Programme

The arrival day of the conference is Monday, September 25th, 2017. The conference takes place from the morning of Tuesday, September 26th, until lunchtime of Friday, September 29th. Departure is after lunch.
All conference activities are held in the seminar room of the Bellavista Relax Hotel, where the participants are lodged.

Detailed programme

Tuesday, September 26
 9:30-9:45 Opening 9:45-10:45 Corrado De Concini, On some modules of covariants for a reflection group 10:45-11:15 Coffee break 11:15-12:15 Guido Pezzini, Symmetric spaces of Kac-Moody groups 13:00 Lunch 15:00-16:00 Mario Marietti, The Combinatorial Invariance Conjecture for parabolic Kazhdan-Lusztig polynomials 16:10-17:10 Benoît Dejoncheere, Differential operators on complete symmetric spaces of small rank 19:30 Dinner

Wednesday, September 27
 9:30-10:30 István Heckenberger, Borel subalgebras of quantum groups 10:30-10:50 Coffee break 10:50-11:50 Stefan Kolb, Braided module categories via quantum symmetric pairs 12:00-13:00 Vladimir Shchigolev, Categories of Bott-Samelson varieties 13:00 Lunch 14:30 Walk around the Levico lake 19:30 Dinner

Thursday, September 28
 9:30-10:30 Andrea Maffei, The Bruhat order on abelian nilradical and hermitian symmetric varieties 10:30-11:00 Coffee break 11:00-12:00 Jacopo Gandini, Spherical Nilpotent orbits and abelian subalgebras in isotropy representations 13:00 Lunch 15:00-16:00 Valentina Kiritchenko, Newton-Okounkov polytopes and Schubert calculus 16:10-17:10 Anna Melnikov, Orbital varieties with a dense Borel orbit 19:30 Dinner

Friday, September 29
 9:00-10:00 Bernhard Krötz, The discrete spectrum of a real spherical space 10:10-11:10 Aleksy Tralle, Hirzebruch proportionality principle and amenable Clifford-Klein forms of symmetric spaces 11:10-11:30 Coffee break 11:30-12:30 Friedrich Knop, The dual group of a spherical variety 12:30 Lunch

Abstracts

Corrado De Concini, On some modules of covariants for a reflection group

This is joint work with Paolo Papi. Let $$W$$ be an irreducible finite reflection group, $$\mathfrak{h}$$ its (complexified) reflection module $$\mathcal{H}=\mathbb{C}[\mathfrak{h}]/I$$, where $$I$$ is the ideal generated by polynomial invariants of positive degree. $$A=(\bigwedge(\mathfrak{h})\otimes \mathcal{H})^W$$ is an exterior algebra and we completely determine the $$A$$−module structure of $$N:= \mathrm{hom}_W (\mathfrak{h},\bigwedge(\mathfrak{h})\otimes \mathcal{H})$$.
When $$\mathfrak{h}$$ is the Cartan subalgebra of a simple Lie algebra $$\mathfrak{g}$$, we know that $$A$$ is canonically isomorphic to $$(\bigwedge(\mathfrak{g}))^\mathfrak{g}$$ and we verify that $$N\cong \mathrm{hom}_\mathfrak{g} (\mathfrak{g}, \bigwedge(\mathfrak{g}))$$ as an $$A$$−module.
Finally if $$V$$ is an irreducible $$\mathfrak{g}$$−module whose zero weight space we denote by $$V_0$$, we construct a degree preserving map $\mathrm{hom}_\mathfrak{g} (V, \bigwedge(\mathfrak{g}))\to \mathrm{hom}_W (V_0,\bigwedge(\mathfrak{h})\otimes \mathcal{H})$ which we conjecture to be injective. This conjecture implies a well know conjecture by Reeder.

