Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer

A.A. Schaeffer Fry; Jay Taylor; C. Ryan Vinroot

doi:10.1016/j.jalgebra.2024.10.017

Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer

A.A. Schaeffer Fry
, Jay Taylor
, C. Ryan Vinroot

Pure Mathematics

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Abstract

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne–Michel’s unique Jordan decomposition in the connected center case.

Original language	English
Pages (from-to)	123-149
Journal	Journal of Algebra
Volume	664
Issue number	Part B
Early online date	21 Oct 2024
DOIs	https://doi.org/10.1016/j.jalgebra.2024.10.017
Publication status	Published - 15 Feb 2025

Keywords

Jordan decomposition of characters
Deligne–Lusztig theory
Finite reductive group
Group of Lie type

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10.1016/j.jalgebra.2024.10.017

SVT19_JordanConCentralizersAccepted author manuscript, 444 KBLicence: CC BY

Cite this

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title = "Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer",

abstract = "We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne–Michel{\textquoteright}s unique Jordan decomposition in the connected center case.",

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AU - Schaeffer Fry, A.A.

AU - Taylor, Jay

AU - Vinroot, C. Ryan

PY - 2025/2/15

Y1 - 2025/2/15

N2 - We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne–Michel’s unique Jordan decomposition in the connected center case.

AB - We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne–Michel’s unique Jordan decomposition in the connected center case.

KW - Jordan decomposition of characters

KW - Deligne–Lusztig theory

KW - Finite reductive group

KW - Group of Lie type

U2 - 10.1016/j.jalgebra.2024.10.017

DO - 10.1016/j.jalgebra.2024.10.017

M3 - Article

SN - 0021-8693

VL - 664

SP - 123

EP - 149

JO - Journal of Algebra

JF - Journal of Algebra

IS - Part B

ER -

Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer

Abstract

Keywords

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