Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 21 (2011), No. 4, 847--860Copyright Heldermann Verlag 2011 Unitary Representations and the Heisenberg Parabolic Subgroup Hongyu He Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. hongyu@math.lsu.edu [Abstract-pdf] We study the restriction of an irreducible unitary representation $\pi$ of the universal covering $\widetilde{Sp}_{2n}(\R)$ to a Heisenberg maximal parabolic subgroup $\tilde P$. 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 contrast with the GL$_n(\R)$ 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 $O^*(2n)$. Keywords: Parabolic subgroups, Heisenberg group, Mackey analysis, branching formula, unitary representations, Kirillov Conjecture, symplectic group, highest weight module. MSC: 22E45, 43A80 [ Fulltext-pdf  (340  KB)] for subscribers only.