Cohomology of infinitesimal algebraic groups

Abstract

In this paper we study the cohomology H'(Br, K) of a higher Frobenius kernel Br, r > 1, of a Borel subgroup B of a reductive algebraic group G over an algebraically closed field K of a positive characteristic p. For the first Frobenius kernel B 1 the cohomology has been completely determined by H.H. Andersen and J.C. Jantzen [3] (in case p > h the Coxeter number of G). In low degrees the cohomology is independent of r > 1, and their results allow us to compute the first different case H zp-I(Br, K) for S Lt+I (3.6). In higher degrees, however, we have to resort to a different method. Although H ' ( B 2, K) has already been computed for SL z also in [3], it is a considerably harder problem for SL 3. We apply algebraic Steenrod operations and the transgression theorem to the Lyndon-HochschildSerre or Cartan-Eilenberg spectral sequence arising from a central series of the unipotent radical U of B to find H~ K) for SL 3 (5.1). The method we employ here was originally developed by M. Tezuka and N. Yagita [21] in order to investigate H ' (U(Fq) , K), q a power ofp. In fact, the results of Sections 4 and 5 were first conceived from the corresponding results of H'(U(Fq), K). Aside from the interests in relation to the cohomology of finite groups the work was also motivated by a close connection of the Bl-cohomology to G. Lusztig's conjectural character formula for the irreducible modules of the reductive group G, which is described in Section 2. Hoping that the representation theory may shed more light on some topological problems, we intend to take up the Steenrod algebra and the A-algebras in a subsequent paper.