Domains of holomorphy for irreducible unitary representations of simple Lie groups

Irreducible characters of $3'$-degree of finite symmetric, general linear and unitary groups

Let $G$ be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to $3$. We construct a canonical correspondence between irreducible characters of degree coprime to $3$ of $G$ and those of $N_{G}(P)$, where $P$ is a Sylow $3$-subgroup of $G$. Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.

Restriction of irreducible unitary representations of Spin(N,1) to parabolic subgroups

In this paper, we obtain explicit branching laws for all irreducible unitary representations of $\Spin(N,1)$ restricted to a parabolic subgroup $P$. The restriction turns out to be a finite direct sum of irreducible unitary representations of $P$. We also verify Duflo's conjecture for the branching law of tempered representations of $\Spin(N,1)$ with respect to a parabolic subgroup $P$. That is to show: in the framework of the orbit method, the branching law of a tempered representation is determined by the behavior of the moment map from the corresponding coadjoint orbit. A few key tools used…

计数一般线性群与酉群的不可约表示

Counting irreducible representations of general linear groups and unitary groups

本文研究实一般线性群与实酉群中给定无穷小特征及相伴簇的不可约 Casselman-Wallach 表示计数问题，并利用涂色与指定 Young 图给出精确的组合表达。

Positive Energy Unitary Irreducible Representations of the Superalgebras $osp(1|2n,R)$ and Character Formulae

We continue the study of positive energy (lowest weight) unitary irreducible representations of the superalgebras $osp(1|2n,R)$. We update previous results and present the full list of these UIRs. We give also some character formulae for these representations.

Lie群 $SO_0(2,m)$ 的具有非零相对李代数上同调的不可约酉表示

Irreducible unitary representations with non-zero relative Lie algebra cohomology of the Lie group $SO_0(2,m)$

本文针对 Lie 群 $SO_0(2,m)$，确定了具有非零 $(\ rak{g},K)$-上同调的不可约酉表示的等价类数量、上同调 Poincaré 多项式，并识别出其中的离散序列表示与全纯离散序列表示。

Unitary Representations of Lattices of Free Nilpotent Lie Groups of Step-Two

Using a theorem proved by Bekka and Driutti, we show that if $\mathfrak{f}$ is a freely generated nilpotent Lie algebra of step-two, then almost every irreducible representation of the corresponding Lie group restricted to some lattice $Γ$ is an irreducible representation of $Γ$ if the dimension of the Lie algebra is odd. However, if the dimension of the Lie algebra is even, then almost every unitary irreducible representation of the Lie group restricted to $Γ$ is reducible.

Unitary (g,K) modules of SU(2,1)

Let G=SU(2,1). In this paper we parametrize irreducible unitary (g,K) modules of G. The parametrization is done in two steps. Firstly, we parametrize irreducible (g,K) modules (Theorem 3). In the second step we find unitary (g,K) modules (Theorem 5). One can compare our results with results of Kraljevic.

Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$. Let ${\mathcal U}_n(A)$ be the subgroup of $\mathrm{GL}_n(A)$ of all upper triangular matrices with 1's along the main diagonal. Let $P=H_n(A)\rtimes {\mathcal U}_n(A)$, where ${\mathcal U}_n(A)$ acts on $H_n(A)$ by $*$-congruence transformations. We may view $P$ as a unipotent subgroup of either a symplectic group $\mathrm{Sp}_{2n}(A)$, if $*=1_A$ (in which c…

Unitary representations of nilpotent super Lie groups

We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.

Irreducibility criterion for representations induced by essentially unitary ones (case of non-archimedean GL(n,A)

Let A be a finite dimensional central division algebra over a local non-archimedean field F. Fix any parabolic subgroup P of GL(n,A) and a Levi factor M of P. Let πbe an irreducible unitary representation of M and ϕ(not necessarily unitary) character of M. We give an explicit necessary and sufficient condition for the parabolically induced representation Ind(ϕπ) of GL(n,A) to be irreducible.

Branching problems of unitary representations

The irreducible decomposition of a unitary representation often contains continuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs discretely with finite multiplicity (admissible restrictions). Basic theory and new perspectives of admissible restrictions are presented from both analytic and algebraic view points. We also discuss some applications of admissible restrictions to modular varieties and $L^p$-harmonic analysis.

