2.2 Representation Theory, Operator Algebras and Complex Analysis

Description of the programme.

The cornerstones of this programme are Lie theory, special functions, analytic aspects of modern mathematical physics, and complex analysis. Interrelations between these areas are emphasized in the programme. In particular, there are important links from Lie theory to special functions and to mathematical physics.
Topics in Lie theory range from analytic to algebraic and comprise analysis on semisimple Lie groups, semisimple symmetric spaces and quantum groups. Study of (q)-special functions living on (quantum) groups naturally links to multivariable functions associated with root systems like Macdonald polynomials and Heckman-Opdam hypergeometric functions.
Study of quantum groups connects via deformation quantization and quantum groupoids to mathematical physics.
Work in the programme on one-variable special functions and orthogonal polynomials uses many techniques from classical analysis, and also emphasizes the development of computer algebra algorithms.
Work in complex analysis, mainly in several complex variables, is in interplay with function algebras and functional analysis.

Status of the programme.

Lie theory, i.e. the mathematics of continuous symmetries, has a central place in mathematics and its applications, in particular theoretical physics. The more recent theories of Kac-Moody algebras and quantum groups gave vigorous new impulses to the field. The work of this year's Fields Medal winner Richard Borcherds nicely illustrates the unifying power of Lie theory in bringing many fields together. Lie theory in the Netherlands, spread over the three research schools Stieltjes, MRI en EIDMA, but closely cooperating, is very active on an international level, with important contributions by Heckman, Opdam, Van den Ban, A.M. Cohen, G. van Dijk, Koornwinder, Koelink, G. Helminck and others.

To a large extent, Dutch work on Special functions is inspired by Lie theory. In addition, Dutch work on classical one-variable special function theory is internationally renowned, in particular Temme's work on asymptotics. Computer algebra methods in special functions, becoming increasingly important, will be further emphasized in this programme.

Methods in modern mathematical physics close to analysis (Lie theory, C*-algebras, non-commutative geometry, K-theory, quantization) are successfully pursued in the present programme by Landsman. See also the Stieltjes Geometry Programme for the interaction between (algebraic) geometry and mathematical physics.
Complex analysis, notably in several variables, is another central theme in mathematics and its application, playing a role in many break-throughs. Expertise in this area (Wiegerinck) is important for this programme.
Parts of this programme are (and have been) supported by grants fromNWO (SWON, FOM).

Research staff (situation at January 1, 2007)

• Permanent staff
• Dr. M.G. de Bruin (TUD)
• Prof.dr. G. van Dijk (UL)
• Dr. E. Hendriksen (UvA)
• Dr. M.F.E. de Jeu (UL) (from programme 2.1)
• Dr. R. Koekoek (TUD)
• Dr. H.T. Koelink (TUD)
• Prof.dr. T.H. Koornwinder (UvA)
• Prof.dr. J. Korevaar (UvA)
• Prof.dr. E.M. Opdam (UvA)
• Dr. P.J.I.M. de Paepe (UvA)
• Dr. J.A. Sanders (VUA)
• Dr. J.V. Stokman (UvA)
• Dr. J.J.O.O. Wiegerinck (UvA)
• Dr. R.A. Zuidwijk (EUR)
• Post Docs
• Dr. G. Carlet (VUA)
• Dr. W.G.M. Groenevelt (UvA)
• Dr. S. Lombardo (VUA)
• Ph.D. students
• E. Asadi (VUA)
• Drs. F.J. van de Bult (UvA)
• Drs. N. Kowalzig (UvA-NWO)
• Drs. S. el Marzguioui (UvA)
• Drs. M. van Meer (UvA)
• Drs. R.I. van der Veen (UvA)
• Drs. M.S. Solleveld (UvA-NWO)
• P.C. Svensson (UL)
• CWI participants
• Dr. N.M. Temme
• Dr. M. Hazewinkel