Introduction to Representation Theory, 1/3

• Birgit Speh (Cornell University)

Room: Amphitheater Darboux The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading. Lecture 1. We will introduce discuss the basic important concepts: Admissible representations, differentiable vectors, equivalence of representations, irreducible representations, unitary representations, (g, K)-modules. Time permitting we will also introduce Cartan subgroups, the Iwasawa decomposition and parabolic subgroups.

Branching Problems in Representation Theory, 1/3

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches. Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods. In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible. A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Different Aspects of Rankin–Cohen Operators, 1/2 (online)

• Michael Pevzner

Room: Amphitheater Darboux Rankin-Cohen brackets provide a fundamental example of “non-elementary” differential symmetry breaking operators. We will explore the combinatorial structure of these operators through multiple perspectives and discuss several approaches to find their higher dimensions analogues within the representation-theoretical perspective of branching problems.

Introduction to Representation Theory, 2/3

• Birgit Speh (Cornell University)

Room: Amphitheater Darboux The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading. Lecture 2. We will discuss the representations of SU(2), SL(2, R) and of SL(2, C).

Different Aspects of Rankin–Cohen Operators, 2/2 (online)

• Michael Pevzner

Room: Amphitheater Darboux Rankin-Cohen brackets provide a fundamental example of “non-elementary” differential symmetry breaking operators. We will explore the combinatorial structure of these operators through multiple perspectives and discuss several approaches to find their higher dimensions analogues within the representation-theoretical perspective of branching problems.

Branching Problems in Representation Theory, 2/3

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches. Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods. In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible. A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Differential Symmetry Breaking Operators and F-method

• Toshihisa Kubo (Ryukoku University)

Room: Amphitheater Darboux Let X be a smooth manifold and Y a smooth submanifold of X. Take G'⊂G to be a pair of Lie groups that act on Y⊂X, respectively. We call a differential operator D between the space of smooth sections for a G-equivariant vector bundle over X and that for a G'-equivariant vector bundle over Y a differential symmetry breaking operator (differential SBO for short) if D is G'-intertwining. The F-method is a powerful machinery on the classification and construction of differential SBOs. It allows one to classify and construct differential SBOs by solving a certain system of partial differential equations. The key idea is the Fourier transform of the generalized Verma modules. In this talk we shall overview the F-method in an elementary manner.

Branching Laws and a Generalized Penrose Transform (online)

• Hideko Sekiguchi (University of Tokyo)

Room: Amphitheater Darboux I plan to discuss a generalized Radon-Penrose transform and provide examples of explicit branching laws from the perspective of differential equations.

Introduction to Representation Theory, 3/3

• Birgit Speh (Cornell University)

Room: Amphitheater Darboux The 3 lectures are introduction to representation theory of reductive real Lie groups at the level of the book ”Representation theory of semisimple Lie groups: An overview based on examples” by A.W. Knapp. All the concepts and theorems will be illustrated by examples, but I will give references for the detailed proofs. In each lecture I will give some suggested homework and reading. Lecture 3. We will discuss principal series representations, intertwining operators and the Langlands classification of irreducible representations.

Geometric and Analytic Aspects of Branching Laws, 1/2

• Bent Ørsted (Aarhus University, Denmark)

Room: Amphitheater Darboux In these lectures will be explained some of the basic mathematical theory of group representations, mostly of reductive Lie groups, and their structure via the theory of branching problems and intertwining operators.

Geometric and Analytic Aspects of Branching Laws, 2/2

• Bent Ørsted (Aarhus University, Denmark)

Room: Amphitheater Darboux In these lectures will be explained some of the basic mathematical theory of group representations, mostly of reductive Lie groups, and their structure via the theory of branching problems and intertwining operators.

Branching Problems in Representation Theory, 3/3

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux The concept of symmetries naturally arises in various areas of mathematics and science, including geometry, number theory, differential equations, and quantum mechanics. The more symmetries an object possesses, the better we can understand it through group-theoretic approaches. Branching problems investigate how large symmetries break down into smaller ones, such as fusion rules, using mathematical formulations based on the language of representations and their restrictions. These problems have been studied for over 80 years. In recent years, there has been a surge of research focused on the restriction of continuous symmetries in infinite-dimensional cases, leading to the development of new geometric and analytic methods. In my three lectures, I plan to provide an introduction to the branching problems of infinite-dimensional representations of real reductive groups, such as GL(n, R), using plenty of elementary examples to make the basic concepts and key ideas more accessible. A major goal of my lectures is to capitalize on this momentum and bring together young students and postdocs from diverse mathematical disciplines, encouraging them to develop an interest in this promising field. If time permits, I will also present some open questions in the area.

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

Workshop: Intertwining operators and geometry Room: Amphithéâtre Hermite See: https://indico.math.cnrs.fr/event/10856/timetable

The Baum-Connes conjecture for real reductive groups 1/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux K-theory for C*-algebras A second review of K-theory for C*-algebras: Bott periodicity, stability and long-exact sequences. Really basic C*-algebras: characterization and K-theory. Continuous fields of C*-algebras, and the K-theory groups of a continuous field. Mapping cone continuous fields. Open subsets of the spectrum of a C*-algebra and ideals. Examples related to SL(2,R).

The Baum-Connes conjecture for real reductive groups 2/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux The Cartan motion group and the Mackey deformation family The Cartan motion group. The unitary dual and the C*-algebra of the motion group. Structure in the unitary dual from the restricted root system. A filtration of the group C*-algebra by ideals. The deformation construction associated to a Lie group and a maximal compact subgroup. The associated continuous field of group C*-algebras. The Connes-Kasparov morphism in C*-algebra K-theory. Relation to the Baum-Connes assembly map. The example of SL(2,R).

Introduction to the Baum-Connes conjecture 1/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux The Baum-Connes conjecture in a nutshell A corollary: the Kaplansky-Kadison conjecture on idempotents

Introduction to the Baum-Connes conjecture 2/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux K-theory for C*-algebras, again K_0, K_1, stability, Bott periodicity, 6-terms exact sequence

Cuntz’ K-theoretic amenability revisited

• Pierre Julg (Université d'Orléans)

Room: Amphitheater Darboux In the early 1980’s, the question was raised of comparing the K-theory groups of the full and reduced C*-algebras of a group $G$. J.Cuntz has described a condition (K-amenability) implying that they are isomorphic. Note that K-amenability is incompatible with Kazhdan’s property T, and is implied by the Haagerup property, a strong negation of property T. In this talk we shall explain that Cuntz’ condition relies on the construction of a $G$-Fredhom module. We shall give such a module in the example of $\mathrm{SL}_2$ on the fields of $p$-adics (Julg-Valette), of complex numbers (Kasparov) and of real numbers (Fox-Haskel and Julg-Kasparov).

Introduction to the Baum-Connes conjecture 3/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Kasparov theory Hilbert C*-modules, operators, compact operators; KK-cycles

Introduction to the Baum-Connes conjecture 4/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Equivariant Kasparov theory KK_G-cycles, the ring R(G), the descent map

Index theory for manifolds with corners

• Florian Thiry (Institut de mathématiques de Toulouse)

Room: Amphitheater Darboux In this talk we define the boundary index of a singular manifold in a general C*-algebraic framework following the Lescure and Carrillo Rouse approach. Then we explain how it encodes geometrical obstruction for symbols to be associated to nicer operators (Fredholm operators in the classical case). Eventually we extend Lescure and Carrillo Rouse's strategies to compute the associated K-theory groups in the case of families of manifolds with corners using iterated blow-ups.

