Cocharacter-closure and spherical buildings

Michael Bate, Sebastian Herpel, Benjamin Martin and Gerhard Röhrle

Vol. 279 (2015), No. 1-2, 65–85

DOI: 10.2140/pjm.2015.279.65

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 $G\left(k\right)$-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 $G\left(k\right)$-orbit in $V$. In the present paper we employ building-theoretic techniques to derive analogous results.

In memory of Robert Steinberg

Keywords

Affine $G$-variety, cocharacter-closed orbit, rationality, spherical building, the centre conjecture

Mathematical Subject Classification 2010

Primary: 20G15

Secondary: 14L24

Milestones

Received: 1 June 2015

Revised: 27 September 2015

Accepted: 27 September 2015

Published: 21 December 2015