On simple polynomial $G_{r}T$-modules

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 $G={\mathrm{GL}}_n$.

13 pages; revised version

References in corpus (1)

• Schur-Weyl duality for orthogonal groups

Cited by in corpus (1)

• Almost split sequences for polynomial $G_{r}T$-modules and polynomial parts of Auslander-Reiten components