Complete reducibility and separable field extensions

Let G be a connected reductive linear algebraic group. The aim of this note is to settle a question of J-P. Serre concerning the behaviour of his notion of G-complete reducibility under separable field extensions. Part of our proof relies on the recently established Tits Centre Conjecture for the spherical building of the reductive group G.Comment: 5 pages; to appear in Comptes rendus Mathematiqu

The strong Centre Conjecture: an invariant theory approach

The aim of this paper is to describe an approach to a a strengthened form of J. Tits' Centre Conjecture for spherical buildings. This is accomplished by generalizing a fundamental result of G. R. Kempf from Geometric Invariant Theory and interpreting this generalization in the context of spherical buildings. We are able to recapture the conjecture entirely in terms of our generalization of Kempf's notion of a state. We demonstrate the utility of this approach by proving the Centre Conjecture in some special cases.Comment: 30 pages, minor changes, new subsection on rationality; v3 updated bibliography and affiliation of second autho

G-complete reducibility in non-connected groups

In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G^0 is G^0 -cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.Comment: 14 page

Cocharacter-closure and spherical buildings

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.Comment: 16 pages; v 2 17 pages, exposition improved; to appear in the Robert Steinberg Memorial Issue of the Pacific Journal of Mathematic