Projective modules for Frobenius kernels and finite Chevalley groups

Abstract

Let G be a connected reductive algebraic group with a Frobenius morphism F: G → G defined over a finite field F p r . The main result of the paper is to prove that any rational G-module M which is projective when restricted to the Frobenius kernel Gr = Ker(F) is also projective over the split and twisted finite Chevalley groups. In 1987, Parshall conjectured this statement for r = 1 in the split case. The authors verified this in 1999 with the possible exclusion of primes 2 and 3 in non-simply laced cases. The converse of the main result is also discussed for split groups in this paper.