Journal of Lie Theory

no. 1
Restrictions of Certain Degenerate Principal Series of the Universal Covering of the Symplectic Group
Hongyu He1
1Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.
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

Hongyu He  1

1 Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. 
Hongyu 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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     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},
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     doi = {10.5802/jolt.585},
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