Analysis on homogeneous spaces and operator algebras

24 mars 2025, 09:00 → 28 mars 2025, 18:00 Europe/Paris

Amphithéâtre Hermite (Institut Henri Poincaré)

Representation Theory and Noncommutative Geometry

Workshop: Analysis on homogeneous spaces and operator algebras

March 24 to 28, 2025 - IHP, Paris

Summary

Harmonic analysis on homogeneous spaces is a fundamental area of research that simultaneously generalizes classical harmonic analysis on groups and on Riemannian symmetric spaces. It naturally relates to many areas of mathematics, playing a central role in representation theory and the theory of automorphic forms. 

    
This workshop will be an occasion to introduce recent developments in some of these areas.  It will also aim to explore new connections between them and extend the fruitful interactions between C*-algebras, harmonic analysis and representation theory beyond the classical setting of groups  to the general setting of homogeneous spaces.

Topics will include: 

• $C^{\ast }$-algebraic approaches to the tempered dual of non-riemannian symmetric spaces;
• Harmonic analysis and Plancherel theory for spherical spaces
• Connections with the Langlands program and periods of automorphic forms
• Recent approaches to the theta correspondence via $C^{\ast }$-algebras

List of speakers:

• Alexandre Afgoustidis
• David Ben-Zvi
• Juliette Coutens
• Wee Teck Gan
• Bernhard Krötz
• Omar Mohsen
• Shintaro Nishikawa
• Yoshiki Oshima
• Yiannis Sakellaridis
• Eitan Sayag
• Haluk Sengün
• Yanli Song
• Polyxeni Spilioti

Organising Committee:

• Anne-Marie Aubert
• Haluk Sengün  

 

Scientific Committee:

• Raphaël Beuzart-Plessis
• Nadya Gurevich
• Nigel Higson
• Gestur Olafsson

 

Support acknowledgement:

The workshop is organised in partnership with the Clay Mathematical Institute.

 

 

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lundi 24 mars ¶

09:00 → 09:30 Welcome coffee

09:30 → 10:30 Reductive groups and C∗-algebras: why and how¶

This talk is a review of aspects, old and new, of the connection between harmonic analysis on real reductive groups and the structure of certain $C^{\ast }$-algebras attached to them.
Topics will include the topology of the tempered dual, the Baum--Connes--Kasparov conjecture in K-theory, and the Mackey bijection between the tempered dual of a real reductive group and the unitary dual of its Cartan motion group.
The talk is intended to serve as an introduction to work in progress on similar questions for harmonic analysis on symmetric spaces. I will discuss what happens when one views the group as a symmetric space in this context. Later in the week, Shintaro Nishikawa will talk about some of our joint work, also joint with Nigel Higson, Peter Hochs and Yanli Song, about $C^{\ast }$-algebras for other symmetric spaces.

Orateur: Alexandre Afgoustidis (CNRS & Institut Élie Cartan de Lorraine)

10:30 → 11:00 Coffee break

11:00 → 12:00 C∗-algebras for real reductive symmetric spaces and K-theory¶

To a real reductive symmetric space $G/H$, we may associate one and often two $C^{\ast }$-algebras. The first corresponds to the support of the Plancherel measure for the regular representation on $L^2(G/H)$, while the second corresponds to the subset of the support consisting of irreducible representations that admit $H$-fixed distributions. The latter $C^{\ast }$-algebra exists for favorable classes of symmetric spaces.

We investigate the structure and properties of these $C^{\ast }$-algebras, leveraging the established Plancherel theory for $G/H$: the Plancherel decomposition developed by Erik van den Ban, Patrick Delorme, and Henrik Schlichtkrull, as well as the theory of discrete series representations, as studied by Flensted-Jensen, Oshima--Matsuki, Schlichtkrull, and others. We also discuss subtle aspects that seem not immediate from these results.

This is joint work with A. Afgoustidis, N. Higson, P. Hochs, and Y. Song.

Orateur: Shintaro Nishikawa

14:00 → 15:00 Theta correspondence and C*-algebras¶

I will discuss recent works with Bram Mesland and Magnus Goffeng in which we have shown that in many cases, theta correspondence, both local and global, lends itself well to a C*-algebraic formulation.

