The Bessel-Plancherel theorem and applications

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 $s$.