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.
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
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