Chong Zhang
Let $(G,H)$ be a symmetric pair over a $p$-adic field and $π$ a discrete series representation of $G$. In this paper, for some type of symmetric pairs $(G,H)$, we show that local periods in $Hom_H(π,\mathbb{C})$ can be constructed by integrating the matrix coefficients of $π$ over $H$.
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We have determined composition series of a class of induced representations appearing in Moeglin Tadić classification of discrete series. The result is further used to determine composition series of certain representations induced from Langlands quotients. This should provide more information on decomposing standard representations as well as Jacquet modules of discrete series, which has application in automorphic forms.
Borel-de Siebenthal discrete series and associated holomorphic discrete series
设G_0是单连通非紧实单李群，其极大紧子群K_0.设秩（G_0）=秩（K_0），使得G_0具有离散级数表示.若G_0/K_0是Hermite对称的，则存在G_0的一个相对简单的离散级数，称为全纯离散级数。现在假设G_0/K_0不是厄米对称的。在这种情况下，我们可以定义类似于全纯离散级数的Borel-de Siebenthal离散级数。考虑K_0的某个循环子群，其中心化子L_0使得K_0/L_0是一个不可约的紧致Hermite对称空间.设（K_0）* 是K_0关于L_0的对偶。则（K_0）*/L_0是与K_0/L_0对偶的不可约非紧Hermite对称空间。对于G_0的每一Borel-de Siebenthal离散级数，我们可以对应一个（K_0）* 的全纯离散级数。本文讨论了在一定条件下这两个离散序列之间存在公共L_0型的问题。
Cuspidal discrete series for projective hyperbolic spaces
在[1]中，我们提出了半单对称空间$G/H$上尖点形式的定义，并引入了Radon变换和Abel变换的概念。对于实非黎曼双曲空间，我们证明了存在无穷多个尖点离散级数，至多存在无穷多个非尖点离散级数，特别是球面离散级数.对于射影空间，球面离散级数是唯一的非尖点离散级数。下面，我们将这些结果推广到其他双曲空间，我们还研究了Schwartz函数的Abel变换何时又是Schwartz函数的问题。
Cuspidal discrete series for semisimple symmetric spaces
在半单对称空间上提出了尖点形式的概念。然后，我们详细研究了实双曲空间，证明了尖点和非尖点离散级数都存在。特别地，我们证明了所有的球面离散级数都是非尖点的。
In this paper we compute explicitly the norm of the vector-valued holomorphic discrete series representations, when its $K$-type is "almost multiplicity-free." As an application, we discuss the properties of highest weight modules, such as unitarizability, reducibility and composition series.
Horospherical model for holomorphic discrete series and horospherical Cauchy transform
本文在Hermitian型仿射对称空间中定义了一个复准球面变换，并证明了它在全纯离散级数上没有核。
In a joint paper P. Pandžić and D. Renard proved that holomorphic and antiholomorphic discrete series representations can be constructed via algebraic Dirac induction. The group $SU(2,1)$, except for those two types, also has a third type of discrete series representations that are neither holomorphic nor antiholomorphic. In this paper we show that nonholomorphic discrete series representations of the group $SU(2,1)$ can also be constructed using algebraic Dirac induction.
We explain by elementary means why the existence of a discrete series representation of a real reductive group $G$ implies the existence of a compact Cartan subgroup of $G$. The presented approach has the potential to generalize to real spherical spaces.
Based on the material in an old article of Wallach a short proof of the Harish-Chandra condition is given.
Discrete Components of Some Complementary Series
我们证明了SO（n，1）的补级数表示包含SO（m，1）的离散补级数，只要连续参数足够接近约化的第一点，并且Levi的紧部分的表示是一个足够小的基本表示.作为结果，我们证明了SO（n，1）的度$i$小于m/2的上同调表示包含离散的SO（m，1）的上同调表示。
Discrete series characters for affine Hecke algebras and their formal degrees
引入仿射Hecke代数的不可约离散级数表示的通有中心特征标。利用这个不变量，我们给出了所有具有任意正参数的抽象仿射Hecke代数（E型除外）的不可约离散级数特征标的一个新的分类，并证明了它们的形式度的一个显式乘积公式（在所有情况下）.
