Yehuda Shalom
The article establishes a long list of rigidity properties of lattices in G = SO(n,1) with n>=3 and G = SU(n,1) with n>=2 that are analogous to superrigidity of lattices in higher-rank Lie groups. The arguments are set in the context of strongly L^p unitary representations of G.
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We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, $SU(np,p), Sp(2n+2,\mathbb R), SO^*(2n+2), SO(2n,2)$. This reproves or generalizes some results in \cite{GM, KKP, Klingler-inv, Pozzetti}.
We consider spherical principal series representations of the semisimple Lie group of rank one $G=SO(n, 1; \mathbb K)$, $\mathbb K=\br, \bc, \bh$. There is a family of unitarizable representations $π_ν$ of $G$ for $ν$ in an interval on $\mathbb R^+$, the so-called complementary series, and subquotient or subrepresentations of $G$ for $ν$ being negative integers. We consider the restriction of $(π_ν, G)$ under the subgroup $H=SO(n-1, 1; \mathbb K)$. We prove the appearing of discrete components. The corresponding results for the exceptional Lie group $F_{4(-20)}$ and its subgroup $Spin(8,1)$ ar…
We consider the spherical complementary series of rank one Lie groups $H_n=\SO_0(n, 1; \mathbb F)$ for $\mathbb F=\mathbb R, \mathbb C, \mathbb H$. We prove that there exist finitely many discrete components in its restriction under the subgroup $H_{n-1}=\SO_0(n-1, 1; \mathbb F)$. This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of $G_n=SU(n, 1)$, $SU(n, 1)\times SU(n, 1)$ and $SU(2n, 2)$ and by the branching of holomorphic representations under the corresponding subgroup $G_{n-1}$.
We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple group is classical, we prove that the new theory is essentially equivalent to Howe's theory of N-rank.
Let $G = {\rm U}(2m, {\mathbb F}_{q^2})$ be the finite unitary group, with $q$ the power of an odd prime $p$. We prove that the number of irreducible complex characters of $G$ with degree not divisible by $p$ and with Frobenius-Schur indicator -1 is $q^{m-1}$. We also obtain a combinatorial formula for the value of any character of ${\rm U}(n, {\mathbb F}_{q^2})$ at any central element, using the characteristic map of the finite unitary group.
We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every $q$-deformation of a higher rank compact Lie group has central property ${\rm (T)}$. We also determine the unitary dual of $SL_q(n,\mathbb{C})$.
In the 1970's, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey's analogy into a one-to-one correspondence for connected complex semisimple groups, and in doing so obtained a novel verification of the Baum-Connes conjecture with trivial coefficients for such groups. Here we extend Higson's results to any Lie group with finitely many connected components whose connected component of the identity is complex semisimple. Our m…
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。
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。
Decay estimates for matrix coefficients of unitary representations of semisimple Lie groups
本文针对有限中心连通半单李群的酉表示矩阵系数，利用 $K$ 双侧均方平均建立由最小正球面函数控制的全局衰减估计，并给出其与逐点估计的等价性及应用。
Let $M$ be compact negatively curved manifold, $Γ=π_1(M)$ and $\tilde{M}$ be its universal cover. Denote by $B =\partial \tilde{M}$ the geodesic boundary of $\tilde{M}$ and by $ν$ the Patterson-Sullivan measure on $X$. In this note we prove that the associated unitary representation of $Γ$ on $L^2(B,ν)$ is irreducible. We also establish a new rigidity phenomenon: we show that some of the geometry of $M$, namely its marked length spectrum, is reflected in this $L^2$-representations.
We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a semisimple Lie group $G$ and a semisimple subgroup $G'$, and between their composition factors. Our method describes the restriction of these operators to the $K'$-isotypic components, $K'\subseteq G'$ a maximal compact subgroup, and reduces the representation theoretic problem to an infinite system of scalar equations of a combinatorial nature. For rank one orthogonal and unitary groups and spherical principal series representations we calcu…
On the unitary cohomology of semisimple groups
本文研究半单李群在任意酉表示下的连续（上）同调，将可约表示情形分解为Hausdorff部分与挠部分，分别由不可约上同调表示的内含性与弱内含性刻画，并给出秩1群及SO°(n,1)的完整结果，同时证明挠上同调首阶在性质(T)下受对称空间维数平方根下界限制。
A compact Lie group G and a faithful complex representation V determine a Sato-Tate measure, defined as the direct image of Haar measure on G with respect to the character of V. We give a necessary and sufficient condition for a Sato-Tate measure to be an isolated point in the set of such measures, regarded as a subset of the space of distributions. In particular we prove that the Sato-Tate measure of a connected and semisimple group with respect to an irreducible representation is an isolated point.
Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups
本文为高阶半单群中的格建立渐近上同调理论，证明第二上同调消失可推出关于有限维酉几乎表示的一致稳定性，从而推广了 Kazhdan 及 Burger–Ozawa–Thom 的刚性结果。
The purpose of this paper is to give a construction of representations of the group of currents for semisimple groups of rank greater than one. Such groups have no unitary representations in the Fock space, since the semisimple groups of this form have no nontrivial cohomology in faithful irreducible representations. Thus we first construct cohomology of the semisimple groups in nonunitary representations. The principal method is to reduce all constructions to Iwasawa subgroups (solvable subgroups of the semisimple groups), with subsequent extension to the original group. The resulting represe…
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.
Rigidity of non-negligible objects of moderate growth in braided categories
本文研究了辫状范畴中非可忽略对象的刚性性质，并探讨了其在半单范畴和对数共形场论中的应用。
Let $\Goo$ be a semisimple real Lie group with unitary dual $\Ghat$. The goal of this note is to produce new upper bounds for the multiplicities with which representations $π\in \Ghat$ of cohomological type appear in certain spaces of cusp forms on $\Goo$.
An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element $s$ in a dual group such that $s^2=1$. We prove that there is a bijection between, on the one hand the set of quadratic unipotent characters of $GL(n,q)$ or $U(n,q)$ for all $n \geq 0$ and on the other hand, the set of quadratic unipotent characters of $Sp(2n,q)$ for all $n \geq 0$. We then extend this correspondence to $\ell$-blocks for certain $\ell$ not dividing $q$.
Using a new definition of rank for representations of semisimple groups sharp results are proved for the decay of matrix coefficients of unitary representations of two types of non-split $p$-adic simple algebraic groups of exceptional type. These sharp bounds are achieved by minimal representations. It is also shown that in one of the cases considered, the minimal representation is isolated in the unitary dual.
Non-uniform higher-rank lattices are character rigid
该研究证明了特征不为2的半单群中非均匀高阶不可约格的特征刚性，由此推出保概率测度作用的稳定子刚性与不变随机子群刚性，全面证实了Stuck和Zimmer的相关猜想。
Hurwitz-Radon numbers and proper actions of semisimple Lie groups
研究半单李群在伪黎曼对称空间上的proper作用，证明此类作用必须同构于$Spin(n,1)$，并确定Hurwitz-Radon数为最大允许$n$值。
We construct, for any symplectic, unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic, a type for each Bernstein component of the category of smooth representations, using Bushnell--Kutzko's theory of covers. Moreover, for a component corresponding to a cuspidal representation of a maximal Levi subgroup, we prove that the Hecke algebra is either abelian, or a generic Hecke algebra on an infinite dihedral group, with parameters which are, at least in principle, computable via results of Lusztig.
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。
Strong $C^*$-rigidity of the Heisenberg groups
本文证明海森堡群具有强 $C^*$-刚性，可由酉对偶空间或群 $C^*$-代数的 Morita 等价类将其与其他单连通李群区分。
We provide a uniform construction of $L^2$-models for all small unitary representations in degenerate principal series of semisimple Lie groups which are induced from maximal parabolic subgroups with abelian nilradical. This generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite, and hence the previously used Jordan algebra methods have to be generalized to Jordan pairs. The crucial ingredients for the construction of the $L^2$-models are the Lie algebra action and the spherical vector. Working in the so-called Fourier …
In this paper, we consider representations of $p$-adic classical groups parabolically induced from the products of shifted Speh representations and unitary representations of Arthur type of good parity. We describe how to compute the socles (the maximal semisimple subrepresentations) of these representations. As a consequence, we can determine whether these representations are reducible or not. In particular, our results produce many unitary representations, which are called complementary series.
We study a contraction of the principal series representations of a noncompact semisimple Lie group to the unitary irreducible representations of its Cartan motion group by means of the Berezin-Weyl quantization on the coadjoint orbits associated with these representations.
We show that stationary characters on irreducible lattices $Γ< G$ of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $Γ< G$, the left regular representation $λ_Γ$ is weakly contained in any weakly mixing representation $π$. We prove that for any such irreducible lattice $Γ< G$, any uniformly recurrent subgroup (URS) of $Γ$ is finite, answering a questio…
In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that the decomposition is controlled by a reductive homogeneous space G'/H'. Our procedure establishes a correspondence between certain unitary representations of G and those of G'. This extends the usual theta--correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E_7 and F_4.
