### Plenary talks

**Anton Alekseev**(Université de Genève)

**The Goldman-Turaev Lie algebra and the Kashiwara-Vergne problem**
Let S be an oriented surface of genus g with n boundary components, and let K be a field of characteristic zero. The K-linear span A_g,n of conjugacy classes in the fundamental group of S carries a canonical Lie bialgebra structure defined in terms of intersections of curves on the surface, and it has a filtration induced by the central series in the fundamental group. Hence, one can ask whether A_g,n is formal (that is, isomorphic to its associated graded).**Alexander Alldridge**(University of Cologne)

**Schur Q functions and Capelli identities for the Lie supergroup Q(n)**
We report on joint work with Hadi Salmasian (Ottawa) and Siddhartha Sahi (Rutgers). Schur Q-functions are symmetric functions introduced by I. Schur in the study of spin representations of the symmetric group. The definition of their factorial versions was suggested by Okounkov. We show that factorial Schur Q-functions appear as eigenvalue polynomials for polynomial differential operators invariant under the Lie supergroup Q(n), and the Schur Q-functions, their homogeneous top degree part, appear as spherical polynomials. These results refine the higher Capelli identities for Q(n) of Nazarov.
**Dorothea Bahns**(Mathematisches Institut, Georg-August-Universität Göttingen)

**On an infinite-dimensional Lie algebra in string theory**
When quantizing strings - or more correctly, immersed extremal surfaces
- in the framework proposed by Klaus Pohlmeyer 30 years ago, a certain
infinite dimensional graded Lie algebra occurs as an auxiliary tool.
This Lie algebra is given in terms of generators and relations on the
indecomposable elements of a certain graded commutative connected Hopf
algebra, but little is known about its structure. I will report on some
ongoing research and first steps towards understanding this algebra.
**Wolfgang Bertram**(Université de Lorraine)

**Lie calculus, groupoids, and loops**

Slides from this talk
I intend to report on ongoing work having its distant roots in joint work with K.-H. Neeb and H. Glöckner on general differential calculus. My recent approach to these topics is via **José F. Cariñena**(Universidad de Zaragoza)

**Recent advances on Lie systems and applications**

Slides from this talk
After a quick presentation of the theory of Lie systems from a geometric perspective, recent progresses on their applications when compatible geometric structures exist
will be described with an special emphasis in the particular case of admissible Kähler structures, and therefore with applications in Quantum Mechanics. The more general cases
of quasi-Lie systems and bundle Lie systems will also be presented.
**Jacques Faraut**(Université Pierre et Marie Curie, Paris)

**Orbital measures and spline functions**

Slides from this talk
Consider a Hermitian $n\times n$-matrix $X$ with eigenvalues λ**Alfred M. Grundland**(Université du Quebec, CRM Université de Montreal)

**On the Fokas-Gel’fand theorem for integrable systems**

Slides from this talk
The Fokas-Gel’fand theorem on the immersion formula of 2D-surfaces is related to the study
of Lie symmetries of an integrable system. A rigorous proof of this theorem is presented which may help
to better understand the immersion formula of 2D-surfaces in Lie algebras. It is shown, that even under
weaker conditions, the main result of this theorem is still valid. A connection is established between three
different analytic descriptions for immersion functions of 2D-surfaces, corresponding to the following three
types of symmetries: gauge symmetries of the linear spectral problem, conformal transformations in the
spectral parameter and generalized symmetries of the integrable system. The theoretical results are applied
to the $\mathbb{C}P^{N−1}$ sigma model and several soliton surfaces associated with these symmetries are constructed.
It is shown that these surfaces are linked by the gauge transformations.
**Madeleine Jotz Lean**(University of Sheffield)

**On ideals in Lie algebroids**
I will describe a notion of ideals in Lie algebroids, named
'infinitesimal ideal systems' and motivate from several point of views
why this is in my opinion the 'right' notion of symmetry in the Lie
algebroid setting: I will discuss quotients by infinitesimal ideal
systems, their equivalence to multiplicative foliations on Lie
groupoids, and how they define sub-representations of Lie algebroids adjoint
representations (up to homotopy). Then I will sketch a first
obstruction to the existence of an infinitesimal ideal system
structure on a Lie pair.
**Peter W. Michor**(University of Vienna)