Introduction to the Baum-Connes conjecture 5/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Proper actions and the classifying space for proper actions

Introduction to the Baum-Connes conjecture 6/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux The assembly map and the BC conjecture (with and without coefficients)

The Baum-Connes conjecture for real reductive groups 3/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux The imaginary part of the infinitesimal character Definition of the infinitesimal character. The infinitesimal character of a parabolically induced representation. Continuity of the infinitesimal character. The imaginary part of the infinitesimal character. A filtration of the reduced C*-algebra of a real reductive group by ideals. Some low-real-rank examples. A theorem of Bruhat and Harish-Chandra about irreducible parabolic induction. Characterization of the fibers of the imaginary part of the infinitesimal character using tempiric representations. Discrete series representations versus tempiric representations. Characterization of the subquotients for the filtration of the reduced group C*-algebra by ideals. The Mackey bijection.

The Baum-Connes conjecture for real reductive groups 4/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux More on the discrete series Harish-Chandra’s Schwartz space. Tempered representations. The Harish-Chandra-Langlands principle. Asymptotic expansions of matrix coefficients and idempotents in the reduced group C*-algebra. Speculations about a C*-algebraic approach to the discrete series and the Harish-Chandra principle.

The Baum-Connes conjecture for real reductive groups 5/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Proof of the Bruhat-Harish-Chandra theorem Distributions from intertwiners between parabolically induced representations. Bruhat decomposition. Bounding the dimensions of the spaces of equivariant distributions. Proof of the theorem.

The Baum-Connes conjecture for real reductive groups 6/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux The Connes-Kasparov isomorphism from the perspective of representation theory The Mackey deformation family and continuity of the rescaled imaginary part of the infinitesimal character. A filtration of the deformation continuous field by ideals. Rescaling automorphisms, embedding morphisms, and computation of the subquotient continuous fields. Reformulation of the Connes-Kasparov isomorphism. Vogan’s theorem.

Introduction to the Baum-Connes conjecture 7/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Status of the BC conjecture

Introduction to the Baum-Connes conjecture 8/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux BC vs CK

Introduction to the Baum-Connes conjecture 9/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux The "Dirac-dual Dirac" method and the gamma-element

Introduction to the Baum-Connes conjecture 10/10

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux Nishikawa's recent approach

Admissibility of irreducible representations

• Michael Cowling (University of New South Wales)

Room: Amphitheater Darboux This is an account of joint work with Francesca Astengo (Genova) and Bianca Di Blasio (Milano Bicocca). Harish-Chandra showed that irreducible unitary representations of semisimple Lie groups are admissible; these are easier to understand than general representations. But, as Pierre Julg and others have shown, there are good reasons to look at representations that are not unitary, so it is natural to ask whether ``all'' irreducible representations on (say) Banach spaces are admissible. We show that the answer to this question is closely related to a classical question in functional analysis, the invariant subspace problem. https://sites.google.com/view/julgfest

Groupoids as cyclic sets

• Georges Skandalis (Université Paris Diderot)

Room: Amphitheater Darboux Associated to a groupoid is a natural cyclic set. Actually, one can give several characterisations of groupoids based on this set. It also gives a nice interpretation of well known notions of on groupoids: symplectic groupoids, duals of VB groupoids and in particular the Weinstein groupoid... Joint work with Claire Debord. https://sites.google.com/view/julgfest

Reciprocal hyperbolic elements in $\operatorname{PSL}_2(\mathbb{Z})$

• Alain Valette (Université de Neuchâtel)

Room: Amphitheater Darboux An element $A$ in $\operatorname{PSL}_2(\mathbb{Z})$ is hyperbolic if $|\operatorname{Tr}(A)|>2$. The maximal virtually abelian subgroup of $\operatorname{PSL}_2(\mathbb{Z})$ containing $A$ is either infinite cyclic or infinite dihedral; say that $A$ is reciprocal if the second case happens ($A$ is then conjugate to its inverse). We give a characterization of reciprocal hyperbolic elements in $\operatorname{PSL}_2(\mathbb{Z})$ in terms of the continued fractions of their fixed points in $\operatorname{P}^1(\mathbb{R})$ (those are quadratic surds). Doing so we revisit results of P. Sarnak (2007) and C.-L. Simon (2022), themselves rooted in classical work by Gauss and Fricke \& Klein. https://sites.google.com/view/julgfest

Tempiric representations, Connes-Kasparov, and pseudodifferential operators on symmetric spaces

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Let $G$ be a real reductive group, connected for simplicity, with maximal compact subgroup $K$. The Connes-Kasparov isomorphism attaches a single parameter to most, but not all, of the components of the tempered dual of a real reductive group. That parameter is a shifted version of a highest weight for $K$, and every such parameter is attached to a unique component in the tempered dual. On the other hand, Vogan’s theory of minimal $K$-types associates to every component of the tempered dual of $G$ a finite collection of highest weights for $K$. These collections are disjoint, and every highest weight appears in one of them. Most of Vogan’s collections are singletons, but not all of them. The Connes-Kasparov and Vogan correspondences seem to be more similar than they are different, and in the online precursor to the current thematic program at IHP, Vogan asked whether they can be reconciled by adjusting the definition of the reduced C*-algebra of $G$? I shall discuss one answer, involving pseudodifferential operators, and some of the new questions that arise from that answer. This is joint work with Peter Debello. https://sites.google.com/view/julgfest

On pseudodifferential calculi for subelliptic operators

• Robert Yuncken (Université de Lorraine)

Room: Amphitheater Darboux Many constructions in representation theory and index theory are based upon elliptic differential operators. The work of Julg and Kasparov shows that subelliptic differential operators, such as the Berstein-Gelfand-Gelfand complex, are often required to make certain key constructions. In this talk, we will describe the tangent groupoid associated a very general class of such operators, namely operators of Helffer-Nourrigat type. Generalising ideas of Connes and Debord-Skandalis then allows us to prove the analytic properties of such operators. (Joint work with Iakovos Androulidakis, Omar Mohsen and Erik Van Erp.) https://sites.google.com/view/julgfest

Index theory on manifolds with a tangent Lie structure

• Gennadi Kasparov (Vanderbilt University)

Room: Amphitheater Darboux In recent years there was a significant progress in the theory of pseudo-differential operators on filtered manifolds. I will introduce in my talk a wider class of manifolds which I call manifolds with a tangent Lie structure. I will explain a coarse approach to pseudo-differential theory which gives a simplified pseudo-differential calculus containing only operators of order 0 and negative order. This calculus easily leads to the Atiyah-Singer type index theorem for operators of order 0 on manifolds with a tangent Lie structure. For filtered manifolds this calculus agrees with the known H\"ormander and van Erp - Yuncken calculi, which allows to extend the index theorem to operators of any order. https://sites.google.com/view/julgfest

From class field theory to zeta spectral triples

• Alain Connes (IHES)

Room: Amphitheater Darboux J'exposerai les résultats récents (< 1 an) de ma collaboration avec C. Consani et H. Moscovici sur la compréhension conceptuelle de l'espace des classes d'adèles comme extension de la théorie du corps de classe et la construction de triplets spectraux donnant les zéros de zêta https://sites.google.com/view/julgfest

The Baum-Connes conjecture for real reductive groups 7/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Kasparov’s gamma element for SL(2,R) Practical person’s guide to the Baum-Connes conjecture from the point of view of Kasparov theory: Kasparov’s representation ring (as an abelian group), the gamma element, Kasparov’s theorem for connected Lie groups, reduction to $\gamma=1$. First examples where $\gamma$ is equal to 1. Examples when $\gamma$ is not equal to 1. Julg’s proposal regarding uniformly bounded representations. Kasparov’s homotopy for SL(2,R).

