Geordie’s Seminars: Geordie Williamson -- Langlands correspondence and Bezrukavnikov’s equivalence

 
Geordie Williamson will offer the following one semester course starting on the 8th of 
March 2019, Friday 10:00--12:00, Carslaw room 830.

Title: Langlands correspondence and Bezrukavnikov’s equivalence 

This course will be in two parts.  

1) I will attempt to give a picture of what the Langlands correspondence is about, from
an arithmetical point of view.  I will start of with some basic questions (e.g.
counting points of varieties over finite fields) and show how they lead to interesting
L-functions which should have an automorphic incarnation.  Explaining this will involve
rather a lot of algebraic number theory, which I will try to go over briefly.  I will
then pass to the local case.  I will review the structure theory of local fields and
their Galois groups and state the main theorems of local class field theory.  I will
then explain what the local Langlands correspondence should be, and we will see that it
boils down to class field theory if the GL_1 case.  I will sketch a beautiful heuristic
argument for the local Langlands correspondence for GL_2 when p is not 2, which can be
turned into a proof which I won’t go into (so-called Jacquet-Langlands
correspondence).  

2) The second half of the course will focus on affine Hecke algebras and their
categorifications.  Roughly speaking this part of the course is about the local
Langlands correspondence at the easiest “layers of difficulty”, which is still
extremely rich.  I will explain the two simplest instances of the local Langlands
correspondence for a general G, namely the case of unramified representations (where the
Langlands correspondence boils down to the Satake isomorphism), and the case of tamely
ramified representations (so-called Deligne-Langlands conjecture).  The
Deligne-Langlands conjecture was proved by Kazhdan and Lusztig (and later Ginzburg).  I
will try to outline their proof.  I will then explain how categorifying Kazhdan and
Lusztig’s proof leads to a remarkable equivalence conceived by Bezrukavnikov in the
late 90s but only recently written down.  If time permits I will try to outline the
proof of his theorem.  

This is an ambitious course in terms of scope, and, unless you have significant
background, will require work and reading outside of the lectures.  I will certainly not
prove everything, however I will try to make everything as explicit as I can for GL_1
and GL_2.  There is some chance it will spill over into second semester, depending on
the interest of the participants and the stamina of the lecturer!