We introduce the notion of relative pseudocoefficient for relative discrete series of real spherical homogeneous spaces of reductive groups. We prove that such relative pseudocoefficient does not exist for semisimple symmetric spaces of type G(C)/G(R) and construct strong relative pseudocoefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H.Jacquet for compact discrete quotient Γ\G. This allows us to prove that a relative discrete series which admits strong pseudocoefficient with sufficiently small support occurs in the spectral decomposition of L…
A simplified approach to the holomorphic discrete series
本文提出Harish-Chandra条件的简单新证明，并附有Hermitian对称空间、全纯归纳及解析工具的阐述。
We give an example of a semisimple symmetric space $G/H$ and an irreducible representation of $G$ which has multiplicity 1 in $L^2(G/H)$ and multiplicity 2 in $C^\infty(G/H)$.
Hua operators, Poisson transform and relative discrete series on line bundle over bounded symmetric domains
设$Ω=G/K$是一个有界对称区域，$S=K/L$是其Shilov边界.本文研究了$G$在$Ω$上齐次线丛截面上的作用以及相应的$G$不变微分算子的本征空间.泊松变换将$S$上的超函数映射到特征空间。我们的形象扭曲华算子的特点。对于某些特殊参数，Poisson变换是Szegö型映射到相应的离散级数中，我们计算了离散级数中相应的元素。
For the discrete series representations of ${\rm GL}(n)$ over a non-archimedean local field $F$, we define a notion of functions similar to "zonal spherical functions" for unramified principal series. We prove the existence of such functions in the level $0$ case. As for unramified principal series, they give rise to explicit coefficients. We deduce a local proof of Matringe's criterion of distinction of discrete series, in the level $0$ case, for the Galois symmetric space ${\rm GL}(n,F)/{\rm GL}(n,F_0 )$, for any unramified quadratic extension $F/F_0$. We also exhibit explicit test vectors w…
Compatible Discrete Series
将Borel子代数的交换理想理论、D. Peterson将先前的理论与仿射外尔群的组合学以及离散级数理论联系起来，这在B.科斯坦特在本文中，我们提供的证明，大部分Kostant的结果将它们扩展到$ad$ -幂零理想理想和发展的一个方向Kostant的调查，兼容的离散系列。
The notion of algebraic Dirac induction was introduced by P. Pandžić and D. Renard. This is a construction which gives representations with prescribed Dirac cohomology. They proved that all holomorphic discrete series representations can be constructed via Dirac induction. All discrete series representations of the group $SO_{e}(4,1)$ are nonholomorphic. In this paper we prove that they can also be constructed using algebraic Dirac induction.
Let $Z=G/H$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of $G$ on $L^2(Z)$. It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $L^2(Z)$, have infinitesimal characters which are real and belong to a lattice. Moreover, let $K$ be a maximal compact subgroup of $G$. Then each irreducible representation of $K$ occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the dis…
Discrete series Whittaker functions on $Spin(2n,2)$
研究了$Spin(2n, 2)$上的离散级数Whittaker函数.代数和连续Whittaker模型的空间的尺寸是明确确定的。它们由$Spin(2n-3, 2)$的不可约表示的维数之和来描述。得到了与极小$K$型向量相关的Whittaker函数的Mellin-Barnes型积分公式.
Restriction des séries discrètes de SU(2,1) à un sous-groupe exponentiel maximal et à un sous-groupe de Borel
我们给出了SU（2，1）的离散级数表示对博雷尔子群和极大指数可解子群的限制，并在轨道方法、Hamilton几何和“$\text{Spin}_{c}$“量子化的框架下解释了我们的结果.我们的结果对SU（2，1）的Duflo猜想给出了肯定的回答.
