I will report on my current (and ongoing) work on understanding Hecke correspondences on the flag variety.
Hecke correspondences are generalizations of incidence correspondences on P^1 to general flag varieties. Since the pioneering work of Beilinson-Bernstein and Kashiwara-Brylinski (on D-modules), Goresky-Macpherson and Beilinson-Bernstein-Deligne's work (on intersection cohomology), and Lusztig-Kazhdan and Lusztig-Vogan's work (in representation theory), it has been understood that the representation theory of reductive groups is controlled by the geometry and combinatorics of these Hecke correspondences. I will present my own contributions to this study.
Time permitting I will explain how this ties into efforts by Nat Thiem and myself to understand and generalize Lusztig's theory of character sheaves.
On Hecke Correspondences