Abstracts
Ilka Agricola (Marburg)
Invariant Spinors on Homogeneous Spheres
(Joint work with Jordan Hofmann and Marie-Amelie Lawn, London)
The classification of transitive sphere actions induces nine different homogeneous realizations of the sphere $S^n$. We explain what a homogeneous spin structure is and describe which homogeneous spheres admit one (which is then unique). In each of the cases we determine the dimension of the space of invariant spinor fields, give their explicit description, and study the underlying related geometric structures depending on the metric. We recover some known results in the Sasaki and 3-Sasaki cases and find several new examples: in particular we give the first known examples of generalized Killing spinors with four distinct eigenvalues. We will sketch how the highly non-trivial computations are based on a rather unusual description of the spin representation in terms of exterior forms.
Erik van den Ban (Utrecht)
Uniform temperedness of Whittaker integrals
In 1982 Harish-Chandra announced a precise explicit formulation of the so-called Whittaker-Plancherel decomposition for a real reductive Lie group. In this decomposition the building blocks are the so-called Whittaker integrals. These are the appropriate analogues of the Eisenstein integrals appearing in the Plancherel decomposition for a real reductive group.
When the details of Harish-Chandra's work finally appeared posthumously, in 2018, the given proof of the Whittaker-Plancherel decomposition turned out to be incomplete, mainly because it lacked a complete Fourier theory for the appropriate space of Schwartz functions. In the talk we will explain the derivation of estimates which are indispensable for such a theory.
Daniel Beltita (Bucharest)
On the primitive ideal spectrum of solvable Lie groups
We discuss topological properties of the Pukánszky correspondence between the primitive ideal spectrum and the generalized coadjoint orbit space of solvable Lie groups. In the general case, no natural topology was yet constructed on the generalized orbit space, which would turn the Pukánszky correspondence into a homeomorphism, as in the special case of the Kirillov correspondence for nilpotent Lie groups. We address this problem for a class of non-nilpotent Lie groups that contains several interesting examples of non-type-I groups, such as the Mautner groups or certain Lie groups whose regular representation is a factor representation. The presentation is based on joint work with Ingrid Beltita.
Aleksandra Garazha (Moscow)
Kronecker’s method and complete systems of functions in bi-involution on classical Lie algebras
The search for completely integrable systems is one of the important problems of Hamiltonian mechanics. In most cases, the integrability of Hamiltonian systems is closely related to their bi-Hamiltonian nature, meaning there are not one but two compatible Poisson brackets. We will study the simplest case when one bracket is linear and the other one is constant. Namely, we will consider a classical simple Lie algebra g and define two Poisson brackets: the classical Lie-Poisson bracket and the bracket "with frozen argument A", which can be constructed for any A \in g.
If A is regular, the corresponding completely integrable bi-Hamiltonian system can be obtained by the Mishchenko-Fomenko argument shift method. We will show how to generalize this method to the case of an arbitrary element A using an algebraic approach.
Joachim Hilgert (Paderborn)
Quantization in fibering polarizations, Mabuchi rays and geometric Peter–Weyl theorem
We use techniques of geometric quantization to give a geometric interpretation of the Peter–Weyl theorem.
We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact semi-simple groups $K$. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of $K\times K$-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action.
Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for $K$ with the Borel–Weil–Bott description of irreducible representations of $K$.
Madeleine Jotz Lean (Würzburg)
The geometrisation of $\mathbb N$-manifolds of degree n
This talk explains the notions of Lie algebroids, Courant algebroids and Lie n-algebroids. While Lie and Courant algebroids have a geometric flavor, the more general notion of Lie n-algebroids, or dq-manifolds, have a more algebraic nature. This talk describes these points of views and establishes the geometrisation of Lie 2-algebroids as linear Courant algebroids -- i.e. the correspondence between the 'algebraic' structures given by Lie 2-algebroids with the geometric linear Courant algebroids.
The core of this correspondence is an equivalence between the category of positively graded manifolds of degree 2 and double vector bundles equipped with an indirect involution. The geometrisation of a positively graded manifold of arbitrary degree n ("[n]-manifold" for short) is then understood as an n-fold vector bundle equipped with a (signed) S_n-symmetry. This talk explains more precisely how symmetric n-fold vector bundle cocycles are the same objects as [n]-manifold cocycles. This is the groundwork for understanding a possible geometrisation of Lie n-algebroids.
