Symmetry Breaking for Representations of Rank One Orthogonal Groups II

Abstract

For a pair $(G,G')=(O(n+1,1), O(n,1))$ of reductive groups, we investigate intertwining operators (symmetry breaking operators) between principal series representations $I_\delta(V,\lambda)$ of $G$, and $J_\epsilon(W,\nu)$ of the subgroup $G'$. The representations are parametrized by finite-dimensional representations $V,W$ of $O(n)$ respectively of $O(n-1)$, characters $\delta$, $\varepsilon$ of O(1), and $\lambda, \nu \in C$. The multiplicty [V:W] of W occurring in the restriction $V|_{O(n-1)}$ is either 0 or 1. If $[V:W] \ne 0$ then we construct a holomorphic family of symmetry breaking operators and prove that dim $Hom_{G'}(I_{\delta}(V, \lambda)|_{G'}, J_{\epsilon}(W, \nu))$ is nonzero for all the parameters $\lambda$, $\nu$ and $\delta$, $\epsilon$, whereas if [V:W] = 0 there may exist sporadic differential symmetry breaking operators. We propose a "classification scheme" to find all matrix-valued symmetry breaking operators explicitly,and carry out this program completely when V and W are exterior tensor representations. In conformal geometry, our results yield the complete classification of conformal covariant operators from differential forms on a Riemannian manifold X to those on a submanifold Y in the model space $(X, Y) = (S^n, S^{n-1})$. We use these results to determine symmetry breaking operators for any pair of irreducible representations of G and the subgroup $G'$ with trivial infinitesimal character. Furthermore we prove the multiplicity conjecture by Gross and Prasad for tempered principal series representations of $(SO(n+1,1),SO(n,1))$ and also for 3 tempered representations $\Pi, \pi, \varpi$ of $SO(2m+2,1)$, $SO(2m+1,1)$ and $SO(2m,1)$ with trivial infinitesimal character. In connection to automorphic form theory, we apply our main results to find "periods" of irreducible representations of the Lorentz group having nonzero (g, K)-cohomologies.