Vortrag im Rahmen des Friedrich-Hirzebruch-Kolloquiums

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 groups and give an overview of what we know about the representations of these groups. We will focus on two crucial aspects: First, I will survey what we know about the construction of all so called supercuspidal representations, which we will see are the building blocks for all representations. Most recent advances include joint work in progress with David Schwein.
Second, we will study the structure of the whole category of representations of p-adic groups in terms of these supercuspidal representations, and I will explain how two recent preprints with Adler, Mishra and 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.