Symmetry breaking for representations of rank one orthogonal groups II

Abstract

For a pair $(G,G^{\prime })=(O(n+1,1),O(n,1))$ 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 $G^{\prime }$. The representations are parametrized by finite-dimensional representations $V,W$ of $O(n)$ respectively of $O(n-1)$, characters $\delta$, $\epsilon$ 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 $Ho{m}_{G^{\prime }}(I_{\delta }(V,\lambda ){|}_{G^{\prime }},J_{\epsilon }(W,\nu ))$ 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 $G^{\prime }$ 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.

366 pages

References in corpus (2)

• Multiplicity one theorem for (GL(n+1,R),GL(n,R))
• Symmetry breaking for orthogonal groups and a conjecture by B. Gross and D. Prasad

Cited by in corpus (18)

• Another Approach to Juhl's Conformally Covariant Differential Operators from $S^n$ to $S^{n-1}$
• Construction of Intertwining Operators between Holomorphic Discrete Series Representations
• Knapp-Stein Type Intertwining Operators for Symmetric Pairs II. -- The Translation Principle and Intertwining Operators for Spinors
• Symmetry breaking operators for real reductive groups of rank one
• Conformally covariant bi-differential operators for differential forms
• Residue families, singular Yamabe problems and extrinsic conformal Laplacians
• Shift operators, residue families and degenerate Laplacians
• Multiplicity in restricting minimal representations
• Branching problems in reproducing kernel spaces
• Symmetry breaking operators for strongly spherical reductive pairs
• Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups
• An extension problem related to the fractional Branson-Gover operators
• Conformally covariant differential symmetry breaking operators for a vector bundle of rank 3 over S^3
• Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups
• Rankin-Selberg periods for spherical principal series
• Computation of weighted Bergman inner products on bounded symmetric domains and restriction to subgroups II
• Topics on global analysis of manifolds and representation theory of reductive groups
• Recent advances in branching problems of representations