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Jordan correspondence and block distribution of characters

Radha Kessar, Gunter Malle

TL;DR

The paper resolves the $\,\ell$-block distribution problem for quasi-simple exceptional groups, including bad primes, by developing a Jordan-block framework that links unipotent blocks in centralisers to global blocks via the map $\bar{J}_t^{\mathbf{G}}$ and by handling $t$-twins in $E_8$. It extends KM’s $e$-Harish-Chandra theory to connected-centre settings, establishes parametrisations of blocks through $e$-cuspidal data, and proves a compatibility between Lusztig induction and Jordan decomposition across Levi subgroups, aided by BR-correspondence and inductive arguments. The work also corrects and completes earlier results (En00, KM13, KM15), provides a descent to adjoint types $E_6$ and $E_7$, and gives a comprehensive treatment of isolated blocks and specific $5$-blocks in $E_8$, culminating in a proof of Robinson's conjecture for all blocks. Overall, the methods yield a complete, symmetry-respecting description of the modular representation theory for these groups and establish block-level compatibilities essential for broader conjectures in finite group representation theory.

Abstract

We complete the determination of the $\ell$-block distribution of characters for quasi-simple exceptional groups of Lie type up to some minor ambiguities relating to non-uniqueness of Jordan decomposition. For this, we first determine the $\ell$-block distribution for finite reductive groups whose ambient algebraic group defined in characteristic different from $\ell$ has connected centre. As a consequence we derive a compatibility between $\ell$-blocks, $e$-Harish-Chandra series and Jordan decomposition. Further we apply our results to complete the proof of Robinson's conjecture on defects of characters.

Jordan correspondence and block distribution of characters

TL;DR

The paper resolves the -block distribution problem for quasi-simple exceptional groups, including bad primes, by developing a Jordan-block framework that links unipotent blocks in centralisers to global blocks via the map and by handling -twins in . It extends KM’s -Harish-Chandra theory to connected-centre settings, establishes parametrisations of blocks through -cuspidal data, and proves a compatibility between Lusztig induction and Jordan decomposition across Levi subgroups, aided by BR-correspondence and inductive arguments. The work also corrects and completes earlier results (En00, KM13, KM15), provides a descent to adjoint types and , and gives a comprehensive treatment of isolated blocks and specific -blocks in , culminating in a proof of Robinson's conjecture for all blocks. Overall, the methods yield a complete, symmetry-respecting description of the modular representation theory for these groups and establish block-level compatibilities essential for broader conjectures in finite group representation theory.

Abstract

We complete the determination of the -block distribution of characters for quasi-simple exceptional groups of Lie type up to some minor ambiguities relating to non-uniqueness of Jordan decomposition. For this, we first determine the -block distribution for finite reductive groups whose ambient algebraic group defined in characteristic different from has connected centre. As a consequence we derive a compatibility between -blocks, -Harish-Chandra series and Jordan decomposition. Further we apply our results to complete the proof of Robinson's conjecture on defects of characters.
Paper Structure (24 sections, 46 theorems, 37 equations, 11 tables)

This paper contains 24 sections, 46 theorems, 37 equations, 11 tables.

Table of Contents

  1. Introduction
  2. On Jordan decomposition and block distribution
  3. Notation and background results
  4. Generalities on Jordan decomposition
  5. Compatibility with central products
  6. Generalities on block distribution
  7. On the $\ell$-block distribution of $\ell'$-characters
  8. Parametrisation of $\ell$-blocks in connected centre groups
  9. $e$-Cuspidal pairs below $({\mathbf{G}}^F,s)$
  10. The block distribution
  11. The map ${\bar{J}}_t^{\mathbf{G}}$
  12. $t$-Twin blocks
  13. Inductive arguments
  14. Theorem \ref{['thm:thmA']} when $t$ is central
  15. Theorem \ref{['thm:thmA']} for good primes
  16. ...and 9 more sections

Key Result

Theorem 1

Let ${\mathbf{X}}$ be a connected reductive group in characteristic $p$ with connected centre and simple, simply connected derived subgroup, with a Frobenius map $F:{\mathbf{X}}\rightarrow{\mathbf{X}}$. Let ${\mathbf{G}}$ be an $F$-stable Levi subgroup of ${\mathbf{X}}$, let $\ell\ne p$ be a prime a

Theorems & Definitions (100)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 90 more