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

Eknath Ghate; Arindam Jana

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arXiv · 2308.10246

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

Eknath Ghate·Arindam Jana

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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.

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BibTeX RIS

Eknath Ghate, Arindam Jana. 2023-08-20. Modular representations of $\mathrm{GL}_2({\mathbb F}_q)$ using calculus. https://arxiv.org/abs/2308.10246

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