Polar orthogonal representations of real reductive algebraic groups

TL;DR

This paper establishes a connection between polar orthogonal representations of real reductive algebraic groups and isotropy representations of pseudo-Riemannian symmetric spaces, extending structural theory.

Contribution

It proves the equivalence of closed orbits in these representations and develops a partial structural theory generalizing real reductive Lie algebra results.

Findings

Polar orthogonal representations share closed orbits with isotropy representations of pseudo-Riemannian symmetric spaces.

A partial structural theory for polar orthogonal representations is developed.

The work generalizes existing results in the structural theory of real reductive Lie algebras.

Abstract

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive Lie algebras.