Journal of Lie Theory

no. 4
Unitary Representations and the Heisenberg Parabolic Subgroup
Journal of Lie Theory, Volume 21 (2011) no. 4, pp. 847-860
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 $O^*(2n)$.
DOI: 10.5802/jolt.652
Classification: 22E45, 43A80
Keywords: Parabolic subgroups, Heisenberg group, Mackey analysis, branching formula, unitary representations, Kirillov Conjecture, symplectic group, highest weight module 
@article{JOLT_2011_21_4_a5,
     author = {H. He},
     title = {Unitary {Representations} and the {Heisenberg} {Parabolic} {Subgroup}},
     journal = {Journal of Lie Theory},
     pages = {847--860},
     year = {2011},
     volume = {21},
     number = {4},
     doi = {10.5802/jolt.652},
     zbl = {1236.22005},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.652/}
}
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H. He. Unitary Representations and the Heisenberg Parabolic Subgroup. Journal of Lie Theory, Volume 21 (2011) no. 4, pp. 847-860. doi: 10.5802/jolt.652

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