Unitary Representations and Heisenberg Parabolic Subgroup

Details Unitary Representations and Heisenberg Parabolic Subgroup Unitary Representations and Heisenberg Parabolic Subgroup

2010-05-28

arxiv.org/abs/1005.5211v2

Mathematics

Representation Theory

11 pages

Scientific paper

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 $O^*(2n)$.

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