**Old and new on diffeomorphism groups too**

Slides from this talk
Groups of diffeomorphisms of a manifold $M$ have many of the properties of
finite dimensional Lie groups, but also differ in surprising ways.
I will review some (or all or more) of the following properties or I do something
else:
- No complexification.
- Exponential mappings are defined but are not locally surjective of injective.
- Right invariant Riemannian metrics might have vanishing geodesic distance.
- Many famous PDE's arise as geodesic equations on diffeomorphism groups.
- There are topological groups of diffeomorphisms which are smooth manifolds but only right translations are smooth.
- There are diffeomorphism groups which are smooth in a certain sense (Some Denjoy-ultradifferentiable class) but not better (not real analytic).
**Gestur Ólafsson**(Louisiana State University)

**Harmonic Analysis with respect to Jack Polynomials**
In an influential though unpublished manuscript I. G. Macdonald developed the theory of multivariate hypergeometric functions depending on a parameter. In this talk we will discuss several conjectures formulated in by Macdonald. Among other we will discuss estimates for the exponential kernel that allow us to establish a rigorous theory of the Fourier and Laplace transforms. We then discuss some of the applications.
**Bent Ørsted**(Aarhus University)

**Branching laws and elliptic boundary value problems**

Slides from this talk
For a unitary representation of a Lie group it is a basic problem
to understand the restriction to a closed subgroup, i.e. the branching law.
For the conformal group of Euclidian space and the conformal group
of a hyperplane we shall explain how branching is related to some
natural elliptic boundary value problems. This lecture is based on joint
work with Jan Möllers and Yoshiki Oshima.
**Angela Pasquale**(Université de Lorraine)

**Resonances and singular integrals**

Slides from this talk
Let Δ be the Laplacian on a Riemannian symmetric space of the noncompact type $X=G/K$, and let σ(Δ) denote its spectrum. The resolvent R(z)=(Δ-z)^{-1} is a holomorphic function on $\C\setminus σ(Δ)$, with values in the space of bounded operators on $L^2(X)$.If we consider $R$ as a map from $C_c^\infty(X)$ to $(C_c^\infty(X))^*$, then a meromorphic continuation of $R$ on a Riemann surface above $\C \setminus σ(Δ)$ is possible. The poles of the meromorphically extended resolvent are called the resonances and the image of the residue operator at a resonance is a $G$-module. The main problems are the existence and the localization of the resonances as well as the study of the (spherical) representations of $G$ so obtained.**Anke Pohl**(MPIM Bonn)

**Sup-norm bounds for Siegel-Maass forms**
Given a Riemannian locally symmetric space, bounds for
eigenfunctions of the Laplace operator or for joint eigenfunctions of
the whole algebra of isometry-invariant differential operators are of
great interest in several areas. Methods from analysis allow to provide
bounds (nowadays called `generic') which are sharp for certain spaces.
However, if the symmetric space and the eigenfunctions enjoy certain
additional symmetries it is expected that the generic bounds can be
improved. We will discuss our joint work with Valentin Blomer, which
provides the first example of such a subconvexity bound for a higher
rank setup.
**Hadi Salmasian**(University of Ottawa)

**Smooth vectors, smoothing operators, and applications**
Let $(\pi,H)$ be a unitary representation of a possibly infinite dimensional Lie group $G$. A bounded operator on $H$ is called a smoothing operator if its image lies inside the subspace of smooth vectors of $(\pi,H)$. We give various characterisations for smoothing operators and also for a more restricted class of operators called Schwartz operators. We use this characterization to obtain characterization of smooth vectors of a semibounded unitary representation of $G$, and to construct $C^*$ algebras that act as host algebras for unitary representations of Lie supergroups. **Henrik Schlichtkrull**(University of Copenhagen)

**Real spherical spaces**
A homogeneous space G/H of a real reductive Lie group G
is called spherical if a minimal parabolic subgroup of G admits an
open orbit on G/H. All symmetric spaces are spherical, but the property
is shared also by other spaces. In the talk I shall discuss the geometry
of such spaces, and a recent classification for the case with G simple
and H reductive. **Karl Strambach**(FAU Erlangen-Nürnberg)

**The origins and developments of the seminar Sophus Lie****Ernest Vinberg**(Moscow State University)

**Short SL(3)-structures on Lie algebras**

Slides from this talk
Let G be a connected complex algebraic group and \g=Lie G. A short
SL(3)-structure on \g is a subgroup L<G locally isomorphic to SL(3)
(or, equivalently, a subalgebra \l<\g isomorphic to \sl(3)) such that
all irreducible components of its adjoint representation in \g/\l
are three- or one-dimensional. Any simple Lie algebra \g but C_n
admits such a structure, and it is unique up to an automorphism of \g.

