Classification of reductive real spherical pairs II. The semisimple case

TL;DR

This paper classifies all reductive real spherical pairs where the ambient Lie algebra is semisimple but not simple, extending previous classifications from the simple case and complex spherical scenarios.

Contribution

It provides a comprehensive classification of semisimple non-simple real spherical pairs with reductive subalgebras, generalizing earlier results.

Findings

Complete classification of semisimple non-simple real spherical pairs.

Extension of previous simple and complex spherical pair classifications.

Generalization of Brion and Mikityuk's results to the real 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.