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Journal of Lie Theory 20 (2010), No. 1, 031--048
Copyright Heldermann Verlag 2010

Restrictions of Certain Degenerate Principal Series of the Universal Covering of the Symplectic Group

Hongyu He
Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.


\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).

Keywords: Complementary series, degenerate principal series, symplectic groups, universal covering, branching formula.

MSC: 22E45, 43A85

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