Abstract induced modules for reductive algebraic groups with Frobenius maps

2019/07/01 by Xiaoyu Chen, Chen, Xiaoyu, Junbin Dong +1 · 2 citations
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.1907.00741

openalex publication_date 2019/07/01 · openalex created_date 2021/09/13 · openalex updated_date 2026/07/28

Abstract

Let \bf G be a connected reductive algebraic group defined over a finite field \mathbbF_q of q elements, and \bf B be a Borel subgroup of \bf G defined over \mathbbF_q. Let \Bbbk be a field and we assume that \Bbbk=\mathbbF_q when char \Bbbk=char \mathbbF_q. We show that the abstract induced module \mathbbM(θ)=\Bbbk\bf G⊗_\Bbbk\bf Bθ (here \Bbbk\bf H is the group algebra of \bf H over the field \Bbbk and θ is a character of \bf B over \Bbbk) has a composition series (of finite length) if char \Bbbk≠ char \mathbbF_q. In the case \Bbbk=\mathbbF_q and θ is a rational character, we give a necessary and sufficient condition for the existence of a composition series (of finite length) of \mathbbM(θ). We determine all the composition factors whenever a composition series exists. Thus we obtain a large class of abstract infinite-dimensional irreducible \Bbbk\bf G-modules.

Cited by

• Some conjectures on the quotients of the tensor products in the category \mathscrX
• The block decomposition of the principal representation category of reductive algebraic groups with Frobenius maps

Related

• The Permutation Module on Flag Varieties in Cross Characteristic
• The Decomposition of Permutation Module for Infinite Chevalley Groups
• The block decomposition of the principal representation category of reductive algebraic groups with Frobenius maps
• On the extensions of certain representations of reductive algebraic groups with Frobenius maps
• On the principal series representations of semisimple groups with Frobenius maps