Orateur: Haluk Sengun (University of Sheffield)

15:00 → 15:30 Coffee break

15:30 → 16:30 A trace Paley-Wiener theorem for GL(n,F)∖GL(n,E)¶

This talk is related to the relative Langlands' program, which aims to extend the classical Langlands' program to spherical varieties. In the classical case, a well-known trace Paley-Wiener theorem was given by Bernstein, Deligne and Kazhdan in 1986. It gives a characterization of the functions

$$\pi \mapsto \mathrm{Tr}(\pi (f)),$$ with $G$ a reductive $p$-adic group, $\pi$ ranges over isomorphism classes of smooth irreducible representations of $G$ and $f\in C_c^{\mathrm{\infty }}(G)$. We will explain how to extend this to the relative case. That is when $E/F$ is a quadratic extension of $p$-adic fields, the theorem is a scalar Paley-Wiener theorem for relative Bessel distributions on ${\mathrm{GL}}_n(F)\mathrm{\setminus }{\mathrm{GL}}_n(E)$. These distributions are relative character of the form $$\pi \mapsto I_{\pi }(f),f\in C_c^{\mathrm{\infty }}({\mathrm{GL}}_n(E)),$$ as $\pi$ ranges over ${\mathrm{GL}}_n(F)$-distinguished irreducible tempered representations, and are constructed from a ${\mathrm{GL}}_n(F)$-invariant functional and a Whittaker functional. We will explain how by using the local Langlands correspondence, and the base-change from a unitary group, the relative characters can be described as elements of the "generic" Bernstein center of the unitary group $U(n)$.

Orateur: Juliette Coutens

mardi 25 mars ¶

09:30 → 10:30 Harmonic analysis on p-adic spherical varieties - 1/2¶

The first lecture will present a general formalism for theorems in harmonic analysis, which applies not only to spherical varieties but also to other spaces of polynomial growth (such as finite-volume quotients of semisimple Lie groups). We will discuss the variation of discrete series, asymptotic cones (boundary degenerations), and how to build the $L^2$ and Harish-Chandra Schwartz spaces out of those ingredients.

The second lecture will focus on arithmetic aspects of harmonic analysis. We will discuss how (local) $L$-functions show up in scattering operators and Plancherel densities, and conjectures about the parametrization of the spectrum by means of the "dual group" of a spherical variety.

Orateur: Yiannis Sakellaridis (Johns Hopkins University)

2025IHP.pdf

10:30 → 11:00 Group photo and coffee break

11:00 → 12:00 On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces - 1/2¶

(Joint with Kuit and Schlichtkrull) We give an overview of the Plancherel theory for Riemannian symmetric spaces $Z=G/K$. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel theorem for $Z$ can be proven from these methods.

Orateur: Bernhard Krötz

14:00 → 15:00 Existence of discrete series for homogeneous spaces and coadjoint orbits¶

When a Lie group $G$ acts transitively on a manifold $X$, an irreducible subrepresentation of the unitary representation $L^2(X)$ is called a discrete series representation of $X$. The discrete series plays an important role in the study of harmonic analysis for symmetric spaces. In this talk, we would like to give sufficient conditions for the existence of discrete series for general homogeneous spaces of real reductive groups and also for the case of equivariant line bundles in terms of coadjoint orbits.

Orateur: Yoshiki Oshima (The University of Tokyo)

15:00 → 15:30 Coffee break

15:30 → 16:30 Density of spherical characters¶

Spherical characters are distributions on homogeneous spaces that play an important role in the relative trace formula. Natural density problems regarding these distributions lead to some open problems in harmonic analysis. In a joint work with A. Aizenbud (Weizmann) and J. Bernstein (Tel-Aviv), we introduce some algebraic methods to tackle some of these density problems in the $p$-adic case.

Orateur: Eitan Sayag (Ben-Gurion University)

mercredi 26 mars ¶

09:30 → 10:30 Harmonic analysis on p-adic spherical varieties - 2/2¶

The first lecture will present a general formalism for theorems in harmonic analysis, which applies not only to spherical varieties but also to other spaces of polynomial growth (such as finite-volume quotients of semisimple Lie groups). We will discuss the variation of discrete series, asymptotic cones (boundary degenerations), and how to build the $L^2$ and Harish-Chandra Schwartz spaces out of those ingredients.

The second lecture will focus on arithmetic aspects of harmonic analysis. We will discuss how (local) $L$-functions show up in scattering operators and Plancherel densities, and conjectures about the parametrization of the spectrum by means of the "dual group" of a spherical variety.

Orateur: Yiannis Sakellaridis (Johns Hopkins University)

10:30 → 11:00 Coffee break

11:00 → 12:00 On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces - 2/2¶

(Joint with Kuit and Schlichtkrull) We give an overview of the Plancherel theory for Riemannian symmetric spaces $Z=G/K$. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel theorem for $Z$ can be proven from these methods.

