Unitary Representations and Heisenberg Parabolic Subgroup

TL;DR

This paper investigates how irreducible unitary representations of the universal covering of Sp(2n,R) behave when restricted to a Heisenberg maximal parabolic subgroup, revealing conditions for irreducibility and discrete decompositions.

Contribution

It characterizes the restrictions of such representations, showing they are either highest or lowest weight modules, and details their discrete decompositions, contrasting with the GL_n(R) case.

Findings

Irreducible restriction implies highest or lowest weight modules.

Highest or lowest weight modules decompose discretely upon restriction.

Results extend to groups U(p,q) and O^*(2n).

Abstract

In this paper, we study the restriction of an irreducible unitary representation $\pi$ of the universal covering $\widetilde{Sp}_{2n}(\mb R)$ to a Heisenberg maximal parabolic group $\tilde P$. We prove that if $\pi|_{\tilde P}$ is irreducible, then $\pi$ must be a highest weight module or a lowest weight module. This is in sharp constrast with the $GL_n(\mathbb R)$ case. In addition, we show that for a unitary highest or lowest weight module, $\pi|_{\tilde P}$ decomposes discretely. We also treat the groups $U(p,q)$ and $O^*(2n)$.