Clebsch-Gordan and the theta filtration for modular representations of $\mathrm{GL}_2({\mathbb F}_q)$

Abstract

Let $p$ be a prime. We solve two problems in the mod $p$ representation theory of $\mathrm{GL}_2(\mathbb{F}_{q})$ where $q=p^f$. We first prove a Clebsch-Gordan decomposition theorem for the tensor product of two mod $p$ representations of $\mathrm{GL}_2(\mathbb{F}_{q})$. As an application, we use this to guess the structure of quotients of symmetric power representations of $\mathrm{GL}_2(\mathbb{F}_{q})$ by submodules in the theta filtration. We then give a direct proof of this structure showing that such quotients are built out of principal series representations.