Lie Minisymposium
University of Sydney, 12 - 14 November 2003

Quantum Groups & Braid Groups
and other topics in Lie Theory

Titles and abstracts.
All talks will be held in Carslaw 175

Main speakers

François Digne Amiens Wednesday 11:30-12:30
Thursday 10:30-11:30
Presentations of braid groups
Recently new presentations for braid groups and their generalizations associated to arbitrary finite Coxeter groups (Artin-Tits groups of spherical type) were discovered. These new "dual" presentations, as well as the classical ones, lead to interesting submonoids of the Artin-Tits groups which are lattices for the order given by (left or right) divisibility (the "Garside property"). The dual presentations are also useful for computing centralizers. In the case of the general Artin groups associated to arbitrary Coxeter groups the classical monoids do not have the Garside property. It is hoped that the dual monoid behaves better with respect to divisibility. Currently this is known only for affine type Ãn.

In these two lectures we will describe the general setup for such presentations and explain some of the results and their consequences, reviewing work by Dehornoy, Bessis, Michel, Paris, Picantin and the speaker.

Anthony Joseph Weizmann Institute Wednesday 10:00-11:00
Thursday 9:00-10:00
Friday 9:00-10:00
Primitive ideals

  1. We start by reviewing the developments in primitive ideal theory. This includes, notably, Duflo's theorem, the theory of (left) cells and Goldie rank polynomials. Then we describe some recent work with Walter Borko describing the decomposition of the primitive spectrum into sheets. Finally, a number of open problems will be discussed.
  2. Primitive ideals and the Springer correspondence.
    We describe some remarkable and unexpected relationships between primitive ideal theory and the geometry of the flag variety. An earlier result relating Goldie rank polynomials of primitive quotients to characteristic polynomials of orbital varieties is reviewed. We then describe recent results obtained with Vladimir Hinich relating left cells of geometric cells. A notable point is that in both cases the objects in questions are computed in the same fashion but with different starting data. This rationalizes in an elegant fashion the slight difference between the representation theory and the geometry
  3. Crystals and Demazure flags.
    We describe a Demazure module and it origins in geometry. We briefly outline the theory of crystals and note that this leads to the most comprehensive proof of the Demazure character formula. Finally we describe recent work concerning the existence of Demazure flags (excellent filtrations) for appropriate tensor products in all characteristics and for all Kac Moody algebras in the symmetric simply laced case. This relies heavily on Kashiwara's globalization technology as well as on a positivity result of Lusztig.

Invited speakers

Peter Bouwknegt University of Adelaide Friday 3:30-4:30
T-duality, in its simplest form, is the R to 1/R symmetry of String Theory compactified on a circle of radius R. It can be generalised to manifolds which admit circle actions (e.g. circle bundles) or, more generally, torus actions. In the case of nontrivial circle bundles, T-duality relates manifolds of different topology and in particular provides isomorphisms between the twisted cohomologies and twisted K-theories of these manifolds. In this talk we will discuss these developments as well as provide some examples for torus bundles over flag manifolds.

Michael Cowling UNSW Friday 10:30-11:30
The fundamental theorem of projective geometry is also a theorem in analysis
The fundamental theorem of projective geometry states that maps on a spherical Tits building come from actions of the associated semisimple algebraic group. The aim of this talk is to convince the audience that in the real case, there is a local version of the theorem, which is based on analytic methods.

Alexei Davydov University of Macquarie Friday 2:00-3:00
Braids and symplectic groups
It is well known that braid groups act naturally on (powers of) objects of a braided monoidal category. We describe a braided monoidal category giving rise to braid group representations by symplectic matrices studied in (B. Wajnryb, A braidlike presentation of Sp(n,p) Israel J. Math. 76 (1991), no. 3, 265--288). In contrast to the "standard" examples of braided monoidal categories, such as categories of representations of quantum groups, the monoidal structure in our example is given by a sum of vector spaces rather than a tensor product. The braiding is given by a simple formula which allows easy generalizations leading to new finite quotients of braid groups.