The Baum-Connes conjecture for real reductive groups 8/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Lafforgue’s Banach version of KK-theory The C*-algebra of a group, versus other convolution Banach algebras. Homotopies of unitary representations versus homotopies of Banach representations. Examples and action of the reduced C*-algebra on $L^p$ spaces. Review of the role of compact operators in Kasparov theory. Dual pairs of Banach spaces and operators of compact type. Lafforgue’s version of Kasparov’s representation ring.

The Baum-Connes conjecture for real reductive groups 9/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Banach modifications of the gamma element The problem with Banach convolution algebras from the point of view of representation theory. The problem with tensor products and unconditional norms. Unconditional norms. Karoubi density. Reduction of the Baum-Connes conjecture to $\gamma=1$. A further technical reduction. The rapid decay property for the Harish-Chandra Schwartz algebra.

The Baum-Connes conjecture for real reductive groups 10/10

• Nigel Higson (Penn State University)

Room: Amphitheater Darboux Rapid decay property and Lafforgue’s proof of the Baum-Connes conjecture The gamma element from Witten’s perturbation of the de Rham complex. Transfer to weighted $L^2$ spaces. A homotopy from the gamma element to 1, almost. Putting everything together.

The noncommutative geometry of the space of positive elements in a C*-algebra

• Roberto Steven Vargas Peña (Estudiante de Doctorado)

Room: Amphitheater Darboux This talk delves into the fascinating differential geometry of positive elements of a $C^*$-algebra, highlighting its complexity, principal results, and depth. It also proposes an intriguing connection between this geometry and Connes' non-commutative approach, opening up new avenues for exploration in this field.

Quantum groups meet graphs

• Pegah Pournajafi (Collège de France)

Room: Amphitheater Darboux Quantum groups and graph theory may seem like distant fields, yet intriguing connections arise when they intersect. After a gentle introduction to the topic, we explore these interactions through the study of quantum automorphism groups of block graphs. We conclude with some open problems and future directions. These results are parts of a common project with Amaury Freslon and Paul Meunier.

First-order differential calculi and Laplacians on $q$-deformations of compact semisimple Lie groups

• Heon Lee (Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China)

Room: Amphitheater Darboux In this talk, we suggest a simple definition of Laplacian on a compact quantum group (CQG) associated with a first-order differential calculus (FODC) on it. Applied to the classical differential calculus on a compact Lie group, this definition yields classical Laplacians, as it should. Moreover, on the CQG $K_q$ arising from the $q$-deformation of a compact semisimple Lie group $K$, we can find many interesting linear operators that satisfy this definition, which converge to a classical Laplacian on $K$ as $q$ tends to 1. In light of this, we call them $q$-Laplacians on $K_q$ and investigate some of their operator theoretic properties. This work is based on the preprint arXiv:2410.00720.

Symmetry Breaking under Translations

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux Joint IHP - AIM RTNCG lecture Abstract: I will discuss symmetry breaking for pairs of real forms of (GL(n, C), GL(n − 1, C)). We introduce the concept of "fences for the interlacing pattern," which refines the usual notion of "walls for Weyl chambers." We then present a theorem stating that multiplicity remains constant unless we cross these "fences." This approach is illustrated with examples of both tempered and non-tempered representations, along with a non-vanishing theorem of period integrals for pairs of reductive symmetric spaces.

Unitary representations and spectral synthesis on contraction groups

• Max Carter (Université catholique de Louvain)

Room: Amphitheater Darboux The solution to Hilbert’s fifth problem dictates that every connected locally compact group is an inverse limit of connected Lie groups, and thus Lie theoretic techniques can be used in the study of connected locally compact groups. In contrast, totally disconnected locally compact (tdlc) groups cannot be approximated by Lie groups or algebraic group over tdlc fields, and their structure is not well understood, generally speaking. Modern research in topological group theory is largely focussed on understanding the class of tdlc groups. Furthermore, our understanding of the (unitary) representation theory and harmonic analysis of tdlc groups lags far behind that of connected locally compact groups. In this talk, I will discuss recent progress on the harmonic analysis of (tdlc) contraction groups. Contraction groups can be viewed, in some sense, as analogues / generalisations of unipotent groups in the theory of tdlc groups. The focus will be on discussing recent progress on the unitary representation theory of these groups, and time pending, I will say a bit about spectral synthesis of certain convolution algebras on these groups.

Lie groupoids and differential operators

• Paul Le Breton (Paris cité)

Room: Amphitheater Darboux In numerous PDE problems, differential operators naturally appear with some geometrical constraints. For instance, if $(M, \mathcal{F})$ is a foliated manifold one may be interested in operators acting longitudinally on the leaves, if $M$ has borders (or corners) one may ask operators to be well-behaved around the borders, on a manifold endowed with an action of a Lie group $G$ one may consider $G$-equivariant operators... The aim of this talk is to explain how Lie groupoids provide a unified framework to study the analytic behaviour of all of these operators. This point of view allows to adapt the notions of principal symbols and ellipticity to these contexts and to define a well behaved pseudodifferential calculus in order to build parametrix. One can then use these operators to build classes in the K-theory of the $C^*$ algebras of the groupoids, as a first step towards Atiyah-Singer type theorems.

Deformation to the Normal Cone, zoom action and distributions

• Moudrik Chamoux

Room: Amphitheater Darboux The $C^*$-analysis of Lie groupoids provides powerful tools for understanding analysis on smooth manifolds and Lie groups. Deformations to the Normal Cone appear in the construction of fundamental instances of such Lie groupoids. In this talk we will discuss deformations to the normal cone, the zoom action and homogeneous distributions on a DNC. If time permits, we will also discuss applications to analysis on manifolds such as the pseudodifferential calculus \emph{à la} Van Erp and Yuncken.

Automorphisms of pseudodifferential operators on Lie groupoids

• Lucas Lemoine (Université Paris-Est Créteil)

Room: Amphitheater Darboux The study of pseudodifferential operators on Lie groupoids offers a natural extension of classical analysis on smooth manifolds, encompassing singular spaces. Moreover, pseudodifferential operators can be generalized in the classical setting by Fourier integral operators. A key result in this context is Egorov’s theorem: conjugation by a Fourier integral operator preserves pseudodifferential operators. Duistermaat and Singer later established a converse: any automorphism of the filtered algebra of classical pseudodifferential operators can be realized as conjugation by a Fourier integral operator. In this talk, I will discuss a generalization of this result in the setting of Lie groupoids. This involves a detailed examination of the symplectic structure of the cotangent groupoid, which is fundamental for constructing Fourier integral operators in this context.

Conférence carte blanche « Quand la symétrie prédit »

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

This talk will be delivered in French at Maison Henri Poincaré. Entrance is free of charge, but registration is mandatory, at: https://billetterie-maison-poincare.ihp.fr/evenements/conferences-cartes-blanche/quand-la-symetrie-predit Dans la vie courante comme en sciences, l'idée de symétrie évoque l’harmonie, l'équilibre. Depuis deux siècles, en mathématiques, les symétries se quantifient : des outils mathématiques précis, les groupes, permettent de mieux comprendre les symétries qui nous sont familières et d’en découvrir de nouveaux types. Depuis un siècle, on sait utiliser ces outils pour aller au-delà de la contemplation de l’harmonie : les groupes, et ce qu’on appelle leurs représentations, ont permis de faire des prédictions concrètes et inattendues dans des domaines variés — de la physique quantique à la théorie des nombres. Cette conférence évoque quelques épisodes de l’histoire des représentations de groupes, et esquisse quelques espoirs que les scientifiques d’aujourd’hui placent en elles.

