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 $G$ 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 $A$ is the additive group of a finite-dimensional vector space $V$ over a locally finite field $F$ of prime characteristic $p$, and the automorphism group $G$ is a simple infinite absolutely irreducible subgroup of $GL(V)$. Thus $G$ is isomorphic to an infinite simple periodic group of Lie type, and $G$ is realized in $GL(V)$ via a twisted tensor product $\phi$ of infinitesimally irreducible representations. If $S$ is a Sylow $p$-subgroup of $G$ and if $\langle v\rangle$ is the unique line in $V$ stabilized by $S$, then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus $T_G$ on $\langle v\rangle$. At present, we are able to handle the case where $\phi$ is a rational representation with character field equal to $F$.