Victor Batyrev
(Universität Tübingen, Deutschland)
"On generalisations of LosevManin moduli spaces for classical root systems."
Abstract:
Losev and Manin introduced fine moduli spaces
L_{n} of stable
npointed chains of projective lines. The moduli space
L_{n+1}
is isomorphic to the toric variety X(A_{n})
associated with the root system A_{n}, which is part of a general
construction to associate with a root system R of rank n an
ndimensional smooth projective toric variety X(R). We investigate
generalisations of the LosevManin moduli spaces for the other families of
classical root systems. Our main idea is to consider stable
(2n+1)pointed resp. 2npointed chains of projective lines with an
involution.
(This is a joint work with Mark Blume.)
Paolo Bravi
(Università di Roma "La Sapienza", Italia)
"Spherical systems and quotients."
Abstract:
Spherical systems are combinatorial invariants associated to the
socalled wonderful varieties and defined axiomatically in terms of roots
systems. They somewhat generalize the restricted root systems associated
to symmetric varieties. In the general context of spherical systems there
is a notion of quotient (which plays no role in the context of restricted
root systems). We will focus on it and illustrate some of its crucial
combinatorial properties which for the moment we are able to prove only by
classification arguments.
Nicoletta Cantarini
(Università di Padova, Italia)
"On linearly compact nLie superalgebras."
Abstract:
We shall generalize the notion of Lie superalgebras to that of nLie
superalgebras and describe the classification
of simple linearly compact nLie superalgebras. This classification is based on
a bijective correspondence between
nonabelian nLie superalgebras and transitive
Zgraded Lie superalgebras of the
form L=\oplus_{j=1}^{n1} L_{j}, where
dim L_{n1}=1, L_{1} and L_{n1}
generate L, and [L_{j}, L_{nj1}]=0 for all j,
thereby reducing it to the known
classification of simple linearly compact Lie superalgebras and their
Zgradings.
Stéphanie CupitFoutou
(Universität zu Köln, Deutschland)
"A geometrical realization of wonderful varieties."
Abstract:
Wonderful varieties are generalizations of De ConciniProcesi
compactifications of symmetric spaces. Luna conjectured that
such varieties could be classified by spherical systems, combinatorial objects
built on Dynkin diagrams. The uniqueness part was positively answered by Losev.
Until recently, only partial positive results about existence (and uniqueness)
were known all obtained by casebycase considerations following Luna's Lie
theoretical approach initiated for acting groups of type A.
In my talk, I will present a geometric approach inspired by Brion's work on Cox
rings of wonderful varieties. I will explain how
by means of invariant Hilbert schemes introduced by Alexeev and Brion, I am able
to answer Luna's conjecture in full generality.
Antonio Di Scala
(Politecnico di Torino, Italia)
"Kähler immersions of homogeneous Kähler manifolds into complex
space forms."
Abstract:
Let (M,Ω) be a homogeneous Kähler manifolds and let
(S,s) be a complex space form. An immersion Φ : M → S is called
a Kähler immersion if Φ is holomorphic and isometric, i.e.
Φ^{*}s = Ω.
In this talk we will discuss such Kähler immersions when S is
l^{2}(C) or
CH^{∞} equipped with the standard flat and hyperbolic Kähler forms.
This generalize previous results when M is a Symmetric Space.
Friedrich Knop
(Universität ErlangenNürnberg, Deutschland)
"Automorphisms of multiplicity free Hamiltonian manifolds."
Abstract:
A Hamiltonian Kmanifold (K a compact connected Lie group) is
multiplicity free if all of its symplectic reductions are points. Among
Hamilonian manifolds they are analogous to spherical varieties. Delzant
conjectured that a (compact) multiplicity free Hamiltonian manifold is
uniquely characterized by two data: the image of the moment map and the
principal isotropy group. We are going to report on the proof of this
conjecture. It proceeds in two parts. First, Losev has shown that
Delzant's conjecture is true locally (over the coadjoint
representation). The second part is to show that the sheaf of
automorphisms has vanishing higher cohomology. In the talk I'll explain
how to compute the automorphisms of a multiplicity free Hamiltonian
manifold. Moreover, I'll discuss an extension of Delzant's conjecture to
quasiHamiltonian manifolds and Hamiltonian manifolds for loop groups.