We consider compact locally symmetric spaces $Γ\backslash G/H$ where $G/H$ is a non-compact semisimple symmetric space and $Γ$ is a discrete subgroup of $G$. We discuss some features of the joint spectrum of the (commutative) algebra $D(G/H)$ of invariant differential operators acting, as unbounded operators, on the Hilbert space $L^2(Γ\backslash G/H)$ of square integrable complex functions on $Γ\backslash G/H$. In the case of the Lorentzian symmetric space $SO_0(2,2n)/SO_0(1,2n)$, the representation theoretic spectrum is described explicitly. The strategy is to consider connected reductive Li…
Quantized semisimple Lie groups
本文从酉表示理论角度介绍了半单李群的q-形变，通过聚焦$\mathrm{sl}(2,\mathbb{C})$及其相关李群，探讨了量子群、量子化代数及Plancherel公式量子类似物等核心主题。
Density and unitarity of the Burau representation from a non-semisimple TQFT
本文利用四次单位根处的非半单 TQFT 构造 Burau 表示上的 Squier Hermitian 形式，并证明辫群在可能不定的酉表示空间中像稠密，同时探讨其拓扑量子计算应用。
Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B…
On Kazhdan-Yom Din asymptotic Schur orthogonality for K-finite matrix coefficients
本文基于Kazhdan与Yom Din的前期工作，利用不可约酉表示的容许性，证明了局部域上半单群tempered不可约酉表示的K-有限矩阵系数在阿基米德情形下的渐近Schur正交性，补充了非阿基米德情形的已有结论。
Wehrl-type entropy problem for compact connected semisimple Lie groups
本文通过二阶变分法证明紧致连通半单李群中相干投影子唯一极小化Wehrl熵并最大化Husimi幂矩，将问题归结为最高权向量的极矩性质。
The $L_p$-dual space of a semisimple Lie group
该研究刻画了半单李群在$p\neq2$的$L_p$空间上的不可约表示，证明其等价于经抛物子群酉特征扭化的齐性空间$L_p$空间上的自然表示，并给出实秩一半单李群此类表示的完整分类。
Algebraic Phase Theory II: The Frobenius Heisenberg Phase and Boundary Rigidity
本文提出代数相理论中弗罗贝尼乌斯海森堡相的刚性理论，通过代数方法证明其在有限弗罗贝尼乌斯环上的表示结构。
We show that the universal unitary completion of certain locally algebraic representation of $G:=\GL_2(\Qp)$ with $p>2$ is non-zero, topologically irreducible, admissible and corresponds to a 2-dimensional crystalline representation with non-semisimple Frobenius via the $p$-adic Langlands correspondence for $G$.
Higher Kazhdan property and unitary cohomology of arithmetic groups
本文研究了算术群的更高Kazhdan性质，通过改进稳定性范围并允许任意单位元表示，推广了Borel和Borel-Yang的定理，并引入了几何群论和Clozel的工作方法。
Oscillators from non-semisimple walled Brauer algebras
研究非半单壁布拉uer代数在非稳定区域的表示理论，通过限制Bratteli图计算维度修正，并揭示其生成函数与无限谐振子塔的关联。
Zariski density of discrete subgroups via critical exponents and unitary representations
该研究利用酉表示理论，证明临界指数足够接近对称空间体积增长熵的离散子群具有Zariski稠密性，并确定了三类双曲空间等距群下的临界指数最优值，推广了Borel稠密性定理。
We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup $P_0$ and a one-parameter group ${\mathbb R {}}^*_+=\{r:r>0\}$ of automorphisms of $P_0$. This construction is determined by a a faithful unitary representation of $P_0$ (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as $r\to 0$. We apply this construction to the groups of currents of the maximal parabolic subgroups of the groups of motions of the $n$-dim…
This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori-Hecke algebras. This is commonly referred to as Kottwitz's Conjecture. The reductive groups associated to the PEL local models under consideration are unramified unitary similitude groups with even di…
A non-unitary approach to the $q$-deformation of $\mathrm{SL}(2,\mathbb{R})$
本文提出了一种非单位化方法，用于研究SL(2,R)的q-变形的表示论，通过类比经典理论，分类了非单位化不可约表示并展示了其与经典理论的趋同。
Prescribed Graphon Symmetries and Flavors of Rigidity
该研究将Frucht定理推广到图灵范畴，定义并证明了紧群图灵刚性与像刚性的等价性，明确了紧李群图灵刚性的判定条件。
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