The Whittaker Plancherel formula for a real reductive group $G$ concerns the unitary direct integral decomposition of the induced representation $L^2(G/N: \chi).$ Here $N$ is an Iwasawa $N$ and $\chi$ is a regular unitary character of $N.$

The Plancherel decomposition in terms of Whittaker integrals was explicitly described in an announcement by Harish-Chandra in 1982. The precise details of this description are now available in the posthumous Volume 5 of his collected works. The Maass-Selberg relations play a crucial role, but the suggested proof of them is very scarce in details. In the talk, I will describe a proof by reduction of the basic case of adjacent maximal parabolics which was inspired by the theory for symmetric spaces. I will also describe Harish-Chandra’s beautiful proof for the basic case, which I was able to reconstruct from the text in the posthumous volume.

The compactified Ruijsenaars–Schneider systems are completely integrable Hamiltonian systems on smooth moduli spaces of flat SU(n) connections on the one-holed torus. The monodromy around the hole is constrained to the smallest non-central conjugacy classes of SU(n), which can be labelled by a parameter $0<x<1$. Depending on the value of x, the systems arise in two drastically different forms: in type (i) these are toric systems, while in the type (ii) cases they possess globally continuous action variables that generate a Hamiltonian torus action (only) on a dense open subset of the phase space of dimension 2(n − 1). After reviewing background material, we report our recent study of the momentum polytopes and the fibers of the action map (alias the momentum map) that are contained in the complement of the domain of the densely defined torus action in the type (ii) cases. We have shown that these fibers are smooth connected isotropic submanifolds, diffeomorphic to the 3-dimensional sphere in the simplest cases, similarly to the fibers occurring, e.g., in the classical Gelfand-Cetlin systems.

The talk is based on joint work with Holger Dullin (https://arxiv.org/abs/2604.18023) and earlier publications.

Nichols algebras emerged first in various studies of Hopf algebras, but are used in the meanwhile also elsewhere. In the talk I will first discuss established structural results connecting Nichols algebras and Lie theory. Then I will explain some recent advances regarding the classification of Nichols algebras over groups, as well as some characterizations of noetherianity and finite Gelfand-Kirillov dimension.

We discuss a functorial framework for the convergence of Drinfeld’s Universal Deformation Formula on spaces of analytic or entire vectors. Algebraically, this type of deformation is based on a Drinfeld twist

$F = \sum_{n=0}^{\infty} \hbar^n \cdot F_n \in \bigl( \operatorname{U} (\mathfrak{g}) \otimes \operatorname{U} (\mathfrak{g}) \bigr)[[\hbar]]$

It constitutes a distinguished, and rather elusive, formal power series with coefficients in two copies of the universal enveloping algebra $\operatorname{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$. Drinfeld’s principal idea is then that the defining axioms of a twist induce formal deformations of any associative algebra our Lie algebra $\mathfrak{g}$ acts on by derivations. Equipping the representation space with a locally convex topology, we overcome the formal character of this deformation, i.e. pass from a formal parameter $\hbar$ to a complex number $\hbar\in \mathbb{C}$. This is achieved by matching an equicontinuity condition on the action of the components F_n of the twist with the order of analytic vectors of the representation. Finally, we demonstrate the effectiveness of our machinery by applying it to the explicit non-abelian Drinfeld twists constructed by Giaquinto and Zhang.

This is joint work with Chiara Esposito and Stefan Waldmann.

The Lie algebra of divergence-free vector fields is one of the four classical infinite-dimensional Lie algebras studied by Élie Cartan, the other three being the Lie algebras of symplectic vector fields, of contact vector fields, and the Lie algebra of all vector fields. In much the same way as Hamiltonian vector fields are the commutator subalgebra of the Lie algebra of symplectic vector fields, the exact divergence-free vector fields are the commutator subalgebra of the Lie algebra of divergence-free vector fields. We show that the Lie algebra of exact divergence-free vector fields has a universal central extension: the Lie algebra of (n-2)-forms modulo exact forms — this is joint work with Leonid Ryvkin and Cornelia Vizman. If time permits I will also comment on central extensions of the other three classical Lie algebras of Cartan.

The smooth deformations of an ideal $\mathfrak{i}$ in a Lie algebra $\mathfrak{g}$ differentiate to cohomology classes in the cohomology of $\mathfrak{g}$ with values in its adjoint representation on $\operatorname{Hom}(\mathfrak{i}, \mathfrak{g}/\mathfrak{i})$. This cohomology associated with the ideal $\mathfrak{i}\trianglelefteq\mathfrak{g}$ is compared with other Lie algebra cohomologies defined by $\mathfrak{i}$, such as the cohomology defined by $\mathfrak{i}$ as a Lie subalgebra of $\mathfrak{g}$ in (Richardson 1969), and the cohomology defined by the Lie algebra epimorphism $\mathfrak{g} \to \mathfrak{g}/\mathfrak{i}$.

