# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1703.08048