Some applications of theta correspondence to branching laws

分支律与对偶原理，第一部分

Branching laws and a duality principle, Part I

本文针对满足等秩条件的半单李群的离散序列表示，结合积分、微分算子与此前提出的对偶原理，研究其限制到同类型子群时的分支律，给出了一般性结果及全纯离散序列的详细结论。

Branching laws for the metaplectic cover of ${\rm GL}_{2}$

Let $F$ be a non-Archimedian local field of characteristic zero and $E/F$ a quadratic extension. The aim of the present article is to study the multiplicity of an irreducible admissible representation of ${\rm GL}_2(F)$ occurring in an irreducible admissible genuine representation of non-trivial two fold covering $\widetilde{\rm GL}_2(E)$ of ${\rm GL}_2(E)$.

Ext-analogues of Branching laws

We consider the Ext-analogues of branching laws of representations for a group to its subgroups in the context of $p$-adic groups.

Branching laws for discrete series of some affine symmetric spaces

In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $Π$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representati…

Stability of Branching Laws for Highest Weight Modules

We say a representation V of a group G has stability if its multiplicities m^{G}_{V}(λ) is dependent only on some equivalence class of λfor a sufficiently large parameter λ. In this paper, we prove that the restriction of a holomorphic discrete series representation with respect to any holomorphic symmetric pairs has stability. As a corollary, we give a necessary and sufficient condition on multiplicity-freeness of the branching laws in this setting. This condition is same as the sufficient condition given by the theory of visible actions. We prove a general theorem before we show the stabilit…

Branching laws for tensor modules over classical locally finite Lie algebras

Let g' and g be isomorphic to any two of the Lie algebras gl(infty), sl(infty), sp(infty), and so(infty). Let M be a simple tensor g-module. We introduce the notion of an embedding of g' into g of general tensor type and derive branching laws for triples g', g, and M, where the embedding of g' into g is of general tensor type. More precisely, since M is in general not semisimple as a g'-module, we determine the socle filtration of M over g'. Due to the description of embeddings of classical locally finite Lie algebras given by Dimitrov and Penkov, our results hold for all possible embeddings o…

椭圆轨道的投影与分支律

Projection of Elliptic Orbits and Branching Laws

本文以 U(p,q) 到 U(p,q-1) 的 Gan-Gross-Prasad 分支律为例，探讨轨道方法在分支问题中的应用，并讨论分支律与波前集的相关结果。

$p$进一般线性群的非缓增 Ext 分支律

Non-tempered Ext Branching Laws for the $p$-adic General Linear Group

本文研究 $p$ 进一般线性群中 Arthur 型表示的 Ext 分支律，并在离散序列表示及其 Aubert–Zelevinsky 对偶乘积的情形下建立局部非缓增 Gan–Gross–Prasad 猜想的 Ext 类比。

Branching laws for the Steinberg representation: the rank 1 case

Let $G/H$ be a reductive symmetric space over a $p$-adic field $F$, the algebraic groups $G$ and $H$ being assumed semisimple of relative rank $1$. One of the branching problems for the Steinberg representation $\St_G$ of $G$ is the determination of the dimension of the intertwining space ${\rm Hom}_H (\St_G ,π)$, for any irreducible representation $π$ of $H$. In this work we do not compute this dimension, but show how it is related to the dimensions of some other intertwining spaces ${\rm Hom}_{K_i} ({\tilde π} ,1)$, for a certain finite family $K_i$, $i=1,...,r$, of anisotropic subgroups of …

Branching laws for discrete Wallach points

We consider the (projective) representations of the group of holomorphic automorphisms of a symmetric tube domain $V\oplus iΩ$ that are obtained by analytic continuation of the holomorphic discrete series. For a representation corresponding to a discrete point in the Wallach set, we find the decomposition under restriction to the identity component of $GL(Ω)$. Using Riesz distributions, an explicit intertwining operator is constructed as an analytic continuation of an integral operator. The density for the Plancherel measure involves quotients of $Γ$-functions and the $c$-function for a symmet…

分支律的同调方面

Homological aspects of branching laws

本文综述了p-adic群表示分支律的同调性质，强调限制表示在多数Bernstein块上为投射表示，并指出其简单结构。

Branching laws for minimal holomorphic representations

In this paper we study the branching law for the restriction from $SU(n,m)$ to $SO(n,m)$ of the minimal representation in the analytic continuation of the scalar holomorphic discrete series. We identify the the group decomposition with the spectral decomposition of the action of the Casimir operator on the subspace of $S(O(n) \times O(m))$-invariants. The Plancherel measure of the decomposition defines an $L^2$-space of functions, for which certain continuous dual Hahn polynomials furnish an orthonormal basis. It turns out that the measure has point masses precisely when $n-m>2$. Under these c…

