Abstract
Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for
G=
GL
n
, we consider polynomial representations of
G
r
T
for an arbitrary closed reductive subgroup scheme
G⊆
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=
GL
n
possible and apply this to recover the corresponding result for
GL
n
with a different proof, extending it to symplectic similitude groups, Levi subgroups of
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=
GL
n
.