On simple polynomial GrT-modules

Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for G=GLn, we consider polynomial representations of GrT for an arbitrary closed reductive subgroup scheme G⊆GLn and a maximal torus T of G in positive characteristic. We give sufficient conditions on G making a classification of simple polynomial GrT-modules similar to the case G=GLn possible and apply this to recover the corresponding result for GLn with a different proof, extending it to symplectic similitude groups, Levi subgroups of GLn and, in a weaker form, to odd orthogonal similitude groups. We also consider orbits of the affine Weyl group and give a condition for equivalence of blocks of polynomial representations for GrT in the case G=GLn.