Generic cohomology for twisted groups

Abstract

Let G be a simple algebraic group defined and split over ko = Fp, and let a be a surjective endomorphism of G with finite fixed-point set Go. We give conditions under which cohomology groups of G are isomorphic to cohomology groups of Go. Let G be a simple algebraic group defined and split over k = Fp, and, for q = pm, let G(q) be the subgroup of Fq-rational points. For a finite dimensional rational G-module V and a nonnegative integer e, let V(e) be the G-module obtained by twisting the original G-action on V by the Frobenius endomorphism x > X1P of G. Cline, Parshall, Scott, and van der Kallen proved in [2] that, for sufficiently large q and e (depending on V and n), the restriction map induces an isomorphism from the rational cohomology group H'(G, V(e)) to H'(G(q), V(e)). This implies that, as q increases, the groups H'(G(q), V(e)) have a stable or generic value H en(G, V). In this paper, we prove an analogous theorem with G(q) replaced by Go for a surjective endomorphism a of G having finite fixed point set. The first section of the paper summarizes the basic results on endomorphisms of algebraic groups required for the proof. Some arithmetic facts are established in the second section, and the main theorem is proved in the third section. The author is grateful to Leonard L. Scott for several helpful conversations, and to the referee for comments on an earlier version of this paper. 1. Endomorphisms of algebraic groups. We briefly review here some results on endomorphisms of algebraic groups which will be needed later. We refer the reader to [4] and [5] for more details. Let k be the algebraic closure of the prime field ko = Fp and let G be a simple algebraic group defined and split over ko. If a is a surjective rational endomorphism of G having finite fixed-point set G0, a stabilizes a Borel subgroup B and a maximal torus T Ua is a T-equivariant isomorphism, then axa(u) = x (cauq(a)) for some Received by the editors May 5, 1980 and, in revised form, November 12, 1980. 1980 Mathematics Subject Classification Primary 20G10.