SCIENTIFIC AFTERNOON: REPRESENTATION THEORY

Institut Élie Cartan de Lorraine (IECL, UMR CNRS 7502) Metz, December 18, 2025

• Alexandre AFGOUSTIDIS (CNRS and Université de Lorraine)
  Title: The Mackey bijection as a stratified equivalence
  Abstract:
  Stratified equivalence between algebras was defined in the context of the Bernstein decomposition of the Hecke algebra for a reductive p-adic group, to express the idea that the spectrum of every Bernstein component is assembled from very simple pieces (work of Baum-Nistor, Aubert-Baum-Plylen-Solleveld). The Mackey bijection is a topic in real groups: it relates the whole tempered dual of a real reductive group to the unitary dual of its Cartan motion group. I will explain how the Mackey bijection, and more precisely the Mackey embedding of C^*-algebras recently constructed by Clare, Higson and Román, can be recast as a stratified equivalence in the sense used for p-adic groups. This is joint work with Pierre Clare.
   
  
• Anne-Marie AUBERT (CNRS and Institut de Mathématiques de Jussieu)
  Title: Asymptotic Schur orthogonality relations
  Abstract:
  The asymptotic Schur orthogonality relations generalize the classical Schur orthogonality relations to certain non-compact groups and representations that are not discrete series.
  A general framework has been recently introduced by Kazhdan and Yom Din via the notion of c-temperedness, a property which implies temperedness.
  The main idea is to replace the L²-pairing of matrix coefficients over the entire group with a limit of a pairing over a sequence of bounded subsets, using a sequence of bounded measures.
  For semisimple groups over local fields, Kazhdan and Yom Din conjectured that conversely temperedness implies c-temperedness. For K-finite matrix coefficients, they proved the validity of their conjecture in the nonarchimedean case, and, with La Rosa, we proved it in the archimedean case.
  On the other hand, Mandal and Mondal showed that all unitary irreducible representations of Heisenberg groups over local fields are c-tempered.
  In this talk, I will describe the above results in more details and provide applications.
   
  
• Jessica FINTZEN (Universität Bonn)
  Title: 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 groups 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 an algebraically closed 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.

Aurélie PAULL's Thesis Defense:

This day is linked to the thesis defense of Aurélie PAULL, on December 18, 2025 at 10:00 AM in the Salle ARJ-027 (Pierre et Marie Curie) of the UFR-MIM Building.
Title: The Weil representation over a finite field of characteristic two.

Registration Form

Participation is free but registration before the deadline is mandatory for good planning of attendance (room, coffee/tea break, dinner ...)