[This is a work in progress with Bonnafé-Broué-Malle-Michel-Rouquier.] In this talk I will explain how to construct a non-trivial action of SL2(Z) on the vector space spanned by unipotent characters. The existence of such an action follows from Lusztig's classification of unipotent characters, but I will present a global construction relying on traces of automorphisms of Deligne--Lusztig varieties. The advantage of this approach is that it can be naturally generalized to Spetses, thus giving candidates for Fourier matrices and unipotent character sheaves for Spetses.

This series of lectures will be centered on decomposition numbers for a special class of finite groups such as GL_n(q), SO_n(q),... E_8(q). I will first present what kind of numerical invariants decomposition numbers are, and what representation-theoretic problems they can solve. For finite reductive groups, I will explain how one can use Deligne--Lusztig theory to get basic sets of ordinary characters and to compute decomposition numbers. If time permits, I will mention a few open problems, including the case of small characteristic.

Lecture 1 - Generalities on decomposition numbers

Lecture 2 - Basic sets for finite reductive groups

Lecture 3 - Computing decomposition numbers