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**Special functions associated with the quantum dynamical
Yang-Baxter equation**

**Abstract**

Let **g** be a complex simple Lie algebra (for instance **sl(2,C)**).
Let **h** be a Cartan subalgebra in **g**, let **h ^{*}**
be its linear dual and make a choice of positive roots. Let

With a tensor product of a Verma module and a finite dimensional module
one can associate an *intertwining operator* which generalizes the
notion of Clebsch-Gordan coefficients. From that one can build a *fusion
matrix* and an *exchange matrix* associated with the tensor product
of a Verma module and two finite dimensional modules. The exchange matrix
generalizes the notion of Racah coefficients. The intertwining operator,
fusion matrix and exchange matrix depend rationally on **lambda** in
**h ^{*}**
(or on

In certain cases, certainly for **sl(2)**, the intertwining operator,
fusion matrix and exchange matrix can be computed exlicitly as special
functions, and the general formulas produced by the above theory yield
formulas for these special functions.

A further construction, starting with the intertwining operator, yields
the *weighted trace functions* depending on **lambda** and **mu**
in **h**, while a construction starting from the exchange matrix yields
a commuting family of difference operators. For **sl(n)** this yields
in the quantum case Macdonald polynomials for root system of type **A**,
and also the Macdonald-Ruijsenaars and dual Macdonald-Ruijsenaars eigenvalue
equations for the Macdonald polynomials, as well as a symmetry in **lambda**
and **mu**. The weighted trace functions also satisfy the so-called
qKZB and dual qKZB equation, while there are limit cases to the qKZ and
KZ equations.

The material sketched above found its origin in physics and was developed during the last 15 years by various phycisists and mathematicians. A nice survey can be found in P. Etingof and O. Schiffmann, math.QA/9908064, while more details and proofs concerning the various difference equations satisfied by the weighted trace functions are given in Etingof and Varchenko, math.QA/9907181.

The lecture will survey these results with emphasis on special function aspects. Only a very minor part of this lecture will contain original results by the speaker (and his co-worker N. Touhami).

In particular during the last few years there has been a fast development,
which particularly aims at the further generalization to elliptic quantum
groups and dynamical quantum groups, and to a generalization of Macdonald
polynomials in the elliptic case. These newer developments will be discussed
in the second lecture
by Koelink at this Advanced Study Institute. The present lecture will
be a useful preparation for Koelink's lecture.

(See tex,
dvi,
ps,
and pdf files
of this abstract.)

View the **extended abstract** of Koornwinder's lecture at