Branching laws, some results and new examples

For a connected, noncompact matrix simple Lie group $G$ so that a maximal compact subgroup $K$ has three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contains the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general $G$ such that $G/K$ is an Hermitian $G$-manifold we give a necessary and sufficient condition so that a square integrable representations of the…

On the direct integral decomposition in branching laws for real reductive groups

The restriction of an irreducible unitary representation $π$ of a real reductive group $G$ to a reductive subgroup $H$ decomposes into a direct integral of irreducible unitary representations $τ$ of $H$ with multiplicities $m(π,τ)\in\mathbb{N}\cup\{\infty\}$. We show that on the smooth vectors of $π$, the direct integral is pointwise defined. This implies that $m(π,τ)$ is bounded above by the dimension of the space $\operatorname{Hom}_H(π^\infty|_H,τ^\infty)$ of intertwining operators between the smooth vectors, also called symmetry breaking operators, and provides a precise relation between t…

Dirac 上同调、分支律与 Wallach 模

Dirac cohomology, branching laws and Wallach modules

本文基于 Dirac 上同调方法，针对例外李群 E_{6(-14)} 和 E_{7(-25)}，从 Dirac 上同调恢复 Wallach 模的 K-谱，推广了 2013 年 Huang 等人的分支律结果。

Branching Laws for Some Unitary Representations of SL(4,R)

In this paper we consider the restriction of a unitary irreducible representation of type $A_{\mathfrak q}(λ)$ of $GL(4,{\mathbb R})$ to reductive subgroups $H$ which are the fixpoint sets of an involution. We obtain a formula for the restriction to the symplectic group and to $GL(2,{\mathbb C})$, and as an application we construct in the last section some representations in the cuspidal spectrum of the symplectic and the complex general linear group. In addition to working directly with the cohmologically induced module to obtain the branching law, we also introduce the useful concept of pseu…

商分支定律与一般线性群的Gan-Gross-Prasad相关性

Quotient branching laws and unitary Gan-Gross-Prasad relevance for general linear groups

本文研究一般线性群中不可约光滑表示的分支定律，提出算法确定Gan-Gross-Prasad相关对，并对广义Speh表示进行分类。

Branching laws for Classical Groups: the non-tempered case

This paper generalizes the GGP conjectures which were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the nongeneric L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of A-parameters which governs the branching laws for $GL_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in our earlier work including Bessel models and Fourier-Jacobi models.

A Generalization of Correspondence Theorem and Its Application to Branching Laws

We begin with recalling the correspond theorem of induced modules and global sections of vector bundles. After that, we give a generalization of this theorem. Finally, we apply the result to branching laws, and give some concrete examples.

The Continuous Spectrum in Discrete Series Branching Laws

If $G$ is a reductive Lie group of Harish-Chandra class, $H$ is a symmetric subgroup, and $π$ is a discrete series representation of $G$, the authors give a condition on the pair $(G,H)$ which guarantees that the direct integral decomposition of $π|_H$ contains each irreducible representation of $H$ with finite multiplicity. In addition, if $G$ is a reductive Lie group of Harish-Chandra class, and $H\subset G$ is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of $π|_H$ is constant along `continuous para…

Branching Laws of Generalized Verma Modules for Non-symmetric Polar Pairs

We give branching formulas from $so(7,\mathbb{C})$ to $\mathfrak{g}_2$ for generalized Verma modules attached to $\mathfrak{g}_2$-compatible parabolic subalgebras of $so(7,\mathbb{C})$, and branching formulas from $\mathfrak{g}_2$ to $sl(3,\mathbb{C})$ for generalized Verma modules attached to $sl(3,\mathbb{C})$-compatible parabolic subalgebras of $\mathfrak{g}_2$ respectively, under some assumptions on the parameters of generalized Verma modules.