Tony Dooley UNSW Wednesday 2:00-3:00
Analysis on harmonic extensions of H-type groups
The study of analysis on the upper half plane and its relationship with analysis on the line may be generalised to the study of the nilpotent component N of a semisimple Lie group as the boundary of the symmetric space G/K. In a similar way, a group of Heisenberg type serves as the boundary of a generalised Siegel domain.

Damek and Ricci showed that these spaces are harmonic, although they are not necessarily symmetric spaces, thus resolving an old conjecture of Lichnerowicz. Zhang and I recently gave a characterisation of the positive definite functions on these spaces.

I will give an introduction to the main results and discuss some open problems.

James East University of Sydney Friday 1:00-2:00
The factorizable braid monoid
In this talk we discuss a new braid monoid which we call the factorizable braid monoid, and denote by FBn. This monoid is a preimage of Fn (the monoid of uniform block bijections of an n-set) in the same natural way that the braid group is a preimage of the symmetric group. We will define FBn geometrically, and also describe how it may be constructed from the braid group and Eqn (the join semilattice of equivalence relations on an n-set). This allows us to find a presentation of FBn, and discover connections with the singular braid monoid. We conclude by showing how to define FBW for an arbitrary Coxeter group W. This involves a new interpretation of Eqn in terms of the Coxeter complex of the symmetric group.

Vyachesla Futorny University of Sydney Thursday 1:00-2:00
Representations of affine Lie superalgebras
We discuss the classification problem for the irreducible modules which have non-zero level and finite-dimensional weight spaces of the affine Lie superalgebras and also the classification of all irreducible weakly integrable modules (in the sense of Kac and Wakimoto). This talk is based on joint work with Senapati Eswara Rao.

Anthony Henderson University of Sydney Wednesday 4:30-5:30
Representations of wreath products on cohomology of De Concini-Procesi compactifications
Any finite complex reflection group W acts on the projective hyperplane complement of its reflecting hyperplanes, and thus the cohomology groups of this hyperplane complement are representations of W; their characters have been computed in many cases by Lehrer and others. It is an interesting problem to do the same for the "wonderful" compactification of the hyperplane complement defined by De Concini and Procesi. If W=Sn, this compactification is the moduli space of stable genus 0 curves with n+1 marked points, and the problem was solved by Ginzburg and Kapranov. I will explain the solution when W is the wreath product of Sn with the group of rth roots of unity.

Ngau Lam National Cheng Kung University Thursday 3:30-4:30
Unitarizable representations of Lie superalgebras and their formal characters
The oscillator representations of the classical Lie superalgebras gl(m|n) ops(m|2n) and their direct limits are unitarizable. We decompose the tensor powers of the oscillator representations into unitarizable irreducible highest weight representations. The multiplicities of the irreducible representations are given explicitly, and their formal characters are also obtained. Our study makes essential use of generalized Howe dualities between Lie superalgebras and classical Lie groups.

Yucai Su Shanghai Jiaotong University Thursday 2:00-3:00
Quasifinite representations of Lie algebras related of the Virasoro algebra
Let Γ be a free Abelian group, let L = ∑α in Γ Lα be a Γ-graded Lie algebra over the field of complex numbers (each homogeneous space may be infinite-dimensional) such that L0 is commutative. An L-module V is quasifinite if V = ∑α in ΓVα is Γ-graded such that Vα is finite dimensional and Lα Vβ is contained in Vα+β , for all α, β in Γ.

The purpose of this talk is to give a classification of irreducible quasifinite modules, and to determine the unitary ones, over the higher rank Virasoro algebras and the Lie algebras of Weyl type and Block type, which are related to the Virasoro algebra. In particular, it is obtained that an irreducible quasifinite module over an above-mentioned Lie algebra is a highest/lowest weight module or a module of the intermediate series except it is a higher rank Virasoro algebra.

George Willis University of Newcastle Wednesday 3:00-4:00
The direction of an automorphism of a totally disconnected group
Many totally disconnected locally compact groups have an associated `structure at infinity'. In the case of the automorphism group of a homogeneous tree it is the space of ends of the tree and in the case of a simple p-adic Lie group it is the Bruhat-Tits building at infinity. The talk will describe a extension of these cases to general totally disconnected groups based on the idea of the direction of an automorphism.

This is joint work with Udo Baumgartner.