Weyl groups, the Dirac inequality, and isolated unitary unramified representations

I present several applications of the Dirac inequality to the determination of isolated unitary representations and associated "spectral gaps" in the case of unramified principal series. The method works particularly well in order to attach irreducible unitary representations to the large nilpotent orbits (e.g., regular, subregular) in the Langlands dual complex Lie algebra. The results could be viewed as a $p$-adic analogue of Salamanca-Riba's classification of irreducible unitary $(\mathfrak g,K)$-modules with strongly regular infinitesimal character.

A construction by deformation of unitary irreducible representations of $SU(1,n)$ and $SU(n+1)$

We recover the holomorphic discrete series representations of $SU(1,n)$ as well as some unitary irreducible representations of $SU(n+1)$ by deformation of a minimal realization of $sl(n+1, {\mathbb C})$.

Geometric analysis on small unitary representations of GL(N,R)

The most degenerate unitary principal series representations π_{iλ,δ} (with λ \in R, δ\in Z/2Z) of G = GL(N,R) attain the minimum of the Gelfand-Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction π_{iλ,δ}|_H (branching law) with respect to all symmetric pairs (G,H). For N=2n with n \geq 2, the restriction π_{iλ,δ}|_H remains irreducible for H=Sp(n,R) if λ\neq0 and splits into two irreducible representations if λ=0. The branching law of the restriction π_{iλ,δ}|_H is purely discrete fo…

Unitary Representations and Heisenberg Parabolic Subgroup

In this paper, we study the restriction of an irreducible unitary representation $π$ of the universal covering $\widetilde{Sp}_{2n}(\mb R)$ to a Heisenberg maximal parabolic group $\tilde P$. We prove that if $π|_{\tilde P}$ is irreducible, then $π$ must be a highest weight module or a lowest weight module. This is in sharp constrast with the $GL_n(\mathbb R)$ case. In addition, we show that for a unitary highest or lowest weight module, $π|_{\tilde P}$ decomposes discretely. We also treat the groups $U(p,q)$ and $O^*(2n)$.

Functors for unitary representations of classical real groups and affine Hecke algebras

We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they preserve unitarity. In the case of split classical groups, we thus obtain a functorial inclusion of the real spherical unitary dual (with real infinitesimal character) into the corresponding p-adic sphe…

Arveson's criterion for unitary similarity

This paper is an exposition of W.B. Arveson's complete invariant for the unitary similarity of complex, irreducible matrices.

秩一正交群酉对偶的显式 Hilbert 空间及其应用

Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications

本文在去原点欧氏空间的显式向量值 $L^2$ Hilbert 空间中实现 $\mathrm{SO}_0(n+1,1)$ 的全部不可约酉表示，并借助 Knapp–Stein 算子刻画 Whittaker 向量及抛物子群分解。

复自旋群的真正酉对偶

The genuine unitary dual of complex spin groups

本文研究复自旋群的酉表示分类问题，给出了其全部不可约酉表示的完整分类，即真正酉对偶。

关于quandle表示

On quandle representations

本文研究quandle表示的分解性质，证明有限quandle表示可分解为不可约表示当且仅当图像中每个元素可对角化。并探讨了不可约表示的单位化条件及quandle的包络群的表示性质。

Enumerating representations of general unitary groups over principal ideal rings of length 2

We enumerate the number of complex irreducible representations of each degree of general unitary groups of degree 4 over principal ideal rings of length 2.

The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring

We find all irreducible constituents of the Weil representation of a unitary group $U_m(A)$ of rank $m$ associated to a ramified quadratic extension $A$ of a finite, commutative, local and principal ring $R$ of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when $R$ is a field.

Unitary spherical spectrum for split classical groups

This paper gives a complete classification of the unitary irreducible spherical representations of split real and p-adic groups. The results were obtained around 2000, the changes to the new version are expository.

Unitary Hecke algebra modules with nonzero Dirac cohomology

In this paper, we review the construction of the Dirac operator for graded affine Hecke algebras and calculate the Dirac cohomology of irreducible unitary modules for the graded Hecke algebra of gl(n).

