A Remark on Pillen's Theorem for Projective Indecomposable kG(n)-Modules

Let g be a connected, semisimple and simply connected algebraic group defined and split over the finite field of order p, and let g(n) be the corresponding finite Chevalley group and g(n) the n-th Frobenius kernel. Pillen has proved that for a 3(h - 1)-deep and p(n)-restricted weight lambda, the G-module Q(n)(lambda) which is extended from the G(n)-PIM for lambda has the same socle series as the corresponding kG(n)-PIM U-n(lambda). Here we remark that this fact already holds for lambda being 2(h - 1)-deep.