CLASSIFICATION OF REDUCTIVE REAL SPHERICAL PAIRS II. THE SEMISIMPLE CASE

Abstract

If g$$ \\mathfrak{g} $$ is a real reductive Lie algebra and h⊂g$$ \\mathfrak{h}\\subset \\mathfrak{g} $$ is a subalgebra, then the pair (h,g$$ \\mathfrak{h},\\mathfrak{g} $$) is called real spherical provided that g=h+p$$ \\mathfrak{g}=\\mathfrak{h}+\\mathfrak{p} $$ for some choice of a minimal parabolic subalgebra p⊂g$$ \\mathfrak{p}\\subset \\mathfrak{g} $$. This paper concludes the classification of real spherical pairs (h,g$$ \\mathfrak{h},\\mathfrak{g} $$), where h$$ \\mathfrak{h} $$ is a reductive real algebraic subalgebra. More precisely, we classify all such pairs which are strictly indecomposable, and we discuss (in Section 6) how to construct from these all real spherical pairs. A preceding paper treated the case where g$$ \\mathfrak{g} $$ is simple. The present work builds on that case and on the classification by Brion and Mikityuk for the complex spherical case.