Structure of tempered homogeneous spaces 1/3 - Dynamical approach

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux The three lectures introduce recent theories of tempered spaces, and I plan to provide an overview of these topics, using plenty of elementary examples to make the basic concepts and key ideas more accessible. In the first lecture, I will review basic concepts such as tempered unitary representations of real reductive groups, like GL(n, R), as well as “tempered spaces” and “tempered subgroups”. I will begin with some geometric observations of group actions, including the properness criterion for reductive homogeneous spaces. Subsequently, I will introduce a “quantification” of proper actions and incorporate a dynamical approach into analytic representation theory, including the temperedness criterion for homogeneous spaces, which was developed recently by Y. Benoist and the speaker, drawing on the Cowling-Haagerup-Howe theory and other related ideas.

The Dolbeault-Dirac operator on the irreducible quantum flag manifolds

• Fredy Diaz Garcia (National Autonomous University of Mexico)

Room: Amphitheater Darboux In this talk I will comment about some aspects in the construction of the Dolbeault-Dirac operator d+d* associated to some type of quantum homogeneous spaces generalizing the classical construction of the Rham complex of smooth manifolds. I will introduce a quantum version of the Bernstein-Gelfand-Gelfand resolution of irreducible quantum flag manifolds in order to dualize it in some way to get the Dolbeault complex and define the Dolbeault-Dirac operator. If time permits, I will give the example of the irreducible quantum flag of type B_2 for which the Dolbeault-Dirac operator leads to a spectral triple in the sense of Connes, also I will give some comments on the case of the quantum Grassmanian Gr(2,4) which is an ongoing project joint with E. Wagner.

Parthasarathy formulas and beyond

• Nicolas Prudhon (université de Lorraine)

Room: Amphitheater Darboux Joint IHP - AIM RTNCG lecture Abstract: Dirac operators have played an important role in representation theory, originating in the work Parthasarathy, who proved a formula for the square of a Dirac operator on a Riemannian symmetric space, and used it to realize discrete series representations of semi simple Lie groups as L2 spaces of harmonic spinors. Later on, Huang and Pandzic, observing that the Parthasarathy formula implies that the square of a Dirac operator is central, developed the theory of Dirac cohomology, and derived property of admissible representations that may not be discrete series. In this talk we will first illustrates these results using Parthasarathy formula, to explain how Vogan classification of tempered representations by their lowest K types, fits into the computation the G index of the Dirac operator. We will next introduce an analogue of the Dirac operator for symplectic homogeneous spaces, and state a Parthasarathy formula in this context, when an invariant polarization exists and a symmetry condition holds. Generalizations of the Parthasarathy formula were proved by Kostant, introducing a cubic term, when the quadratic homogeneous space is not symmetric. We will show that no cubic terms may lead to any corresponding formula in the case of the symplectic Dirac operator.

Twisted Ruelle zeta functions, complex-valued analytic torsion and the Fried's conjecture - 1/3

• Polyxeni Spilioti

Room: Amphitheater Darboux In this mini course, we will introduce the dynamical zeta functions of Ruelle and Selberg for hyperbolic manifolds or more generally for locally symmetric spaces of rank one. We will present the so called Fried 's conjecture, i.e., a conjectural equality between the Ruelle zeta function at zero and a spectral invariant, the analytic torsion. In the second part of the mini course, we will talk about this problem, in the setting of a non-unitary representation of the lattice we consider. We will see how one can use the trace formula to study this problem and also we will present the definition of the refined analytic torsion and the Cappell-Miller torsion. Time depending, we will discuss some further problems related with these objects.

Structure of tempered homogeneous spaces 2/3 - Combinatorics approach

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux The criterion for tempered spaces, explained in the first lecture, is computable. In this lecture, I will explain how this criterion leads to the classification theory of non-tempered reductive homogeneous spaces by breaking it down into several steps. The technical methods used in the second lecture differ from the dynamical approach presented in the first lecture. Our approach relies on elementary results from finite-dimensional representations and some combinatorics of convex polyhedral cones.

Twisted Ruelle zeta functions, complex-valued analytic torsion and the Fried's conjecture - 2/3

• Polyxeni Spilioti

Room: Amphitheater Darboux In this mini course, we will introduce the dynamical zeta functions of Ruelle and Selberg for hyperbolic manifolds or more generally for locally symmetric spaces of rank one. We will present the so called Fried 's conjecture, i.e., a conjectural equality between the Ruelle zeta function at zero and a spectral invariant, the analytic torsion. In the second part of the mini course, we will talk about this problem, in the setting of a non-unitary representation of the lattice we consider. We will see how one can use the trace formula to study this problem and also we will present the definition of the refined analytic torsion and the Cappell-Miller torsion. Time depending, we will discuss some further problems related with these objects.

Twisted Ruelle zeta functions, complex-valued analytic torsion and the Fried's conjecture - 3/3

• Polyxeni Spilioti

Room: Amphitheater Darboux In this mini course, we will introduce the dynamical zeta functions of Ruelle and Selberg for hyperbolic manifolds or more generally for locally symmetric spaces of rank one. We will present the so called Fried 's conjecture, i.e., a conjectural equality between the Ruelle zeta function at zero and a spectral invariant, the analytic torsion. In the second part of the mini course, we will talk about this problem, in the setting of a non-unitary representation of the lattice we consider. We will see how one can use the trace formula to study this problem and also we will present the definition of the refined analytic torsion and the Cappell-Miller torsion. Time depending, we will discuss some further problems related with these objects.

Structure of tempered homogeneous spaces 3/3 - Limit algebras

• Toshiyuki Kobayashi (The University of Tokyo)

Room: Amphitheater Darboux Recently, surprising and intriguing connections have been observed between the concept of "tempered spaces" for unitary representations and various other areas of mathematics. In this lecture, we will explore different aspects of tempered spaces from the perspectives of topology and geometry, including limit algebras (collapsing Lie algebras) and geometric quantization.

Workshop: Tempered representations and K-theory Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10857

Workshop: Tempered representations and K-theory Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10857

Workshop: Tempered representations and K-theory Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10857/

Workshop: Tempered representations and K-theory Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10857/

Workshop: Tempered representations and K-theory Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10857/

Workshop: Tempered representations and K-theory Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10857/

Workshop: Tempered representations and K-theory Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10857/

Topos and noncommutative geometry: Two perspectives on spaces and numbers

• Alain Connes (IHES)

Room: Amphithéâtre Hermite 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

Workshop: Tempered representations and K-theory Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10857/

Mini-course: The Bernstein center - 1/3

• David Renard (Ecole Polytechnique)

Room: Amphitheater Darboux In this series of lectures, we will explain J. Bernstein's description of the center of the category of smooth representations of reductive $p$-adic groups as rational functions on the variety of irreducible representations. The course is intended for PhD students and non-specialists.

$\ell$-Modular blocks of $\mathrm{SL}_n$

• Peiyi Cui (Morningside Center of Mathematics)

Room: Amphitheater Darboux Reduction to depth zero is a promising approach for understanding $\ell$-modular blocks of $p$-adic groups when $\ell$ differs from $p$. In this talk, I will introduce $\ell$-modular blocks of $\mathrm{SL}_n$ from this perspective. We will explore the technical challenges in associating an $\ell$-modular block with a depth-zero block and consider a natural candidate for this potential connection. Toward the end, I will also discuss related topics on both the automorphic and Galois sides.

