Positive-depth Deligne–Lusztig varieties and character sheaves

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.