A criterion for discrete branching laws for Klein four symmetric pairs and its application to $E_{6(-14)}$

Let $G$ be a noncompact connected simple Lie group, and $(G,G^Γ)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^Γ)$, there does not exist any unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $(\mathfrak{g}^Γ,K^Γ)$-module. As an application, for $G=\mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^Γ)$ with $G^Γ$ n…

对称对与分支律

Symmetric pairs and branching laws

本文研究紧李群对称对的分支律，利用复约化子群在旗簇上有限轨道的经典结果，给出由轨道空间 $H_C\ackslash F$ 参数化的分支系数公式。

Branching laws on the metaplectic cover of ${\rm GL}_{2}$

Representation theory of $p$-adic groups naturally comes in the study of automorphic forms and one way to understand representations of a group is by restricting to its nice subgroups. D. Prasad studied the restriction for pairs $({\rm GL}_{2}(E), {\rm GL}_{2}(F))$ and $({\rm GL}_{2}(E), D_{F}^{\times})$ where $E/F$ is a quadratic equation and $D_{F}$ is the unique quaternion division algebra, and $D_{F}^{\times} \hookrightarrow {\rm GL}_{2}(E)$. Prasad proved a multiplicity one result and a `dichotomy' relating the restriction for the pairs $({\rm GL}_{2}(E), {\rm GL}_{2}(F))$ and $({\rm GL}_…

高阶局部化与高阶分歧律

Higher localization and higher branching laws

本文利用 h-复形形式体系将局部化函子推广至等变导出范畴，证明了球面约化子群设定下局部化的正则全holonomic性，并几何解释了相对李代数同调与Ext-分歧空间，恢复了部分已知结果。

经典群的新分支公式及其相互关系

New Branching Formulae for Classical Groups and Relations among them

本研究针对多类复经典群嵌套对，推导了适用于任意维数差的广义分支律，给出统一证明并比较了不同群对的分支重数，拓展了经典分支理论的适用范围。

例外情形下超空间上球谐函数的分支律

Branching laws for spherical harmonics on superspaces in exceptional cases

针对超空间R^{m|2n}上球谐函数的分支律问题，本文在正交辛超代数对称性从osp(m|2n)约化为osp(m-1|2n)的场景下，明确给出了例外情形下的分支律，补充了非例外情形下已有结论的空白。

GL_N(F)的所有零深度表示的Zelevinsky段与超特殊分支律

Zelevinsky Segments and Hyperspecial Branching Laws for All Depth-Zero Representations of GL_N(F)

该研究针对非阿基米德局部域上GL_N(F)的不可约零深度表示，结合Bushnell-Kutzko型等工具，确定了超特殊极大紧子群幂幺根基固定向量空间的分解，建立了p进与有限侧对偶的交织关系，得到了完整有效的超特殊分支律。

阿基米德 Bernstein-Zelevinsky 理论与同调分支律

Archimedean Bernstein-Zelevinsky Theory and Homological Branching Laws

本文发展拟分裂实经典群的 Bernstein-Zelevinsky 理论，建立一般线性群的 Euler-Poincaré 特征标公式，并证明高阶扩张群消失、最高导数 Leibniz 律、酉性判据及分裂偶正交群扭曲同调的 Hausdorff 性与正合性。

Reciprocity Algebras and Branching for Classical Symmetric Pairs

We study branching laws for a classical group $G$ and a symmetric subgroup $H$. Our approach is through the {\it branching algebra}, the algebra of covariants for $H$ in the regular functions on the natural torus bundle over the flag manifold for $G$. We give concrete descriptions of (natural subalgebras of) the branching algebra using classical invariant theory. In this context, it turns out that the ten classes of classical symmetric pairs $(G,H)$ are associated in pairs, $(G,H)$ and $(H',G')$, and that the (partial) branching algebra for $(G,H)$ also describes a branching law from $H'$ to $…

On branching laws of Speh representations

In this paper, we consider the branching law of the Speh representation $\mathrm{Sp}(π,n+l)$ of $\mathrm{GL}_{2n+2l}$ with respect to the block diagonal subgroup $\mathrm{GL}_n\times\mathrm{GL}_{n+2l}$ for any irreducible generic representation $π$ of $\mathrm{GL}_2$ over any $p$-adic field. We use the Shalika model of $\mathrm{Sp}(π,n)$ to construct certain zeta integrals, which were defined by Ginzburg and Kaplan independently, and study them. Finally, using these zeta integrals, we obtain a nonzero $\mathrm{GL}_n\times\mathrm{GL}_{n+2l}$-map from $\mathrm{Sp}(π,n+l)$ to $τ\boxtimesτ^\veeχ_π…

Analysis on the minimal representation of O(p,q) -- II. Branching laws

This is a second paper in a series devoted to the minimal unitary representation of O(p,q). By explicit methods from conformal geometry of pseudo-Riemannian manifolds, we find the branching law corresponding to restricting the minimal unitary representation to natural symmetric subgroups. In the case of purely discrete spectrum we obtain the full spectrum and give an explicit Parseval-Plancherel formula, and in the general case we construct an infinite discrete spectrum.