Representations of $p$-adic groups in arbitrary residue characteristic

• Jessica Fintzen (University of Bonn)

Room: Amphitheater Darboux This talk is an introduction to the representation theory of p-adic groups aimed at all participants of the trimester program. I will survey the current state of the art and include new developments of the last six months. We will focus on two crucial aspects: First, I will provide an overview of the construction of all so called supercuspidal representations, which are the building blocks for all representations. For more than 20 years it remained open to extend Yu's general construction to the case p=2, and I will sketch what makes this case so special and how we could overcome the obstacles in my recent joint work with David Schwein. Second, we will study the structure of the whole category of representations of p-adic groups in terms of these supercuspidal representations, and I will explain how two recent preprints with Jeffrey Adler, Manish Mishra and Kazuma Ohara allow us to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or are at least easier to achieve.

Bounding Harish-Chandra characters

• Anna Szumowicz (IMPAN)

Room: Amphitheater Darboux Let $G$ be a connected reductive algebraic group over a $p$-adic local field $F$. We study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular semisimple element of $G(F)$ as $\pi$ varies among supercuspidal representations of $G(F)$. Kim, Shin and Templier conjectured that $\frac{\theta_{\pi}(\gamma)}{\deg(\pi)}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $G(F)$ whose central character is unitary and the formal degree of $\pi$ tends to infinity. I will sketch the proof that for $G$ semisimple the trace character is uniformly bounded on $\gamma$ under the assumption, which is believed to hold in general, that all irreducible supercuspidal representations of $G(F)$ are compactly induced from an open CVcompact modulo center subgroup. If time allows I could also discuss progress on optimizing the bound.

$p$-modular Iwahori-Hecke algebras and their simple modules for the $p$-adic metaplectic group $\widetilde{\mathrm{SL}}_2(F)$

• Ramla ABDELLATIF (LAMFA - Université de Picardie Jules Verne)

Room: Amphitheater Darboux

Asymptotics, mostly for $SL_2$

• Guy Henniart (Université Paris-Saclay)

Room: Amphitheater Darboux Let F be a non Archimedean local field and let p be its residue characteristic. Let R be an algebraically closed field with char(R) different from p. We consider a connected reductive group G over F, and look at irreducible smooth R-representations of the locally pro-p group G=G(F). Classifying all such representations up to isomorphism is very involved, but their behaviour under restriction to small enough open pro-p subgroups is expected to be more uniform. When R=C and char(F)=0, that can be obtained from Harish-Chandra's germ expansion. One can hope for a similar control for general R and char(F)>=0. This has consequences for the asymptotics of fixed points under congruence subgroups of G. We shall survey what is known for G=GL(N) and G=SL(2) (Joint work with Vignéras) and what it suggests for the general case.

Mini-course: The Bernstein center - 2/3

• David Renard (Ecole Polytechnique)

Room: Amphitheater Darboux In this series of lectures, we will explain J. Bernstein's description of the center of the category of smooth representations of reductive $p$-adic groups as rational functions on the variety of irreducible representations. The course is intended for PhD students and non-specialists.

Mini-course: The Bernstein center - 3/3

• David Renard (Ecole Polytechnique)

Room: Amphitheater Darboux In this series of lectures, we will explain J. Bernstein's description of the center of the category of smooth representations of reductive $p$-adic groups as rational functions on the variety of irreducible representations. The course is intended for PhD students and non-specialists.

On the unitary dual of a $p$-adic group and the Aubert involution

• Darija Brajković Zorić (School of Applied Mathematics and Informatics, Josip Juraj Strossmayer University of Osijek)

Room: Amphitheater Darboux A significant aspect of representation theory research is determination of the unitary dual of a reductive algebraic group over a local non-archimedean field. We will look at the unitary dual of $p$-adic group $SO(7)$ with support on minimal parabolic subgroup. There is the conjecture stating that the Aubert involution preserves unitarity confirmed the conjecture for that case. This work is supported (in part) by the Croatian Science Foundation under the project number HRZZ-IP-2022-10-4615.

A multiplicity one theorem for general Spin groups

• Melissa Emory (Oklahoma State University)

Room: Amphitheater Darboux There are many ingredients needed to prove a local Gan-Gross-Prasad conjecture. In this talk, we will focus on the first ingredient: a multiplicity at most one theorem. Aizenbud, Gourevitch, Rallis and Schiffmann proved a multiplicity at most one theorem for restrictions of irreducible representations of certain p-adic classical groups and Waldspurger proved the same theorem for the special orthogonal groups. We will discuss work that establishes a multiplicity at most one theorem for restrictions of irreducible representations for a non-classical group, the general spin group. This is joint work with Shuichiro Takeda.

Non-basic rigid packets for discrete L-parameters

• David Schwein (University of Bonn)

Room: Amphitheater Darboux In this talk I'll propose a new way to organize discrete compound L-packets of a p-adic group G. Our packets include representations not only of inner twists of G but also of its elliptic twisted Levi subgroups, conjecturally realizing some version of loop Deligne–Lusztig induction or Yu's construction. The main idea is to replace Kottwitz's set with the non-basic cohomology of the Kaletha gerbe. This new cohomology set is much richer but turns out to carry many of the same structures as B(G), such as the Kottwitz and Newton maps. On the Galois side, we define a rigid enhancement for L-parameters and show that the resulting refined local Langlands conjectures reduce, à la Bertoloni Meli–Oi, to the basic rigid conjectures of Kaletha. This work is joint with Peter Dillery.

Mini-course: Hochschild homology of reductive $p$-adic groups - 1/3

• Maarten Solleveld (Radboud Universiteit Nijmegen)

Room: Amphitheater Darboux The goal of this mini-course is to explain a description of the Hochschild homology of the Hecke algebra of a reductive $p$-adic group, in spectral terms related to the Bernstein center. The lectures will start quite general, and will gradually specialize towards $p$-adic groups. No prior knowledge of Hochschild homology is required. Lecture 1: Introduction to Hochschild homology and twisted crossed product algebras Lecture 2: Representation theory and Hochschild homology of graded Hecke algebras Lecture 3: Hochschild homology of algebras associated to $p$-adic groups

Mini-course: Hochschild homology of reductive $p$-adic groups - 2/3

• Maarten Solleveld (Radboud Universiteit Nijmegen)

Room: Amphitheater Darboux The goal of this mini-course is to explain a description of the Hochschild homology of the Hecke algebra of a reductive $p$-adic group, in spectral terms related to the Bernstein center. The lectures will start quite general, and will gradually specialize towards $p$-adic groups. No prior knowledge of Hochschild homology is required. Lecture 1: Introduction to Hochschild homology and twisted crossed product algebras Lecture 2: Representation theory and Hochschild homology of graded Hecke algebras Lecture 3: Hochschild homology of algebras associated to $p$-adic groups

Cuspidal $l$-modular representations of $p$-adic $GL(n)$ distinguished by a Galois involution

• vincent secherre (Université de Versailles St-Quentin)

Room: Amphitheater Darboux Let $E/F$ be a quadratic extension of $p$-adic fields. A smooth representation of ${\rm GL}_n(E)$ is said to be distinguished by ${\rm GL}_n(F)$ if it carries a non-zero ${\rm GL}_n(F)$-invariant linear form. Distinguished complex representations have been extensively studied: there is in particular a full classification of distinguished generic complex representations. The case of $l$-modular representations (that is, with coefficients in a field whose characteristic is a prime number $l$ different from $p$) is much less well understood. In this talk, I will discuss the case of cuspidal $l$-modular representations (for $p$ odd). This is a joint work with Robert Kurinczuk and Nadir Matringe.

