Cocharacter-closure and spherical buildings

TL;DR

This paper extends the study of cocharacter-closed orbits in algebraic group actions by employing building-theoretic methods to derive rationality and descent results, complementing previous Galois-based approaches.

Contribution

It introduces building-theoretic techniques to analyze cocharacter-closure and descent properties of orbits, providing new proofs and insights.

Findings

Derived Galois ascent/descent results using building theory

Established Levi ascent/descent results for orbits

Connected cocharacter-closure with spherical building structures

Abstract

Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a rationality condition on the point stabilizer of a $G$-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding $G(k)$-orbit in $V$. In the present paper we employ building-theoretic techniques to derive analogous results.