Invariant ideals of Abelian group algebras and representations of groups of Lie type (Q2719056)

Let \(H\) be a group and let \(K\) be a field. The group algebra \(KH\) has three obvious ideals: the zero ideal, \(KH\) itself, and the augmentation ideal, which has codimension 1 in \(KH\). We refer to these as trivial ideals. A natural question to ask is: when does \(KH\) have only trivial ideals? The question is of interest only when \(H\) is infinite. Furthermore, if \(H\) contains a proper normal subgroup, then \(KH\) has a non-trivial ideal. Thus attention is focused on infinite simple groups. The second author has investigated the question when \(H\) is an infinite locally finite simple group and \(K\) is the field of complex numbers, and good progress has been made in this case.NEWLINENEWLINENEWLINEIn the present paper, the authors extend this investigation by studying the non-trivial ideals of \(KH\) when \(H\) is an Abelian-by-simple group. They reduce their problem to studying ideals of Abelian group algebras \(KA\) invariant under certain simple automorphism groups. The particular \(A\) chosen here is the additive group of a finite dimensional vector space over an infinite field of prime characteristic and the automorphism group \(G\) is a simple infinite absolutely irreducible subgroup of automorphisms of \(V\).NEWLINENEWLINENEWLINEThe main theorem of the paper is as follows. Let \(G\) be an infinite locally finite quasi-simple group of Lie type and let \(\phi\colon G\to\text{GL}(n,F)\) be a rational irreducible representation, where \(F\) is a field of prime characteristic. Suppose that \(F\) is generated by the values \(\chi(g)\) for \(g\in G\), where \(\chi\) is the character of \(\phi\). Let \(V\) be the \(FG\)-module associated with \(\phi\) and let \(A=V^+\) be the additive group of \(V\). If \(K\) is a field of characteristic different from that of \(F\), then \(KA\) and the augmentation ideal are the only non-zero \(G\)-invariant ideals in \(KA\).NEWLINENEWLINENEWLINEThe proof depends on a careful analysis of the rational representations of groups of Lie type. Properties of the weights must be examined in detail, and this part of the paper, independently of its application to group algebras, makes interesting reading.