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, Tome 20 (2010) no. 1, pp. 31-48

Résumé

\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).

Détail

Classification : 22E45, 43A85
Mots-clés : Complementary series, degenerate principal series, symplectic groups, universal covering, branching formula