Benoît Dejoncheere, Differential operators on complete symmetric spaces of small rank

Let $$X$$ be a complex smooth projective algebraic variety. The algebra of global differential operators $$D_X$$ is rather badly understood, except in the case of curves, toric varieties, and flag varieties. In this talk, we will investigate these algebras in the case of some wonderful varieties of small rank. More precisely, if $$X$$ is a wonderful $$G$$-variety, two questions naturally arise, namely the study of the infinitesimal action of the Lie algebra of $$G$$, and when $${\cal L}$$ is an invertible sheaf on $$X$$, the study of the algebra of twisted global differential operators $$D_{X,{\cal L}}$$ on the cohomology groups $$H^i(X,{\cal L})$$. We will give an answer to these questions in some particular cases, keeping in mind that wonderful varieties can be seen as generalizations of flag varieties.

Jacopo Gandini, Spherical Nilpotent orbits and abelian subalgebras in isotropy representations

Let G be a semisimple algebraic group with Lie algebra $$\mathfrak g$$. Let $$K$$ be a symmetric subgroup of $$G$$ and $$B \subset K$$ a Borel subgroup, and let $$\mathfrak p \subset \mathfrak g$$ be the isotropy representation of $$K$$. Expanding on previous work of Panyushev, I will explain some connections between the $$B$$-stable abelian subalgebras of $$\mathfrak g$$ which are contained in $$\mathfrak{p}$$, the spherical nilpotent $$K$$-orbits in $$\mathfrak p$$ and the spherical nilpotent $$G$$-orbits in $$\mathfrak g$$. The talk is based on a joint work with P. Mösender Frajria and P. Papi.

István Heckenberger, Borel subalgebras of quantum groups

Many of the representation theoretical and geometrical constructions in the theory of homogeneous spaces is based on the notion of Borel and parabolic subgroups of Lie or algebraic groups. Despite of this fact, in the theory of quantum groups one mainly works in a related context with the standard Borel subalgebras. In the talk I will discuss recent progress in the theory of right coideal and Borel subalgebras and give some examples. First applications to the representation theory of quantum groups are presented.

Valentina Kiritchenko, Newton-Okounkov polytopes and Schubert calculus

In toric geometry, Newton (or moment) polytopes provide a convenient convex geometric model for intersection theory on smooth toric varieties. It is tempting to use Newton-Okounkov polytopes to build a similar convex geometric model for non-toric varieties. For flag varieties in type $$A$$, this approach yields positive presentations of Schubert cycles by faces of the Gelfand-Zetlin polytope. I will talk about possible extensions of convex geometric Schubert calculus to type $$B$$ and $$C$$.

Friedrich Knop, The dual group of a spherical variety

Let $$X$$ be a spherical variety for a reductive group $$G$$. Deep work of Gaitsgory-Nadler indicates that the Langlands dual group $$G^\vee$$ should contain a reductive subgroup $$G_X^\vee$$ whose Weyl group coincides with the little Weyl group of $$X$$. We show that such a subgroup indeed exists (even for any $$G$$-variety). Moreover we exhibit some functoriality properties of $$G_X^\vee$$. This is joint work with Barbara Schalke.

Stefan Kolb, Braided module categories via quantum symmetric pairs

The theory of quantum symmetric pairs provides coideal subalgebras of quantized enveloping algebras which are quantum group analogs of Lie subalgebras fixed under an involution. The finite dimensional representations of a quantized enveloping algebra form a braided monoidal category $$C$$, and the finite dimensional representations of any coideal subalgebra form a module category $$M$$ over $$C$$. In this talk I will discuss what it means for the braiding of $$C$$ to extend to the module category $$M$$. I will then explain how quantum symmetric pairs give rise to braided module categories over $$C$$.

Bernhard Krötz, The discrete spectrum of a real spherical space

We give a general introduction to the discrete spectrum of a homogeneous space $$Z=G/H$$ attached to a real reductive group $$G$$. Specifically for $$Z$$ real spherical we will explain recently obtained results such as the spectral gap theorem. (Joint with Job Kuit, Eric Opdam and Henrik Schlichtkrull).

Andrea Maffei, The Bruhat order on abelian nilradical and hermitian symmetric varieties

We describe the Bruhat order of $$B$$-orbits in abelian nilradical and hermitian symmetric varieties by proving a conjecture of Panyushev and a conjecture of Richardson and Ryan.

