Tempered representations and K-theory

Feb 24, 2025, 9:00 AM → Feb 28, 2025, 6:00 PM Europe/Paris

Amphithéâtre Hermite (Institut Henri Poincaré)

Representation Theory and Noncommutative Geometry

Workshop: Tempered representations and K-theory

February 24 to 28, 2025 - IHP, Paris

Summary

The classification of tempered irreducible representations for real reductive groups was completed in the 1970s by Knapp and Zuckerman, following Harish-Chandra's work on the Plancherel formula. But some aspects of the subject are now undergoing a re-examination, following the discovery of new perspectives. C*-algebras and K-theory are valuable tools in Representation Theory, as shown, for instance, by the Mackey bijection. Indeed, it was the Connes-Kasparov isomorphism in K-theory that motivated the search for a natural bijection between the tempered dual of a real reductive group and the unitary dual of its Cartan motion group, as initially suggested by Mackey in the 1970s. 

The meeting will focus on recent developments in which K-theoretic ideas have offered new perspectives on the tempered dual for reductive groups or symmetric spaces, and conversely on new approaches to operator-algebraic problems using contemporary tools in representation theory. 

Topics will include: 

• New approaches to the Mackey bijection through pseudodifferential operator theory, which has itself undergone an extensive conceptual redesign in the past decade, thanks again to $C^*$-algebra $K$-theory connections.
• New perspectives on the the Connes--Kasparov isomorphism using Dirac cohomology and cohomological induction;
• Higher orbital intergrals, which make it possible to go beyond the ``noncommutative topology of the tempered dual'', hinting at something like the ``differential geometry'' of this noncommutative space.
• Study of the Casselman--Schwartz algebras and their K-theory via Paley--Wiener theorems, and connections with the Connes--Kasparov isomorphism.
• $C^*$-algebraic analysis of the tempered dual from the point of view of $G$ as a symmetric space for $G{\times}G$, and more generally of the tempered spectrum of symmetric spaces.

 

Speakers:

• Anne-Marie Aubert
• Bachir Bekka
• Yves Benoist
• Jacob Bradd
• Mikael de la Salle
• Axel Gastaldi
• Peter Hochs
• Jing-Song Huang
• Vincent Lafforgue
• Ada Masters
• Roger Plymen
• Angel Román
• Maarten Solleveld
• Xiang Tang
• Robert Yuncken

Organising Committee:

• Alexandre Afgoustidis 
• Pierre Clare 

 

Scientific Committee:

• Tyrone Crisp
• Claire Debord
• Toshiyuki Kobayashi
• Hang Wang

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Monday, February 24

9:00 AM Welcome coffee

1 Biunimodular functions

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 symplectic geometry.

Speaker: Yves Benoist (Université Paris-Saclay)

10:30 AM Coffee break

2 Strong convergence of unitary representations

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 and the work of Haagerup and Thorbjornsen where is it called strong convergence, and which involves convergence of operator norms. I will present examples, counterexamples, applications, and many questions. Based on joint works with Michael Magee from Durham.

Speaker: Mikael de la Salle (CNRS, Institut Camille Jordan)

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

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 $p\neq2$.

Speaker: Bachir Bekka (Université de Rennes)

4:00 PM Coffee break

4 Mackey analogy in periodic cyclic homology

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 crossed product associated to a maximal compact subgroup. In this way we extend the results established by V. Nistor in the early 90' and provide a Mackey analogy in this framework.

Speaker: Axel Gastaldi (Aix-Marseille Université)

Tuesday, February 25

5 A higher index for finite-volume locally symmetric spaces

Let $G$ be a connected real semisimple Lie group, and $K

Speaker: Peter Hochs (Radboud University)

10:30 AM Coffee break

6 Functoriality via Dirac cohomology

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.

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

7 On the crystallisation of semisimple Lie groups

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 easy understanding of basic problems like tensor decompositions and branching rules. In this talk, I will explain a dual phenomenon, namely the crystallisation of the algebra of functions on a compact Lie group.
By the quantum duality principle, this has implications for unitary representations of complex semisimple Lie groups. If time permits, we will discuss the case of $\operatorname{SL}(2,\mathbb{C})$.

(Joint work with Marco Matassa.)

