On two geometric realizations of an affine Hecke algebra

TL;DR

This paper advances the geometric Langlands program by categorifying the isomorphism between affine Hecke algebras and equivariant coherent sheaves, crucial for local Langlands conjectures.

Contribution

It provides a categorification of the affine Hecke algebra's isomorphism with coherent sheaves on the Steinberg variety, building on prior work and technical developments.

Findings

Categorification of the affine Hecke algebra isomorphism.

Connection to the local Langlands correspondence.

Framework for tamely ramified local Langlands conjectures.

Abstract

The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant functions on a semi-simple group over a local non-Archimedian field, and Grothendieck group of equivariant coherent sheaves on Steinberg variety of the Langlands dual group; this isomorphism due to Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of an earlier joint work with S. Arkhipov, it relies on technical material developed in a paper with Z. Yun.