Lie Days in Martina Franca
May 22-27, 2006
Invited Talks

home page of the conference

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Anton ALEKSEEV (Université de Genève)
"The Campbell-Hausdorff series and the Kashiwara-Vergne conjecture"

Abstract:   In 1978, Kashiwara and Vergne (KV) put forward a conjecture on the properties of the Campbell-Hausdorff series. Among other things, the KV conjecture gives a natural proof of the Duflo theorem on the isomorphism between the center of the universal enveloping algebra U(g) and the algebra of invariant polynomials on g^*. I'll report on a recent proof of the KV conjecture based on the deformation quantization techniques and on the earlier results by Torossian.
The talk is based on a joint with with E. Meinrenken.

Karin BAUR (Leicester University)
"Richardson elements for parabolic subgroups"

Abstract:   Let P be a parabolic subgroup of a simple algebraic group (over an algebraically closed field). Let p = l + u be a Levi decomposition of the corresponding Lie algebra. By a fundamental theorem of Richardson, P on u with an open dense orbit, the orbit of Richardson elements (for P). We are able to construct Richardson elements using minimal subsystems of the roots. From these we recover the Bala-Carter label of the dense orbit.

Roman BEZRUKAVNIKOV (Northwestern University)
"Noncommutative resolutions and canonical basis via positive characteristic"

Abstract:   Study of modular representations of simple Lie algebras and Cherednik algebras lead to examples of the following situation: given a well-known resolution of singularities X of some singular space Y we get also a noncommutative resolution A of Y with a derived equivalence between A and X. This also yields a basis in cohomology of X, which often turns out to be the canonical basis in the sense of Kazhdan-Lusztig theory. Conjecturally this scenario applies also to other settings, such as quiver varieties.

Alexander BRAVERMAN (Brown University)
"Pursuing the double affine Grassmannian"

Anders BUCH (Rutgers University)
"Quantum cohomology of isotropic Grassmannians"

Abstract:   The (small) quantum cohomology ring of a homogeneous space is a deformation of the classical cohomology ring, which uses the three point, genus zero Gromov-Witten invariants as its structure constants. I will present structure theorems for the quantum cohomology of isotropic Grassmannians, including a quantum Pieri rule for multiplication with the special Schubert classes, and a presentation of the quantum ring over the integers with the special Schubert classes as the generators. These results are new even for the ordinary cohomology of isotropic Grassmannians, and are proved directly from the definition of Gromov-Witten invariants by applying classical Schubert calculus to the kernel and span of a curve. This is joint work with A. Kresch and H. Tamvakis.

Nicoletta CANTARINI (Università di Padova)
"Infinite dimensional primitive linearly compact Lie superalgebras"

Abstract:   A well-known theorem of E. Cartan gives a classification of infinite-dimensional linearly compact Lie algebras L, which admit a maximal open subalgebra L₀ containing no non-zero ideals of L. Such a Lie algebra is called primitive and the pair (L, L₀) is called a primitive pair. We consider the problem of classification of primitive pairs in the Lie superalgebra case and describe its solution. This talk is based on a joint paper with Victor Kac.

Alberto DE SOLE (Harvard University & INdAM)
"Quantum and classical W-algebras"

Abstract:   I will first introduce the notions of vertex algebra and Poisson vertex algebra, and the relation between them via the so called quasi-classical limit. I will then introduce the Zhu algebra of a vertex algebra, which controls its representations, both in the quantum and in the classical case. As a special case, I will consider the affine and finite W algebras, obtained by the method of quantum Hamiltonian reduction.

Harm DERKSEN (University of Michigan)
"LR-coefficients which are equal to 1"

Abstract:   The triples of partitions (\lambda,\mu,\nu) for which the corresponding Littlewood-Richardson coefficient c_{\lamda,\mu}^\nu is nonzero form a polyhedral cone. Horn's conjecture, now a theorem, recursively provides a list of inequalities which describes this cone. The set of all triples (\lambda,\mu,\nu) for which c_{\lambda,\mu}^\nu = 1 can be described in a similar recursive way.
This is joint work with Jerzy Weyman.

Iain GORDON (Glasgow University)
"Rational Cherednik algebras, quiver varieties and q-Schur algebras"

Abstract:   We will discuss the representation theory of category O for rational Cherednik algebras as studied by Rouquier et al. In this context, category O is a generalisation of the q-Schur algebra to the complex reflection groups of type G(m,1,n) - wreath product of finite cyclic group and symmetric group; it is currently poorly understood. Using noncommutative geometry, we will relate category O to quiver varieties for affine Dynkin diagrams: the variation of parameter for category O will become the variation of GIT quotient for the quiver varieties. In this way, a little of the combinatorics of category O might become visible.

Maria GORELICK (Weizmann Institute of Science)
"On a generic Verma module at the critical level"

Abstract:   I'll describe an algebra of singular vectors of a generic Verma module at the critical level over affine Lie superalgebra with a symmetrizable Cartan matrix. This description implies the Kac-Kazhdan formula for the character of a generic irreducible highest weight module at the critical level.

David HERNANDEZ (CNRS & Université de Versailles)
"Symmetries and crystals of extremal modules"

Abstract:   The crystals of extremal modules of quantum Kac-Moody algebras introduced by Kashiwara are a generalization of crystals of simple highest weight modules. For quantum affine algebras, Kashiwara proved that each fundamental level 0 extremal module V_i admits an automorphism z_i, and that the module V_i  /(z_i - Id) V_i is simple finite dimensional with a crystal basis B_i. One problem which has been recently intensively studied is to describe explicitly the finite crystal B_i.
In a joint work with Nakajima, we give a positive answer to a question of Kashiwara by proposing a new uniform construction of the principal connected component of extremal crystals: for arbitrary symmetrizable Kac-Moody algebra we prove that each connected component of the Nakajima monomial crystal is isomorphic to a connected component of an extremal crystal. Moreover we interpret z_i as a "strong" symmetry property on monomials. As an application we describe explicitly B_i (except for a few nodes of type E₇⁽¹⁾, E₈⁽¹⁾ and E₆⁽²⁾ ), including cases non described so far.