Orateur: Bernhard Krötz

jeudi 27 mars ¶

09:30 → 10:30 Relative Langlands duality - 1/2¶

In these two lectures I will give an overview of the relative Langlands duality conjectures, as they are formulated in my work with Yiannis Sakellaridis and Akshay Venkatesh. The rough idea is that multiplicity-free harmonic analysis problems (spherical varieties and their variants) come in pairs associated to Langlands dual groups, so that representation theoretic questions on one side are matched with questions of spectral geometry on the other. This matching is expected to be realized in each of the settings of the Langlands program (global, local, arithmetic and geometric). The classical setting of harmonic analysis over local fields (``local, arithmetic'') is thus sandwiched between the global arithmetic setting, which pairs period integrals with special values of L-functions, and the local geometric setting, which provides instructions for building the dual geometry out of a sheaf-theoretic form of harmonic analysis.

Orateur: David Ben-Zvi (University of Texas at Austin)

10:30 → 11:00 Coffee break

11:00 → 12:00 Relative cuspidality and theta correspondence¶

In the context of the relative Langlands program, the notion of relative cuspidality was introduced by Kato and Takano based on the support of relative matrix coefficients. In this talk, I will explain how theta correspondence can be used to demonstrate relative cuspidality of non-supercuspidal distinguished representations, based on a relative character identity.

Orateur: Wee Teck Gan (National University of Singapore)

14:00 → 15:00 Resonances and residue operators for pseudo-Riemannian hyperbolic spaces¶

In this talk, we present results about resonances and residue operators for pseudo-Riemannian hyperbolic spaces. In particular, we show that for any pseudo-Riemannian hyperbolic space X, the resolvent of the Laplace--Beltrami operator can be extended meromorphically as a family of operators . Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces. This is joint work with Jan Frahm.

Orateur: Polyxeni Spilioti

15:00 → 16:00 Lefschetz formula for locally symmetric spaces¶

In this talk, we present an analogue of the Atiyah-Singer Lefschetz fixed point theorem for generalized Hecke operators acting on a locally symmetric space of finite volume. We will discuss how the invariant trace formula can be applied to the difference of two heat kernels associated with Dirac operators on locally symmetric spaces. Additionally, we will explore applications, including an extension of the Osborne-Warner multiplicity formula to discrete series representations.

Orateur: Yanli Song

16:00 → 16:30 Coffee break

16:30 → 17:30 Lecture by Jeffrey Adams — The unitary dual¶

This talk will be an update on the Atlas of Lie Groups and Representations project. I will describe an algorithm for computing the unitary dual of a real reductive group, and discuss our computer calculations of $E_7$ and (partially completed) $E_8$. Then I will discuss recent progress on proving Arthur's conjectures about the unitary of Arthur packets for real reductive groups. This work is joint with the members of the Atlas project - Lucas Mason-Brown, Stephen Miller, Marc van Leeuwen, Annegret Paul and David Vogan, as well as Dougal Davis and Kari Vilonen.

Orateur: Jeffrey Adams (University of Maryland)

vendredi 28 mars ¶

09:30 → 10:30 Relative Langlands duality - 2/2¶

In these two lectures I will give an overview of the relative Langlands duality conjectures, as they are formulated in my work with Yiannis Sakellaridis and Akshay Venkatesh. The rough idea is that multiplicity-free harmonic analysis problems (spherical varieties and their variants) come in pairs associated to Langlands dual groups, so that representation theoretic questions on one side are matched with questions of spectral geometry on the other. This matching is expected to be realized in each of the settings of the Langlands program (global, local, arithmetic and geometric). The classical setting of harmonic analysis over local fields (``local, arithmetic'') is thus sandwiched between the global arithmetic setting, which pairs period integrals with special values of L-functions, and the local geometric setting, which provides instructions for building the dual geometry out of a sheaf-theoretic form of harmonic analysis.

Orateur: David Ben-Zvi (University of Texas at Austin)

10:30 → 11:00 Coffee break

11:00 → 12:00 On maximally hypoelliptic differential operators¶

The class of maximally hypoelliptic differential operators is a large class of differential operators which contains elliptic operators as well as Hörmander’s sum of squares. I will present our work where we define a principal symbol generalising the classical principal symbol for elliptic operators which should be thought of as the analogue of the principal symbol in sub-Riemannian geometry. Our main theorem is that maximal hypoellipticity is equivalent to invertibility of our principal symbol, thus generalising the main regularity theorem for elliptic operators and confirming a conjecture of Helffer and Nourrigat. While defining our principal symbol, we will answer the question: What is the tangent space in sub-Riemman geometry in the sense of Gromov? If time permits, I will also talk about the heat kernel of maximally hypoelliptic differential operators. This is partly joint work with Androulidakis and Yuncken.

Orateur: Omar Mohsen (Université Paris Diderot)

14:00 → 17:00 Workshop over, but Bourbaki seminar may be of insterest¶