Reduction to depth-zero for $\bar{\mathbb{Z}}[1/p]$-representations of p-adic groups

Abstract: The category of smooth complex representations of p-adic groups decomposes into Bernstein blocks and by a joint result with Adler, Mishra and Ohara from August 2024 we know that under some minor tameness assumptions each Bernstein block is equivalent to a depth-zero Bernstein block, which are the representations that correspond roughly to representations of finite group of Lie type. This result allows to reduce a lot of problems about representations of p-adic groups and the Langlands correspondence to their depth-zero counterpart that is often easier to solve or already known. For number theoretic applications one likes to have a similar result when working with representations whose coefficients are a more general ring than the complex numbers.

In this talk we present analogous results for R-representations of p-adic groups where R is any ring that contains all p-power roots of unity, a fourth root of unity and the inverse of a square-root of p, for example, R could be a field of characteristic different from p or the ring $\bar{\mathbb{Z}}[1/p]$. This is joint work in progress with Jean-François Dat. While the result is analogous to the result with complex coefficients (except for the “blocks” being “larger”), the proof is of a very different nature. In the complex setting the proof is achieved via type theory and an isomorphism of Hecke algebras, which are techniques not available for general R-representations. We will sketch in the talk how we deal with the category of R-representations instead.

number theoryrepresentation theory

Audience: researchers in the topic

University of Utah Representation Theory / Number Theory Seminar