Analysis on homogeneous spaces and operator algebras

Liste des contributions

1. Reductive groups and $C^{\ast }$-algebras: why and how

Alexandre Afgoustidis (CNRS & Institut Élie Cartan de Lorraine)

24/03/2025 09:30

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...

2. $C^{\ast }$-algebras for real reductive symmetric spaces and $K$-theory

Shintaro Nishikawa

24/03/2025 11:00

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...

10. Theta correspondence and C*-algebras

Haluk Sengun (University of Sheffield)

24/03/2025 14:00

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.

4. A trace Paley-Wiener theorem for $\mathrm{GL}(n,F)\mathrm{\setminus }\mathrm{GL}(n,E)$

Juliette Coutens

24/03/2025 15:30

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...

5. Harmonic analysis on $p$-adic spherical varieties - 1/2

Yiannis Sakellaridis (Johns Hopkins University)

25/03/2025 09:30

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...

7. On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces - 1/2

Bernhard Krötz

25/03/2025 11:00

(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.

9. Existence of discrete series for homogeneous spaces and coadjoint orbits

Yoshiki Oshima (The University of Tokyo)

25/03/2025 14:00

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...

3. Density of spherical characters

Eitan Sayag (Ben-Gurion University)

25/03/2025 15:30

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.

6. Harmonic analysis on $p$-adic spherical varieties - 2/2

Yiannis Sakellaridis (Johns Hopkins University)

26/03/2025 09:30

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...

8. On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces - 2/2

Bernhard Krötz

26/03/2025 11:00

(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.

11. Relative Langlands duality - 1/2

David Ben-Zvi (University of Texas at Austin)

27/03/2025 09:30

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...

13. Relative cuspidality and theta correspondence

Wee Teck Gan (National University of Singapore)

27/03/2025 11:00

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.

14. Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

Polyxeni Spilioti

27/03/2025 14:00

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,...

15. Lefschetz formula for locally symmetric spaces

Yanli Song

27/03/2025 15:00

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...

16. Lecture by Jeffrey Adams — The unitary dual

Jeffrey Adams (University of Maryland)

27/03/2025 16:30

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...

12. Relative Langlands duality - 2/2

David Ben-Zvi (University of Texas at Austin)

28/03/2025 09:30

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...

17. On maximally hypoelliptic differential operators

Omar Mohsen (Université Paris Diderot)

28/03/2025 11:00

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....