On the extensions of certain representations of reductive algebraic groups with Frobenius maps

2024/04/15 by Xiaoyu Chen, Chen, Xiaoyu, Junbin Dong +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2404.09495

openalex publication_date 2024/04/15 · openalex created_date 2024/04/17 · openalex updated_date 2026/07/28

Abstract

Let \bf G be a connected reductive algebraic group defined over the finite field \mathbbF_q with q elements,where q is a power of a prime number p. Let \Bbbk be a field and we study the extensions of certain \bk\bg-modules in this paper. We show that the extensions of any modules in \mathscrO(\bg) by a finite-dimensional \bk\bg-module is zero if p≠ \opchar\bk≥5 or \opchar\bk=0, where \mathscrO(\bg) is the principal representation category defined in \citeD1. We determine the necessary and sufficient condition for the vanishing of extensions between naive induced modules. As an application, we give the condition of the vanishing of extensions between simple modules in \mathscrO(\bf G) for \bg=SL₂(\mathbbF_q).

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• The block decomposition of the principal representation category of reductive algebraic groups with Frobenius maps

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