The Bessel-Plancherel theorem and applications

TL;DR

This paper explores the Bessel-Plancherel theorem for Lie groups of tube type, characterizes Bessel models of induced representations, and applies these results to compute measures and relate them via Howe's dual pairs.

Contribution

It provides a classification of Bessel models, establishes a local multiplicity one theorem, and connects Bessel-Plancherel measures across different groups using Howe's duality.

Findings

Classification of simple Lie groups of tube type.

Characterization of Bessel models for induced representations.

Calculation of Bessel-Plancherel measure for groups of tube type.

Abstract

Let $G$ be a simple Lie Group with finite center, and let $K\subset G$ be a maximal compact subgroup. We say that $G$ is a Lie group of tube type if $G/K$ is a hermitian symmetric space of tube type. For such a Lie group $G$, we can find a parabolic subgroup $P=MAN$, with given Langlands decomposition, such that $N$ is abelian, and $N$ admits a generic character with compact stabilizer. We will call any parabolic subgroup $P$ satisfying this properties a Siegel parabolic. Let $(\pi,V)$ be an admissible, smooth, Fr\'echet representation of a Lie group of tube type $G$, and let $P \subset G$ be a Siegel parabolic subgroup. If $\chi$ is a generic character of $N$, let $Wh_{\chi}(V)={\lambda:V \longrightarrow \mathbb{C} | \lambda(\pi(n)v)=\chi(n)v}$ be the space of Bessel models of $V$. After describing the classification of all the simple Lie groups of tube type, we will give a characterization of the space of Bessel models of an induced representation. As a corollary of this characterization we obtain a local multiplicity one theorem for the space of Bessel models of an irreducible representation of $G$. As an application of this results we calculate the Bessel-Plancherel measure of a Lie group of tube type, $L^2(N\backslash G;\chi)$, where $\chi$ is a generic character of $N$. Then we use Howe's theory of dual pairs to show that the Plancherel measure of the space $L^2(O(p-r,q-s)\backslash O(p,q))$ is the pullback, under the $\Theta$ lift, of the Bessel-Plancherel measure $L^2(N\backslash Sp(m,\mathbb{R});\chi)$, where $m=r+s$ and $\chi$ is a generic character that depends on $r$ and $s$.