## Lunch / Coffee Break

Lunch will be provided at the university restaurant in the same building. There will be a choice of different dishes, including meat or fish, vegetarian options and a salad bar.
Coffee Break: "Übungsraum 4", directly  beside Lecture Hall H12 (1st floor).

## Conference Dinner

The conference dinner will take place Thursday, September 24, 7:00 pm at

the restaurant Schwarzer Bär, Innere Brucker Straße 19, 91054 Erlangen. The restaurant is in Erlangen city centre in walking distance from all hotels (Map).

## Talks

• Eric van den Ban:
Normalizations of Eisenstein integrals for semisimple symmetric spaces

Eisenstein integrals for a semisimple symmetric space G/H are essentially matrix coefficients of parabolically induced representations, pairing K-finite with H-fixed vectors. The parabolic subgroups involved are usually taken $\theta\sigma$-stable in order to avoid a certain degeneracy. However, in the case of the group they are different from Harish-Chandra's Eisenstein integrals which are obtained by induction from different parabolic subgroups. In this talk, based on joint work with Job Kuit, I will describe how Eisentein integrals with different normalizations may be obtained, such that also Harish-Chandra's Eisenstein integrals are covered. The new Eisenstein integrals can be applied in a theory of cusp forms for G/H.

• Sigiswald Barbier:

Polynomial realisations of Lie (super)algebras associated to Jordan (super)pairsWe will give a general method to construct polynomial realisations for Lie superalgebras. We will consider in particular the case when our Lie superalgebra is the Tits-Kantor-Koecher Lie superalgebra associated to a Jordan superpair. For a Jordan pair coming from a semi-simple Jordan algebra, we recover the representations used by Hilgert-Kobayashi-Möllers to construct minimal representations of Lie algebras.

• Daniel Beltita:
Representations of Infinite-Dimensional Linear Lie Groups: Some Specific Constructions and Open Problems
By linear groups we mean groups that can be faithfully represented by invertible operators on Banach spaces.
We discuss unitary irreducible representations and faithful representations of infinite-dimensional Lie groups modeled on Banach spaces, for short Banach-Lie groups. The focus will be on infinite-dimensional versions of compact Lie groups and nilpotent Lie groups, respectively.
1. Unitary groups of C*-algebras.
It is well known that the Banach-Lie groups of this type are linear in the above sense, so the first basic problem in their representation theory is the existence of unitary irreducible representations. In order to address this problem we investigate the applicability in infinite dimensions of the Schur-Weyl method of constructing tensor realizations of irreducible representations of unitary matrix groups. This is joint work with Karl-Hermann Neeb.
2. Nilpotent Banach-Lie groups.
The very first problem on these groups is to prove that they are linear in the above sense. We will show that this is the case and then we will discuss some constructions of unitary irreducible representations for some infinite-dimensional Heisenberg groups. This is joint work with Ingrid Beltita.
• Jaques Faraut:
Analysis of the Minimal Representation for Pseudo-Orthogonal Groups Theory
The minimal representation of O(n; n) can be realized on a Hilbert space of holomorpic functions. This is the Brylinski-Kostant model. It can also be realized on a Hilbert space of homogeneous functions on the isotropic cone. This is the Kobayashi-Oersted model. We will describe a transformation which maps one model to the other. It can be seen as an analogue of the Segal-Bargmann transform.
• Helge Glöckner
Measurable regularity properties of infinite-dimensional Lie groups
As is well known, initial value problems on non-normable locally convex spaces (or manifolds modelled thereon) need not have solutions; and if solutions exist, they need not be unique. The situation improves when integral curves for a time-dependent left-invariant vector field on an infinite-dimensional Lie group G are considered. Then integral curves starting at a given point are necessarily unique. And, surprisingly, inspection showed that all main examples of Lie groups are regular in the sense that integral curves always exist and depend smoothly on the left-invariant vector field. This is known for time-dependent vector fields depending smoothly on t, and, more generally, for vector fields depending continuously on t.In the talk, I'll explain that all main examples of Lie groups have measurable regularity properties, in the sense that the above statements remain valid for left-invariant vector fields which are merely measurable in t and belong to suitably-defined Lebesgue spaces. For example, every Banach-Lie group and all diffeomorphism groups are L^1-regular.
The measurable regularity properties ensure validity of the Trotter product formula and the commutator formula.
• Joachim Hilgert:
Resonances in Lie Theory

In this talk we shall explain several concepts of resonances which occur in analysis on Lie groups and symmetric spaces. In particular we show that the resolvent of the Laplacian on SL(3,R)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of C. The poles of this function are a particular case of resonances. We determine all these resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are infinite dimensional irreducible SL(3,R)-representations whose Langlands parameters can also be read off from the corresponding resonances.
• Roger Howe:
Small representations of finite groups

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimension tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of some of these facts. This talk will discuss a conjectural method for systematically constructing the small representations of finite classical groups. The method is closely related to the theory of theta series, and has an analog over local fields.
• Bas Janssens:
Universal Central extension of the Lie algebra of Hamiltonian vector fields

Motivated by projective representation theory, we construct the universal central extension of the Lie algebra of Hamiltonian vector fields.

• Bernhard Krötz:
Lie algebra homology of Harish-Chandra modules
This talk is an overview talk about comparison theorems between Lie algebra homology of Harish-Chandra modules and their smooth completions.  This theme features a tight relationship with quantative bounds of matrix coefficients which will be explained in an example.