Endoscopic character identities for depth-zero supercuspidal representations of SL_2

• Roger Plymen

Room: Amphitheater Darboux We consider the depth-zero supercuspidal $L$-packets of $\mathrm{SL}_2$. With the aid of the classical character formulas of Sally-Shalika, we prove the endoscopic character identities. Our account fills a gap in the extensive literature on $\mathrm{SL}_2$. Joint work with Anne-Marie Aubert, see arXiv:2410.20183v2

Mini-course: Hochschild homology of reductive $p$-adic groups - 3/3

• Maarten Solleveld (Radboud Universiteit Nijmegen)

Room: Amphitheater Darboux The goal of this mini-course is to explain a description of the Hochschild homology of the Hecke algebra of a reductive $p$-adic group, in spectral terms related to the Bernstein center. The lectures will start quite general, and will gradually specialize towards $p$-adic groups. No prior knowledge of Hochschild homology is required. Lecture 1: Introduction to Hochschild homology and twisted crossed product algebras Lecture 2: Representation theory and Hochschild homology of graded Hecke algebras Lecture 3: Hochschild homology of algebras associated to $p$-adic groups

Rankin-Selberg integrals and double flag varieties

• Lei Zhang (National University of Singapore)

Room: Amphitheater Darboux The Rankin–Selberg method is a classical approach for constructing integral representations of L-functions, facilitating their meromorphic continuation and the establishment of functional equations. In this talk, we present a novel perspective that interprets Rankin–Selberg integrals within the framework of double flag varieties of finite type. We introduce the concept of strongly tempered spherical varieties of Rankin–Selberg type and, leveraging the classification by He, Nishiyama, Ochiai, and Oshima, systematically identify all Rankin–Selberg integrals associated with these varieties. This framework enables a unified treatment of local zeta integrals and the gamma factors for this family. As an application, we can derive complete L-functions corresponding to these Rankin–Selberg integrals and establish local multiplicity formulas via the local relative trace formula for certain strongly tempered spherical varieties of Gan-Gross-Prasad type over p-adic fields.

Geometric analogues to local Arthur packets: $p$-adic classical groups

• Mishty Ray (University of Calgary)

Room: Amphitheater Darboux Arthur packets are sets containing irreps that form constituents of important classes of automorphic representations. ABV-packets, named after Jeff Adams, Dan Barbasch, and David Vogan, are proposed generalizations to local Arthur packets for archimedean groups. They arise out of a geometric perspective on the local Langlands correspondence involving microlocal analysis on moduli spaces of Langlands parameters. Vogan gave an analogous description for the non-archimedean case, which was then adapted by Cunningham et al. The proposal that an ABV-packet for a Langlands parameter of Arthur-type coincides with the corresponding local Arthur packet is stated as Vogan's conjecture by Cunningham et al in their adaptation. This talk aims to report on the progress of Vogan's conjecture for $p$-adic general linear, odd special orthogonal, and symplectic groups.

Progress on the local Gan-Gross-Prasad conjecture

• Cheng Chen (University of Minnesota, Twin Cities)

Room: Amphitheater Darboux The classical branching problem studies the restriction of an irreducible representation to a compact subgroup. The works of Gross-Prasad and Gan-Gross-Prasad generalized this framework into a conjecture for classical groups over local fields of characteristic zero. The first breakthrough was achieved by Waldspurger in the non-Archimedean special orthogonal cases. Since then, various approaches have been developed, leading to the complete proof of the conjecture in all cases. In this presentation, I will introduce an approach that applies to both unitary and non-unitary cases, Archimedean and non-Archimedean, as well as Bessel and Fourier-Jacobi cases. This approach, based on the foundational works of Waldspurger, Mœglin-Waldspurger, and Gan-Ichino, utilizes the trace formula, endoscopy, the multiplicity formula, and the Theta correspondence. The development of this approach includes some of my work, as well as collaborative efforts with Luo, work in collaboration with Chen and Zou, and work in collaboration with Jiang, Liu, and Zhang.

Mini-course: The theta correspondence for classical groups over finite fields of odd characteristic 1/3

• Anne-Marie Aubert (IMJ-PRG CNRS)

Room: Amphitheater Darboux The theta correspondence for classical groups over finite fields of odd characteristic is defined via the Weil representation of a finite symplectic group. It maps a given irreducible representation of a group $G$ to a finite collection of irreducible representations of a group $G'$, such that $(G,G')$ is a dual pair. I will first recall the definition of the correspondence and some its useful properties. Next, I will provide a full and explicit description of the theta correspondence via a conjecture that I have formulated, and proved in the case of linear and unitary dual pairs, with J. Michel and R. Rouquier, and which has been recently established in all cases by Pan and by Ma, Qiu and Zou, independently. I will later explain several ways to extract a bijective correspondence, including the construction by Gurevich and Howe of the eta correspondence, and give some applications. I will also describe the links between the theta correspondence and the finite analogue of the Gan-Gross-Prasad problem, which can be viewed as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh.

Mini-course: Minimal representations and applications 1/3

• Gordan Savin (University of Utah)

Room: Amphitheater Darboux I will define minimal representations of $p$-adic groups and explain how one can work with them. As an application, using minimal representations, I will decompose certain Weil representations attached to multiplicity-free symplectic representations as defined and classified by Knop. These results are consistent with predictions in the relative Langlands program of Ben-Zvi, Sakellaridis and Venkatesh.

Theta Correspondence between principle series and its geometrization

• JIAJUN MA (Xiamen University)

Room: Amphitheater Darboux Consider the reductive dual pairs consisting of a symplectic group and an orthogonal group. The Harish-Chandra series of these two groups correspond to each other under the theta correspondence. In collaboration with Congling Qiu and Jialiang Zou, we explicitly computed this correspondence by analyzing the relevant Hecke algebra bimodules and applying a Tits deformation argument. This approach provides an alternative proof of Aubert-Michel-Rouquier's conjecture, which was initially settled by Shu-Yen Pan. In this presentation, we explore the geometrization of this construction. Consequently, we have derived a new description of the theta correspondence in terms of the Springer theory. This is a joint work with Congling Qiu, Jialiang Zou, and Zhiwei Yun.

Modular local theta correspondence

• Justin Trias (University of East Anglia)

Room: Amphitheater Darboux The local theta correspondence over a non-Archimedean local field of residual characteristic p asserts a bijection between (subsets of) irreducible complex representations of two reductive groups forming a dual pair in a symplectic group. In this talk, I will explain how this theory can be generalized to $l$-modular representations — i.e. when the coefficient field has positive characteristic $l$, distinct from $p$. Provided that l is sufficiently large relative to the size of the dual pair, this generalisation also results in a bijection, which we refer to as the $l$-modular theta correspondence. However, for certain values of~$l$, such a bijection fails to hold.

Theta correspondence as an induction functor for group C*-algebras

• Haluk Sengun (University of Sheffield)

Room: Amphitheater Darboux In recent works, we have exhibited a C*-algebraic approach to theta correspondence. I will try to explain this approach, its connection to Jian-Shu Li's explicit theta lifting method and will try to convince representation theorists that such an approach is interesting. Joint works with B. Mesland (Leiden) and M. Goffeng (Lund).

