# Orbits of real semisimple Lie groups on real loci of complex symmetric spaces

## TL;DR

This paper classifies the orbits of real semisimple Lie groups acting on real loci of complex symmetric spaces using Galois cohomology, extending classical results with a combinatorial approach.

## Contribution

It provides a combinatorial description of real group orbits on symmetric spaces, building on and extending Borel and Ji's foundational work.

## Key findings

- Finite number of orbits of G(ℝ) on X(ℝ)
- Orbit classification via Galois cohomology
- Extension of classical orbit results

## Abstract

Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$-variety. Assume that both $G$, $X$ as well as the $G$-action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$ with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of A.Borel and L.Ji.