• Roman Lavicka:
Howe Duality for Polynomials on Superspace

Recently, the Howe duality has been described for polynomials on superspace $R^{m|2n}$ (that is, for polynomials in $m$ commuting and $2n$ anti-commuting variables). The Howe dual pair is $(osp(m|2n),sl(2))$ and $(osp(m|2n),osp(1|2))$ for scalar and super valued polynomials on $R^{m|2n}$, respectively. It was shown that the space of polynomials has a multiplicity free irreducible decomposition under the joint action of the Howe dual pair unless the superdimension $M=m-2n$ is even and non-positive. In the talk, we focus on this exceptional case and show that, in general, all polynomials decompose into indecomposable (not necessarily irreducible) pieces. This is a joint work with V. Soucek and D. Smid.

• Gandalf Lechner:

R-matrices and integrable quantum field theories
Given a unitary representation of a group G, an R-matrix is an operator-valued function commuting with the tensor square of this representation, which has a number of additional symmetry and analyticity properties (such as the Yang-Baxter equation and crossing symmetry). In this talk I explain how integrable quantum field theories (formulated in terms of a system of von Neumann algebras) with global gauge group G can be constructed with the help of an R-matrix and Tomita-Takesaki modular theory. As two examples, I then focus on G=O(N), related to O(N)-symmetric sigma models, and G=O(d,1), related to quantum field theories with target de Sitter space.

• Ivan Penkov:
Categories of $(\gg,\sl_2)-$modules

Let $\gg$ be a complex semisimple Lie algebra and $\kk \subset \gg$ be a subalgebra isomorphic to $\sl_2$. Admissible $(\gg,\kk)-$modules are an interesting class of generalized Harish-Chandra modules, and are not part of Harish-Chandra module theory unless $\gg$ has rank $2$. One way to construct admissible $(\gg,\kk)-$modules is to apply the Zuckerman functor to a suitable thick parabolic category $\tilde{O}_\pp$. Zuckerman and I conjectured in 2012 that, under a certain restriction on the $\kk-$weights of modules in $\tilde{O}_\pp$, the Zuckerman functor is an equivalence of the respective part of the category  $\tilde{O}_\pp$ with the category of admissible $(\gg,\kk)-$modules satisfying a certain lower bound on the minimal $\kk-$type. Serganova, Zuckerman and I recently proved this conjecture, and I will report on the main ideas of the proof.

• Ana Prlic:
Dirac Induction for Discrete Series Representations
In a joint work P. Pandzic and D. Renard introduced new notions of Dirac cohomology and homology of a Harish-Chandra module $X$. The functor of Dirac cohomology has a left adjoint functor and the functor of Dirac homology has a right adjoint functor, which are both called Dirac induction functors. We will give some examples of discrete series representations constructed via Dirac induction.
• Tudor Ratiu:
Canonical symplectic structure on coadjoint orbits of solvable Lie algebras.
Several important integrable systems have coadjoint orbits as phase spaces. If the Lie algebra in question is solvable, one can hope that some of the coadjoint orbits are cotangent bundles. In this talk, I will discuss the momentum map nature of the Flaschka transformation that is crucial in the study of the finite Toda lattice and present a generalization yielding magnetic cotangent bundles. This is based on certain conditions introduced by Pukanszky and on reduction theorems. There are two types of such conditions. The second group of Pukanszky conditions guarantees that simply connected coadjoint orbits of connected, simply connected, solvable Lie groups are symplectomorphic to the standard symplectic vector space. Time permitting, I will outline the proof of this theorem, due to Pukanszky, based on reduction theory.

• Alexander Schmeding:
Linking Lie Groupoids and Infinite-Dimensional Lie Groups

It is well known that to each Lie groupoid one can associate its group of bisections. In this talk we use this link to connect the geometries of Lie groupoids and their (infinite-dimensional) bisection Lie groups. Namely, two (re-)construction principles for Lie groupoids from candidates for their bisection Lie groups are discussed. Motivation for this is an application to the prequantisation of (pre)symplectic manifolds. This is joint work with Christoph Wockel (Universität Hamburg).
• Friedrich Wagemann:
Deformation Quantization of Leibniz Algebras

Duals of Lie algebras are Poisson manifolds with the Kostant-Kirillov-Souriau (KKS) Poisson bracket. We present a generalized Poisson bracket on the dual of Leibniz algebras which we deformation quantize according to the same lines as the Baker-Campbell-Hausdorff formula quantizes the KKS bracket. This joint work with B. Dhérin and with C. Alexandre, M. Bordemann and S. Rivière.
• Stefan Waldmann:
Convergence of the Gutt Star Product
In this joint work with Chiara Esposito and Paul Stapor we investigate the convergence properties of the Gutt star product on the dual of a Lie algebra. We consider the symmetric algebra of the Lie algebra as model for the polynomial functions on its dual. The Gutt star product is obtained by using the canonical isomorphism to the universal enveloping algebra and turning this into an algebra isomorphism for the rescaled Lie bracket. While for the symmetric algebra everything is convergent (in fact algebraic) we are interested in a locally convex topology on the symmetric algebra such that on one hand the product is continuous, on the other hand the completion becomes large. This way we obtain a nontrivial completion of the universal algebra as well with interesting analytic properties. Our approach works in infinite dimensions as soon as the Lie algebra satisfies a mild analytic property: it has to be an asymptotic estimate Lie algebra.

• Wojciech Wojtyński:
Towards Lie theory of diffeomorphism groups – an introduction to  string Lie theory
The talk will sketch idea of “string Lie theory”. It aims to construct the Lie functor  for a topological group having many one-parameter subgroups and satisfying certain “analyticity conditions”.