ON FREE MODULES FOR FINITE SUBGROUPS OF ALGEBRAIC GROUPS

Abstract

We show that given an affine algebraic group G over a field K and a finite subgroup scheme H of G there exists a finite dimensional G-module V such that V[mid ]_H is free. The problem is raised in the recent paper by Kuzucuo˘glu and Zalesskiiˇ [15] which contains a treatment of the special case in which K is the algebraic closure of a finite field and H is reduced. Our treatment is divided into two parts, according to whether K has zero or positive characteristic. The essence of the characteristic 0 case is a proof that, for given n, there exists a polynomial GL_n(ℚ)-module V of dimension 2ⁿΠ_pp^(n₂), where the product is over all primes less than or equal to n+1, such that V is free as a ℚH-module for every finite subgroup H of GL_n(ℚ). The module V is the tensor product of the exterior algebra Λ*(E), on the natural GL_n(ℚ)-module E, and Steinberg modules St_p, one for each prime not exceeding n+1. The Steinberg modules also play the major role in the case in which K has characteristic p>0 and the key point in our treatment is to show that for a finite subgroup scheme H of a general linear group scheme (or universal Chevalley group scheme) G over K, the Steinberg module St_pⁿ for G is injective (and projective) on restriction to H for n[Gt ]0. A curious consequence of this is that, despite the wild behaviour of the modular representation theory of finite groups, one has the following. Let H be a finite group and V a finite dimensional vector space. Then there exists a (well-understood) faithful rational representation π: GL(V)→GL(W) such that, for each faithful representation ρ: H→GL(V), the composite πορ: H→GL(W) is free, in particular all representations πορ are equivalent.