Mini-course: The theta correspondence for classical groups over finite fields of odd characteristic 2/3

• Anne-Marie Aubert (IMJ-PRG CNRS)

Room: Amphitheater Darboux The theta correspondence for classical groups over finite fields of odd characteristic is defined via the Weil representation of a finite symplectic group. It maps a given irreducible representation of a group $G$ to a finite collection of irreducible representations of a group $G'$, such that $(G,G')$ is a dual pair. I will first recall the definition of the correspondence and some its useful properties. Next, I will provide a full and explicit description of the theta correspondence via a conjecture that I have formulated, and proved in the case of linear and unitary dual pairs, with J. Michel and R. Rouquier, and which has been recently established in all cases by Pan and by Ma, Qiu and Zou, independently. I will later explain several ways to extract a bijective correspondence, including the construction by Gurevich and Howe of the eta correspondence, and give some applications. I will also describe the links between the theta correspondence and the finite analogue of the Gan-Gross-Prasad problem, which can be viewed as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh.

Mini-course: Minimal representations and applications 2/3

• Gordan Savin (University of Utah)

Room: Amphitheater Darboux I will define minimal representations of $p$-adic groups and explain how one can work with them. As an application, using minimal representations, I will decompose certain Weil representations attached to multiplicity-free symplectic representations as defined and classified by Knop. These results are consistent with predictions in the relative Langlands program of Ben-Zvi, Sakellaridis and Venkatesh.

New supercuspidal representations from the Weil representation in characteristic two

• David Schwein (University of Bonn)

Room: Amphitheater Darboux Supercuspidal representations are the basic building blocks in the representation theory of reductive p-adic groups, and there is intense interest in constructing these representations explicitly. The construction problem faces many additional obstacles in residue characteristic two, however, and even for classical groups, our knowledge there is incomplete. In this talk, based on joint work with Jessica Fintzen, I'll explain how to overcome one such obstacle: the exceptional behavior of the Weil representation in characteristic two.

Automorphic kernel and Langlands conjecture

• Dihua Jiang (School of Mathematics, University of Minnesota)

Room: Amphitheater Darboux I will discuss constructions of automorphic kernel functions, with which the integral transforms will conjecturally yield the analytic properties of automorphic L-functions and the certain functoriality as expected by the Langlands conjectures. The known cases includes the theta correspondence and certain Rankin-Selberg global zeta integrals".

Mini-course: The theta correspondence for classical groups over finite fields of odd characteristic 3/3

• Anne-Marie Aubert (IMJ-PRG CNRS)

Room: Amphitheater Darboux The theta correspondence for classical groups over finite fields of odd characteristic is defined via the Weil representation of a finite symplectic group. It maps a given irreducible representation of a group $G$ to a finite collection of irreducible representations of a group $G'$, such that $(G,G')$ is a dual pair. I will first recall the definition of the correspondence and some its useful properties. Next, I will provide a full and explicit description of the theta correspondence via a conjecture that I have formulated, and proved in the case of linear and unitary dual pairs, with J. Michel and R. Rouquier, and which has been recently established in all cases by Pan and by Ma, Qiu and Zou, independently. I will later explain several ways to extract a bijective correspondence, including the construction by Gurevich and Howe of the eta correspondence, and give some applications. I will also describe the links between the theta correspondence and the finite analogue of the Gan-Gross-Prasad problem, which can be viewed as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh.

Mini-course: Local theta correspondence 1/3

• Chen-Bo Zhu (National University of Singapore)

Room: Amphitheater Darboux This mini-course on local theta correspondence will be mainly concerned with the Archimedean theory, and will consists of three lectures. In the first lecture, I will explain its basic theory, including the Howe duality theorem, and the conservation relations. I will also discuss some examples. In the second lecture, I will explain how some of the basic invariants of representations behave under the local theta correspondence. The final lecture will be about applications to unitary representation theory.

The Weil representation for a finite field of characteristic two

• Aurélie Paull (Université de Lorraine)

Room: Amphitheater Darboux The Weil representation is relatively well understood for local fields or finite fields of odd characteristics. In characteristic two, even in the case of a finite field, it is still not understood. In this talk, we will present an explicit construction of the Weil representation for a finite field of characteristic two. After defining the Heisenberg group in this setting, we construct the Weil representation of a two-fold covering of the pseudo-symplectic group. We obtain explicit formulas for this representation, its character and the associated cocycle. The pseudo-symplectic group is an extension of the orthogonal group, which is smaller than the symplectic group. Therefore, for the field $\mathbb{F}_2$, using the ring $\mathbb{Z}/4\mathbb{Z}$, we will provide a second construction of the Weil representation, defined on a covering of the affine symplectic group. All along, we will illustrate our results with the example of a two-dimensional vector space over $\mathbb{F}_2$.

The theta correspondence over $p$-adic fields in terms of Bernstein blocks of enhanced $L$-parameters

• Anne-Marie Aubert (IMJ-PRG CNRS)

Room: Amphitheater Darboux I will describe a partition, obtained with A. Moussaoui and M. Solleveld, of enhanced $L$-parameters for $p$-adic reductive groups in -- Galois analogues of -- Bernstein blocks. Next, in the case of classical groups, I will show the compatibility of the theta correspondence with this partition, and how it allow to obtain an explicit description of the correspondence.

Mini-course: Local theta correspondence 2/3

• Chen-Bo Zhu (National University of Singapore)

Room: Amphitheater Darboux This mini-course on local theta correspondence will be mainly concerned with the Archimedean theory, and will consists of three lectures. In the first lecture, I will explain its basic theory, including the Howe duality theorem, and the conservation relations. I will also discuss some examples. In the second lecture, I will explain how some of the basic invariants of representations behave under the local theta correspondence. The final lecture will be about applications to unitary representation theory.

Mini-course: Minimal representations and applications 3/3

• Gordan Savin (University of Utah)

Room: Amphitheater Darboux I will define minimal representations of $p$-adic groups and explain how one can work with them. As an application, using minimal representations, I will decompose certain Weil representations attached to multiplicity-free symplectic representations as defined and classified by Knop. These results are consistent with predictions in the relative Langlands program of Ben-Zvi, Sakellaridis and Venkatesh.

Symmetry breaking operators and the Howe correspondence for dual pairs with one member compact

• Angela Pasquale (Institut Elie Cartan de Lorraine (IECL, UMR 7502), Metz)

Room: Amphitheater Darboux We consider Howe correspondence for dual pairs with one member compact. Given an irreducible representation $\Pi$ of the compact member, we study the projection operator $P_\Pi$ of the Weil representation onto its $\Pi$-isotopic component. These operators are examples of symmetry breaking operators in the sense of Kobayashi. Guided by Hermann Weyl’s integration formula, we study orbital integrals on the underlying symplectic space. They are needed to compute the Weyl symbols of $P_\Pi$. Our main example will be the dual pair of two compact unitary groups. In this case, we recover by purely analytic methods the existence of Howe correspondence for compact unitary groups, or equivalently, Weyl’s First Fundamental Theorem for Classical Invariant Theory for complex general linear groups. This is joint work with Mark McKee and Tomasz Przebinda (University of Oklahoma).

Unipotent Arthur packets for real classical groups

• Binyong Sun (Zhejiang University)

Room: Amphitheater Darboux Suppose that $G$ is a real symplectic group or a real even special orthogonal group. We explicitly construct all representations in an arbitrary unipotent Arthur packet of $G$, by using parabolic induction and iterated theta lift. The talk is based on a joint work with Dan Barbasch, Jia-Jun Ma and Chen-Bo Zhu, and a joint work with Taiwang Deng, Chang Huang, and Bin Xu.

Howe duality for Lie superalgebras

• Allan Merino (Simmons University)

Room: Amphitheater Darboux As suggested in "Remarks on classical invariant theory”, the Howe duality in the symplectic group should be extendable to the orthosymplectic supergroup, with the Weil representation replaced by the spinor-oscillator representation of the supergroup SpO. In my talk, I will explain what is known about the duality in the supercase, present some of the results obtained recently with H. Salmasian, and give an explicit description of the duality for the dual pair $({\mathrm{SpO}}(2n|1), {\mathrm{osp}}(2|2))$.

