Modular representations of $\mathrm{GL}_2({\mathbb F}_q)$ using calculus

TL;DR

This paper introduces a novel approach to modular representations of igg2(\u211d_q) using calculus, expressing certain induced representations as cokernels of differential operators involving twisted Dickson and Serre polynomials.

Contribution

It provides explicit isomorphisms for modular induced representations of igg2(_q) via differential operators and introduces new twisted Dickson and Serre operators.

Findings

Representation cokernels are expressed using differential operators.

Explicit isomorphisms involve twisted Dickson and Serre polynomials.

Improves periodicity results in theta filtration.

Abstract

We show that certain modular induced representations of $\mathrm{GL}_2({\mathbb F}_q)$ can be written as cokernels of operators acting on symmetric power representations of $\mathrm{GL}_2({\mathbb F}_q)$. When the induction is from the Borel subgroup, respectively the anisotropic torus, the operators involve multiplication by newly defined twisted Dickson polynomials, respectively, twisted Serre operators. Our isomorphisms are explicitly defined using differential operators. As a corollary, we improve some periodicity results for quotients in the theta filtration.