Holomorphic Realization of Unitary Representations of Banach--Lie Groups

In this paper we explore the method of holomorphic induction for unitary representations of Banach--Lie groups. First we show that the classification of complex bundle structures on homogeneous Banach bundles over complex homogeneous spaces of real Banach--Lie groups formally looks as in the finite dimensional case. We then turn to a suitable concept of holomorphic unitary induction and show that this process preserves commutants. In particular, holomorphic induction from irreducible representations leads to irreducible ones. Finally we develop criteria to identify representations as holomorph…

Direct proofs of the Feigin-Fuchs character formula for unitary representations of the Virasoro algebra

Previously we gave a proof of the Feigin--Fuchs character formula for the irreducible unitary discrete series of the Virasoro algebra with 0<c<1. The proof showed directly that the mutliplicity space arising in the coset construction of Goddard, Kent and Olive was irreducible, using the elementary part of the unitarity criterion of Friedan, Qiu and Shenker, giving restrictions on h for c=1-6/m(m+1) with m>2. In this paper we consider the same problem in the limiting case of the coset construction c=1. Using primary fields, we directly establish that the Virasoro algebra acts irreducibly on the…

单位元模 over gap-$p$ 维拉斯洛夫代数

Unitary modules over the gap-$p$ Virasoro algebras

研究gap-$p$ 维拉斯洛夫代数上不可约Harish-Chandra模的单位元条件。

有限一般线性群和酉群的Shalika模型

Shalika Models for Finite General Linear and Unitary Groups

本文完全显式分类了有限一般线性群GL_{2n}(F_q)和有限酉群U_{2n}(F_q)中允许Shalika模型的不可约表示，利用Lusztig的Jordan分解建立了重数公式，并联系了辛模型、线性模型和相对Langlands对偶。

Positive Energy Unitary Irreducible Representations of the Superalgebras osp(1|2n,R)

We give the classification of the positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R).

Lie supergroups, unitary representations, and invariant cones

The main result of this article is an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie supergroups. It also includes an exposition of recent results of the second author on the classification of irreducible unitary representations of nilpotent Lie supergroups using the method of coadjoint orbits.

Unitary Representations with non-zero Dirac cohomology for complex $E_6$

This paper classifies the equivalence classes of irreducible unitary representations with nonvanishing Dirac cohomology for complex $E_6$. This is achieved by using our finiteness result, and by improving the computing method.

Unitary representations of real reductive groups

We present a finite algorithm for computing the set of irreducible unitary representations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant Hermitian form as a deformation of one of the unitary representations in Harish-Chandra's Plancherel formula. The behavior of these deformations was determined to a first approximation in the Kazhdan-Lusztig analysis of irreducible characters; more complete information comes from the Beilinson-Bernstein proof of the Jantzen conjectures. The basic idea of our algor…

关于有限长度主理想局部环上二阶一般线性群和酉群的正则特征标的张量积

On tensor products of regular characters of the general linear and unitary groups of degree two over the principal ideal local rings of finite length

本文研究有限长度主理想局部环上二阶一般线性群和酉群的正则不可约表示的张量积分解，证明不同类型正则表示张量积的重数性质，揭示有限域与一般局部环情形的差异。

A modified Brauer algebra as centralizer algebra of the unitary group

The centralizer algebra of the action of the unitary group on the real tensor powers of its natural module, is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for U(n), and with the decomposition of the above tensor powers into irreducible modules, is considered.

Matrix elements for type 1 unitary irreducible representations of the Lie superalgebra gl(m|n)

Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary generators, on finite dimensional type 1 unitary irreducible representations. We compare our results with existing works that deal with only subsets of the class of type 1 unitary representations, all of which only present explicit matrix elements for elementary generators. Our work therefore provides an important extension to existing methods, and thus highlights the…

幺正与非幺正表示的海森堡-魏尔李代数

Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra

本文研究了海森堡-魏尔李代数的幺正与非幺正表示，分析了张量积和不可分解表示的构造与性质。

Types and unitary representations of reductive p-adic groups

We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.

Branching rules for the Weyl Groups of the Unitary and Orthogonal Lie Groups

This note presents a procedure to determine the reduction of the irreducible and the induced characters of the symmetric group in terms of the irreducible and induced characters of the hyperoctahedral group Key Words: Symmetric Group, Hyperoctahedral group, Representations, Characters, Re- duction.

关于包含U(p, q)的酉最低权表示的A参数

On $A$-parameters containing unitary lowest weight representations of $\mathrm{U}(p, q)$

本文利用Barbasch-Vogan参数化和Trapa算法，确定了实酉群U(p, q)包含不可约酉最低权表示的所有Arthur包，证明每个包最多含一个此类表示，并给出其最低K型的显式公式。

Unitary representations of type B rational Cherednik algebras and crystal combinatorics

We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remov…

Dirac series for $E_{6(-14)}$

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the simple Lie group $E_{6(-14)}$, which is of Hermitian symmetric type. Each FS-scattered Dirac series of $E_{6(-14)}$ is realized as a composition factor of certain $A_{\mathfrak{q}}(λ)$ module. Along the way, we have also obtained all the fully supported irreducible unitary representations of $E_{6(-14)}$ with integral infinitesimal characters.

