Courses in advanced mathematics from basics

Speaker: Tanmay Deshpande (Kavli IPMU)
Title: Etale cohomology, l-adic sheaves and Deligne-Lusztig theory
Date (JST): Thu, Dec 05, 2013, 13:40 - 18:00
Place: Seminar Room B
Abstract: In the first part of my talk I will give a brief introduction to the theory of etale cohomology and the bounded derived category of l-adic complexes on a scheme. The main goal of the talk is to use this geometric theory to study the representation theory of finite groups of Lie type i.e. groups of the form G(F_q) where G is a reductive group defined over a finite field F_q. I will define certain varieties, known as Deligne-Lusztig varieties which are equipped with an action of the finite group G(F_q). Thus we get an induced action of G(F_q) on the l-adic cohomology of these Deligne-Lusztig varieties. Deligne and Lusztig proved that all irreducible representations of G(F_q) occur in the cohomology of the various Deligne-Lusztig varieties. Using this theory it is possible to partition the irreducible representations of G(F_q) into equivalence classes which are parametrized by semisimple conjugacy classes in the dual reductive group. I will try to illustrate these ideas explicitly for G=SL_2.
Remarks: 1st 13:40 - 15:00, 2nd 15:30 - 18:00