Representation Theory Seminar



The Representation Theory Seminar at the UvA regularly takes place on fridays.

Time: 14.00-15.00
Place: To be announced.


Upcoming talks

Definite schedule will be determined along the way, so please check the web-site on a regular basis. The seminar is now being set-up for this academic year, the first upcoming talks will be announced soon.

Previous talks

Date: 21 September 2006
Speaker: Thomas Quella (UvA)
Title: Wess-Zumino-Witten models on Supergroups.
Abstract: Wess-Zumino-Witten models - widely used to describe the propagation of strings on group manifolds - are conformal sigma-models with an infinite-dimensional current algebra symmetry. In this talk we will consider the generalization to supergroups which describe bosonic and fermionic degrees of freedom on an equal footing. Our focus will be on the geometric properties of these models and how harmonic analysis can be used to extract physical information. One of the surprising results is that the Laplacian is not diagonalizable. An essential part of the presentation will be concerned with the representation theory of Lie superalgebras which somewhat differs from that of ordinary Lie algebras.
Date: 28 September 2006
Speaker: Thomas Quella (UvA)
Title: Harmonic analysis on supergroups.
Abstract: We show how the algebra of functions on a (sufficiently well-behaved) supergroup can be decomposed with respect to the left and right regular action of the underlying Lie superalgebra. The analysis will take place in two steps. First an auxiliary problem with a set of reduced differential operators is solved. Afterwards eigenfunctions of the Laplacian related to this auxiliary problem are mapped to generalized eigenfunctions of the full Laplacian. Among our findings is that the complete spectrum consists of projective, not necessarily irreducible, representations only. Also - as indicated by the word "generalized eigenfunctions" - the full Laplacian turns out to be non-diagonalizable in a subsector of the representation space.
Mathematical foundations:
- Y.M. Zou, "Category of finite dimensional weight modules over type I classical Lie superalgebras", J. Alg. 180 (1996), 459-482.
- R.B. Zhang and Y.M. Zou, http://de.arxiv.org/abs/math.RT/0608693 (Hopf superalgebra approach to a similar problem but without true representation theory).
Date: 5 October 2006
Speaker: Wolter Groenevelt (UvA)
Title: Vector valued big q-Jacobi functions.
Abstract: We study an unbounded second order q-difference operator that has certain basic-hypergeometric functions, called big q-Jacobi functions, as eigenfunction. Spectral analysis of the difference operator leads to an integral transform that has two big q-Jacobi functions as a kernel. The difference operator corresponds to the action of the Casimir operator in a certain representation of the quantized universal enveloping algebra of su(1,1).
Date: 19 October 2006
Speaker: Jan Felipe van Diejen (Universidad de Talca, Chili)
Title: Bernstein-Szego Polynomials Associated with Root Systems.
Abstract: We introduce multivariate generalizations of the Bernstein-Szego polynomials, which are associated to the root systems of the complex simple Lie algebras. The multivariate polynomials in question generalize Macdonald's Hall-Littlewood polynomials associated with root systems. For the root system of type A_1 (corresponding to the Lie algebra sl(2;C)) the classic Bernstein-Szego polynomials are recovered.
Date: 26 October 2006
Speaker: Fokko van de Bult (UvA)
Title: The modular double of a quantum group and hyperbolic hypergeometric functions.
Abstract: The classical hypergeometric functions (normal and basic or q-hypergeometric functions) are intimately related to representations of certain algebras. However for the new hyperbolic hypergeometric functions the relevant algebras are not yet well understood. We will introduce the hyperbolic hypergeometric functions and the modular double of a quantum group and then argue that they are related. We will mainly do this by showing an example where Ruijsenaars' R-function appears as matrix-coefficients of a representation of the modular double of Uq(sl2).
Date: 2 November 2006
Speaker: Eric Opdam (UvA)
Title: KZ-twists and fake symmetries.
Abstract: We introduce the so-called fake degrees for representations of complex reflection groups. We discuss a symmetry property of these degrees with respect to a twisting operation on the category of representations of the complex reflection group. We interpret this formula in view of the theory of cyclotomic Hecke algebras. Finally we discuss recent progress due to Yuri Berest and Oleg Chalykh on a related conjecture about these twists.
Date: 9 and 16 November 2006
Speaker: Maarten Solleveld (UvA)
Title: K-theory and representations of affine Hecke algebras.
Abstract: Affine Hecke algebras are defined as q-deformations of the group algebra of a Coxeter group. It is conjectured that the topological K-theory of the C*-completion of an affine Hecke algebra (AHA for short) is independent of the parameter(s) q. In these talks we plan to relate this conjecture to the representation theory of AHA's. The first part will be mainly about noncommutative geometry. We explain some of the ideas behind this relatively recent field of mathematics. We introduce topological K-theory and show that it can be considered as a cohomology theory for non-Hausdorff spaces. In the second part we discuss completions of AHA's, and the representation theory thereof. We describe the spectrum of an AHA and show how topological K-theory may help to understand this space.
Date: 23 November 2006
Speaker: Tonny Springer (UU)
Title: Weyl groups and Hecke algebras (after Chriss-Ginzburg), I.
Date: 30 November 2006
Speaker: Jasper Stokman (UvA)
Title: Difference Harish-Chandra series.
Abstract: I describe the centralizer of the Macdonald difference operator in suitable rings of difference operators. The associated spectral problem gives rise to a system of basic hypergeometric difference equations. Well known examples of solutions are the Macdonald polynomials associated to root systems. In this talk I discuss the Harish-Chandra series solutions of the system. The talk is based on joint with Gail Letzter.
Date: 7 December 2006
Speaker: Tom H. Koornwinder (UvA)
Title: The relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra in the rank 1 case.
Abstract: Zhedanov [1] introduced in 1991 an algebra AW(3) with three generators and three relations in the form of q-commutators, which describes deeper symmetries of the Askey-Wilson polynomials.
One year later Cherednik introduced double affine Hecke algebras associated with root systems. See Macdonald [2] for a nice description of this theory. In the so-called basic representation of a double affine Hecke algebra certain of the operators in this algebra have non-symmetric Macdonald polynomials as their joint eigenfunctions. A suitable symmetrization of these polynomials and of these operators gives the original Macdonald polynomials, together with the operator of which they are eigenfunctions. This theory was also extended to the Macdonald-Koornwinder case (in rank 1 the Askey-Wilson case).
In the lecture I will give for the rank 1 case an embedding of slightly modified AW(3) (one generator commuting with the other generators added) in the double affine Hecke algebra. If time permits, related results concerning so-called structure relations and lowering and raising relations will be discussed.

