Invariant ideals of abelian group algebras and representations of groups of Lie type

This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group is the additive group of a finite-dimensional vector space over a locally finite field of prime characteristic , and the automorphism group is a simple infinite absolutely irreducible subgroup of . Thus is isomorphic to an infinite simple periodic group of Lie type, and is realized in via a twisted tensor product of infinitesimally irreducible representations. If is a Sylow -subgroup of and if is the unique line in stabilized by , then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus on . At present, we are able to handle the case where is a rational representation with character field equal to .