Notes on Stein-Sahi representations and some problems of non $L^2$ harmonic analysis

Notes on Sobolev spaces on compact classical groups and Stein--Sahi representations

We discuss kernels on complact classical groups $G=\U(n)$, $\OO(2n)$, $\Sp(n)$ defined by the formula $K(z,u)=|\det(1-zu^*)|^s$. We obtain the explicit Plancherel formula for these kernels and the interval of positive-definiteness. We also obtain explicit models for Sahi's 'unipotent' representations.

Stein--Sahi complementary series and their degenerations

The aim of the paper is an introduction to Stein--Sahi complementary series, holomorphic series, and 'unipotent representations'. We also discuss some open problems related to these objects. For the sake of simplicity, we consider only the groups U(n,n).

Steinberg representations and harmonic cochains for split adjoint quasi-simple groups

Let $G$ be an adjoint quasi-simple group defined and split over a non-archimedean local field $K$. We prove that the dual of the Steinberg representation of $G$ is isomorphic to a certain space of harmonic cochains on the Bruhat-Tits building of $G$. The Steinberg representation is considered with coefficients in any commutative ring.

Steinberg表示与对称空间的对偶群的区别

Distinction of the Steinberg representation and the dual group of a symmetric space

研究Steinberg表示相对于对称子群的区分问题，通过超图上的调和函数存在性联系，并验证Steinberg表示在p-adic域上的区分性与Langlands参数与对偶群的关系。

Stein's Method and Plancherel Measure of the Symmetric Group

We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic deco…

关于对称对的 Steinberg 表示的区分性

Distinction of the Steinberg representation with respect to a symmetric pair

本文将 Steinberg 表示实现为调和上链空间，研究其关于对称对固定子群的区分空间维数，给出可达上界并应用于一般线性群与特殊正交子群完成分类。