On simple polynomial $G_r T$-modules
Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for $G = \mathrm{GL}_n$, we consider polynomial representations of $G_r T$ for an arbitrary closed reductive subgroup scheme $G \subseteq \mathrm{GL}_n$ and a maximal torus $T$ of $G$ in positive characteristic. We give sufficient conditions on $G$ making a classification of simple polynomial $G_r T$-modules similar to the case $G = \mathrm{GL}_n$ possible and apply this to recover the corresponding result for $\mathrm{GL}_n$ with a different proof, extending it to symplectic similitude groups, Levi subgroups of $\mathrm{GL}_n$ 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 $G_r T$ in the case $G = \mathrm{GL}_n$.