The Cohomology of Small Irreducible Modules for Simple Algebraic Groups

LetG be a quasisimple, connected, and simply connected algebraic group defined and split over the field k of characteristic p > 0. In this paper, we are interested in small modules for G; for us, small modules are those with dimension ≤ p. By results of Jantzen [Jan96] one knows that anyGmodule V with dimV ≤ p is semisimple. (We always understand aG-module V to be given by a morphism of algebraic groupsG → GL(V ); thus V is a rational module). We are interested in the cohomology groups H(G, V ) for i ≥ 1, where V is a rational module for G and dimension ≤ p. It follows from the work of Jantzen just cited thatH(G, V ) = 0. Moreover, by semisimplicity, one may as well focus on the case where V is a simple module. We remark that this result of Jantzen’s is one of a number of recent results studying the semisimplicity of lowdimensional representations of groups in characteristic p. For example, the author has extended Jantzen’s result (for quasisimple G); he has shown that V is semisimple if dimV ≤ rp where r is the rank of G. Guralnick has proved a result like Jantzen’s for the p-modular representations of any finite group G with trivial Op(G). He gets that any module V with dimV ≤ p − 2 is semisimple, and H(G, V ) = 0 if dimV ≤ p − 3. See [Ser94], [McN98], [McN99], [Gur99], [McN00] for these and more results. We obtain in this paper results for the higher cohomology groups at small modules. We prove: