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
$O^*(2n)$.