Cohomology of infinitesimal and discrete groups

Abstract

An infinitesimal algebraic group over a field k is a (connected) algebraic group scheme whose coordinate ring is a finite dimension~al k-algebra. If one begins with a reductive algebraic group G defined over Fp, then one has a close relationship between the representation theory of G, that of its infinitesimal subgroups G, (defined as the kernels of the Frobenius endomorphisms ~rr: G-+ G), and that of its finite (Chevalley) subgroups G(Fq) of Fq-rational points (q = pd). A basic theorem of Cline et al. I-9, Theorem 6.6] relates the cohomology of G in a rational representation to that of the finite groups G(Fq). Related work of Cline et al. [6] and of the authors 1-16] relates the cohomology of G to that of its infinitesimal subgroups. A determination of the cohomology of the first infinitesimal subgroup G 1 of G with trivial coefficients has enabled the authors in [14] to make certain explicit cohomology calculations for the finite groups G(Fq) with certain nontrivial coefficients. In this paper, we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge. In particular, we provide a computation which applies in a range of degrees to Chevalley groups over the prime field. We also investigate the cohomology of infinitesimal subgroups of nilpotent groups, enabling us for example to determine the cohomology of their Lie algebras in certain cases. We apply this Lie algebra cohomology computation to determine the integral cohomology of various torsion free, nilpotent groups modulo "small prime" torsion, thereby extending a theorem of Dwyer [12]. In more detail, the paper begins in Sect. 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup G~ which was previously studied in [14]. From this spectral sequence, it easily follows that H*(G1, M) is a Noetherian H*(G1, k)-module for each finite dimensional rational Gl-module M. We then apply the work of Andersen and Jantzen [1] on the Gl-cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup Gz. Our analysis also permits a determination of the "generic