Local integrability results in harmonic analysis on reductive groups in large positive characteristic

A criterion for integrability of matrix coefficients with respect to a symmetric space

Let $G$ be a reductive group and $θ$ an involution on $G$, both defined over a $p$-adic field. We provide a criterion for $G^θ$-integrability of matrix coefficients of representations of $G$ in terms of their exponents along $θ$-stable parabolic subgroups. The group case reduces to Casselman's square-integrability criterion. As a consequence we assert that certain families of symmetric spaces are strongly tempered in the sense of Sakellaridis and Venkatesh. For some other families our result implies that matrix coefficients of all irreducible, discrete series representations are $G^θ$-integrab…

Harmonic Analysis on Homogeneous Spaces

This article is an expository paper. We first survey developments over the past three decades in the theory of harmonic analysis on reductive symmetric spaces. Next we deal with the particular homogeneous space of non-reductive type, the so called Siegel-Jacobi space that is important arithmetically and geometrically. We present some new results on the Siegel-Jacobi space.

Wave Front Sets of Reductive Lie Group Representations III

Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of $L^2(X)$. Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on $X$. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.

Jordan Decompositions of cocenters of reductive $p$-adic groups

Cocenters of Hecke algebras $\mathcal H$ play an important role in studying mod $\ell$ or $\mathbb C$ harmonic analysis on connected $p$-adic reductive groups. On the other hand, the depth $r$ Hecke algebra $\mathcal H_{r^+}$ is well suited to study depth $r$ smooth representations. In this paper, we study depth $r$ rigid cocenters $\overline{\mathcal H}^{\mathrm{rig}}_{r^+}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\ell\neq p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an exp…

关于正特征中Harish-Chandra积分可积定理的探讨

On Harish-Chandra's integrability theorem in positive characteristic

本文探讨了正特征中Harish-Chandra积分可积定理的适用性，证明了在特定条件下cuspidal表示的可积性及正则性。

Wave Front Sets of Reductive Lie Group Representations

If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $τ$ is a unitary representation of $H$, then the authors give a sufficient condition on $ξ\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^Gτ$. In the special case where $τ$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $π$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $\operatorname{WF}(π)$ in terms of the direct integral decomposition of $π$ int…

约化齐性空间上的猜想

Conjectures on reductive homogeneous spaces

探讨约化齐性空间在经典框架外的对称性分析问题，包括非对称空间的离散级数、Riemannian设定外的非连续群及伪Riemannian局部齐性空间上的分析。

一个Harish-Chandra不等式

An inequality of Harish-Chandra

本文针对Harish-Chandra在《实约化群上的调和分析》中提出的、用于证明含参数关键积分大参数收敛性的不等式，给出了基于《实约化群I》思路拓展的初等证明，同时修正了原著多处关键印刷错误，可作为该书勘误。

Integrability and singularities of Harish-Chandra characters

Let $G$ be a reductive group over a local field $F$ of characteristic $0$. By Harish-Chandra's regularity theorem, the character $Θ_π$ of an irreducible, admissible representation $π$ of $G$ is given by a locally integrable function $θ_π$ on $G$. It is a natural question whether $θ_π$ has better integrability properties, namely, whether it is locally $L^{1+ε}$-integrable for some $ε>0$. It turns out that the answer is positive, and this gives rise to a new singularity invariant of representations $ε_{\star}(π):=\sup\left\{ ε:θ_π\in L_{Loc}^{1+ε}(G)\right\} $, which we explore in this paper. We…

The algebra of K-invariant vector fields on a symmetric space G/K

When $G$ is a complex reductive algebraic group and $G/K$ is a reductive symmetric space, the decomposition of $\C[G/K]$ as a $K$-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of Kostant-Rallis for the linearized problem (the harmonic decomposition of the isotropy representation). To obtain a constructive version of Richardson's results, this paper studies the infinite dimensional Lie algebra $\X(G/K)^K$ of $K$-invariant regular algebraic vector fields using the geometry of $G/K$ and the $K$-spherical representations of $G$. Assume $G$ is sem…

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。

Geometric Dynamics of a Harmonic Oscillator, Arbitrary Minimal Uncertainty States and the Smallest Step 3 Nilpotent Lie Group

The paper presents a new method of geometric solution of a Schrodinger equation by a construction of an equivalent first-order partial differential equation with a bigger number of variables. The equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform. The method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three Lie group---the shear group (also known as quartic group in literature). We obtain a geometric solution for an arbi…