  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 $G = \mathrm{GL}_n$.
