Introduction & Purposes

We might (or might not) be interested in representation theory of p-adic reductive groups because

1.    They act on automorphic forms,

2.    We are interested in the (local) Langlands program, or

3.    We simply find the representation theory interesting.

In either case, the character theory of p-adic reductive groups is a fundamental tool, particularly in the theory of endoscopy and in the study of trace formulas. In this course, we will study the character theory of p-adic reductive groups, partly following the classical approach of Harish-Chandra, but also emphasizing new algebraic tools and results relevant to mod-ℓ representations.

Outline & Descriptions

After introducing some basics of representations and characters of a p-adic reductive group G, we plan to cover a proper subset of the following topics, depending on the interests of the audience:

1.    The theorem of Harish-Chandra and Howe that a complex irreducible admissible character locally looks like the Fourier transform of a distribution on a nilpotent cone.

2.    Further enhancement by Waldspurger, DeBacker and Adler-Korman, using Bruhat-Tits theory, when p is very large.

3.    Harish-Chandra's theorem that a complex irreducible admissible character is represented by a locally-L^1 function that is locally constant on G^{rs}.

4.    The theorem of Rodier and of Mœglin-Waldspurger that relates the aforementioned distribution on the nilpotent cone to degenerate Whittaker models.

Moreover, we plan to re-work some analytical arguments of Harish-Chandra into algebraic arguments, with the additional goals to:

1.    Understand the results for representations and characters over bar{Q}_ell, where analysis is not always applicable, and

2.    Generalize these results to mod-ℓ representations and characters, as in e.g. https://arxiv.org/abs/2510.20509.

We will mostly not discuss applications to the Langlands program.

Grading

Homework 100%, potentially (partly) replaced by presentation if the student wishes.

Prerequisites

The audience are assumed to be familiar with complex representations and characters of finite groups. The audience will be told to assume G is a classical group if they are not familiar with algebraic reductive groups, but it's better that they are.
At some point we plan to use Bruhat-Tits theory a lot, for which the audience are referred to Masao's course happening in the same semester. We will state all necessary Bruhat-Tits theory results so that the audience could be comfortable if they are willing to assume the results, most of which can be directly proved for classical groups.

Credit: 2

Course No.: NCTS5061

Course ID: V41 U1120

Registration