\Sp_{2n}(\C)\ltimes \oH_{2n+1}(\C)$, with their respective subgroups $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\Sp_{2n}(\R)$, $\Sp_{2n}(\C)$.
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Notes on MVW-extensions
We review certain basic geometric and analytic results concerning MVW-extensions of classical groups, following Moeglin-Vigneras-Waldspurger. The related results for Jacobi groups, metaplectic groups, and special orthogonal groups are also included.
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A general form of Gelfand-Kazhdan criterion
We formalize the notion of matrix coefficients for distributional vectors in a representation of a real reductive group, which consist of generalized functions on the group. As an application, we state and prove a Gelfand-Kazhdan criterion for a real reductive group in very general settings.
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On representations of real Jacobi groups
We consider a category of continuous Hilbert space representations and a category of smooth Frechet representations, of a real Jacobi group $G$. By Mackey's theory, they are respectively equivalent to certain categories of representations of a real reductive group $\widetilde L$. Within these categories, we show that the two functors of taking smooth vectors for $G$, and for $\widetilde L$, are consistent with each other. By using Casselman-Wallach's theory of smooth representations of real reductive groups, we define matrix coefficients for distributional vectors of certain representations of $G$. We also formulate Gelfand-Kazhdan criteria for Jacobi groups which could be used to prove the multiplicity one theorem for Fourier-Jacobi models.
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Uniqueness of Ginzburg-Rallis models: the Archimedean case
In this paper, we prove the uniqueness of Ginzburg-Rallis models in the archimedean case. As a key ingredient, we introduce a new descent argument based on two geometric notions attached to submanifolds, which we call metrical properness and unipotent $χ$-incompatibility.
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Dual pairs and contragredients of irreducible representations
Let $G$ be a classical group $\GL(n)$, $\oU(n)$, $\oO(n)$ or $\Sp(2n)$, over a non-archimedean local field of characteristic zero. Let $π$ be an irreducible admissible smooth representation of $G$. It is well known that the contragredient of $π$ is isomorphic to a twist of $π$ by an automorphism of $G$. We prove a similar result for double covers of $G$ which occur in the study of local theta correspondences.
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