First cohomology groups for finite groups of Lie type in defining characteristic

Abstract

Let G be a finite group of Lie type, defined over a field k of characteristic p > 0. We find explicit bounds for the dimension of the first cohomology group for G with coefficients in a simple kG-module. We proceed by bounding the number of composition factors of Weyl modules for simple algebraic groups independently of p and using this to deduce bounds for the 1-cohomology of simple algebraic groups. Finally, we use this to obtain estimations for the growth rate of the maximum dimension {\gamma_l} of these 1-cohomology groups over all groups of Lie type of rank l. We find that log \gamma_l is O(l^3 log l) (or if the Lusztig conjecture holds, O(l^2 log l)).