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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.

On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis–Alvis duality map and Mackey's theorem and the results that can be deduced from it, before moving on to a discussion of Deligne–Lusztig induction and Lusztig's Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples.

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12 - Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters

Summary

Representations of Finite Groups of Lie Type

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12 - Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters

Summary

Representations of Finite Groups of Lie Type