Representations of General Linear Groups in the Verlinde Category

Abstract

In this article, we construct affine group schemes GL(X) where X is any object in the Verlinde category in characteristic p and classify their irreducible representations. We begin by showing that for a simple object X of categorical dimension i, this representation category is semisimple and is equivalent to the connected component of the Verlinde category for SLi. Subsequently, we use this along with a Verma module construction to classify irreducible representations of GL(nL) for any simple object L and any natural number n. Finally, parabolic induction allows us to classify irreducible representations of GL(X) where X is any object in Verp.