Detailed programme
Tuesday, September 26
9:309:45 
Opening 
9:4510:45 
Corrado De Concini, On some modules of covariants
for a reflection group

10:4511:15 
Coffee break

11:1512:15 
Guido Pezzini, Symmetric spaces of KacMoody groups 
13:00 
Lunch 
15:0016:00 
Mario Marietti, The Combinatorial Invariance
Conjecture for parabolic KazhdanLusztig polynomials

16:1017:10 
Benoît Dejoncheere, Differential operators on complete symmetric spaces of small rank

19:30 
Dinner 
Wednesday, September 27
9:3010:30 
István Heckenberger,
Borel subalgebras of quantum groups

10:3010:50 
Coffee break

10:5011:50 
Stefan Kolb, Braided module categories via quantum symmetric pairs

12:0013:00 
Vladimir Shchigolev, Categories of BottSamelson varieties

13:00 
Lunch 
14:30 
Walk around the Levico lake

19:30 
Dinner 
Thursday, September 28
9:3010:30 
Andrea Maffei, The Bruhat order on abelian nilradical and hermitian symmetric varieties

10:3011:00 
Coffee break

11:0012:00 
Jacopo Gandini,
Spherical Nilpotent orbits and abelian subalgebras in
isotropy representations 
13:00 
Lunch 
15:0016:00 
Valentina Kiritchenko,
NewtonOkounkov polytopes and Schubert calculus

16:1017:10 
Anna Melnikov,
Orbital varieties with a dense Borel orbit

19:30 
Dinner 
Friday, September 29
9:0010:00 
Bernhard Krötz, The discrete spectrum of a real spherical space

10:1011:10 
Aleksy Tralle, Hirzebruch proportionality principle and amenable CliffordKlein forms of symmetric spaces

11:1011:30 
Coffee break 
11:3012:30 
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,
NewtonOkounkov 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 NewtonOkounkov polytopes to build a similar convex geometric model for nontoric varieties. For flag varieties in type \(A\), this approach yields positive presentations of Schubert cycles by faces of the GelfandZetlin 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
GaitsgoryNadler 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
KazhdanLusztig polynomials
The DyerLusztig Combinatorial Invariance Conjecture states that a KazhdanLusztig 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 KacMoody groups
In the talk we will report on a research project, joint with Bart Van Steirteghem, aimed at studying symmetric spaces for KacMoody groups. Our goals include defining a structure of indvariety on such symmetric spaces, and studying a ring of functions called "strongly regular", which generalizes the ring of strongly regular functions on a KacMoody 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 KacMoody case.
Vladimir Shchigolev, Categories of BottSamelson varieties
We consider all BottSamelson 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 simplylaced irreducible components by explicitly constructing morphisms between BottSamelson varieties (different from the canonical ones).
Aleksy Tralle, Hirzebruch proportionality principle and amenable
CliffordKlein forms of symmetric spaces
In the talk, I will describe my recent contributions to the problem of compact CliffordKlein forms of pseudoRiemannian homogeneous spaces (together with Maciej Bochenski). In particular, we prove that pseudoRiemannian noncompact symmetric spaces do not admit amenable CliffordKlein forms. This generalizes a well known theorem of Benoist on the nonexistence of nilpotent CliffordKlein forms.