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Binyong Sun

Publications and source records attributed to Binyong Sun.

At least 37 records · Page 2Linked to original sources
arXiv2024

Function spaces on formal manifolds

This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In a previous paper, we introduce the notion of formal manifolds and develop the foundational framework of formal manifolds. In this paper, we study various function spaces on formal manifolds, including generalizations of vector-valued generalized functions and vector-valued distributions on smooth manifolds to the setting of formal manifolds.

math.FA↗
arXiv2024

Period Relations for Standard $L$-functions of Symplectic Type

This article is to understand the critical values of $L$-functions $L(s,Π\otimes χ)$ and to establish the relation of the relevant global periods at the critical places. Here $Π$ is an irreducible regular algebraic cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb A)$ of symplectic type and $χ$ is a finite order automorphic character of $\mathrm{GL}_1(\mathbb A)$, with $\mathbb A$ is the ring of adeles of a number field $\mathrm k$.

math.NT↗
arXiv2023

Irreducible representations of $\textrm{GL}_n(\mathbb{C})$ of minimal Gelfand-Kirillov dimension

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of $G=\textrm{GL}_n(\mathbb{C})$ possessing the minimal Gelfand-Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of $G$ of type $(n-1,1)$. We give the transition matrix between the two bases for the corresponding coherent families.

math.RT↗
arXiv2023

Genuine special unipotent representations of spin groups

We determine all genuine special unipotent representations of real spin groups and quaternionic spin groups, and show in particular that all of them are unitarizable. We also show that there are no genuine special unipotent representations of complex spin groups.

math.RT↗
arXiv2023

On the notion of metaplectic Barbasch-Vogan duality

In analogy with the Barbasch-Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this duality notion with the theory of primitive ideals and extend the notion of special unipotent representations to the real metaplectic groups. We also interpret the duality map in terms of double cells of Weyl group representations.

math.RT↗
arXiv2022

Rankin-Selberg convolutions for $\mathrm{GL}(n)\times \mathrm{GL}(n)$ and $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$ for principal series representations

Let $\mathsf k$ be a local field. Let $I_ν$ and $I_{ν'}$ be smooth principal series representations of $\mathrm{GL}_n(\mathsf k)$ and $\mathrm{GL}_{n-1}(\mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_ν\times I_{ν'}\rightarrow \mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_ν\times I_{ν'}\rightarrow \mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $\mathrm{GL}_n(\mathsf k)\times \mathrm{GL}_n(\mathsf k)$.

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arXiv2021

Real fundamental Chevalley involutions and conjugacy classes

Let $\mathsf G$ be a connected reductive linear algebraic group defined over $\mathbb R$, and let $C: \mathsf G\rightarrow \mathsf G$ be a fundamental Chevalley involution. We show that for every $g\in \mathsf G(\mathbb R)$, $C(g)$ is conjugate to $g^{-1}$ in the group $\mathsf G(\mathbb R)$. Similar result on the Lie algebras is also obtained.

math.RT↗
arXiv2020

Local theta correspondence: the basic theory

We give an elementary introduction to Classical Invariant Theory and its modern extension "Transcending Classical Invariant Theory", commonly known as the theory of local theta correspondence. We explain the two fundamental assertions of the theory: the Howe duality conjecture and the Kudla-Rallis conservation relation conjecture. We give a status report on the problem of explicitly describing local theta correspondence in terms of Langlands-Vogan parameters. We conclude with a discussion on a certain problem of automatic continuity, which manifests unity of the theory in algebraic and smooth settings.

math.RT↗
arXiv2019

Cohomologically induced distinguished representations and cohomological test vectors

Let $G$ be a real reductive group, and let $χ$ be a character of a reductive subgroup $H$ of $G$. We construct $χ$-invariant linear functionals on certain cohomologically induced representations of $G$, and show that these linear functionals do not vanish on the bottom layers. Applying this construction, we prove two archimedean non-vanishing assumptions, which are crucial in the study of special values of L-functions via modular symbols.

math.RT↗
arXiv2019

Low degree cohomologies of congruence groups

We prove the vanishing of certain low degree cohomologies of some induced representations. As an application, we determine certain low degree cohomologies of congruence groups.

math.RT↗
arXiv2017

Godement-Jacquet L-functions and full theta lifts

We relate poles of local Godement-Jacquet L-functions to distributions on matrix spaces with singular supports. As an application, we show the irreducibility of the full theta lifts to $GL_n(F)$ of generic irreducible representations of $GL_n(F)$, where $F$ is an arbitrary local field.

math.RT↗
arXiv2017

Uniqueness of twisted linear periods and twisted Shalika periods

Let $\rk$ be a local field of characteristic zero. Let $π$ be an irreducible admissible smooth representation of $\GL_{2n}(\rk)$. We prove that for all but countably many characters $χ$ of $\GL_n(\rk)\times \GL_n(\rk)$, the space of $χ$-equivariant (continuous in the archimedean case) linear functionals on $π$ is at most one dimensional. Using this, we prove the uniqueness of twisted Shalika models.

math.RT↗
arXiv2016

Kazhdan's orthogonality conjecture for real reductive groups

We prove a generalization of Harish-Chandra's character orthogonality relations for discrete series to arbitrary Harish-Chandra modules for real reductive Lie groups. This result is an analogue of a conjecture by Kazhdan for $\mathfrak p$-adic reductive groups proved by Bezrukavnikov, and Schneider and Stuhler.

math.RT↗
arXiv2016

Coincidence of algebraic and smooth theta correspondences

An "automatic continuity" question has naturally occurred since Roger Howe established the local theta correspondence over $\mathbb R$: does the algebraic version of local theta correspondence over $\mathbb R$ agrees with the smooth version? We show that the answer is yes, at least when the concerning dual pair has no quaternionic type I irreducible factor.

math.RT↗
arXiv2016

Generalized semi-invariant distributions on p-adic spaces

In this paper we investigate some methods on calculating the spaces of generalized semi-invariant distributions on p-adic spaces. Using homological methods, we give a criterion of automatic extension of (generalized) semi-invariant distributions. Based on the meromorphic continuations of Igusa zeta integrals, we give another criteria with purely algebraic geometric conditions, on the extension of generalized semi-invariant distributions.

math.RT↗