Tempered representations and K-theory

Contribution List

1. Biunimodular functions

Yves Benoist (Université Paris-Saclay)

2/24/25, 9:30 AM

A biunimodular function on a cyclic group of prime order is a function with constant modulus whose Fourier transform also has constant modulus. For instance gaussian functions are biunimodular. According to a theorem of Haagerup there are only finitely many biunimodular functions up to scalar. In this talk we will construct new biunimodular functions for all prime $p>5$. The proof relies on...

2. Strong convergence of unitary representations

Mikael de la Salle (CNRS, Institut Camille Jordan)

2/24/25, 11:00 AM

Tempered representations of a group are the unitary representations that are weakly contained in the regular representation. The standard way to say that a representation is close to being tempered is probably to use Fell's topology which involves convergence of matrix coefficients. I will discuss another (in general much stronger) notion of closeness, that originates from random matrix theory...

3. The $L^p$-dual space of a semisimple Lie group

Bachir Bekka (Université de Rennes)

2/24/25, 3:00 PM

Given a group $G$ and a real number $p$, it is natural to study representations of G by linear isometries on $L^p$-spaces. Of course, the case where $p$ is equal to 2 corresponds to the familiar and much studied case of unitary representations of $G$. For a semisimple Lie group $G$, we will give a complete classification of all its irreducible $L^p$-representations, for $p\neq2$.

16. Mackey analogy in periodic cyclic homology

Axel Gastaldi (Aix-Marseille Université)

2/24/25, 4:30 PM

Crossed product algebras are fundamental objects that describe actions of a Lie group G on a Fréchet algebra A. In this talk we will consider the convolution algebra of compactly supported smooth functions on G with values in A. Using geometrical arguments, we will canonically identify the periodic cyclic homology of this crossed product (up to a dimension shift) with the homology of the...

4. A higher index for finite-volume locally symmetric spaces

Peter Hochs (Radboud University)

2/25/25, 9:30 AM

Let $G$ be a connected real semisimple Lie group, and $K<G$ maximal compact. For a discrete subgroup $\Gamma < G$, we have the locally symmetric space $X = \Gamma \backslash G/K$. If $X$ is smooth and compact, then Atiyah-Singer index theory is a source of useful and computable invariants of $X$. One then also has the higher index, with values in the $K$-theory of the $C^*$-algebra of...

5. Functoriality via Dirac cohomology

Jing-Song Huang (The Chinese University of Hong Kong, Shenzhen)

2/25/25, 11:00 AM

Dirac cohomology of a discrete series representation is similar to the highest weight vectors of a highest weight module. We discuss how to formulate the character lifting of discrete series by employing Dirac cohomology. This lifting is closely related to the Langlands functoriality and the Howe dual pair correspondence.

6. On the crystallisation of semisimple Lie groups

Robert Yuncken (Université de Lorraine)

2/25/25, 2:30 PM

In 1990, Kashiwara and Lusztig independently discovered the theory of crystal bases for complex semisimple Lie algebras. This theory says that if we deform the universal enveloping algebra by the Drinfled-Jimbo quantisation procedure, and let the quantisation parameter go to infinity, then the structure theory of the finite-dimensional representations becomes extremely simple. This allows an...

7. Conformal geometry and group actions in unbounded KK-theory

Ms Ada Masters (University of Wollongong)

2/25/25, 4:00 PM

I will outline a puzzle relating to group actions in noncommutative geometry and its resolution by way of conformal geometry. I will discuss the discrepancy between Kasparov's bounded picture of KK-theory and the unbounded picture of KK-theory, including the spectral triples so important to noncommutative geometry, particularly with respect to group equivariance. The discrepancy occurs already...

8. The large time behavior of the heat kernel on homogenous spaces and Bismut's formula

Xiang Tang (Washington University in St. Louis)

2/26/25, 9:30 AM

Let $G$ be a connected linear real reductive group with a maximal compact subgroup $K$. In this talk, we will explain an approach to study the large-time behavior of the heat kernel on the corresponding homogeneous space $G/K$ using Bismut’s formula. We will discuss how Bismut’s formula provides a natural link between the index theory and Vogan's minimal $K$-type theory. In particular, we...

9. Paley-Wiener for the Casselman--Wallach algebra and the Oka principle

Jacob Bradd (Pennsylvania State University)

2/26/25, 11:00 AM

I will talk about how Delorme's techniques used in his proof of the Paley-Wiener theorem can be applied to the Casselman-Wallach Schwartz algebra of rapidly decreasing smooth functions on a real reductive group, and how they can be used to give a sort of structure theorem strong enough to prove a refined version of the Connes-Kasparov isomorphism. If there's time, I will discuss how the...

10. K-Theory and Langlands duality

Roger Plymen

2/27/25, 9:30 AM

Let $G$ be a compact connected semisimple Lie group. We will describe the Langlands dual group~$G^\vee$. We now have two extended affine Weyl groups, one for $G$ and one for $G^\vee$. We will compare the C*-algebras of these two discrete groups, and show that they have the same K-theory. In this sense, Langlands duality is an invariant of K-theory.
With the aid of the...

11. Topological K-theory of reductive $p$-adic groups

Maarten Solleveld (Radboud Universiteit Nijmegen)

2/27/25, 11:00 AM

The goal of this talk is a simple, concrete description of the topological K-theory of the reduced C*-algebra of a reductive p-adic group $G$, in terms coming from representation theory.

We will discuss the structure of several group algebras of $G$, and how the K-theory can be computed from the space of irreducible tempered $G$-representations. The final description is closely related to...

12. Tempered representations with unipotent parahoric restriction: a noncommutative geometry viewpoint

Prof. Anne-Marie Aubert (IMJ-PRG CNRS)

2/27/25, 3:00 PM

The category of smooth representations of a $p$-adic group $G$ admits a decomposition into Bernstein blocks. With Paul Baum, Roger Plymen and Maarten Solleveld, we have formulated a conjecture which relates these blocks to (possibly twisted) extended quotients of complex algebraic varieties by finite groups.

In this talk, I will first introduce $p$-adic groups and some of their...

13. Topos and Noncommutative Geometry: Two Perspectives on Space and Numbers

Alain Connes (IHES)

2/27/25, 4:30 PM

Noncommutative geometry and the notion of topos are two mathematical concepts that provide complementary perspectives on the structure of a space. In this talk, I will begin by explaining, as simply as possible, these two concepts and what makes them unique. The originality of noncommutative geometry can be directly perceived through the existence of an intrinsic time evolution of a...

14. Mackey embedding for reduced group C*-algebras

Angel Roman (Washington University in St Louis)

2/28/25, 9:30 AM

Recently, Nigel Higson and Alexandre Afgoustidis made precise an analogy proposed by George Mackey between some unitary representations of a semisimple Lie group and unitary representations of its associated semidirect product group. In this talk, I will show a construction of an embedding of the reduced group C-algebra of the Cartan motion group into the reduced group C-algebra of the...

15. Schur multipliers, spectral radius and Baum-Connes conjecture

Vincent Lafforgue (CNRS et Universite Paris Cite)

2/28/25, 11:00 AM