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

• Alexandre Afgoustidis (CNRS & Institut Élie Cartan de Lorraine)

Room: Amphithéâtre Hermite 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.

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

• Shintaro Nishikawa

Room: Amphithéâtre Hermite 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.

Theta correspondence and C*-algebras

• Haluk Sengun (University of Sheffield)

Room: Amphithéâtre Hermite 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.

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

• Juliette Coutens

Room: Amphithéâtre Hermite 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 $U(n)$.

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

• Yiannis Sakellaridis (Johns Hopkins University)

Room: Amphithéâtre Hermite 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.

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

• Bernhard Krötz

Room: Amphithéâtre Hermite (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.

Existence of discrete series for homogeneous spaces and coadjoint orbits

• Yoshiki Oshima (The University of Tokyo)

Room: Amphithéâtre Hermite 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.

Density of spherical characters

• Eitan Sayag (Ben-Gurion University)

Room: Amphithéâtre Hermite 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.

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

• Yiannis Sakellaridis (Johns Hopkins University)

Room: Amphithéâtre Hermite 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.

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

• Bernhard Krötz

Room: Amphithéâtre Hermite (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.

Relative Langlands duality - 1/2

• David Ben-Zvi (University of Texas at Austin)

Room: Amphithéâtre Hermite 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.

Relative cuspidality and theta correspondence

• Wee Teck Gan (National University of Singapore)

Room: Amphithéâtre Hermite 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.

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

• Polyxeni Spilioti

Room: Amphithéâtre Hermite 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.

Lefschetz formula for locally symmetric spaces

• Yanli Song

Room: Amphithéâtre Hermite 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.

Lecture by Jeffrey Adams — The unitary dual

• Jeffrey Adams (University of Maryland)

Room: Amphithéâtre Hermite 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.

Relative Langlands duality - 2/2

• David Ben-Zvi (University of Texas at Austin)

Room: Amphithéâtre Hermite 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.

On maximally hypoelliptic differential operators

• Omar Mohsen (Université Paris Diderot)

Room: Amphithéâtre Hermite 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.