Knapp-Stein type intertwining operators for symmetric pairs

Abstract

For a symmetric pair $(G,H)$ of reductive groups we construct a family of intertwining operators between spherical principal series representations of $G$ and $H$ that are induced from parabolic subgroups satisfying certain compatibility conditions. The operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation. We further discuss uniqueness of intertwining operators, and for the rank one cases $$(G,H)=(SU(1,n;\mathbb{F}),S(U(1,m;\mathbb{F})\times U(n-m;\mathbb{F}))),\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O},$$ and for the pair $$(G,H)=(GL(4n,\mathbb{R}),GL(2n,\mathbb{C}))$$ we show that for a certain choice of maximal parabolic subgroups our operators generically span the space of intertwiners.

44 pages, added another detailed example in Section 5

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Cited by in corpus (3)

• Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups
• On boundary value problems for some conformally invariant differential operators
• Upper bounds for geodesic periods over hyperbolic manifolds