Representations of Finite Groups of Lie Type

An up-to-date and self-contained introduction based on a graduate course taught at the University of Paris.

François Digne (Author), Jean Michel (Author)

9781108481489, Cambridge University Press

Hardback, published 5 March 2020

264 pages, 6 tables
23.4 x 15.6 x 1.9 cm, 0.49 kg

'… clearly written; there are useful examples, motivational comments, and exercises scattered throughout the text.' Mark Hunacek, The Mathematical Gazette

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.

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 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.

Subject Areas: Algebra [PBF]