Book contents
- Frontmatter
- Contents
- Introduction to the Second Edition
- From the Introduction to the First Edition
- 1 Basic Results on Algebraic Groups
- 2 Structure Theorems for Reductive Groups
- 3 (B,N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-simple Elements
- 4 Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem
- 5 Harish-Chandra Theory
- 6 Iwahori–Hecke Algebras
- 7 The Duality Functor and the Steinberg Character
- 8 ℓ-Adic Cohomology
- 9 Deligne–Lusztig Induction: The Mackey Formula
- 10 The Character Formula and Other Results on Deligne–Lusztig Induction
- 11 Geometric Conjugacy and the Lusztig Series
- 12 Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters
- 13 Green Functions
- 14 The Decomposition of Deligne–Lusztig Characters
- References
- Index
12 - Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters
Published online by Cambridge University Press: 14 February 2020
Book contents
- Frontmatter
- Contents
- Introduction to the Second Edition
- From the Introduction to the First Edition
- 1 Basic Results on Algebraic Groups
- 2 Structure Theorems for Reductive Groups
- 3 (B,N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-simple Elements
- 4 Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem
- 5 Harish-Chandra Theory
- 6 Iwahori–Hecke Algebras
- 7 The Duality Functor and the Steinberg Character
- 8 ℓ-Adic Cohomology
- 9 Deligne–Lusztig Induction: The Mackey Formula
- 10 The Character Formula and Other Results on Deligne–Lusztig Induction
- 11 Geometric Conjugacy and the Lusztig Series
- 12 Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters
- 13 Green Functions
- 14 The Decomposition of Deligne–Lusztig Characters
- References
- Index
Summary
We prove the existence of regular semi-simple elements, then of regular unipotent elements. We give properties of Gelfand–Graev representations and their duals. We compute their values on regular unipotent elements using Gauss sums. We decompose Gelfand–Graev representations into sums of regular characters and use them to prove the disjunction of Deligne–Lusztig characters corresponding to distinct rational semi-simple classes. The chapter ends with the character table of SL2.
- Type
- Chapter
- Information
- Representations of Finite Groups of Lie Type , pp. 196 - 224Publisher: Cambridge University PressPrint publication year: 2020