Symmetric Spaces and their Generalisations

Titles and Abstracts of the talks

Victor Batyrev (Universität Tübingen, Deutschland)
"On generalisations of Losev-Manin moduli spaces for classical root systems."
Losev and Manin introduced fine moduli spaces Ln of stable n-pointed chains of projective lines. The moduli space Ln+1 is isomorphic to the toric variety X(An) associated with the root system An, which is part of a general construction to associate with a root system R of rank n an n-dimensional smooth projective toric variety X(R). We investigate generalisations of the Losev-Manin moduli spaces for the other families of classical root systems. Our main idea is to consider stable (2n+1)-pointed resp. 2n-pointed 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."
Spherical systems are combinatorial invariants associated to the so-called 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 n-Lie superalgebras."
We shall generalize the notion of Lie superalgebras to that of n-Lie superalgebras and describe the classification of simple linearly compact n-Lie superalgebras. This classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive Z-graded Lie superalgebras of the form L=\oplus_{j=-1}^{n-1} Lj, where dim Ln-1=1, L-1 and Ln-1 generate L, and [Lj, Ln-j-1]=0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras and their Z-gradings.

Stéphanie Cupit-Foutou (Universität zu Köln, Deutschland)
"A geometrical realization of wonderful varieties."
Wonderful varieties are generalizations of De Concini-Procesi 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 case-by-case 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."
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 l2(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 Erlangen-Nürnberg, Deutschland)
"Automorphisms of multiplicity free Hamiltonian manifolds."
A Hamiltonian K-manifold (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 quasi-Hamiltonian 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_m-graded semisimple Lie algebras."
We propose a method to classify homogeneous nilpotent elements in a real Zm-graded semisimple Lie algebra g. Using this we describe the set of orbits of homogeneous elements in a real Z2-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on R8 can be given using this method.

Paul Levy (Lancaster University, UK)
" Vinberg's θ-groups in positive characteristic and Popov's conjecture."
It is well known that symmetric spaces share many invariant-theoretic 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 θ-groups and will explain some of the steps required to generalise them to positive characteristic. Subsequently I will discuss a fruitful approach to tackling a long-standing 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."
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."
A pair (s,k), where s is a finite-dimensional 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 Kac-Moody 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 Kostant-Peterson theory of abelian ideals in Borel subalgebras and to Kac-Wakimoto 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 1-dimensional representations of quantized Slodowy slices."
In my talk, I'll review the latest results on 1-dimensional representations of the endomorphism algebras of the generalised Gelfand-Graev modules over finite dimensional simple Lie algebras.

Fulvio Ricci (SNS Pisa, Italia)
"Analytic properties of the spherical transform on nilpotent Gelfand pairs."
Let K be a compact subgroup of a Lie group G. The pair (G,K) is called a Gelfand pair if the algebra of G-invariant 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."
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 (Ben-Gurion University, Israel)
"Invariant measures, decay of smooth vectors, and lattice counting."
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 G-space 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 (Duke-Rudnick-Sarnak) and using Ergodic theoretic methods (Eskin-Mcmullen). 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."
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 G-invariant symplectic structure. If the G-action 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 2-form <Ω(X,Y),μ(f)> is non-degenerate 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 non-compact type, and their locally symmetric compact quotients, homogeneous quaternionic-Kähler manifolds, and some other important classes. (pdf-version.)

Vladimir Zhgoon (Moscow State University, Russia)
"On the equivariant geometry of the cotangent vector bundle of quasiprojective varieties."
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. (pdf-version.)