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

TL;DR

This work extends mod $p$ Clebsch-Gordan theory for $\mathrm{GL}_2$ from $\mathbb{F}_{p}$ to general finite fields $\mathbb{F}_q$, providing an explicit decomposition for tensor products of symmetric powers with determinant twists. It then uses these results to analyze the theta filtration on symmetric power representations, proving that quotients $V_r/V_r^{(m+1)}$ decompose into principal series, with a precise count of constituents given by $(m+1)^f$ and $(m+1)^f-m^f$ for the successive layers. In the particularly explored case $f=2$, the paper demonstrates a diamond arrangement of four principal-series factors inside $V_r/V_r^{**}$, and it generalizes this pattern to arbitrary $f$ by exhibiting a directed hypercube structure for the extensions among principal-series constituents. The findings illuminate the modular representation theory of $\mathrm{GL}_2(\mathbb{F}_q)$ and have potential implications for reductions of Galois representations and related arithmetic geometry problems.

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.

Theorems & Definitions (42)

• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4
• Corollary 2.5
• proof
• Lemma 2.6
• proof