We answer this question in the positive for n>0. For genus 0, it reduces to the Kashiwara-Vergne problem on the properties of the Campbell-Hausdorff series. For genus 1, we define the elliptic Kashiwara-Vergne problem and solve it using the theory of elliptic associators. The general case follows by combining genus 0 and 1 results.

The formality problem has interesting applications to geometry of moduli spaces of flat connections.

The talk is based on a joint work with N. Kawazumi, Y. Kuno and F. Naef.

*groupoids*: for the purpose of this talk, I will call it "Lie calculus", since its distinctive feature is to describe calculus itself by a certain Lie groupoid (in fact, a generalized version of Connes'

*tangent groupoid*). Thus foundations of calculus and of Lie theory appear to be merged with each other from the very beginning on. Seen this way, a Lie group is a group plus a groupoid structure, leading to a double groupoid (or even to a threefold groupoid when formally taking account of its manifold structure), and higher order calculus leads to n-fold groupoids. Passing from Lie groups to general affinely connected manifolds, the group structure is replaced by a family of

*loop*structures, and one is lead to investigate the interaction between groupoid and loop structures. A non-technical overview about these topics can be found on my homepage.

*References*:

[1] J.F. Cariñena, J. Grabowski and J. de Lucas,

*Quasi-Lie schemes: theory and applications*, J. Phys. A

**42**, 335206 (2009).

[2] J.F. Cariñena, J. Grabowski, J. de Lucas and C. Sardón,

*Dirac--Lie systems and Schwarzian equations*, J. Diff. Eqns

**257**, 2303--2340 (2014).

[3] J.F. Cariñena, J. Grabowski and G. Marmo,

*Lie--Scheffers systems: a geometric approach*, Bibliopolis, Naples, 2000.

[4] J.F. Cariñena, J. Grabowski and G. Marmo,

*Superposition rules, Lie Theorem and partial differential equations*, Rep. Math. Phys.

**60**, 237--258 (2007).

[5] J.F. Cariñena, J. de Lucas and C. Sardón,

*Lie--Hamilton systems: theory and applications*, Int. J. Geom. Methods Mod. Phys.

**10**, 09129823 (2013).

[6] P.G. Estèvez, F.J. Herranz, J. de Lucas and C. Sardón,

*Lie symmetries for Lie systems: applications of ODEs and systems of PDEs*, Appl. Math. Comp.

**273**, 435--452 (2016).

[7] S. Lie and G. Scheffers,

*Vorlesungen uber continuierliche Gruppen mit geometrischen und anderen Anwendungen*, Teubner, Leipzig, 1893.

_{1}≤ ... ≤ λ

_{n}, and the projection $Y$ of $X$ on the $(n-1)\times (n-1)$ upper left corner. Rayleigh Theorem says that the eigenvalues μ

_{1}≤ ... ≤ μ

_{n-1}of $Y$ interlace those of $X$:

_{1}≤ μ

_{1}≤ λ

_{2}≤ ...≤ λ

_{n-1}≤ λ

_{n}.

_{1}≤ ... ≤ μ

_{n-1}is described by a formula due to Baryshnikov. More generally the eigenvalues of the projection of $X$ on the $k\times k$ upper left corner (1 ≤ k ≤ n-1) is distributed according to a determinantal formula due to Olshanski. This formula involves spline functions.

Recently analogous results have been obtained by Zubov in case of the action of the orthogonal group on the space of real skew-symmetric matrices. For the action of the orthogonal group on the space of real symmetric matrices much less is known.

We will also consider unbounded orbital measures for the action of the noncompact group $U(p,q)$. In this case there is an analogue of Rayleigh Theorem and Baryshnikov formula.

This is partly based on joint work with Ortiz and Drummond.

In this talk we will consider the case of Riemannian symmetric spaces of rank >1. The search for resonances is then connected with the analysis of certain singular integrals on the complex sphere. This talk is based on joint works with Joachim Hilgert (Universität Paderborn) and Tomasz Przebinda (University of Oklahoma).

