Equivariant coherent sheaves, Soergel bimodules, and categorification of affine Hecke algebras

Abstract

In this thesis, we examine three different versions of "categorification" of the affine Hecke algebra and its periodic module: the first is by equivariant coherent sheaves on the Grothendieck resolution (and related objects), the second is by certain classes on bimodules over polynomial rings, called Soergel bimodules, and the third is by certain categories of constructible sheaves on the affine flag manifold (for the Langlands dual group). We prove results relating all three of these categorifications, and use them to deduce nontrivial equivalences of categories. In addition, our main theorem allows us to deduce the existence of a strict braid group action on all of the categories involved; which strengthens a theorem of Bezrukavnikov-Riche.