Hong Van Le
(Institute of Mathematics of the Academy of Sciences, Czech Republic)
"Nilpotent orbits in real Z_mgraded semisimple Lie algebras."
Abstract:
We propose a method to classify homogeneous nilpotent elements
in a real Z_{m}graded semisimple Lie algebra g. Using this we describe
the set of orbits of homogeneous elements in a real Z_{2}graded semisimple
Lie algebra. A classification of 4vectors (resp. 4forms) on R^{8} can be
given using this method.
Paul Levy
(Lancaster University, UK)
" Vinberg's θgroups in positive characteristic and Popov's conjecture."
Abstract:
It is well known that symmetric spaces share many
invarianttheoretic properties with the adjoint representation of a
complex reductive group G. A useful notion in the symmetric space
setting is that of a Cartan subspace, which plays the role played by the
Cartan subalgebra in the adjoint representation and allows one to
describe the invariants via an analogue of the Chevalley restriction
theorem. In a seminal 1976 paper, Vinberg showed that one can extend
much of that analysis to an arbitrary periodic automorphism of G. The
new feature here is that the Weyl group associated to a Cartan subspace
is in general a complex reflection group.
In this talk I will outline the main results in the theory of Vinberg's
θgroups and will explain some of the steps required to generalise
them to positive characteristic. Subsequently I will discuss a fruitful
approach to tackling a longstanding conjecture of Popov on the
existence of a slice analogous to Kostant's slice to the regular
nilpotent orbit in the adjoint representation.
Pierluigi Moseneder
(Politecnico di Milano, Italia)
"Denominator formulas for superalgebras."
Abstract:
The Weyl denominator identity is one of the most intriguing identities
in the character ring of a complex finite dimensional simple Lie
algebra. In this talk we are presenting expressions for the analog of
the denominator identity in the case of a basic classical Lie superalgebra.
Unlike the Lie algebra case, the denominator identity depends on the choice
of the positive system. Kac and Gorelik provided formulas for a special
class of positive systems. In our talk we will explore a case that is
opposite to the ones studied by Kac and Gorelik: the so called
distinguished case. Connections with Howe theory of dual pairs are also made.
Yury Neretin
(Universität Wien, Österreich)
" Multiplications on quotient spaces and multivariate characteristic
functions"
Paolo Papi
(Università di Roma "La Sapienza", Italia)
"Conformal pairs associated to symmetric spaces and the algebraic
Dirac Operator."
Abstract:
A pair (s,k), where s is a finitedimensional simple complex Lie
algebra and k is a semisimple subalgebra of s is called a conformal
pair if there exists an integrable highest weight module V over the affine
KacMoody algebra s^ which decomposes finitely w.r.t. k^. We shall discuss
the explicit decompositions for a remarkable class of conformal pairs
associated to infinitesimal symmetric spaces. This problems turns out to
be connected to generalizations of KostantPeterson theory of abelian
ideals in Borel subalgebras and to KacWakimoto analysis of the modular
properties of characters of affine algebras. Finally we will discuss the
problem in the framework of Kostant's cubic Dirac operator.
Alexander Premet
(University of Manchester, UK)
"On 1dimensional representations of quantized Slodowy slices."
Abstract:
In my talk, I'll review the latest results on
1dimensional representations of the endomorphism algebras
of the generalised GelfandGraev modules over
finite dimensional simple Lie algebras.
Fulvio Ricci
(SNS Pisa, Italia)
"Analytic properties of the spherical transform on nilpotent Gelfand pairs."
Abstract:
Let K be a compact subgroup of a Lie group G. The pair (G,K) is called a
Gelfand pair if the algebra of Ginvariant differential operators on G/K is
commutative. A nilpotent Gelfand pair is one in which K is a group of
automorphisms of a nilpotent Lie group N and G=K\ltimes N.
The spherical transform is an important tool in the analysis of operators
acting on function spaces on G/K, and it plays the same rôle as the
Fourier transform does in classical analysis.
