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 $O^*(2n)$.
Categories: math.RT