On simple polynomial $G_r T$-modules

TL;DR

This paper extends the classification of simple polynomial modules from general linear groups to a broader class of reductive groups, providing new conditions and applications in positive characteristic.

Contribution

It generalizes the polynomial representation framework for reductive groups beyond GL_n, including symplectic and orthogonal similitude groups, with new classification conditions and block equivalence results.

Findings

Classification of simple polynomial G_r T-modules under new conditions

Extension of results to symplectic and orthogonal similitude groups

Conditions for block equivalence in polynomial representations

Abstract

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 $G = \mathrm{GL}_n$.