(Université Jean Monnet de Saint-Etienne, France)
"The commuting variety of semisimple symmetric Lie algebras."
It is known for more than 30 years that the commuting variety of semisimple Lie algebras is
irreducible. However, the analogue variety in the symmetric Lie algebra case may have several
irreducible components. Following [D. Panyushev, 1994], several authors tried to determine the
(ir)reducibility of this variety in the different cases of the classification. I will present
a result describing the irreducible components that are generically reduced. Those components
are in one-one correspondence with some computable data called rigid pairs. A classification
of these pairs is still lacking.
(Università di Padova, Italia)
"Spherical orbits and Weyl group elements."
Spherical conjugacy classes in a reductive algebraic group G can be characterized in two different ways by means of the Bruhat decomposition. Lu has proved that the first characterization generalizes to twisted conjugacy classes. We will provide a twisted analogue of the second one.
We will also show how these characterizations allow a combinatorial description of the map recently introduced by Lusztig from the set of conjugacy classes in the Weyl group to the set of unipotent conjugacy classes in G in the spherical case.
(Part of the results have been obtained jointly with M. Costantini.)
(Higher School of Economics, Moscow, Russia)
"Degenerate flag varieties."
Let G be a simple Lie group. Degenerate flag varieties for G are
certain degenerations of the classical
flag varieties, defined via the PBW filtration on the irreducible
finite dimensional G modules.
These degenerations in the known cases are (typically singular) normal
irreducible projective algebraic
varieties. They have
many nice properties similar to that of the Schubert varieties. The
degenerate flag varieties are
not spherical, but are acted upon by an abelian unipotent group, which
has an open dense orbit.
In the talk we will describe the case G=SL(n). It turns out that in
this case there exists a group, containing
the abelian unipotent subgroup, whcih acts on the degenerate flag
varieties with finite number of orbits.
Moreover, these orbits are affine cells each containing exactly one
torus fixed point.
We will describe combinatorics and geometry related to this cellular
(Universität Erlangen-Nürnberg, Deutschland)
"On the projective normality of model wonderful varieties."
Wonderful varieties (for a semisimple complex group G) are a remarkable class of G-varieties which generalise flag varieties as well as the complete symmetric varieties introduced by De Concini and Procesi. Among these, a special class is that of the model wonderful varieties, which were introduced by Luna to classify those quasi-affine homogeneous spaces for G whose algebra of regular functions contains every irreducible representation exactly once. Given a wonderful variety M and a couple of globally generated line bundles on it, an important problem is the study of the surjectivity of the associated multiplication of sections. This problem is trivial in the case of a flag variety, whereas in the symmetric case it received a positive answer by Chirivì and Maffei in 2004. After giving a general strategy to reduce the study of the surjectivity to a particular class of "fundamental couples" of line bundles on M, we will show how to apply this in the case of a model wonderful variety. (The talk is based on a joint work with Bravi and Maffei.)
Wilberd van der Kallen
(Universiteit Utrecht, The Netherlands)
"Cohomological finite generation and some homogeneous spaces."
About ten years ago I conjectured the following:
Let G be a reductive linear algebraic group over a field k.
Let A be a k-algebra of finite type on which G acts algebraically.
Then H*(G,A) is a finitely generated k-algebra.
This is now a theorem of Touzé. In his proof one studies the homological
algebra of a category of strict polynomial bifunctors from finite dimensional vector spaces
to vector spaces. We will focus instead on the way various homogeneous spaces
enter the picture before one ends up with the problem that Touzé solves.
(Universität Basel, Switzerland)
"On the nullcone of representations of reductive groups",
(Université Libre de Bruxelles, Belgium)
"Riemannian foliations of symmetric spaces", abstract.
(Institute for Information Transmission Problems, Moscow, Russia)
"Commuting involutions and degenerations of isotropy representations."
(Università di Roma "La Sapienza", Italia)
"Maximal abelian Borel stable subspaces and the CSDW conjecture for
We will discuss a generalization to infinitesimal symmetric spaces of results
of Suter and Panyushev on maximal abelian ideals of Borel subalgebras
obtained jointly with Cellini, Mosender Frajra and Pasquali.
