"The commuting variety of semisimple symmetric Lie algebras."Abstract: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.
"Spherical orbits and Weyl group elements."Abstract: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.)
"Degenerate flag varieties."Abstract: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 decomposition.
"On the projective normality of model wonderful varieties."Abstract: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 ChirivĂ¬ 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.)
"Cohomological finite generation and some homogeneous spaces."Abstract: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.
"On the nullcone of representations of reductive groups",
abstract
"Riemannian foliations of symmetric spaces", abstract.
"Commuting involutions and degenerations of isotropy representations."
"Maximal abelian Borel stable subspaces and the CSDW conjecture for
symmetric spaces."Abstract: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).
"A generalization of Delzant polytopes for certain non-abelian
multiplicity-free Hamiltonian
actions."Abstract: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.
"Multiplicity-free actions induced by Dual pairs."Abstract: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 superalgebras.
"Graded Lie algebras and representations of p-adic groups."
"On certain spherical embeddings in characteristic p."Abstract: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 G _{0}-embeddings,
where G_{0} 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.
"Double centralizers of unipotent elements in semisimple algebraic groups",
abstract.
"Hamiltonian characteristic
classes and cohomology of the group of hamiltonian symplectomorphisms of
some homogeneous spaces."Abstract: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).
"On the complexity of the invariant Lagrangian subvarieties in the symplectic G-varieties."Abstract: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.) |