We introduce a notion of elliptic fake degrees for unipotent elliptic representations of a semisimple p-adic group. We conjecture, and verify in some cases, that the relation between the formal degrees of unipotent discrete series representations of a semisimple p-adic group and the elliptic fake degrees is given by the exotic Fourier transform matrix introduced by Lusztig in the study of representations of finite groups of Lie type.
The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula
本文研究了管型Hermitian李群中全纯离散级数对广义Whittaker Plancherel公式的贡献，构造了广义Whittaker向量嵌入并给出了其形式维数的显式公式及Hardy型空间解释。
For T $Ω$ a Hermitian symmetric tube-type domain, a family ($π$ $μ$) $μ$$\in$C of holomorphic vector-valued representations is studied. The corresponding Wallach set is determined. The main tool is a realization of the representations as weighted L 2-spaces on the cone $Ω$ through the Laplace transform.
In this note we determine the irreducible square integrable representations of a simple group which admits an admissible restriction to a subgroup $H$ locally isomorphic to $SL_2(\mathbb R).$ We show such representation is holomorphic and we determine the essentially unique $H$ with this property as well as multiplicity formulae.
Discrete series characters and the Lefschetz formula for Hecke operators
本文给出了实约化群上离散级数常数的组合公式，包括稳定的和不稳定的。它还进行了一个步骤的比较Hecke算子的拓扑迹公式与Arthur的Hecke算子的迹公式，其中拓扑项的组合方式，稳定的离散系列字符出现的步骤。
In this paper, we study extensions of graded affine Hecke algebra modules. In particular, based on an explicit projective resolution on graded affine Hecke algebra modules, we prove a duality result for Ext-groups. This duality result with an Ind-Res resolution gives an algebraic proof of the fact that all higher Ext-groups between discrete series vanish.
Symmetry breaking differential operators and Discrete Series
本文研究半单李群离散序列表示限制到同类型子群时的对称性破缺算子结构，利用再生核和对偶原理揭示其微分算子性质及法向微分形式。
Correspondance de Jacquet-Langlands et distinction : cas de certaines series discretes non cuspidales de niveau 0
设K/F是非阿基米德局部域的二次扩张.设V是群G = GL 2（R）的不可约零级离散级数表示，其中R是以K为中心的r阶除代数.假设V不是超尖点表示。设D是中心为F，指标为2r的可除代数.在接下来的工作中，我们试图回答这个问题：是表示VD ^{*}-可区别的当且仅当它的图像由Jacquet-Langlands对应是GL 2（F）-可区别的。为了回答这个问题，我们使用了P. Schneider和U. Stuhler和J. Silberger和E. W.辛克
On the geometric approach to the discrete series
本文通过几何方法建立了离散系列表示的参数化与特征公式的对应关系，并推导出Blattner猜想。
This article is devoted to branching problems for holomorphic discrete series representations of a conformal group $G$ of a tube domain $T_Omega$ over a symmetric cone $Ω$. More precisely, we analyse restrictions of such representations to the conformal group $G'$ of a tube domain $T_{Ω'}$ holomorphically embedded in $T_Ω$. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. To do so, a stratification space for a symmetric cone is introduced. This structure put the light on a new functional model, called the \emph{s…
Transfer of characters for discrete series representations of the unitary groups in the equal rank case via the Cauchy-Harish-Chandra integral
本文证明在等秩情况下，通过Cauchy-Harish-Chandra积分可实现双对中离散级数表示的特征转移。
Restricting holomorphic discrete series representations to a compact dual pair
本文研究了简单欧几里得若尔当代数共形群的全纯离散序列表示限制到子群$\operatorname{PSL}_2(\mathbb{R})\times\operatorname{Aut}(V)$的分支问题，利用向量值$L^2$函数实现与分层模型将其与紧群表示分解及正交多项式相联系。
Zelevinsky's classification theory of discrete series of $p$-adic general linear groups has been well known. Mœglin and Tadic gave the same kind of theory for $p$-adic classical groups, which is more complicated due to the occurrence of nontrivial structure of L-packets. Nonetheless, their work is independent of the endoscopic classification theory of Arthur (also Mok in the unitary case), which concerns the structure of L-packets in these cases. So our goal in this paper is to make more explicit the connection between these two very different types of theories. To do so, we reprove the result…
Quantization commutes with reduction in the non-compact setting: the case of holomorphic discrete series
本文证明了全纯离散级数表示相对于约化子群的重数满足“量子化与约化互换”的信条。
Ring structures for holomorphic discrete series and Rankin-Cohen brackets
本文讨论了Cayley型因果对称空间$G/H$的全纯离散级数集合上的两种不同的环结构，并根据全纯离散级数表示的张量积分解中产生的交织算子给出了Rankin-Cohen括号的一种新的解释.