Dirac series of $GL(n, \mathbb{R})$

The unitary dual of $GL(n, \mathbb{R})$ was classified by Vogan in the 1980s. Focusing on the irreducible unitary representations of $GL(n, \mathbb{R})$ with half-integral infinitesimal characters, we find that Speh representations and the special unipotent representations are building blocks. By looking at the $K$-types of them, and by using a Blattner-type formula, we obtain all the irreducible unitary $(\mathfrak{g}, K)$-modules with non-zero Dirac cohomology of $GL(n, \mathbb{R})$, as well as a formula for (one of) their spin-lowest $K$-types. Moreover, analogous to the $GL(n,\mathbb{C})$ …

On Uniform Admissibility of Unitary and Smooth Representations

Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth representations of $G$ are uniformly admissible. We also show that the latter property is inherited by finite-index subgroups and overgroups of $G$.

Classification of the finite-dimensional unitary representations of type B rational Cherednik algebras

We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary representations of type $B$ rational Cherednik algebras to show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica.

Weil representations of unitary groups over ramified extensions of finite local rings with odd nilpotency length

We find the irreducible decomposition of the Weil representation of the unitary group $\mathrm{U}_{2n}(A)$, where $A$ is a ramified quadratic extension of a finite, commutative, local, principal ideal ring $R$ and the nilpotency degree of the maximal ideal of $A$ is odd. We show in particular that this Weil representation is multiplicity free. Restriction to the special unitary group $\mathrm{SU}_{2n}(A)$ preserves irreducibility and multiplicity freeness provided $n>1$.

Branching laws of unitary representations associated to minimal elliptic orbits for indefinite orthogonal group O(p,q)

We give a complete description of the discrete spectra in the branching law $Π|_{G'}$ with respect to the pair $(G,G')=(O(p,q), O(p',q') \times O(p'',q''))$ for irreducible unitary representations $Π$ of $G$ that are "geometric quantization" of minimal elliptic coadjoint orbits. We also construct explicitly all holographic operators and prove a closed Parseval-type formula.

不可约单位ary模的分类 over $\mathfrak{u}(p,q|n)$

Classification of irreducible unitary modules over $\mathfrak{u}(p,q|n)$

本文基于Howe对偶性和二次不变量，对$\mathfrak{u}(p,q|n)$的不可约最高权单位ary模进行分类，并通过duality和李超代数同构分类了最低权单位ary模和$\mathfrak{u}(n|q,p)$的不可约单位ary模。

A reducibility problem for even unitary groups: The depth zero case

We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for $E/F$ a quadratic extension of p-adic fields the associated unitary group $G=\mathrm{U}(n,n)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E)$. Let $π$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $ι_P^G π$ is reducible.

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。

晶体学群的酉对偶的拓扑

The Topology of the Unitary Dual of Crystallography Groups

本文提出生成任意维晶体学群不可约表示的程序，并利用矩阵序列研究其酉对偶拓扑，以三维晶体学群90为例验证了方法有效性。

不可约特征标与半单共轭轨道

Irreducible Characters and Semisimple Coadjoint Orbits

本文研究了半单共轭轨道对应表示的几何特征公式，结合上同调诱导法和抛物面诱导法，为正交表示提供了新的理论框架。

秩一自旋群的张量积分解 I：酉主序列表示

Tensor product decomposition for rank-one spin groups I : unitary principal series representations

本文针对秩一自旋群 Spin(n,1)，在 π1 为酉主序列表示、π2 为任意不可约酉表示时，给出了张量积表示的显式直接积分分解。

Endoscopic Classification of Representations: Inner Forms of Unitary Groups

We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is known for quasi-split unitary groups. The classification of automorphic representations is given in terms of automorphic representations of general linear groups. The classification of irreducible admissible representations is given in terms of Langlands parameters.

Explicit constructions of unitary transformations between equivalent irreducible representations

Irreducible representations (irreps) of a finite group $G$ are equivalent if there exists a similarity transformation between them. In this paper, we describe an explicit algorithm for constructing this transformation between a pair of equivalent irreps, assuming we are given an algorithm to compute the matrix elements of these irreps. Along the way, we derive a generalization of the classical orthogonality relations for matrix elements of irreps of finite groups. We give an explicit form of such unitary matrices for the important case of conjugated Young-Yamanouchi representations, when our g…