Number Theory Seminar

Let G be a complex reductive group. A celebrated theorem of Kazhdan-Lusztig establishes an isomorphism between the extended affine Hecke algebra of G and certain equivariant K-group of the Steinberg variety of the Langlands dual group of G. This isomorphism plays a crucial role in Kazhdan-Lusztig's proof of the Deligen-Langlands conjecture. In the equal characteristic setting, Bezrukavnikov studied the categorification of this isomorphism and proved an equivalence between two geometric realizations of the affine Hecke algebra, which can be seen as the tamely ramified local geometric Langlands correspondence. In mixed characteristics, Bando and Yun-Zhu independently used implicit approaches to derive Bezrukavnikov's equivalence from the equal characteristic setting. In this talk, I will discuss an explicit approach to establish Bezrukavnikov's equivalence building on a previous joint work with Anschütz, Lourenço, and Wu. This talk is based on an ongoing project with Bando, Gleason, and Lourenço.