Modular representations of GL₂(\mathbb F_q) using calculus

2023/08/20 by Eknath Ghate, Ghate, Eknath, Arindam Jana +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2308.10246

Abstract

We show that certain modular induced representations of GL₂(\mathbb F_q) can be written as cokernels of operators acting on symmetric power representations of GL₂(\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.

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