Journal of Lie Theory

no. 1
Restrictions of Certain Degenerate Principal Series of the Universal Covering of the Symplectic Group
Journal of Lie Theory, Volume 20 (2010) no. 1, pp. 31-48
\def\R{{\mathbb{R}}} Let $\widetilde{Sp}(n,\R)$ be the universal covering of the symplectic group. In this paper, we study the restrictions of the degenerate unitary principal series $I(\epsilon,t)$ of $\widetilde{Sp} (n,\R)$ onto $\widetilde{Sp}(p,\R) \widetilde{Sp}(n-p,\R)$. We prove that if $n \geq 2p$, $I(\epsilon, t)|_{\widetilde{Sp}(p,\R) \widetilde{Sp}(n-p,\R)}$ is unitarily equivalent to an $L^2$-space of sections of a homogeneous line bundle $L^2(\tilde{Sp}(n-p,\R) \times_{\widetilde{GL}(n-2p) N} \mathbb C_{\epsilon,t+\rho})$ (see Theorem 1.1). We further study the restriction of complementary series $C(\epsilon, t)$ onto $\tilde{U}(n-p) \widetilde{Sp}(p,\R)$. We prove that this restriction is unitarily equivalent to $I(\epsilon,t)|_{\tilde{U}(n-p)\widetilde{Sp}(p,\R)}$ for $t\in i\R$. Our results suggest that the direct integral decomposition of $C(\epsilon, t)|_{\widetilde{Sp}(p,\R) \widetilde{Sp}(n-p, \R)}$ will produce certain complementary series for $\widetilde{Sp}(n-p, \R)$ (H. He, Certain Induced Complementary Series of the Universal Covering of the Symplectic Group, submitted 2009).
DOI: 10.5802/jolt.585
Classification: 22E45, 43A85
Keywords: Complementary series, degenerate principal series, symplectic groups, universal covering, branching formula 
@article{JOLT_2010_20_1_a3,
     author = {H. He},
     title = {Restrictions of {Certain} {Degenerate} {Principal} {Series} of the {Universal} {Covering} of the {Symplectic} {Group}},
     journal = {Journal of Lie Theory},
     pages = {31--48},
     year = {2010},
     volume = {20},
     number = {1},
     doi = {10.5802/jolt.585},
     zbl = {1225.22010},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.585/}
}
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%J Journal of Lie Theory
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H. He. Restrictions of Certain Degenerate Principal Series of the Universal Covering of the Symplectic Group. Journal of Lie Theory, Volume 20 (2010) no. 1, pp. 31-48. doi: 10.5802/jolt.585

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