Multiplicities of the discrete series
本文的目的是表明，一个离散的系列表示相对于一个紧凑的子群的重数可以“计算”几何，在预测的方式由“量化与减少”的Guillemin-Sternberg原则。
Nearly Holomorphic Functions and Relative Discrete Series of Weighted $L^2$-Spaces on Bounded Symmetric Domains
我们将有界对称域上的加权$L^2$ -空间的相对离散级数实现为不变Cauchy-Riemann算子的核，从而实现为近似全纯函数的空间.
We obtain an explicit integral representation of Siegel-Whittaker functions on Sp(2,R) for the large discrete series representations. We have another integral expression different from that of Miyazaki [7].
We obtain a classification of discrete series representations of odd general spin groups, generalizing the Mœglin-Tadić classification for classical groups. Using mostly algebraic methods, available in both classical and the odd general spin case, we provide a simplified and shorter version of the classification of non-cuspidal discrete series in terms of the admissible triples. The presented classification can potentially be directly applied to many other reductive $p$-adic groups.
We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
We study holographic operators associated with Rankin-Cohen brackets which are symmetry breaking operators for the restriction of tensor products of holomorphic discrete series of SL2(R). Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.
A binomial-coefficient identity arising from the middle discrete series of SU(2,2)
本文的目的是给出二项式上某些恒等式的初等证明，并回答Takahiro Hayata，Harutaka Koseki，and Takayuki Oda，Matrix coefficients of the middle discrete series of SU（2，2），J. Funct. Anal. 185（2001），297-341.
This note provides new criteria on a unimodular group $G$ and a discrete series representation $(π, \mathcal{H}_π)$ of formal degree $d_π > 0$ under which any lattice $Γ\leq G$ with $\text{vol}(G/Γ) d_π \leq 1$ (resp. $\text{vol}(G/Γ) d_π \geq 1$) admits $g \in \mathcal{H}_π$ such that $π(Γ) g$ is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.
Density conditions with stabilisers for lattice orbits of discrete series representations
本研究针对离散系列表示格轨道中框架和里斯序列的必要密度条件提供初等证明，涉及向量射影稳定子，扩展并简化了半单李群中伯格曼核和格的已知密度条件。
Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients $\neq 0$, of integrable discrete series in families of congruence subgroups. In the case of $G=Sp_{2n}(\mathbb R)$, we relate our work to that of Li [15]. For $\mathcal G$ quasi--split over $\mathbb Q$, …
Let V be an Harish-Chandra discrete series representation of a real semi-simple Lie group G' and let G be a semi-simple subgroup of G'. In this paper, we give a geometric expression of the G-multiplicities in V when the representation V is supposed to be G-admissible.
