A generalization of Pillen's theorem for principal series modules II

Let G   be a connected, semisimple and simply connected algebraic group and G(q)G(q) the corresponding finite Chevalley group over the finite field of order q=prq=pr. In a recent paper the author determined a direct sum decomposition of the kG(q)kG(q)-submodule generated by a highest weight vector of a certain Weyl module when q is not too small, which is a generalization of Pillen's result in 1997. In this article, we claim that the result does not need the assumption on q.