Multiplicities of the discrete series
Paul-Emile Paradan
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。
The purpose of this paper is to show that the multiplicities of a discrete series representation relatively to a compact subgroup can be "computed" geometrically, in the way predicted by the "qantization commutes with reduction" principle of Guillemin-Sternberg.
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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。
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.
Associated symmetric pair and multiplicities of admissible restriction of Discrete Series
设$(G, H)$是实半单李群$G$的对称对，$(G, H_0)$是它的伴随对.对于每个不可约平方可积〔的〕表示$π$的$G$，使其限制$H$是允许的，我们发现一个不可约平方可积〔的〕表示$π_0$的$H_0$，它允许计算的Harish-Chandra参数的每个不可约$H-$子表示的$π$以及它的多重性。该计算是基于对$π_0$到$H_0.$的极大紧子群的限制的谱分析
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。
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。
A general proposition is proved relating multiplicities (of restriction of a representation of a group to a subgroup) under basechange, and used to calculate some multiplicities for cuspidal representations which become principal series representations under basechange for which multiplicities can be calculated by geometric methods.
$K$-type multiplicities in degenerate principal series via Howe duality
本文通过Howe对偶和跷跷板互惠性，推导出复经典群到块对角子群的分支重数公式，并解释为退化主序列中的K_R-型重数，在最小情形给出表格理论解释。
Global rigid inner forms and multiplicities of discrete automorphic representations
给定一个数域，我们在其上构造一个规范伽罗瓦格贝，并证明其上同调提供了arXiv：1304.3292中引入的改进的局部内窥镜和经典的全局内窥镜之间的桥梁。作为特殊的应用，我们表示的典型adelic转移因子，支配稳定的阿瑟-塞尔伯格迹公式作为产品的归一化的本地转移因子，我们给出了一个椭圆L-包与相应的S-群之间配对的一个显式构造（根据当地环境中的文化配对）这是必要的成分，在描述的离散自守谱的约化群，我们给出了一个证明的一些期望亚瑟。
Kirillov's orbit method: the case of discrete series representations
设V是实半单李群G'的Harish-Chandra离散级数表示，G是G'的半单群.本文给出了当表示V是G-可容许的时V中G-重数的一个几何表达式。
The Jordan-Hoelder series for nearby cycles on some Shimura varieties and affine flag varieties
我们研究的约旦-Hoelder系列附近的周期上某些志村品种和Rapoport-津克本地模型，并在有限维件贝林森的变形的仿射格拉斯曼的仿射旗品种（和他们的p-adic类似物）。本文给出了不可约组分在某些上同调群中重数的计算公式，并给出了在仿射Hecke代数中计算重数的算法.
The geometry of tilting composition series via Richardson varieties
研究翻转sheaves在旗形簇上的(加权)乔丹-霍尔乘数通过理查德森品种在朗兰兹对偶旗形簇上的超上同调几何解释，并提供基于ℓ-卡兹德-卢斯滕布希多项式的显式公式。
Moduli spaces of representations of quivers with multiplicities via non-reductive GIT
该研究利用非约化群的相对仿射GIT与新稳定性条件，构造了截断幂级数环上带重数的箭图表示模空间（含带标架模空间及Nakajima箭图簇类似物），并证明部分光滑模空间上同调纯净。
We study local multiplicities associated to the so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan $L$-packet and that they are related to local exterior square $L$-functions.
The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level $1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and provide numerical applications in absolute rank $\leq 8$. Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) whose motivic weight is $\leq 24$. In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existenc…
Analysis on Symmetric and Locally Symmetric Spaces (Multiplicities, Cohomology and Zeta functions)
课程的目标是回顾主要由M. Olbrich和第一作者。考虑半单李群$G$的离散余紧子群$Γ$.我们涉及的$Γ$的群上同调系数的最大全球化的表示$G$的酉表示的$G$在$L^2(G/Γ)$的多重性。在$G=SL(2,R)$情形下给出了显式计算。在秩一的情况下，我们状态的Selberg迹公式的一个版本，讨论的Selberg zeta函数的奇异性，并说明它们的关系与$Γ$的上同调的主列表示。
Limit multiplicities for principal congruence subgroups of GL(n) and SL(n)
研究数域上约化群的主同余子群的离散谱的极限行为。虽然这个问题在协紧情况下是很好理解的（即，当群以中心为模是各向异性的时），我们处理秩无界的群。对于群GL（n）和SL（n），我们证明了适当归一化的谱收敛到Plancherel测度（极限多重性）。对于一般的约化群，我们得到了问题的实质性约化。我们的主要工具是最近的改进阿瑟的迹公式在[FLM11，FL11]，这使我们能够表明，对于GL（n）和SL（n）的连续谱的贡献是可以忽略不计的限制。
Stability theorems for multiplicities in graded $S_n$-modules
本文证明对称群自然表示与 Weyl 模不变量重数的稳定性定理，并据此将多项式环余不变量 Hilbert 级数的截断猜想推广到更多变量，同时给出 Gröbner 基等价猜想。
The Jordan-Hölder series of the locally analytic Steinberg representation
我们确定的合成因子的约旦-Hölder系列，包括局部解析斯坦伯格表示的多重性。为了这个目的，我们证明了非循环性的评价当地分析山雀复杂的斯坦伯格表示。此外，我们描述了一些类似的Jacquet函子适用于不可约的主列表示构建的Orlik和Strauch。
Explicit multiplicities in the cuspidal spectrum of SU(n,1)
本文利用 Selberg 迹公式推导了实李群 SU(n,1) 主同余子群商的可积离散序列表示重数显式公式，并应用于计算局部对称空间的尖点上同调及证明 Dirichlet $L$-函数取值的有理性。
On reducibility points beyond ends of complementary series of p-adic GL(n)
本文从Speh表示出发，研究p-adic域上一般线性群的余级数端点以外的约化点.我们明确地描述了在这些约化点的表示的合成列。它们是多重性的表示，并且它们可以是任意长度。给出了所有不可约子元的Langlands参数。
Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be compact. Under a condition on $K$, which holds in particular if $K$ is maximal compact, we give a geometric expression for the multiplicities of the $K$-types of any tempered representation (in fact, any standard representation) $π$ of $G$. This expression is in the spirit of Kirillov's orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of $π|_K$ obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and fo…
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。
Norm computation and analytic continuation of vector valued holomorphic discrete series representations
本文给出了向量值全纯离散级数表示在其$K$型为“几乎无乘法”时的范数的显式计算。“作为应用，我们讨论了最高权模的性质，如酉化性，可约性和合成列。
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。
Let $N$ be a normal subgroup of a finite group $G$ and $V$ be a fixed finite-dimensional $G$-module. The Poincaré series for the multiplicities of induced modules and restriction modules in the tensor algebra $T(V)=\oplus_{k \geq 0}V^{\otimes k}$ are studied in connection with the McKay-Slodowy correspondence. In particular, it is shown that the closed formulas for the Poincaré series associated with the distinguished pairs of subgroups of $\mathrm{SU}_2$ give rise to the exponents of all untwisted and twisted affine Lie algebras except ${\rm A}_{2n}^{(1)}$.
Algebraic Dirac induction for nonholomorphic discrete series of SU(2, 1)
P.Pandžić和D.雷纳德证明了全纯和反全纯离散级数表示可以通过代数狄拉克归纳法来构造。群$SU(2,1)$，除了这两种类型，还有第三种类型的离散级数表示，既不是全纯的也不是反全纯的。在本文中，我们表明，非全纯离散级数表示的组$SU(2,1)$也可以构造使用代数狄拉克归纳法。
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。
The Continuous Spectrum in Discrete Series Branching Laws
设$G$是Harish-Chandra类的约化李群，$H$是对称子群，$π$是$G$的离散级数表示，给出了关于$(G,H)$对的一个条件，保证了$π|_H$的直接积分分解包含$H$的每个有限重不可约表示.另外，若$G$是Harish-Chandra类的约化李群，$H\subset G$是Harish-Chandra类的闭约化子群，证明了$π|_H$的直接积分分解的重数函数沿"连续参数"为常数.在获得这些结果，作者开发了一种新的技术，通过卷积与哈里什-钱德拉字符的限制$π|_H$的多重性研究。这种技术的优点是，对于研究连续谱和离散谱都是有用的。
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）.
Local periods for discrete series representations
设$(G,H)$是$p$ -adic域上的对称对，$π$是$G$的离散级数表示。本文证明了对于某些对称对$(G,H)$，$Hom_H(π,\mathbb{C})$中的局部周期可以通过对$H$上的$π$的矩阵系数积分来构造.
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.
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.
The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding of irreducible subquotients of a reducible principal series representation of $GL_n(F)$, $F$ a $p$-adic field (without specifying their multiplicities which is done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposal of a similar kind for principal series representations of $GL_n({\mathbb R})$. Our investigation on principal series representations naturally led us to consider the Steinberg representation for real groups, which has curiuosly not been paid much attention to in the subject (unlike the $…
Restriction des séries discrètes de SU(2,1) à un sous-groupe exponentiel maximal et à un sous-groupe de Borel
我们给出了SU（2，1）的离散级数表示对博雷尔子群和极大指数可解子群的限制，并在轨道方法、Hamilton几何和“$\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.
Theta correspondence and Springer correspondence
本文通过Springer对应推导出偶正交群与辛群之间，以及一般线性群在有限域上的unipotent主系列表示的theta对应公式，并研究了相关Hecke范畴的模块结构。
Formal degrees of unipotent discrete series representations and the exotic Fourier transform
我们引入了半单p-adic群的幂幺椭圆表示的椭圆伪度的概念。我们猜想并在某些情况下证明了半单p-adic群的幂幺离散级数表示的形式度与椭圆假度之间的关系是由Lusztig在研究有限群表示时引入的奇异傅里叶变换矩阵给出的.
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.
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.
An explicit integral representation of Siegel-Whittaker functions on Sp(2,R) for the large discrete series representations
对于大离散级数表示，我们得到了Sp（2，R）上Siegel-Whittaker函数的一个显式整表示.我们得到了与Miyazaki [7]不同的另一种整式。
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.
Discrete series of simple Lie groups with admissible restriction to a $SL_2(\mathbb R)$
本文确定了一个单群的不可约平方可积〔的〕表示，该单群允许对一个局部同构于$SL_2(\mathbb R).$的子群$H$有一个容许限制。我们证明了这种表示是全纯的，并确定了具有这个性质的本质上唯一的$H$以及重数公式。
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。
Duality for Ext-groups and extensions of discrete series for graded Hecke algebras
本文研究了分次仿射Hecke代数模的扩张。特别地，基于分次仿射Hecke代数模上的一个显式投射分解，证明了Ext-群的一个对偶结果.这个对偶结果与Ind-Res决议给出了一个代数证明的事实，即所有高的Ext-群之间的离散系列消失。
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.
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