This talk is based on a joint project with K.-H. Neeb, G. van Dijk, and C. Zellner.

The talk is based on joint work with F. Knop, B. Krötz and T. Pecher.

__:__

**EMS Distinguished Speaker**For any short SL(3)-structure on a Lie algebra \g, there is a "section", a subalgebra \s<\g intersecting every minimal L-invariant subspace of \g in a one-dimensional subspace. Under the additional condition that \s\cap\l consists of semisimple elements, a section is unique up to a conjugation by L. For the simple Lie algebras of types A_n, B_n, D_n, G_2, F_4, E_6, E_7, E_8 such sections are reductive Lie algebras of types A_{n-2}+T_1, B_{n-2}+A_1, D_{n-2}+A_1, A_1, C_3, A_5, D_6, E_7, respectively. They appear together with a "very short even \sl_2-structure", an sl(2)-subalgebra such that all irreducible components of its adjoint representation in \s are three- or one-dimensional. These algebras can be characterized axiomatically and classified apriori. They are related to some remarkable cubic forms.

There is an inverse procedure reconstructing a simple Lie algebra from its section. This gives some models for the exceptional simple Lie algebras, in terms of which one can better understand some of their embeddings.

All this matter is closely related to previous works of Freudenthal, Tits, Rosenfeld, I.Kantor, Allison, Seligman and myself devoted to constructing exceptional simple Lie algebras in terms of some non-associative algebras (the algebra of octonions etc.) However, the approach presented in the talk permits to avoid considering non-associative algebras other than Lie algebras.

### Contributed lectures

**Daniel Beltita**(Institute of Mathematics "Simion Stoilow" of the Romanian Academy)

**Groupoids, coadjoint dynamical systems of solvable Lie groups, and their $C^*$-algebras**

Slides from this talk
Coadjoint actions of Lie groups are natural examples of transformation groups, hence locally compact
groupoids, whose spectra have a quite complicate topological structure. We show how that structure can be studied in the case of exponential solvable Lie groups, by restricting the coadjoint action to suitable invariant subsets of the corresponding Lie-Poisson spaces. As an application of our groupoid techniques we produce examples of Lie groups whose $C^*$-algebras admit faithful irreducible representations. By classical results in representation theory, such Lie groups can be neither reductive nor nilpotent, but it is still an open problem if they can be exponential solvable Lie groups. This talk is based on joint work with Ingrid Beltita.
**Javier de Lucas**(University of Warsaw)

**A Lie systems approach to the Riccati hierarchy and partial differential equations**

Slides from this talk
A general change of variables is provided, allowing the mapping of the members of the Riccati hierarchy, the so-called Riccati chain equations, into projective Riccati equations. This leads us to characterize Riccati chain equations in terms of the projective vector fields of a Riemannian flat metric and to derive their associated superposition rules. Next, we give necessary and sufficient conditions under which it is possible to map second-order Riccati chain equations into conformal Riccati equations through a local diffeomorphism. This allows one to determine superposition rules for particular higher-order Riccati chain equations depending on fewer particular solutions than in the previous case. Finally, the use of nonlocal transformations enables us to apply the derived results to the study of relevant partial differential equations, such as the Kaup-Kupershmidt and Sawada-Kotera equations.
**Helge Glöckner**(University of Paderborn)

**L**^{1}-regularity of Banach-Lie groups and diffeomorphism groups

Slides from this talk
A Lie group $G$ modelled on a (sufficiently complete) locally convex space is called
L**Stéphane Korvers**(University of Luxembourg)

**Formal and non formal deformation quantizations of bounded symmetric domains**
We introduce an explicit construction leading to a realization of the space of all invariant deformation
quantizations on an arbitrary bounded symmetric domain of $\mathbb{C}^n$. This approach unifies existing methods giving such deformation quantizations and it extends new methods initiated by Bieliavsky and his collaborators in the 2000's. The method used in this work is lying at a crossing point between Lie theory, harmonic analysis and noncommutative geometry.
**Jan Möllers**(FAU Erlangen-Nürnberg)

