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The values of unipotent characters at unipotent elements for groups of type $E_8$ and ${^2\!E}_6$

Jonas Hetz

Abstract

In order to tackle the problem of generically determining the character tables of the finite groups of Lie type $\mathbf{G}(q)$ associated to a connected reductive group $\mathbf{G}$ over $\overline{\mathbb F}_p$, Lusztig developed the theory of character sheaves in the 1980s. The subsequent work of Lusztig and Shoji in principle reduces this problem to specifying certain roots of unity. The situation is particularly well understood as far as character values at unipotent elements are concerned. We complete the computation of the values of unipotent characters at unipotent elements for the groups $\mathbf{G}(q)$ where $\mathbf{G}$ is the simple group of type $E_8$, by specifying the aforementioned roots of unity for all prime powers $q$. We also resolve this task for the groups ${^2\!E}_6(q)$ when $q$ is a power of $p=2$. Our results thus conclude the project of computing the values of unipotent characters at unipotent elements for the simple exceptional groups of Lie type.

The values of unipotent characters at unipotent elements for groups of type $E_8$ and ${^2\!E}_6$

Abstract

In order to tackle the problem of generically determining the character tables of the finite groups of Lie type associated to a connected reductive group over , Lusztig developed the theory of character sheaves in the 1980s. The subsequent work of Lusztig and Shoji in principle reduces this problem to specifying certain roots of unity. The situation is particularly well understood as far as character values at unipotent elements are concerned. We complete the computation of the values of unipotent characters at unipotent elements for the groups where is the simple group of type , by specifying the aforementioned roots of unity for all prime powers . We also resolve this task for the groups when is a power of . Our results thus conclude the project of computing the values of unipotent characters at unipotent elements for the simple exceptional groups of Lie type.
Paper Structure (8 sections, 6 theorems, 105 equations, 6 tables)

This paper contains 8 sections, 6 theorems, 105 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Good unipotent elements
  3. Ree's formula and (generalised) Green functions
  4. The case pgeq5
  5. The case p=3
  6. The case p=2
  7. Representatives for the good unipotent GF-classes
  8. Groups of type 2E6

Key Result

Theorem 2.3

Let ${\mathcal{O}}$ be a unipotent conjugacy class of $\mathbf G$ which does not appear in the following list: Then ${\mathcal{O}}^F$ contains exactly one good $\mathbf G^F$-conjugacy class.

Theorems & Definitions (15)

  • Definition 2.2: cf. HDiss
  • Theorem 2.3
  • Remark 2.4
  • Corollary 2.5
  • Remark 2.6
  • Remark 3.3
  • Proposition 3.10
  • Remark 6.9
  • Lemma 7.2: see HDiss; cf. LuWeylUni
  • proof
  • ...and 5 more