After a choice of complement of the ideal $\mathfrak{i}$ in the Lie algebra $\mathfrak{g}$, its deformation complex is enriched to the differential graded Lie algebra that controls its deformations, in the sense that its Maurer-Cartan elements are in one-to-one correspondence with the (small) deformations of the ideal. Furthermore, the $L_{\infty}$-algebra that simultaneously controls the deformations of $\mathfrak{i}$ and of the ambient Lie bracket is identified. Under appropriate assumptions on the low degrees of the deformation cohomology of a given Lie ideal, the (topological) rigidity and  stability of ideals are  described, as well as obstructions to deformations of ideals of Lie algebras.

This is a joint work with Ilias Ermeidis.

We consider the family of Gibbs probability distributions introduced by Souriau in the context of statistical mechanics, on coadjoint orbits of Lie groups. The parameter spaces are convex domains in the Lie algebra and are naturally equipped with a Riemannian metric: the Fisher-Rao metric.

For the nilpotent orbits of $\mathfrak{sl}(2, \mathbb{R})$, we show that the resulting Riemannian geometry is isometric to the symmetric cone $\operatorname{Sym}_2^{++}(\mathbb{R})$. According to a construction by Koszul, $\operatorname{Sym}_{2n}^{++}(\mathbb{R})$ is also the natural parameter space for a different family of probability distributions, for which the symmetric metric is again the Fisher-Rao metric. We explain how this correspondence extends to the minimal nilpotent orbits of $\mathfrak{sp}(2n, \mathbb{R})$ and the symmetric cones $\operatorname{Sym}_{2n}^{++}(\mathbb{R}).$

Based on a joint work with Pierre Bieliavsky and Guillaume Neuttiens.

Based on the methods developed in the framework of the small quantum group, a description of Hochschild cohomology of the restricted unrolled quantum group at a root of unity can be derived. The unrolled quantum group presents interest in particular because of the relation with representation category over a certain vertex algebra, as pointed out in the work of Arakawa, Creutzig and Kawasetsu. In particular I will discuss the structure of the center and its action on the weight representations of the restricted unrolled quantum group in the case $\mathfrak{g}=\mathfrak{sl}_2.$

A theorem of Monod-Py states that unlike finite dimensional settings, there are continuous irreducible representations of PO(1,n) in PO(1,∞). These representations are called exotic. In this talk, we will use the Debin-Fillastre hyperbolic model for convex bodies to give a different construction of non-totally geodesic quasi-isometric embedding of the hyperbolic plane in the infinite dimensional hyperbolic space, which will induce an exotic representation of PSL(2,$\mathbb{R}$). With the classical results from convex geometry, we show that this exotic representation is convex-cocompact, and that the quotient of the minimal PSL(2,$\mathbb{R}$)-invariant closed convex set by its action is homeomorphic to the oriented Banach-Mazur compactum. Moreover, we also compute the dimension of this convex set and the Hausdorff dimension of the PSL(2,$\mathbb{R}$)-limit set.

This is a joint work with François Fillastre and David Xu.

This talk is centered around our longtime project with Karl-Hermann Neeb and others on homogeneous spaces, representation theory and ideas derived from Algebraic Quantum Field Theory. We start with short discussion on causal symmetric spaces and how they lead naturally to Lie algebra elements with eigenvalues $\{0,\pm 1\}$. Those elements play a special role in the construction of wedges in the symmetric spaces, real subspaces in representation theory, the connection to the Tomita–Takesaki modular theory and realization of representations in spaces of distributions via boundary values. We then discuss some resent work with K.-H. Neeb and V. Morinelli on classification of orthogonal Euler elements.

