The structure of Humphreys–Verma modules for projective spaces

Let E be a finite dimensional linear space over an algebraically closed field k of positive characteristic, G=GL(E), and P a parabolic subgroup of G such that G/P is a projective space over k. Let G1 be the Frobenius kernel of G and T a maximal torus of P. We will determine the G1T-structure on all G1P-modules induced from 1-dimensional P-modules. They are, in particular, all multiplicity-free and uniserial.