Closed orbits of real reductive representations

TL;DR

This paper proves that in real reductive representations, the set of closed orbits is dense and contains an open subset in the real Zariski topology if it has non-empty interior, revealing topological properties of orbit closures.

Contribution

It establishes the density and Zariski openness of the set of closed orbits in real reductive representations, a new topological insight.

Findings

The set of closed orbits is dense in the representation space.

If the set of closed orbits has non-empty interior, it contains an open subset in the Zariski topology.

Closed orbits form a topologically significant subset in real reductive representations.

Abstract

We prove that the set of closed orbits in a real reductive representation contains a subset which is open with respect to the real Zariski topology if it has non-empty interior. In particular the set of closed orbits is dense.