We consider representations of the affine Lie algebra $\hat{\mathfrak{g}}$ of a simple Lie algebra $\mathfrak{g}$ at a fixed level $k\in\mathbb{C}$. For $k+h^\vee\in\mathbb{C}\setminus\mathbb{Q}_{\geq 0}$, where $h^\vee$ is the dual Coxeter number of $\mathfrak{g}$, Kazhdan and Lusztig showed that a certain category $KL_k(\mathfrak{g})$ of $\hat{\mathfrak{g}}$-modules at level $k$ is a braided tensor category which is braided tensor equivalent to a category of finite-dimensional modules for a quantum group of $\mathfrak{g}$ at a parameter corresponding to $k$. For $k+h^\vee\in\mathbb{Q}_{>0}$, there is more than one way to define the Kazhdan-Lusztig category. We define $KL_k(\mathfrak{g})$ to be the category of ordinary modules for the simple affine vertex operator algebra (VOA) associated to $\hat{\mathfrak{g}}$ at level $k$. We explain how to use recent results on braided tensor categories to show that $KL_k(\mathfrak{g})$ is a semisimple rigid braided tensor category. For certain $k$, we conjecture that $KL_k(\mathfrak{g})$ is braided tensor equivalent to a certain minor modification of the category of finite-dimensional $\mathfrak{g}$-modules. If this is true, it is then possible to construct many new VOAs by extending tensor products of affine VOAs associated to $\mathfrak{g}$ at different levels.

Hardy spaces arise naturally as normal forms for unitary one-parameter groups and, more generally, for representations of involutive semigroups. Motivated by this representation-theoretic viewpoint, we introduce an abstract notion of Hankel operators associated with unitary representations of involutive semigroups. For suitable choices of semigroups, this framework recovers the classical Hankel operators on Hardy spaces. This perspective also leads to an investigation of the unit group of the algebra of bounded holomorphic functions on a simply connected domain. Although the corresponding Hardy spaces are all isomorphic as Banach algebras by the Riemann Mapping Theorem, different domains naturally single out different one-parameter semigroups, leading to different group-theoretic realizations of the same Banach algebra.

In the classical theory of Lie groups, Kirillov’s coadjoint orbit method establishes a profound link between algebra and geometry, equipping orbits in the dual of a Lie algebra with a canonical symplectic structure. In this talk, we explore a parallel construction developed for Jordan algebras and their $\mathbb{Z}_2$-graded extensions, Jordan superalgebras.

We demonstrate how the Jordan product can be used to define a generalized distribution on the dual space, inducing canonical pseudo-Riemannian metrics on the resulting orbits. For formally real (Euclidean) Jordan algebras, we show that these orbits possess a natural Riemannian structure that recovers fundamental objects of information geometry: specifically, the Fisher–Rao metric in the classical setting and the Bures–Helstrom metric in the quantum setting.

This is based on joint work with Florio Ciaglia, Shuhan Jiang and Jürgen Jost

In the unitary representation theory of a connected semisimple Lie group $G=K\exp(\mathfrak{p})$, one studies the matrix coefficients $\pi^{v,w}(g)=\langle v,\pi(g)w\rangle$, for irreducible unitary representations $(\pi,\mathcal{H})$ and $K$-finite vectors $v,w\in \mathcal{H}^{[K]}$, which define matrix-spherical functions. Restrictions of matrix-spherical functions to a maximal abelian hyperbolic torus $A$ in $G$ satisfy the so-called matrix-spherical hypergeometric system of partial differential equations. This system is a meromorphic system of differential equations on the complexified torus $A_\mathbb{C}$ with poles along an algebraic hypersurface $D:=A_\mathbb{C}\setminus A_\mathbb{C}^{\rm reg}$.

In this talk, we briefly recall the theory of radial terms of $K$-invariant left invariant differential operators on $G$ acting on generalized spherical functions as well as known results about the singularities of the hypergeometric system of partial differential equations.

In the case of $K$-fixed vectors, results by Heckman and Opdam show that this system has regular singularities along the singular locus $D$. We report on our results generalizing these results to the $K$-finite case and briefly discuss the methods developed to prove this results.

This work is part of the PhD-thesis of the speaker.

For a simple Lie algebra $\mathfrak{g}$, the principal $\mathcal{W}$-algebra admits a coset realization in terms of affine vertex algebras. In 2020, Creutzig and Linshaw showed that these coset constructions fit into a broader framework of trialities, which also generalizes the well-known Feigin-Frenkel duality. For Lie superalgebras, analogous duality phenomena are also conjectured, but remain open. For $\mathfrak{g}=\mathfrak{sl}_{n+1|n}$, the coset realization of the corresponding principal $\mathcal{W}$-algebra was conjectured by Ito in 1992, with the relation between the levels taking a form analogous to Feigin-Frenkel duality. The case $n=1$ was known earlier, where both sides are isomorphic to the $\mathcal{N}=2$ Neveu-Schwarz vertex algebra. More recently, in joint work with my collaborators, we proved Ito’s conjecture, making essential use of its supersymmetric structure. In this talk, I will introduce Ito’s conjecture and explain $\mathcal{N}=2$ supersymmetry.