Langlands correspondence and Bezrukavnikov's equivalence

TL;DR

This paper provides an introduction to the Langlands correspondence and explores Bezrukavnikov's equivalence, connecting arithmetic and geometric representation theory through categorification of affine Hecke algebra realizations.

Contribution

It offers a comprehensive overview linking the Langlands program with geometric representation theory, highlighting Bezrukavnikov's equivalence as a categorification of affine Hecke algebra realizations.

Findings

Clarifies the Langlands correspondence from an arithmetical perspective

Explains Bezrukavnikov's equivalence in geometric representation theory

Connects categorification with affine Hecke algebra realizations

Abstract

These are lecture notes (by the first author) from a course (by the second author) given over two extended semesters at the University of Sydney. The first part provides an introduction to the Langlands correspondence from an arithmetical point of view. The second part gives enough background in geometric representation theory to understand Bezrukavnikov's equivalence, which is a categorification of Kazhdan and Lusztig's two realizations of the affine Hecke algebra.