Syu KATO (Tokyo University)
"An exotic Deligne-Langlands correspondence for symplectic group"

Abstract:   The Deligne-Langlands-Lusztig conjecture (proved by Kazhdan-Lusztig, Ginzburg) asserts that each irreducible module of an affine Hecke algebra corresponds to some geometric datum. An affine Hecke algebra of type C admits a natural two-parameter deformation H (and this is best possible in some sense).
In this talk, we realize H as the equivariant K-group of a certain variety, which we refer as the exotic Steinberg variety. This enables us to present a Deligne-Langlands type classification of simple H-modules when the values and ratios of deformation parameters are not too bad.

Allen KNUTSON (University of California - San Diego)
"Joseph polynomials, the Brauer algebra, statistical mechanics, and the commuting variety"

Abstract:   Joseph associated a polynomial (really, an equivariant cohomology class) to each component of the intersection of a nilpotent orbit with a Borel subalgebra, and proved that these give a basis for a (in fact Springer) representation of the Weyl group. I'll explain that a richer theory is available by looking at the normal cone to this intersection, which has many extra components that were invisible to Spaltenstein and Springer.
In the case of the orbit of matrices whose square is zero, the components afford a representation of the Brauer algebra, already of interest in statistical mechanics. As an application of this example I'll give a formula for the degree of the commuting variety, previously calculated up to 4x4 matrices; our formula is effective to 11x11.
This work is joint with Paul Zinn-Justin of Paris-Sud.

Guido PEZZINI (Università di Roma "La Sapienza")
"Simple immersions of wonderful varieties"

Abstract:   Wonderful varieties are projective algebraic varieties endowed with an action of a semisimple connected algebraic group G, having certain properties which have been inspired by the know compactifications of symmetric homogeneous spaces given by De Concini and Procesi. Wonderful varieties turn out to have a significative role in the theory of spherical varieties, which are a class of G-varieties representing a common generalization of flag varieties and toric varieties. In this talk we answer a question raised in the 90's in an article by Michel Brion: given a wonderful G-variety X, we study whether X can be realized as a closed subvariety of the projective space of a simple G-module.

Olivier SCHIFFMANN (École Normale Superieure - Paris)
"Elliptic Hall algebra and Cherednik algebra"

Abstract:   We give a realization of the spherical double affine Hecke algebra of tye A in terms of a convolution algebra of perverse sheaves on the moduli space of vector bundles on an elliptic curve. This yields a "canonical basis" for the (spherical) DAHA in type A, as well as an interpretation of Macdonald polynomials in terms of some Hecke eigensheaves.

Boris SHOIKHET (Glasgow University)
"On the associated graded Lie algebra of a free associative algebra"

Abstract:   We consider the lower central filtration of the Lie algebra associated with the free associative algebra. We study the associated graded Lie algebra. We prove, that after a quotient by the center, this Lie algebra has an action of the Lie algebra of vector fields W_n, and the bracket is W_n-equivariant
(joint work in progress with Boris Feigin and Eric Rains).

Eric SOMMERS (University of Massachusetts - Amherst)
"Equivalence classes of Borel-stable subspaces"

Abstract:   An equivalence relation is defined on the set of subspaces of the nilradical which are stable under the Borel subgroup. A partial order is defined and studied on the equivalence classes of this relation, and connections are made to a partial order defined by Achar on a related set.

Catharina STROPPEL (Glasgow University)
"Serre functors, symmetric algebras and Lie theory"

Abstract:   The talk will be on Serre functors in representation theory. Serre functors were introduced by Orlov and Kapranov, motivated by the classical Serre duality. Motivated by a question of Kapranov we will give a criteria for functors being a Serre functor. This will be applied to categories arising in representation theory (of Lie algebras, rational Cherednik algebras, Schur algebra etc. ) to obtain an algebraic description of the corresponding Serre functors. As an application we verify a conjecture of Khovanov which generalises Soergel's endomorphism theorem and provides a conjectural Lie theoretic description of Khovanov homology.

Alexis TCHOUDJEM
(Université de Lyon "Claude Bernard")
"Cohomology of line bundles over wonderful varieties"

Abstract:   The wonderful varieties are compactifications of some homogeneous spaces for a linear semi-simple group and which, in some sense, are made a bit like toric varieties. Flag varieties and the compactifications of symmetric spaces built by De Concini and Procesi are examples of wonderful varieties. For some wonderful varieties one can prove a Borel-Weil-Bott type theorem, i.e. a description in any degree of the cohomology groups of their line bundles.

Valerio TOLEDANO LAREDO
(Université de Paris VI "Pierre et Marie Curie")
"Quasi-Coxeter algebras, Dynkin diagram cohomology and quantum Weyl groups"

Abstract:   I will describe in this talk the proof of an analogue of the Kohno-Drinfeld theorem which had been independently conjectured by De Concini and myself. In this analogue, R-matrices are replaced by quantum Weyl group operators and the Knizhnik-Zamolodchikov equations by the Casimir equations, a flat connection on the generalised configuration spaces associated with root systems. One of the key ingredients in the proof is the notion of quasi-Coxeter algebras which are to generalised braid groups what Drinfeld's quasitriangular, quasi-Hopf algebras are to Artin's braid groups. In turn, the definition of quasi-Coxeter algebras is inspired by De Concini and Procesi's wonderful compactifications of these generalised configuration spaces.