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

François Digne is Emeritus Professor at the Université de Picardie Jules Verne, Amiens. He works on finite reductive groups, braid and Artin groups. He has also co-authored with Jean Michel the monograph Foundations of Garside Theory (2015) and several notable papers on Deligne-Lusztig varieties.

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.