Speaker: Robert Yuncken (Université de Lorraine)

3:30 PM Coffee break

8 Conformal geometry and group actions in unbounded KK-theory

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 when considering group actions on the 'patient zero' of noncommutative geometry, the Dirac spectral triple of a Riemannian manifold. In the bounded picture, conformal group actions are allowed but, in the unbounded picture, only isometries are naturally permitted. This extra freedom in the bounded picture is quite consequential; for instance for Kasparov's construction of the $\gamma$-element for the Lorentz groups. I will discuss a general framework solving this problem, making use of a new multiplicative perturbation theory for abstract differential operators. I will also briefly explain how the same technology can be used to analyse the $\mathrm{SL}_q(2)$-equivariance of the Podle\'s quantum sphere and to give new meaning to examples of twisted spectral triples in the literature. This is joint work with Adam Rennie and appears in a recent preprint.

Speaker: Ms Ada Masters (University of Wollongong)

Wednesday, February 26

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

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 will show that Vogan's $\lambda$-map plays a central role in the large time asymptotics analysis about the trace of the heat kernel. This talk is based on joint works with Shu Shen and Yanli Song.

Speaker: Xiang Tang (Washington University in St. Louis)

10:30 AM Coffee break

10 Paley-Wiener for the Casselman--Wallach algebra and the Oka principle

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 Connes-Kasparov isomorphism (in the formulation I use) fits into an "Oka principle" philosophy for Baum-Connes (which started with Bost), and I will discuss further ideas in this direction.

Speaker: Jacob Bradd (Pennsylvania State University)

Thursday, February 27

11 K-Theory and Langlands duality

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 equivariant Chern character of Baum-Connes, we will compute this K-theory for $\mathrm{SU}(n)$ and the exceptional Lie group $E_6$. As an application, we will compute the K-theory of the Iwahori-spherical C-algebra of the $p$-adic version of $E_6$. The spectrum of this C*-algebra comprises irreducible tempered representations of $E_6$ which admit a nonzero Iwahori-fixed vector. From the point of view of noncommutative geometry, we are computing the K-theory of this spectrum.

Speaker: Roger Plymen

10:30 AM Coffee break

12 Topological K-theory of reductive $p$-adic groups

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 conjectures by Aubert-Baum-Plymen-Solleveld.

Speaker: Maarten Solleveld (Radboud Universiteit Nijmegen)

13 Tempered representations with unipotent parahoric restriction: a noncommutative geometry viewpoint

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 representations. Next, I will describe the conjecture in the case of tempered representations with unipotent parahoric restriction, focusing on its links with the generalized Springer correspondence for complex disconnected groups and its K-theoretical aspects. As an application, I will show how it can be used to describe the theta correspondence.

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

4:00 PM Coffee break

14 Topos and Noncommutative Geometry: Two Perspectives on Space and Numbers

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 noncommutative space. The originality of toposes can similarly be perceived through the intuitionistic logic associated with a topos. It is the metric structure, embodied by a representation—as operators in Hilbert space—of coordinates and the length element, that allows noncommutative geometry to engage with reality, namely the structure of space-time at the infinitesimally small scale as revealed by contemporary physics through the Standard Model. As for toposes, it is the additional structure of a sheaf of algebras that enables geometry to manifest beyond topology.

In the second part of the talk, I will explain how the spectrum of the ring of integers can be understood through these two geometric lenses. The connection between these two approaches rests on an extension of class field theory that sheds light on the analogy established by Mumford and Mazur between knots and prime numbers. The spectral perception of the ring of integers naturally emerges from the study of the zeros of the Riemann zeta function, thereby revealing deep structures at the interface of arithmetic, topology, and geometry.

Speaker: Alain Connes (IHES)

Friday, February 28

15 Mackey embedding for reduced group C*-algebras

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 reductive group. This can then be used to characterize the Mackey bijection. We shall discuss the case of the complex reductive group before proceeding to discuss the difficulty behind the construction for a real reductive group.

Speaker: Angel Roman (Washington University in St Louis)

10:30 AM Coffee break

16 Schur multipliers, spectral radius and Baum-Connes conjecture

Speaker: Vincent Lafforgue (CNRS et Universite Paris Cite)