Affine Hecke category and noncommutative Springer resolution
I will start with a few comments on the proof of the equivalence presented in the previous talks. Then I will focus on the description of the abelian category of perverse sheaves on the affine flag variety on the coherent side, where the answer is given in terms of the so called noncommutative Springer resolution. This is a key ingredient in applications to representations of semisimple Lie algebras, leading to the proof (with Mirkovic) of Lusztig's 1999 conjecture for large pp, as well as applications to cells in affine Weyl groups and asymptotic Hecke algebras and potential applications to characters of modules over an affine Lie algebra at the critical level and (Kac - De Concini) quantum groups at a root of unity. The story can be viewed as upgrading a (very) special case of geometric Langlands duality to a statement involving abelian, rather than triangulated, categories.