426 research outputs found
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 be a field, let be a reductive -group and an affine
-variety on which acts. In this note we continue our study of the notion
of cocharacter-closed -orbits in . In earlier work we used a
rationality condition on the point stabilizer of a -orbit to prove Galois
ascent/descent and Levi ascent/descent results concerning cocharacter-closure
for the corresponding -orbit in . 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
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