Mini-course: Local theta correspondence 3/3

• Chen-Bo Zhu (National University of Singapore)

Room: Amphitheater Darboux This mini-course on local theta correspondence will be mainly concerned with the Archimedean theory, and will consists of three lectures. In the first lecture, I will explain its basic theory, including the Howe duality theorem, and the conservation relations. I will also discuss some examples. In the second lecture, I will explain how some of the basic invariants of representations behave under the local theta correspondence. The final lecture will be about applications to unitary representation theory.

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 1/4 (pt.1)

• Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Room: Amphitheater Darboux Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied. We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$. Lecture 1: Nets of operator algebras and AQFT. We start with the translation from nets of operator algebras to nets of real subspaces, based on modular theory. We introduce real standard subspaces, discuss the Tomita-Takesaki Theorem as a key result from the modular theory of operator algebras and then describe axioms for nets of real subspaces ${\sf H}({\mathcal O})$ in a unitary representation of a Lie group. These are structures than can be explored completely from the perspective of the geometry of homogeneous spaces and unitary representations. Lecture notes are available under: https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 1/4 (pt.2)

• Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Room: Amphitheater Darboux Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied. We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$. Lecture 1: Nets of operator algebras and AQFT. We start with the translation from nets of operator algebras to nets of real subspaces, based on modular theory. We introduce real standard subspaces, discuss the Tomita-Takesaki Theorem as a key result from the modular theory of operator algebras and then describe axioms for nets of real subspaces ${\sf H}({\mathcal O})$ in a unitary representation of a Lie group. These are structures than can be explored completely from the perspective of the geometry of homogeneous spaces and unitary representations. Lecture notes are available under: https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Problem session

• Tobias Simon (Friedrich-Alexander Universität Erlangen-Nürnberg (FAU))

Room: Amphitheater Darboux This session will be devoted to exercises and complements related to the mini-course "Algebraic Quantum Field Theory and causal homogeneous spaces" by K.-H. Neeb.

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 2/4 (pt.1)

• Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Room: Amphitheater Darboux Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied. We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$. Lecture 2: Euler elements and causal homogeneous spaces. We explore which specific structures we need on the homogeneous space $M = G/H$ and the Lie group $G$, so that a rich supply of nets may exist. In particular, we explain how Euler elements of Lie algebras (elements defining 3-gradings) enter the picture as candidates of generators of modular groups. This leads to several families of causal homogeneous spaces such as compactly and non-compactly causal symmetric spaces and causal flag manifolds. Lecture notes are available under: https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 2/4 (pt.2)

• Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Room: Amphitheater Darboux Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied. We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$. Lecture 2: Euler elements and causal homogeneous spaces. We explore which specific structures we need on the homogeneous space $M = G/H$ and the Lie group $G$, so that a rich supply of nets may exist. In particular, we explain how Euler elements of Lie algebras (elements defining 3-gradings) enter the picture as candidates of generators of modular groups. This leads to several families of causal homogeneous spaces such as compactly and non-compactly causal symmetric spaces and causal flag manifolds. Lecture notes are available under: https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Problem session

• Tobias Simon (Friedrich-Alexander Universität Erlangen-Nürnberg (FAU))

Room: Amphitheater Darboux This session will be devoted to exercises and complements related to the mini-course "Algebraic Quantum Field Theory and causal homogeneous spaces" by K.-H. Neeb.

Standard subspaces and distribution vectors

• Gestur Olafsson (Louisiana State University)

Room: Amphitheater Darboux Standard subspaces in a Hilbert space are closely related to the modular and Tomita--Takesaki theory theory, two cornerstones in Algebraic Quantum Field Theory (AQFT). In this talk we start with a short overview of the theory of standard subspaces and the connection to the Tomita--Takesaki theory. We will then discuss Euler elements and causal symmetric spaces and then connect those two subjects via representation theory, analytic continuation to the crown and then limit processes to realize standard subspaces in spaces of distribution vectors.

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 3/4 (pt.1)

• Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Room: Amphitheater Darboux Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied. We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$. Lecture 3: Analytic continuation of orbit maps and crown domains. The construction of interesting nets of real subspaces rests on the existence of holomorphic extension of orbit maps in unitary representations. For semisimple groups, complex crowns of Riemannian symmetric spaces $G/K$ provide a natural context for this extension process. We explain how this can be set up on general Lie groups whose Lie algebra contains an Euler element. Lecture notes are available under: https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 3/4 (pt.2)

• Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Room: Amphitheater Darboux Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied. We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$. Lecture 3: Analytic continuation of orbit maps and crown domains. The construction of interesting nets of real subspaces rests on the existence of holomorphic extension of orbit maps in unitary representations. For semisimple groups, complex crowns of Riemannian symmetric spaces $G/K$ provide a natural context for this extension process. We explain how this can be set up on general Lie groups whose Lie algebra contains an Euler element. Lecture notes are available under: https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Problem session

• Tobias Simon (Friedrich-Alexander Universität Erlangen-Nürnberg (FAU))

Room: Amphitheater Darboux This session will be devoted to exercises and complements related to the mini-course "Algebraic Quantum Field Theory and causal homogeneous spaces" by K.-H. Neeb.

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 4/4 (pt.1)

• Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Room: Amphitheater Darboux Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied. We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$. Lecture 4: Constructing nets of real subspaces. Finally, we arrive at rather general characterizations of unitary representations and homogeneous spaces for which a rich supply of nets exists. Many classification results are still open and more bridges to Physics have to be built, but the overall structure of the theory takes shape. Lecture notes are available under: https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Mini-course: Algebraic Quantum Field Theory and causal homogeneous spaces 4/4 (pt.2)

• Karl-Hermann Neeb (Friedrich-Alexander-University Erlangen-Nuremberg)

Room: Amphitheater Darboux Lorentzian manifolds and their conformal compactifications provide the most symmetric models of spacetimes. The structures studied on such spaces in Algebraic Quantum Field Theory (AQFT) are so-called nets of operator algebras, i.e., to each open subset ${\mathcal O}$ of the space-time manifold one associates a von Neumann algebra ${\mathcal M}({\mathcal O})$ in such a way that a certain natural list of axioms is satisfied. We report on an ongoing project concerned with the construction of such nets on general causal homogeneous spaces $M = G/H$. Lecture 4: Constructing nets of real subspaces. Finally, we arrive at rather general characterizations of unitary representations and homogeneous spaces for which a rich supply of nets exists. Many classification results are still open and more bridges to Physics have to be built, but the overall structure of the theory takes shape. Lecture notes are available under: https://en.www.math.fau.de/wp-content/uploads/sites/3/2025/03/qft-lect.pdf

Workshop: Analysis on homogeneous spaces and operator algebras See: https://indico.math.cnrs.fr/event/10858/

Workshop: Analysis on homogeneous spaces and operator algebras Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10858/

Workshop: Analysis on homogeneous spaces and operator algebras Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10858/

Workshop: Analysis on homogeneous spaces and operator algebras Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10858/

Workshop: Analysis on homogeneous spaces and operator algebras Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10858/

Workshop: Analysis on homogeneous spaces and operator algebras Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10858/

Workshop: Analysis on homogeneous spaces and operator algebras Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10858/

Workshop: Analysis on homogeneous spaces and operator algebras Room: Amphitheater Darboux See: https://indico.math.cnrs.fr/event/10858/