On the Vanishing Ranges for the Cohomology of Finite Groups of Lie Type

Abstract

Abstract. The computation of the cohomology for finite groups of Lie type in the describing characteristicis a challenging and difficult problem. In [BNP], the authors constructed an induction functor which takesmodules over the finite group of Lie type, G(F q ), to modules for the ambient algebraic group G. In particularthis functor when applied to the trivial module yields a natural G-filtration. This filtration was utilized in[BNP] to determine the first non-trivial cohomology class when the underlying root system is of type A n orC n . In this paper the authors extend these results toward locating the first non-trivial cohomology classesfor the remaining finite groups of Lie type (i.e., the underlying root system is of type B n , C n , D n , E 6 , E 7 ,E 8 , F 4 , and G 2 ) when the prime is larger than the Coxeter number. 1. Introduction1.1. Let Gbe a simple algebraic group scheme over a field kof prime characteristic pwhich is definedand split over the prime field F p , and F : G→ Gdenote the Frobenius map. The fixed points of therth iterate of the Frobenius map, denoted G(F