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, it enables progress on various different forms of the Langlands program. In this write-up of the author's ECM 2024 colloquium-style talk, we will introduce p-adic groups and explain how the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. We also provide an overview of what we know about the structure of these Bernstein blocks including a sketch of recent results of the author with Adler, Mishra and Ohara that allow 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. Moreover, we provide an overview of what is known about the construction of supercuspidal representations, which are the building blocks of all smooth representations and whose construction is also the key to obtain the above results about the structure of the whole category of smooth representations. We will, in particular, focus on recent advances which include the work of the author mentioned in the EMS prize citation as well as a hint towards her recent joint work with David Schwein.