Invariant Ideals and Matsushima’s Criterion

Let G be a reductive algebraic group and H a closed subgroup of G. Explicit constructions of G-invariant ideals in the algebra K[G/H] are given. This allows to obtain an elementary proof of Matsushima’s criterion: a homogeneous space G/H is an affine variety if and only if H is reductive. 1. Algebraic homogeneous spaces Let G be an affine algebraic group over an algebraically closed field K. A Gmodule V is said to be rational if any vector in V is contained in a finite-dimensional rational G-submodule. Below all modules are supposed to be rational. By V G denote the subspace of G-fixed vectors in V . The group G × G acts on G by translations, (g1, g2)g := g1gg −1 2 . This action induces the action on the algebra of regular functions on G: (G×G) : K[G], ((g1, g2)f)(g) := f(g −1 1 gg2). For any closed subgroup H of G, Hl, Hr denote the groups of all left and right translations of K[G] by elements of H . Under these actions, the algebra K[G] becomes a rational Hl(and Hr-) module. By Chevalley’s Theorem, the set G/H of left H-cosets in G admits a structure of a quasi-projective algebraic variety such that the projection p : G → G/H is a surjective G-equivariant morphism. Moreover, a structure of an algebraic variety on G/H satisfying these conditions is unique. It is easy to check that the morphism p is open and the algebra of regular functions on G/H may be identified with the subalgebra K[G]r in K[G]. We refer to [6, Ch. IV] for details. 2. Matsushima’s criterion Let G be a reductive algebraic group and H a closed subgroup of G. It is known that the homogeneous space G/H is affine if and only if H if reductive. The first proof was given over the field of complex numbers and used some results from algebraic topology, see [8] and [9, Th. 4]. An algebraic proof in characteristic zero was obtained in [2]. A characteristic-free proof that uses the Mumford conjecture proved by W.J. Haboush is given in [11]. Another proof based on the MorozovJacobson Theorem may be found in [7]. Below we give an elementary proof of Matsushima’s criterion in terms of representation theory. The ground field K is assumed to be algebraically closed and of characteristic zero. 1991 Mathematics Subject Classification. 14M17, 14L30; 13A50, 14R20. Supported by DFG Schwerpunkt 1094, CRDF grant RM1-2543-MO-03, RF President grant MK-1279.2004.1, and RFBR 05-01-00988. 1