Distinguished representations of SO(n+1,1) x SO(n,1), periods and branching laws

Abstract

Given irreducible representations $\Pi$ and $\pi$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G^{\prime }=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $\theta$-stable parameter necessary and sufficient conditions so that \[ \operatorname{Hom}_{G'}(Π|_{G'}, π)\not = \{0\}. \] In the special case that both $\Pi$ and $\pi$ are tempered, this implies the Gross--Prasad conjectures for tempered representations of $SO(n+1,1)\times SO(n,1)$ which are nontrivial on the center. We apply these results to construct nonzero periods and distinguished representations. If both $\Pi$ and $\pi$ have the trivial infinitesimal character $\rho$ then we use a theorem that the periods are nonzero on the minimal $K$-type to obtain a nontrivial bilinear form on the $(\mathfrak{g},K)$-cohomology of the representations.

33 pages, Minor changes. To appear in Simons Proceedings, Springer