Mario Marietti, The Combinatorial Invariance Conjecture for parabolic Kazhdan-Lusztig polynomials

The Dyer-Lusztig Combinatorial Invariance Conjecture states that a Kazhdan-Lusztig polynomial is determined by the underlying poset structure. We discuss the problem of combinatorial invariance in the parabolic setting.

Anna Melnikov, Orbital varieties with a dense Borel orbit

Let $$G$$ be a complex reductive group and $$\mathfrak g$$ its Lie algebra. Let $$B$$ be Borel subgroup of $$G$$, $$\mathfrak B=Lie(B)$$ and $$\mathfrak n$$ its nilradical. $$G$$ acts on $$\mathfrak g$$ by adjoint action.The intersection of a nilpotent $$G$$-orbit with $$\mathfrak n$$ is reducible in general and in this case it is equidimesional. Its components are called orbital varieties. Although an orbital variety is stable under the action of $$B$$, in general it does not admit a dense $$B$$-orbit. In this talk we show that, in the case where $$G$$ is classical, every nilpotent $$G$$-orbit contains at least one orbital variety with a dense $$B$$-orbit. The existence of a dense $$B$$-orbit does not provide in general that an orbital variety is a union of a finite number of $$B$$-orbits. Moreover, there are nilpotent orbits in which there are no orbital varieties with finite number of $$B$$-orbits. However, if $$G$$ is classical there is sphericity'' type phenomenon for the intersection of a nilpotent orbit and $$\mathfrak n$$, namely if each orbital variety in the intersection admits a dense $$B$$-orbit then this intersection is a union of finite number of $$B$$-orbits. D. Panyushev constructed the classification of spherical $$G$$-orbits for all simple Lie algebras. We provide the full classification of orbits with finite number of $$B$$-orbits in the intersection with $$\mathfrak n$$ for $$G$$ classical.

Guido Pezzini, Symmetric spaces of Kac-Moody groups

In the talk we will report on a research project, joint with Bart Van Steirteghem, aimed at studying symmetric spaces for Kac-Moody groups. Our goals include defining a structure of ind-variety on such symmetric spaces, and studying a ring of functions called "strongly regular", which generalizes the ring of strongly regular functions on a Kac-Moody group defined by Kac and Peterson. The multiplication on this ring has properties similar to the classical (finite dimensional) case, and this suggests the possibility of developing a theory of embeddings of symmetric spaces in the Kac-Moody case.

Vladimir Shchigolev, Categories of Bott-Samelson varieties

We consider all Bott-Samelson varieties $$BS(s)$$ for a fixed connected semisimple complex algebraic group with maximal torus $$T$$ as the class of objects of some category. The class of morphisms of this category is an extension of the class of canonical (inserting the neutral element) morphisms $$BS(s)\to BS(s')$$ where $$s$$ is a subsequence of $$s'$$. Every morphism of the new category induces a map between the $$T$$-fixed points but not necessarily between the whole varieties. We construct a contravariant functor from this new category to the category of graded $$H^\bullet_T(pt)$$-modules coinciding on the objects with the usual functor $$H_T^\bullet$$ of taking $$T$$-equivariant cohomologies. We also discuss the problem how to define a functor to the category of $$T$$-spaces from a smaller subcategory. The exact answer is obtained for groups whose root systems have simply-laced irreducible components by explicitly constructing morphisms between Bott-Samelson varieties (different from the canonical ones).

Aleksy Tralle, Hirzebruch proportionality principle and amenable Clifford-Klein forms of symmetric spaces

In the talk, I will describe my recent contributions to the problem of compact Clifford-Klein forms of pseudo-Riemannian homogeneous spaces (together with Maciej Bochenski). In particular, we prove that pseudo-Riemannian non-compact symmetric spaces do not admit amenable Clifford-Klein forms. This generalizes a well known theorem of Benoist on the non-existence of nilpotent Clifford-Klein forms.