Departmental Colloquium

Title

An introduction to representations of p-adic groups

Abstract

An explicit understanding of the (category of all smooth, complex) representations of p-adic groups provides an important tool not just within representation theory. It also has applications to number theory and other areas, and in particular enables progress on various different forms of the Langlands program.

In this talk, I will introduce p-adic numbers and p-adic groups and provide an overview of what we know about the representations of these groups including recent developments. We will focus on two crucial aspects:

First, I will survey what we know about the construction of all so called supercuspidal representations, which are the building blocks for all representations. For more than 20 years it remained open to extend a general construction to the case p=2, and I will sketch what makes this case so special and how we could overcome the obstacles in recent joint work with David Schwein.

Second, we will also briefly sketch the structure of the whole category of representations of p-adic groups in terms of these supercuspidal representations. Two recent preprints with Jeffrey Adler, Manish Mishra and Kazuma Ohara allow us to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or are at least easier to achieve.