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Generic weights for finite reductive groups

Zhicheng Feng, Gunter Malle, Jiping Zhang

TL;DR

The paper develops a generalized Harish-Chandra framework by introducing $(e,\ell)$-generalized-cuspidality and $(e,\ell)$-Jordan-generalized-cuspidality, and defines generic weights to connect Alperin’s weight conjecture with the representation theory of finite reductive groups in non-defining characteristic. It constructs an equivariant bridge between generic weights and Alperin weights, proves a type-$\mathsf{A}$ equivariant bijection for GL$_n$ and SL$_n$, and reduces the generic-weight problem to quasi-isolated blocks to enable inductive arguments. The work shows that, under mild hypotheses (abelian Sylow subgroups, abelian defect groups, or good primes with appropriate extendibility), the numbers of weights align with defect-zero/Weyl-group data, supporting the inductive AW framework. It also proposes and provides evidence for a local-global correspondence at the level of relative Weyl groups, with explicit bijections established for unipotent blocks and certain quasi-isolated blocks in exceptional groups. Overall, the paper offers a coherent strategy to advance towards a complete proof of Alperin’s weight conjecture using generic weights and generalized Harish-Chandra theory, particularly for groups of Lie type at good primes.

Abstract

This paper is motivated by the study of Alperin's weight conjecture in the representation theory of finite groups. We generalize the notion of $e$-cuspidality in the $e$-Harish-Chandra theory of finite reductive groups, and define generic weights in non-defining characteristic. We show that the generic weights play an analogous role as the weights defined by Alperin in the investigation of the inductive Alperin weight condition for simple groups of Lie type at most good primes. We hope that our approach will constitute a step towards an eventual proof of Alperin's weight conjecture.

Generic weights for finite reductive groups

TL;DR

The paper develops a generalized Harish-Chandra framework by introducing -generalized-cuspidality and -Jordan-generalized-cuspidality, and defines generic weights to connect Alperin’s weight conjecture with the representation theory of finite reductive groups in non-defining characteristic. It constructs an equivariant bridge between generic weights and Alperin weights, proves a type- equivariant bijection for GL and SL, and reduces the generic-weight problem to quasi-isolated blocks to enable inductive arguments. The work shows that, under mild hypotheses (abelian Sylow subgroups, abelian defect groups, or good primes with appropriate extendibility), the numbers of weights align with defect-zero/Weyl-group data, supporting the inductive AW framework. It also proposes and provides evidence for a local-global correspondence at the level of relative Weyl groups, with explicit bijections established for unipotent blocks and certain quasi-isolated blocks in exceptional groups. Overall, the paper offers a coherent strategy to advance towards a complete proof of Alperin’s weight conjecture using generic weights and generalized Harish-Chandra theory, particularly for groups of Lie type at good primes.

Abstract

This paper is motivated by the study of Alperin's weight conjecture in the representation theory of finite groups. We generalize the notion of -cuspidality in the -Harish-Chandra theory of finite reductive groups, and define generic weights in non-defining characteristic. We show that the generic weights play an analogous role as the weights defined by Alperin in the investigation of the inductive Alperin weight condition for simple groups of Lie type at most good primes. We hope that our approach will constitute a step towards an eventual proof of Alperin's weight conjecture.

Paper Structure

This paper contains 25 sections, 55 theorems, 66 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Radical subgroups and weights
  5. A generalization of cuspidality and generic weights
  6. Embeddings between reductive groups
  7. Blocks of finite reductive groups
  8. $e$-Jordan-generalized-cuspidal pairs
  9. Generic weights
  10. Reduction to quasi-isolated blocks
  11. Weights of finite reductive groups
  12. Groups of Lie type with abelian Sylow subgroups
  13. Blocks with abelian defect groups
  14. Centers of radical subgroups
  15. Groups of type $\mathsf{A}$
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $G$ be a finite group with $G=HZ$, where $H\le G$, $Z\le \operatorname{Z}(G)$. Then $S\mapsto S Z_\ell$ gives a bijection from $\Re^0(H)$ to $\Re^0(G)$ with inverse $R\mapsto R\cap H$. In addition, this induces bijections $\Re(H)\to\Re(G)$ and $\Re_w(H)\to\Re_w(G)$.

Theorems & Definitions (119)

  • Lemma 2.1
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Definition 3.5
  • ...and 109 more