**The compact picture of symmetry breaking operators for rank one orthogonal and unitary groups**

Slides from this talk
Symmetry breaking operators are intertwining operators from a
representation of a group $G$ to a representation of a subgroup $G'$,
intertwining for the subgroup. For spherical principal series of
$G=O(1,n+1)$ and $G'=O(1,n)$ such operators have been classified recently by
Kobayashi-Speh in the smooth category. We study symmetry breaking
operators in the category of Harish-Chandra modules, recovering the
results by Kobayashi-Speh in this setting, and thus providing the
compact picture of their operators. We further extend their main results
to the case of unitary groups $G=U(1,n+1)$ and $G'=U(1,n)$.
**Andriy Panasyuk**(University of Warmia and Mazury)

**On classification of Lie pencils**

Slides from this talk
I will discuss the problem of classification of pencils of Lie algebra structures, one of which is semisimple, and the approach to it presented in my paper "Compatible Lie brackets: Towards a classification"(Journal of Lie Theory, 24(2014), 561-623). Any such pencil is determined by a linear operator which is defined up to the addition of a derivation. A special fixing of this operator is introduced to get rid of this ambiguity and the operators preserving the root decomposition with respect to a Cartan subalgebra are considered. The classification leads to two disjoint classes of pairs depending on the symmetry properties of the corresponding operator with respect to the Killing form.
**Norbert Poncin**(University of Luxembourg)

**Higher algebra over the Leibniz operad**

Slides from this talk
The lecture would be based on the four (related) papers whose abstracts you find below.**Tomasz Przebinda**(University of Oklahoma)

**The character and the wave front set correspondence in the stable range**

Slides from this talk
We relate the distribution characters and the wave front sets of unitary representation for real reductive dual pairs of type I in the stable range.
**Mikołaj Rotkiewicz**(University of Warsaw, Department of Mathematics)

**A note on actions of some monoids**

Slides from this talk
Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot)$ of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid $(\mathbb{R},\cdot)$. Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps $(\mathbb{R},0)\rightarrow (\mathbb{R},0)$, smooth action of the monoid of real 2 by 2 matrices and smooth actions of multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.**Valdemar Tsanov**(Universität Göttingen)

**Variations of geometric invariant theory on flag varieties**
We consider an embedding of semisimple complex Lie groups $G'$ in $G$. A fundamental problem in representation theory is to understand the space of $G'$-invariants in a given simple $G$-module.
We address this problem in the framework of Geometric Invariant Theory (GIT). By the Borel-Weil
theorem, every simple $G$-module can be obtained as the space of
sections of a suitable line bundle $L$ on the flag variety $G/B$. The $G'$-nullcone
$N$ of $L$ is the common zero locus of all $G'$-invariants in the homogeneous coordinate ring of $L$. The GIT-quotient of the complement $(G/B)\setminus N$
carries a line bundle, whose space of sections is isomorphic to the $G'$-invariant
sections of $L$. **Maarten van Pruijssen**(Universität Paderborn)

**Vector valued orthogonal polynomials**

Slides from this talk
The relation between Jacobi polynomials and matrix coefficients of compact symmetric pairs is also available for vector valued functions. Indeed, if we impose a multiplicity freeness condition, then the generalized spherical functions of fixed type have the structure of a free module over the polynomial ring of zonal spherical functions. In this way we obtain families of vector valued orthogonal polynomials in several variables, that are simultaneous eigenfunctions of a commutative algebra of differential operators.

^{1}-

*regular*if each L

^{1}-curve γ: [0,1] → $\mathfrak g$ in its Lie algebra arises as the left logarithmic derivative of an absolutely continuous curve Evol(γ): [0,1]→ $G$ starting at the neutral element, and the map

^{1}([0,1],\mathfrak g) → AC([0,1],G), γ → Evol(γ)

to the Lie group of absolutely continuous $G$-valued curves is smooth. Many classes of Lie groups are L

^{1}-regular, and the latter property has interesting consequences like validity of the Trotter Product Formula and Commutator Formula for one-parameter groups [1], which are important in the representation theory of infinite-dimensional Lie groups [2] (and difficult to verify by other means).

In the talk, I'll outline the general framework and then explain the L

^{1}-regularity for two major classes of examples, namely (a) Lie groups modelled on Banach spaces and (b) Lie groups of $C^\infty$-diffeomorphisms of finite-dimensional smooth manifolds. In contrast to other examples, the discussion of (a) and (b) is mainly based on a study of integral curves to ODEs, flows and parameter-dependence in the classical setting of Banach manifolds (albeit for right-hand sides which are merely measurable in time).