Discrete series for the graded Hecke algebra of type $H_{4}$
本文证明 $H_4$ 型分次仿射 Hecke 代数的离散序列中心特征与 Heckman–Opdam 中心特征一致，并借助校准模和极小诱导模完成正参数情形的全部分类。
Spin_c quantization and the K-multiplicities of the discrete series
本文证明了Guillemin-Sternberg的“量子化与归约交换”原理对于实连通半单李群的离散级数的参数化余伴随轨道成立。
Quaternionic discrete series for Sp(1,1)
本文研究了对称空间$G/K:=Sp(1,1) /(Sp(1) \times Sp(1))$上齐次向量丛中平方可积〔的〕截面空间的子空间$G=Sp(1,1)$群的离散级数表示的解析实现。利用Szegö映射给出了表示空间中的$K$型对子流形$AK/K$的限制表达式。
Discrete series representations and K multiplicities for U(p,q). User's guide
本文档是Maple程序的同伴：离散级数和U（p，q）的K型可在：http://www.math.jussieu.fr/~vergne上获得我们解释了一种计算U（p，q）的离散级数中U（p，q）x U（q）的不可约表示的重数的算法。它是基于Blattner的公式。我们回顾了一般的数学背景计算Kostant分拆函数通过多维残留物，我们概述了我们的算法。在Paradan结果的基础上，我们还指出了描述重数的分段多项式函数的一些性质.
Reducibility of induced discrete series representations for affine Hecke algebras of type B
最近Delorme和Opdam将R-群的理论推广到了具有不等标号的仿射Hecke代数。我们应用他们的结果的情况下，仿射Hecke代数是B型，诱导离散级数表示与真正的中心字符。我们计算了这样一个诱导表示的R-群，并证明了它将多重自由分解为2^d不可约和项。功率d可以组合地计算。
Let $F$ be a nonarchimedean local field of characteristic $0$ and residue field of order not divisible by $2$. We show how to calculate the product of the covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimension of a discrete series representation of $GL(2,F)$. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered …
Wehrl inequalities for matrix coefficients of holomorphic discrete series
该研究针对G的向量值全纯离散级数矩阵系数，证明了偶数阶p=2n的Wehrl型L²(G)-Lᵖ(G)不等式，给出以Harish-Chandra形式次数表示的最优常数，并确定再生核为其最大化子。
Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $σ$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $σ.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.
In this paper we construct smooth cuspidal automorphic forms related to integrable discrete series of a connected semisimple Lie group with finite center for classical and adelic situation as an application of the theory of Schwartz spaces for automorphic forms developed by Casselman. In the classical situation, smooth cuspidal automorphic forms are constructed via an explicit continuous map from the Frech\' et space of smooth vectors of a Banach realization inside $L^1(G)$ of an integrable discrete series into the space of smooth vectors of a strong topological dual of an appropriate Schwartz…
A finite Linear Dependence of Discrete Series Multiplicities
本文证明离散系列重数可由有限个重数决定，从而精细化强重数一结果，并应用于尖点形式空间维数的相等性判定。
Une remarque sur le degre formel d'une serie discrete d'un groupe lineaire general p-adique
我展示的例子，一般线性群，如何可以推导出从我以前的工作“分解谱德联合国集团reductif $p$ -adique”（出现在杂志的数学研究所的Jussieu）精确的信息正式程度的平方可积〔的〕表示的$p$ -adic组。
Commuting Toeplitz Operators on Bounded Symmetric Domains and Multiplicity-Free Restrictions of Holomorphic Discrete Series
对任意给定的有界对称域，证明了Toeplitz算子作用于任意加权Bergman空间生成的交换$C^*$ -代数的存在性.生成这种代数的Toeplitz算子的符号由定义域的双全纯群的适当子群下的基本上有界函数不变的定义。这些子群包括双全纯的极大紧群。我们证明了交换性的Toeplitz运营商考虑Bergman空间作为基础空间的全纯离散系列，然后应用已知的多重性的结果限制某些子群的全纯离散系列。在紧的情况下，我们完全特征的子群定义不变的符号，产生交换Toeplitz运营商的多重性自由的财产。
Quaternionic symplectic model for discrete series representations
该研究通过四元数辛模型，证明了离散系列表示的Sp_n(D)-区分性当且仅当其为超 cuspidal 表示，并完成了相关分类。
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