Appears in collection : Geometric representation theory in the Langlands program - Thematic month week 2 / Théorie géométrique des représentations dans le programme de Langlands - Mois thématique sem. 2

Representation theory and the geometry of flag varieties are deeply intertwined. For finite groups of Lie type, Deligne and Lusztig's breakthrough work in 1976 defined Frobenius-twisted versions of flag varieties whose cohomology realizes all representations of these groups. Lusztig's theory of character sheaves further revolutionized the subject in the 1980s, yielding a perverse-sheaf-theoretic basis of the vector space of class functions. In the last quarter-century, generalizations of Deligne-Lusztig varieties and character sheaves have allowed us to study representations of p-adic groups using explicit geometric methods. I will describe recent advances in this subject and their relationship to the Langlands program.