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 2 n Π p p ( n 2 ) , where the product is over all primes less than or equal to n +1, such that V is free as a ℚ H -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 p n 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.