This work is partly joint with Malte Heuer.
Job Kuit (Paderborn)
The most continuous part of the Plancherel decomposition for a real spherical space
Let Z be a homogeneous space of a real reductive group G. The Plancherel decomposition of Z is the decomposition of the space L^2(Z) of square integrable functions into a direct integral of irreducible unitary representations of G. In general this decomposition has a mixed discrete and continuous nature.
In this talk we consider real spherical homogeneous spaces Z. In particular, we will focus on the most continuous part of L^2(Z), i.e., the closed subspace of L^2(Z) that decomposes in the largest continuous families. Recently, Eitan Sayag and I gave a precise description of the Plancherel decomposition of the most continuous part. I will discuss some of the ingredients and key ideas that went into the proof.
Gandalf Lechner (Erlangen)
Borchers triples and their deformations in QFT
In quantum field theory one is interested in systems of (von Neumann) algebras that carry an action of the Poincaré group compatible with the basic axioms of locality and covariance. In this talk I will review the notion of a Borchers triple, consisting of a single von Neumann algebra, a positive energy representation of the Poincaré group in suitable relative position, and an invariant vector. I will explain that any such triple gives rise to a QFT. Simple triples, corresponding to interaction-free theories, can be defined entirely in terms of a positive energy representation. More complicated triples, corresponding to interacting theories, arise from free ones by a deformation procedure known as warped convolution, which is intimitaly connected to Rieffel's deformation procedure for C*-algebras by automorphic R^n-actions. Joint work with Detlev Buchholz, Harald Grosse, and Stephen Summers.
Milan Niestijl (Delft)
Holomorphic induction beyond the norm-continuous setting
In the context of possibly infinite-dimensional Lie groups, we discuss a recently obtained extension of holomorphic induction of unitary representations to the situation where the representation that is induced from need not be norm-continuous. We determine the (expected) intimate relationship between holomorphic induction and so-called positive energy representations and explain why holomorphic induction can be a useful tool for the task of classifying irreducible positive energy representations. If time permits, we will also consider some concrete examples.
Angela Pasquale (Lorraine)
Symmetry breaking operators for reductive dual pairs with one member compact
A symmetry-breaking operator (SBO) is an intertwining operator from a representation of a group to an irreducible representation of a subgroup. Symmetry-breaking operators are intrinsically related to branching problems, which justifies their name. In Howe’s theory, the space of the symmetry-breaking operators from the Weil representation to a representation \PI\otimes\Pi’, where \Pi and \Pi’ are in Howe’s duality, is one dimensional. The explicit construction of the SBO sheds additional light on Howe's correspondence.
In this talk, which is based on an ongoing project with Mark McKee and Tomasz Przebinda (University of Oklahoma), we study the SBOs corresponding to irreducible reductive dual pairs with one member compact. They are pseudo-differential operators, and we compute their Weyl symbols. We present some results, examples and applications.
Coadjoint orbits for singular vortex configurations and dual pairs
We describe a few classes of coadjoint orbits of the group of compactly supported area preserving diffeomorphisms on the plane, and we explain their relation to singular vorticities for ideal fluids. We show how some of these coadjoint orbits arise via symplectic reduction in dual pairs of momentum maps: the Marsden-Weinstein ideal fluid dual pair and a new variant of the Holm-Marsden EPDiff dual pair.
Chengbo Zhu (Singapore)
Counting irreducible representations
I will explain how to count irreducible representations with a given infinitesimal character and a given bound of the complex associated variety. The main conceptual tool for the counting is coherent families of representations. For the category of Harish-Chandra modules, the concept was developed by Schmid, Zuckerman, Speh-Vogan, and Vogan. Putting together all coherent families (based on a translate of the root lattice), one obtains a representation of a certain integral Weyl group. The key to the counting lies in understanding cell structures of this representation (called Harish-Chandra cells), and this is where Kazhdan-Lusztig theory, more specifically the theory of primitive ideas (as developed by Joseph and Barbasch-Vogan), double cells and special representations (in the sense of Lusztig), comes in to play. The end result is a counting inequality on irreducible representations, which is shown to be an equality in important cases (e.g. all classical groups). This is joint work with Dan Barbasch, Jia-Jun Ma and Binyong Sun, and is Part I of our project to construct and classify special unipotent representations of all classical groups (with unitarity as a direct consequence).