We present some recent results on the mapping properties on the spherical
transform on nilpotent Gelfand pairs, and the aspects of representation
theory that intervene.
(This is joint work with V. Fischer and O. Yakimova.)
Simon Salamon
(Politecnico di Torino, Italia)
"Geometry and topology of Wolf spaces."
Abstract:
For each compact simple Lie group G there is a Riemannian symmetric space
G/H with holonomy H a subgroup of Sp(n)Sp(1). These spaces are long known
to provide models for a quaternionic version of Kähler geometry and
twistor theory (Wolf 1964, Alekseevsky 1968) but reign supreme in the
realm of positive Ricci curvature. In this talk I shall survey their
properties and focus on open problems.
Eitan Sayag
(BenGurion University, Israel)
"Invariant measures, decay of smooth vectors, and lattice
counting."
Abstract:
We study the decay of smooth vectors of the regular representation of
a reductive groupon the Banach space of integrable functions on a
unimodular Homogeneous Gspace X.
The key to our results is a bound on the measure of certain distorted
balls in X.
We connect this to the problem of counting lattice points, a problem with
origins in Gauss's circle problem. In more recent times the counting of
lattice points in symmetric spaces was studied both using Spectral
methods (DukeRudnickSarnak) and using Ergodic theoretic methods
(EskinMcmullen).
We review these approaches and present our result on the invariant
measure on Homogeneous spaces that allows us to give a refinement of
spectral method.
(Joint work with B. Krotz and H. Schlichtkrull.)
Aleksy Tralle
(University of Warmia and Mazury, Poland)
"On curvature constructions of symplectic forms over symmetric spaces."
Abstract:
In symplectic geometry, it is important to have a method of constructing
symplectic forms on total spaces of bundles in a way, that the constructed form
restricts symplectically on the fibers (note that it is not required that the
base of the bundle is symplectic as well). Such forms constitute a particular
class of coupling forms which have numerous applications in mathematical
physics. One of such methods was proposed by Sternberg and Guillemin. Let there
be given a fiber bundle F → E → B
such that its structure group is a Lie group G. Assume that F admits a
Ginvariant symplectic structure. If the Gaction on F is hamiltonian,
with moment map μ, then, the following assumption allows one to construct a
symplectic form on E:
there exists a connection in the associated principal bundle
G → P → B
such that the connection form Ω restricted to the horizontal distribution
of the connection, has the property that the 2form
<Ω(X,Y),μ(f)>
is nondegenerate for all horizontal vector fields X,Y, and all covectors from
μ(F) (which is a subset of g^*).
However, such connections are scarce, and there are obstructions to their
existence. Nevertheless, Lerman constructed such connections in fiber bundles
which are bundles of coadjoint orbits of compact Lie groups over coadjoint
orbits. In the present paper we give a full solution to the following problem:
find all fiber bundles of the form
H/V → K/V → K/H
for compact Lie groups K such that the canonical invariant connection has the
required property. In particular, new homogeneous bases with the required
property have been found, among them locally symmetric spaces of noncompact type, and their
locally symmetric compact quotients, homogeneous quaternionicKähler manifolds,
and some other important classes. (pdfversion.)
Vladimir Zhgoon
(Moscow State University, Russia)
"On the equivariant geometry of the cotangent vector bundle of quasiprojective
varieties."
Abstract:
Let G be a connected reductive group acting on an irreducible normal algebraic
variety X. The aim of this talk is to generalize the result of E.B.Vinberg who
constructed the rational Galois covering of T^{*}X for quasiaffine X by the cotangent
bundle to the variety of horospheres. The Galois group of this rational covering is
equal to a little Weyl group of the variety X. We notice that this result could not be
directly generalized to quasiprojective varieties since the set of generic horospheres
s not good enough for this purpose, that can be seen in the case when X is a flag
variety. In this talk we give the construction of a family of degenerate horospheres and
he variety Hor parametrizing them, such that there is a rational covering of the
cotangent vector bundles T^{*}Hor > T^{*}X. The Galois group of this rational
covering is equal to a little Weyl group of the variety X.
(pdfversion.)
