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Journal of Lie Theory 21 (2011), No. 4, 847--860
Copyright 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.


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

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