TCC: The local Langlands correspondence for GL_2
Summer Term, Thursdays 111
Dr. David Helm
672 Huxley
dhelm@imperial.ac.uk
Useful Background
A basic familiarity with algebraic number theory and the theory of local fields will be very helpful. Local class field theory is
definitely a prerequisite, but we will review what we need early in the course. We will also make occasional use of some basic
representation theory of finite groups (nothing beyond the level of the first few chapters of Serre's Representation Theory of finite groups.
We will make occasional references to other topics in number theory (Artin Lfunctions, modular forms, etc.) for the purposes of motivation
but these topics will not be needed to understand the main thrust of the course.
Primary Reference
The main reference for the course is BushnellHenniart's The local Langlands conjecture for GL(2) which contains almost everything
we will cover in the course.
Syllabus
This is a rough list of the topics we will cover. Note that this plan is on the ambitious side and some pieces will only be sketched.
Motivation: Lfunctions and the Langlands correspondence
Local fields and local class field theory
The local Langlands correspondence
Twodimensional representations of W_F
Smooth representations of locally profinite groups
Frobenius reciprocity and Hecke algebras
Parabolic induction and restriction
Cuspidal representations in depth zero
The naive tame correspondence for GL_2
Cuspidal representations in higher depth (sketch)
The naive local Langlands correspondence for GL_2
Zeta integrals and local L and epsilon factors
Rectifiers and local Langlands for GL_2
Converse theorems and the uniqueness of the correspondence
Lectures
Lectures will be on Thursdays from 11 till 1. In the final week of the course there will also be a lecture on Tuesday from 111.
Exercises
Exercises will be posted here occasionally. They are optional, though if you require assessment I may ask you to do some. There are likely
to be typos; please let me know if you find any!
Exercises on representations of W_F
Exercises on representations of GL_2
Assessment
If you need any form of assessment of credit for the course, please contact me and let me know, so we can discuss both how you will be assessed (most
likely via some subset of the exercises above!) and what I need to do to get you credit for the course.
Further Reading
Much is known about the local Langlands correspondence for groups well beyond GL_2; here are a few good places to learn more.
 For background reading on the representation theory of G(F), for G a reductive algebraic group, there are
Bernstein's lecture notes as well
as the article
Representations of the group GL(n,F) where F is a local nonarchimedian field.

For the structure of parabolic induction for GL_n(F), one has the articles
Induced representations of reductive padic groups I and Induced representations of reductive padic groups II by
BernsteinZelevinsky and Zelevinsky, respectively. Beyond the case of GL_n this is still not fully understood in general, even over the complex
numbers!

The theory of strata and their associated characters generalizes to an arbitrary reductive group; this can be found in MoyPrasad's 1994
Inventiones paper Unrefined minimal Ktypes for padic group.

By contrast, it is still unclear how to generalize the full type theory to an arbitrary reductive group. BushnellKutzko's book The admissible
dual of GL_n via compact open subgroups explains how to do this for GL_n, and there is now (very recently!) also a welldeveloped theory
for classical groups due to Shaun Stevens and his collaborators. There is also a more general construction of types due to J.K. Yu, but it
is not wellunderstood at present.

A good starting point for the modular representation theory of padic groups, modulo primes different from p, is Vigneras' book
Representations lmodulaires d'un groupe reductif padique avec l ≠ p.