Unitary Representations and Heisenberg Parabolic Subgroup

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 $G{L}_n(\mathbb{R})$ 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 $O^{\ast }(2n)$.

arXiv comment11 pages