Global branching laws by global Okounkov bodies

Let $G'$ be a complex semisimple group, and let $G \subseteq G'$ be a semisimple subgroup. We show that the branching cone of the pair $(G, G')$, which (asymptotically) parametrizes all pairs $(W, V)$ of irreducible finite-dimensional $G$-representations $W$ which occur as subrepresentations of a finite-dimensional irreducible $G'$-representation $V$, can be identified with the pseudo-effective cone, $\overline{\mbox{Eff}}(Y)$, of some GIT quotient $Y$ of the flag variety of the group $G \times G'$. Moreover, we prove that the quotient $Y$ is a Mori dream space. As a consequence, the global Ok…

Distinguished representations of SO(n+1,1) x SO(n,1), periods and branching laws

Given irreducible representations $Π$ and $π$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $θ$-stable parameter necessary and sufficient conditions so that \[ \operatorname{Hom}_{G'}(Π|_{G'}, π)\not = \{0\}. \] In the special case that both $Π$ and $π$ are tempered, this implies the Gross--Prasad conjectures for tempered representations of $SO(n+1,1) \times SO(n,1)$ which are nontrivial on the center. We apply these results to construct nonzero periods and distinguished representations. If both …

II型中的\(\ell -\)模局部西塔对应与部分置换

The $\ell$-modular local theta correspondence in type II and partial permutations

研究在非阿基米德域\(F\)上II型的\(\overline{\mathbb{F}}_\ell\)-模西塔对应中出现的重数，通过对称群对部分置换的作用及相关表示，将问题归结为对称群模表示理论中的分支问题。

Stein 互补序列与 $\operatorname{GL}(2n,\mathbb{R})$ 的 Speh 表示的分支律

Branching laws for Stein's complementary series and Speh representations of $\operatorname{GL}(2n,\mathbb{R})$

本文给出 GL(2n,R) 的 Stein 互补序列和 Speh 表示限制到 GL(2n-1,R) 的显式直接积分分解，分解由极大抛物子群的酉诱导表示构成，证明利用了两族对称破缺算子。

Askey-Wilson 多项式与分支律

Askey-Wilson Polynomials and Branching Laws

本文推导了非对称 Askey-Wilson 多项式参数平移的连接系数公式，并利用该结果重新证明了对称情形下 Askey 和 Wilson 的经典结论。

Geometry of coadjoint orbits and multiplicity-one branching laws for symmetric pairs

Consider the restriction of an irreducible unitary representation $π$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $ν$ occurring in the restriction $π|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $π$ and $ν$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results…

Branching laws for Verma modules and applications in parabolic geometry. I

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, http://dx.doi.org/10.1007/s00031-012-9180-y {Transf. Groups (2012)}], we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in genera…

Symplectic branching laws and Hermitian symmetric spaces

Let $G$ be a complex simple Lie group, and let $U \subseteq G$ be a maximal compact subgroup. Assume that $G$ admits a homogenous space $X=G/Q=U/K$ which is a compact Hermitian symmetric space. Let $\mathscr{L} \rightarrow X$ be the ample line bundle which generates the Picard group of $X$. In this paper we study the restrictions to $K$ of the family $(H^0(X, \mathscr{L}^k))_{k \in \N}$ of irreducible $G$-representations. We describe explicitly the moment polytopes for the moment maps $X \rightarrow \fk^*$ associated to positive integer multiples of the Kostant-Kirillov symplectic form on $X$,…

A hidden symmetry of a branching law

We consider branching laws for the restriction of some irreducible unitary representations $Π$ of $G=O(p,q)$ to its subgroup $H=O(p-1,q)$. In Kobayashi (arXiv:1907.07994), the irreducible subrepresentations of $O(p-1,q)$ in the restriction of the unitary $Π|_{O(p-1,q) }$ are determined. By considering the restriction of packets of irreducible representations we obtain another very simple branching law, which was conjectured in Orsted-Speh (arXiv:1907.07544).

Restriction for general linear groups: the local non-tempered Gan-Gross-Prasad conjecture (non-Archimedean case)

We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.

A program for branching problems in the representation theory of real reductive groups

We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G' (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are {\it{…

p进群(GL_{n+1}, GL_n)的商分支律I：广义Gan-Gross-Prasad相关对

Quotient branching law for $p$-adic $(\mathrm{GL}_{n+1}, \mathrm{GL}_n)$ I: generalized Gan-Gross-Prasad relevant pairs

本文针对非阿基米德局部域上的GL_{n+1}与GL_n，给出并证明了二者不可约光滑表示间同态非零的充要条件，解决了商分支律问题，还将结果推广到A型仿射Hecke代数并细化了分支律。