Mumford's conjecture for the general linear group

Abstract

In this paper, we show that GL(n, K) is geometrically reductive over any field k. We begin with a short discussion of the history and significance of the problem. There are two central problems in the qualitative theory of invariants: Hilbert’s 14th problem and the qualitative study of quotient varieties under the action of an algebraic group. Hilbert’s 14th problem asks: If G is a subgroup of GL(n, k) and G acts on R = k[x, ,..., XJ by substitution of variables, is the subring of R left fixed by G finitely generated over k ? Nagata has shown that the answer is negative, even in characteristic zero [5], therefore, the determination of the groups for which the answer is positive becomes a substitute for the original problem. In characteristic zero, the theory is quite well developed due to the efforts of Hilbert and many others, among them H. Weyl and D. Mumford. Let us say that an algebraic group G is linearly reductive if every rational G-module is semisimple. For such groups, Hilbert’s 14th problem has a positive solution and the theory of quotient varieties is particularly well developed. In characteristic zero, it is a classical theorem of Weyl [12, Chap. 71 that a semisimple group G is linearly reductive. More generally, if the radical of G is a torus, then G is linearly reductive. Groups whose radical is a torus are called reductive, but in characteristic p > 0, they are not linearly reductive in general. In fact, Nagata [6] has shown that in characteristic p > 0 the only linearly reductive groups are extensions of a torus by a finite group of order prime to p. Therefore, a weaker notion than linearly reductive has been introduced.