Classification of reductive real spherical pairs II. The semisimple case

Abstract

If $\mathfrak{g}$ is a real reductive Lie algebra and $\mathfrak{h}<\mathfrak{g}$ is a subalgebra, then $(\mathfrak{g},\mathfrak{h})$ is called real spherical provided that $\mathfrak{g}=\mathfrak{h}+\mathfrak{p}$ for some choice of a minimal parabolic subalgebra $\mathfrak{p}\subset \mathfrak{g}$. In this paper we classify all real spherical pairs $(\mathfrak{g},\mathfrak{h})$ where $\mathfrak{g}$ is semi-simple but not simple and $\mathfrak{h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where $\mathfrak{g}$ is simple (see arXiv:1609.00963) and generalizes the results of Brion and Mikityuk in the (complex) spherical case.

arXiv commentExtended revised version. Section 6 and Appendix B are new. To appear in Transformation Groups. 40p

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• Finite multiplicity theorems for induction and restriction
• Classification of smooth affine spherical varieties
• Classification of finite-multiplicity symmetric pairs
• Simple compactifications and polar decomposition of homogeneous real spherical spaces
• Volume growth, temperedness and integrability of matrix coefficients on a real spherical space
• The local structure theorem for real spherical varieties

Cited by in corpus (3)

• Symmetry breaking operators for real reductive groups of rank one
• Symmetry breaking operators for strongly spherical reductive pairs
• The tempered spectrum of a real spherical space