*References*:

[1] H.G.,

*Measurable regularity properties of infinite-dimensional Lie groups*, preprint, arXiv:1601.02568

[2] Neeb, K.-H. and H. Salmasian,

*Differentiable vectors and unitary representations of Fréchet-Lie supergroups*, Math. Z.

**275**(2013) no. 1--2, 419--451.

*The supergeometry of Loday algebroids*(Journal of geometric mechanics, American Institute of Mathematical Sciences, Volume 5, Number 2, June 2013 - with J. Grabowski and D. Khudaverdyan)

A new concept of Loday algebroid (and its pure algebraic version – Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. Loday algebroids are interpreted as homological vector fields on a 'supercommutative manifold' associated with a shuffle product and the corresponding Cartan calculus is introduced. Several examples, including Courant algebroids, Grassmann-Dorfman and twisted Courant-Dorfman brackets, as well as algebroids induced by Nambu-Poisson structures, are given.

*Free Courant and derived Leibniz pseudoalgebras*(Journal of Geometric Mechanics (2016), 8(1), 71-97 - with B. Jubin and K. Uchino)

We introduce the category of generalized Courant pseudoalgebras and show that it admits a free object on any anchored module over 'functions'. The free generalized Courant pseudoalgebra is built from two components: the generalized Courant pseudoalgebra associated to a symmetric Leibniz pseudoalgebra and the free symmetric Leibniz pseudoalgebra on an anchored module. Our construction is thus based on the new concept of symmetric Leibniz algebroid. We compare this subclass of Leibniz algebroids with the subclass made of Loday algebroids, which were introduced by Grabowski, Khudaverdyan and Poncin, as geometric replacements of standard Leibniz algebroids. Eventually, we apply our algebro-categorical machinery to associate a differential graded Lie algebra to any symmetric Leibniz pseudoalgebra, such that the Leibniz bracket of the latter coincides with the derived bracket of the former.

*On the infinity category of homotopy Leibniz algebras*(Theory and Applications of Categories, Vol. 29, 2014, No. 12, pp 332-370 - with D. Khudaverdyan and J. Qiu)

We discuss various concepts of infinity-homotopies, as well as the relations between them, focusing on the Leibniz type. In particular infinity-n-homotopies appear as the n-simplices of the nerve of a complete Lie infinity-algebra. In the nilpotent case, this nerve is known to be a Kan complex. We argue that there is a quasi-category of infinity-algebras and show that for truncated infinity-algebras, i.e. categorified algebras, this infinity-categorical structure projects to a strict 2-categorical one. The paper contains a shortcut to (infinity,1)-categories, as well as a review of Getzler's proof of the Kan property. We make the latter concrete by applying it to the 2-term infinity-algebra case, thus recovering the concept of homotopy of Baez and Crans, as well as the corresponding composition rule given by Schreiber and Stasheff. We also answer a question of Shoikhet about composition of infinity-homotopies of infinity-algebras.

*A Tale of Three Homotopies*(Applied Categorical Structures, pp 1-29 - with V. Dotsenko)

For a Koszul operad P, there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy P-algebras. Some of those approaches are known to give rise to the same concept. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfer theorem for homotopy cooperads due to Drummond-Cole and Vallette.

The talk is based on a joint work with Michał Jóźwikowski.

Under some hypotheses, we derive an explicit geometric description of the $G'$-nullcone $N$ for an arbitrary ample line bundle $L$ on $G/B$. We compute the Picard groups of the quotients, and deduce some interesting properties, e.g. being Mori dream spaces. We also prove the existence of $L$ such that the associated GIT-quotient carries the information about the $G'$invariants in all $G$-modules, not only the one associated to $L$. This allows us to deduce some global properties of multiplicities of invariants.

This is joint work with Henrik Seppänen.

We briefly discuss this construction together with the classification of the data for which this construction is possible. We also indicate how the families of polynomials in low dimensional examples of rank one can be deformed and allow for shift operators.

Finally, we discuss some details for the symmetric pair $( SU(n+1) \times SU(n+1), diag(SU(n+1)) )$: the polynomials in this case depend on n variables and take values in a $(\binom{n+k}{n}-1)$-dimensional space, for $k\ge 1$. The weight matrix that determines the pairing for the polynomials can be made explicit by a generalization of a miraculous formula of Tom Koornwinder.

This is joint work with Erik Koelink and Pablo Román.