An introduction to representations of p-adic groups
Jessica Fintzen
TL;DR
The paper surveys the representation theory of $p$-adic groups with a focus on the category $\mathrm{Rep}(G)$ of smooth representations and its Bernstein block decomposition $\mathrm{Rep}(G)=\prod_{(M,\sigma)/\sim} \mathrm{Rep}(G)_{[M,\sigma]}$, highlighting the role of parabolic induction and supercuspidal building blocks. It explains how Bernstein blocks can be modeled by Hecke algebras, and how depth-zero reductions reduce many questions to representations of finite groups of Lie type via equivalences such as $\mathrm{Rep}(G)_{[M,\sigma]}\simeq \mathrm{Rep}(G^0)_{[M^0,\sigma_0]}$ and $\mathbb{C}[\Omega,\mu]\ltimes \mathcal{H}_{\mathrm{aff}}$-mod descriptions. The work of AFMO and related developments connect block theory to explicit algebraic structures, enabling tractable descriptions of blocks and their modules. On the construction side, the Moy–Prasad filtration and Yu-type constructions (with recent refinements) provide broad exhaustion results for supercuspidal representations under mild $p$-restrictive conditions, including new results for small primes (epipelagic cases and $p=2$ in FS25), with representations typically realized as $\mathrm{c\text{-}ind}_K^G V_\rho$. Collectively, these advances reduce deep questions about $p$-adic representations to finite-group and Hecke-algebra problems, advancing the local Langlands program and automorphic applications.
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.