Book contents
- Frontmatter
- Contents
- Preface
- List of tables
- Notation
- PART I LINEAR ALGEBRAIC GROUPS
- PART II SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS
- 11 BN-pairs and Bruhat decomposition
- 12 Structure of parabolic subgroups, I
- 13 Subgroups of maximal rank
- 14 Centralizers and conjugacy classes
- 15 Representations of algebraic groups
- 16 Representation theory and maximal subgroups
- 17 Structure of parabolic subgroups, II
- 18 Maximal subgroups of classical type simple algebraic groups
- 19 Maximal subgroups of exceptional type algebraic groups
- 20 Exercises for Part II
- PART III FINITE GROUPS OF LIE TYPE
- Appendix A Root systems
- Appendix B Subsystems
- Appendix C Automorphisms of root systems
- References
- Index
14 - Centralizers and conjugacy classes
from PART II - SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- List of tables
- Notation
- PART I LINEAR ALGEBRAIC GROUPS
- PART II SUBGROUP STRUCTURE AND REPRESENTATION THEORY OF SEMISIMPLE ALGEBRAIC GROUPS
- 11 BN-pairs and Bruhat decomposition
- 12 Structure of parabolic subgroups, I
- 13 Subgroups of maximal rank
- 14 Centralizers and conjugacy classes
- 15 Representations of algebraic groups
- 16 Representation theory and maximal subgroups
- 17 Structure of parabolic subgroups, II
- 18 Maximal subgroups of classical type simple algebraic groups
- 19 Maximal subgroups of exceptional type algebraic groups
- 20 Exercises for Part II
- PART III FINITE GROUPS OF LIE TYPE
- Appendix A Root systems
- Appendix B Subsystems
- Appendix C Automorphisms of root systems
- References
- Index
Summary
We now consider properties like generation, conjugacy, classification, connectedness and dimension of centralizers in connected reductive groups. It turns out that the situation is easiest for semisimple elements.
Semisimple elements
Recall from Corollary 6.11(a) that every semisimple element of a connected group lies in some maximal torus. More precisely we have:
Proposition 14.1Let G be connected, s ∈ G semisimple, T ≤ G a maximal torus. Then s ∈ T if and only if T ≤ CG(s)°. In particular, s ∈ CG(s)°.
Proof As T is abelian, s ∈ T if and only if T ≤ CG(s), which is equivalent to T ≤ CG(s)° as T is connected.
We remark that in contrast, for u ∈ G unipotent, u may not be in CG(u)°.
See Exercise 20.10 for an example in Sp4 over a field of characteristic 2.
We now determine the structure of centralizers of semisimple elements:
Theorem 14.2Let G be connected reductive, s ∈ G semisimple, T ≤ G a maximal torus with corresponding root system Φ. Let s ∈ T and Ψ ≔ {α ∈ Φ | α(s) = 1}. Then:
(a) CG(s)° = 〈T,Uα; | α ∈ Ψ〉.
(b) CG(s) = 〈T,Uα,ẇ | α ∈ Ψ, w ∈ W with sw = s〉.
Moreover, CG(s)° is reductive with root system Ψ and Weyl group W1 = 〈sα | α ∈ Ψ〉.
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- Information
- Linear Algebraic Groups and Finite Groups of Lie Type , pp. 112 - 120Publisher: Cambridge University PressPrint publication year: 2011