Introduction to smooth representation theory

Lectures

• Monday 2-4pm, WSC-N-U-4.04

• Wednesday 2-4pm, WSC-S-U-4.02

Moodle page

There is an electronic classroom for the seminar on the moodle platform. Read backward, the enrolment key needed to subscribe is

4232yroehtperhtooms

Lecture notes and exercises are available here

Course description

Let p denote a prime number and let GL(n) denote the group of invertible nxn matrices over the field of p-adic numbers. A so-called p-adic group is a Zariski closed subgroup of GL(n). Such groups are canonically equipped with a topology coming from the p-adic topology, and we view them as topological groups in this way. In order to study p-adic groups in a purely algebraic (i.e. non-topological) way, one investigates how they act continuously on vector spaces equipped with the discrete topology. Vector spaces endowed with such an action are called smooth representations and their study is at the heart of the local Langlands program.

When a p-adic group G is reductive (which is a technical condition satisfied for example when G=GL(n)) and when working with the complex numbers as coefficient field, the smooth representation theory of G was studied intensively in the 1970's by Borel, Bernstein and Zelevinsky among others, and it continues to be an active area of research to this day.

The aim of this course is to give an introduction to the smooth representation theory of p-adic groups. In order to avoid having to introduce too much machinery, we will restrict ourselves to studying representations of GL(n) for some of the more advanced result. In the first half of the course, I will cover the following:

• Review of useful background: general representation theory of groups, valuations and local fields.

• Smooth and admissible representations of locally profinite groups, (compact) induction, Haar measures.

• Parabolic induction, Jacquet modules and (super)cuspidal representations.

• Proof that every smooth irreducible representation of GL(n) is admissible.

In the second half of the course, I will cover some (but not all) topics chosen among the following:

• Hecke algebras and representations generated by their Iwahori fixed vectors.

• The Bernstein centre and the second adjunction theorem.

• The Satake isomorphism.

• L-functions associated to smooth representations of GL(n).

Prerequisites

The course will be largely independent from the Algebraic Number Theory courses in Essen. That being said, some familiarity with local fields and representation theory will be useful, although not strictly indispensable (I plan to recall the necessary facts at the start of the course). I expect the participants to be well acquainted with the techniques of abstract algebra and basic group theory, as provided by an introductory course on these topics. The lectures will be given in English.

Literature/references

Background material on representation theory

• All the notions we need are contained in the first three sections of these lecture notes (by Simon Wadsley).

Representations of p-adic groups

• D. Renard: Représentations des groupes réductifs p-adiques. Paris: Société Mathématique de France, 2010. Chapter VI contains most of the relevant material.

• C. Bushnell and G. Henniart: The local Langlands conjecture for GL(2). Vol. 335. Springer Science & Business Media, 2006. Chapter 1 gives a nice introduction to smooth representations.

• W. Casselman: Introduction to admissible representations of p-adic reductive groups. Unpublished notes.

For those interested in further developments of the theory, here is some additional reading that goes far beyond what we'll cover:

• J. Fintzen: Representations of p-adic groups. Survey article. General overview of the theory leading to recent developments in the construction of supercuspidal representations for arbitrary reductive groups.

• M.-F. Vignéras: Représentations l-modulaires d'un groupe réductif p-adique avec l≠p. Progress in Math 137, Birkhaüser, 1996. A book covering the theory working more generally over coefficient fields of characteristic not equal to p.

• F. Herzig: The mod-p representation theory of p-adic groups. Course notes. The theory over coefficient fields of characteristic p is considerably harder and still largely mysterious, but these notes give an introduction to some of what is known.