We will also discuss a speculative connection with the
Cachazo-Douglas-Seiberg-Witten conjecture in the version for symmetric
spaces (due to Kumar).
(Universität Erlangen-Nürnberg, Deutschland)
"A generalization of Delzant polytopes for certain non-abelian
A celebrated theorem of Delzant states that a multiplicity-free
hamiltonian action of an abelian
compact Lie group K is uniquely determined by the associated moment
polytope. Admissible polytopes
are the well-known Delzant polytopes.
This uniqueness property, thanks to works of Losev and Knop, has been
generalized to the case where
K is non-abelian, but in this case the problem of a purely
combinatorial characterization of all
admissible polytopes remains open.
This is also equivalent to characterize combinatorially the weight
monoids of smooth affine complex
algebraic varieties that are spherical under the action of a reductive
group. In this talk we will describe a solution to this problem,
under the assumption that the monoids are free and saturated.
(Universität Paderborn, Deutschland)
"Multiplicity-free actions induced by Dual pairs."
Let V be a finite dimensional representation of a reductive group G. The geometric problem,
whether there is a Zariski
open B-orbit in V (i.e., if V is spherical) is equivalent to the multiplicity-freeness of
the symmetric algebra of (the dual of) V. It is also natural to consider the question when the
exterior algebra of V is multiplicity-free. Unlike in the symmetric case, there is no
geometric counterpart of this condition. However, the lists of multiplicity-free symmetric and
exterior algebras are intimately related to each other.
The purpose of this talk is to give an explanation of this phenomenon by tracing back (skew)
multiplicity-free actions to realizations of Howe's dual pairs. We also discuss further
applications of this approach to harmonic analysis and multiplicity-freeness results for Lie
(Boston College, USA)
"Graded Lie algebras and representations of p-adic groups."
(Trinity College Dublin, Ireland )
"On certain spherical embeddings in characteristic p."
This talk is based on my paper
"On embeddings of certain spherical homogeneous spaces in prime characteristic".
Let G be a reductive group and let G/H be a spherical homogeneous space.
The G-equivariant (normal) embeddings of G/H have been the object of study
for quite some time. In characteristic p much more care is required.
I will discuss some results on embeddings that are induced from G0-embeddings,
where G0 is a smaller reductive group.
I will discuss results in the literature, generalities on parabolic induction,
existence of G-equivariant rational resolutions by toroidal embeddings,
and formula's for the canonical divisor.
(École Polytechnique Fédérale de Lausanne, Switzerland)
"Double centralizers of unipotent elements in semisimple algebraic groups",
(University of Warmia and Mazury, Poland)
classes and cohomology of the group of hamiltonian symplectomorphisms of
some homogeneous spaces."
In the talk I review my recent results on the cohomology of the group of
hamiltonian symplectomorphisms of some coadjoint orbits. The results are
contained in 2 my recent papers:
1) On the algebraic independence of the hamiltonian characteristic classes,
J. Symplectic Geom. 9(2011), 1-9 (with S. Gal and J. Kedra);
2) On the non-degenerate coupling forms, J. Geom. Physics, 61(2011),462-475
(with J. Kedra and A. Woike).
(Scientific Research Institute for System Studies of RAS, Moscow, Russia)
"On the complexity of the invariant Lagrangian subvarieties in the symplectic G-varieties."
Let G be a reductive group over an algebraically closed field of
characteristic zero, and let X be a symplectic G-variety equipped with a
moment map. We prove that all G-invariant Lagrangian subvarieties of X have
the same complexity and rank. We also give a description of the closure of
the image of the moment map that generalizes well-known results on the
description of the moment map for the cotangent bundles of G-varieties. We
note that this is a generalization of a result of D.I.Panyushev, who proved
that for a G-invariant subvariety Y of a G-variety X the conormal bundle of Y
in X has the same complexity as X.
(Joint work with D.A.Timashev.)