[1] A.S. Zhedanov, "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.
[2] I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge University Press, 2003.
Date: 15 February 2007
Speaker: Jasper Stokman (UvA)
Title: Holonomic systems of difference equations.
Abstract: I will discuss:
1. The construction of holonomic systems of difference equations using either R-matrices or one-cocycles of affine Weyl groups.
2. The construction of one-cocycles using the double affine Hecke algebra.
3. Solutions of holonomic systems of difference equations.
4. Connection to the spectral problem of the Macdonald difference operators.
This talk is based on my ongoing efforts to distillate the core of Cherednik's book on double affine Hecke algebras.
Date: 22 February 2007
Speaker: Yuri Berest (Cornell University and I.H.E.S.)
Title: Ideals of rings of differential operators on algebraic curves.
Abstract: There is a simple geometric classification of ideals of the first complex Weyl algebra in terms of the Calogero-Moser algebraic manifolds. In this talk I will describe a global version of this construction, replacing the Weyl algebra by the ring of regular differential operators on an arbitrary smooth algebraic curve.

This is joint work with George Wilson (Oxford).
Date: 8 March 2007
Speaker: Fokko van de Bult (UvA)
Title: Limits of hyperbolic hypergeometric integrals.
Abstract: Starting with the hyperbolic version of the multivariate BCn type hypergeometric integrals, I consider what integrals we can obtain as limits. An entire degeneration scheme is obtained and by viewing the integrals as limits we obtain their symmetries and transformation formulas immediately. This is joint work with Eric Rains.
Date: 15 March 2007
Speaker: Arthemy Kiselev (I.H.E.S.)
Title: Pre-Hamiltonian structures for integrable nonlinear systems.
Abstract: I consider pre-Hamiltonian differential operators in total derivatives; they are defined by the property that their images are subalgebras of the Lie algebra of evolutionary vector fields. This construction is naturally related to the Lie algebroids over infinite jet spaces. I assign a class of these operators to integrable KdV-type hierarchies of symmetry flows on hyperbolic Euler-Lagrange Liouville-type systems (e.g., 2D Toda lattices associated with semi-simple Lie algebras).

The talk is based on a joint paper with J.W.van de Leur.
Date: 22 March 2007
Speaker: Roland van der Veen (UvA)
Title: The Volume Conjecture.
Abstract: The volume conjecture relates two very different ways of studying knots. The first is by using quantum groups to calculate polynomial invariants from a diagram of the knot. The second is by considering the geometry and topology on the complement of the knot. In its simplest form the volume conjecture states that the quantum invariant constructed from the N-dimensional irreducible representation of U_q(sl_2), evaluated at the primitive N-th root of unity, grows exponentially as N goes to infinity and that the growth rate is exactly the volume of the complement of the knot with respect to the unique metric of constant curvature -1.
Date: 5 April 2007
Speaker: Erik Koelink (TUD)
Title: The dual of the quantum group analogue of the normalizer of SU(1,1) in SL(2,C).
Abstract: Abstract: There is a very nice theory of locally compact quantum groups on the level of von Neumann algebras due to Kustermans and Vaes including a Pontryagin duality theorem. An important example is the quantum group analogue of the normalizer of SU(1,1) in SL(2,C). The purpose is to describe the abstractly defined von Neumann algebra for the dual locally compact quantum group in an explicit way using the quantized universal algebra of su(1,1).
Date: 26 April 2007
Speaker: Maarten Solleveld (UvA)
Title: Ext functors for affine Hecke algebras.
Abstract: Let H be an affine Hecke algebra, S its Schwartz completion and W the underlying affine Weyl group. We would like to determine all the discrete series representations of H. Using higher Ext functors we introduce a pairing on S-modules, for which the discrete series are orthonormal. Then we compare extensions of S-modules with extensions of H-modules and W-modules. We apply these techniques to affine Hecke algebras of type Bn with three independent positive parameters. It follows that there is a bijection between the collection of discrete series and bipartitions of n.
Date: 10 May 2007
Speaker: Tonny Springer (UU)
Title: Kato's exotic nilpotent cone.
Date: 24 May 2007
Speaker: Bruno Schapira (UvA)
Title: New results in the theory of Heckman and Opdam.
Abstract: In the talk I will present some new estimates of the hypergeometric functions of Heckman and Opdam. The proof uses in an essential way, the non symmetric versions of these functions, which were introduced by Opdam after the discovery of the Cherednik operators. If time allows I will speak about some applications to the study of the associated stochastic processes.

For further information, please contact one of the organizers Eric Opdam, Jasper Stokman.