The Steinberg tensor product theorem for general linear group schemes in the Verlinde category

Abstract

The Steinberg tensor product theorem is a fundamental result in the modular representation theory of reductive algebraic groups. It describes any finite-dimensional simple module of highest weight λ over such a group as the tensor product of Frobenius twists of simple modules with highest weights the weights appearing in a p-adic decomposition of λ, thereby reducing the character problem to a finite collection of weights. In recent years this theorem has been extended to various quasi-reductive supergroup schemes. In this paper, we prove the analogous result for the general linear group scheme $GL\left(X\right)$ for any object X in the Verlinde category ${\mathrm{Ver}}_p$.