Koelink, Erik (ed.) et al., Orthogonal polynomials and special functions. Notes for the lectures of the summer school, Leuven, Belgium, August 12-16, 2002. Berlin: Springer. Lect. Notes Math. 1817, 93-135 (2003).
These notes give an introduction to the analytic theory of so-called rational Dunkl operators which are modifications of the partial derivatives on and which depend on some fixed root system on and additional multiplicity parameters . The generate a commutative algebra of differential-reflection operators; the joint eigenfunctions lead to the Dunkl kernel which forms a generalization of the exponential function . Motivated by the case , the symmetrizations of with respect to the reflection group generated by may be seen as multi-dimensional Bessel functions. These functions and the Dunkl operators are related to recent investigations of integrable particle systems of Calogero-Moser-Sutherland-type and admit, for certain multiplicities, an interpretation as spherical functions.
The main topics of these notes are as follows: Due to the work of Dunkl, Opdam, and others, there is a unique intertwining operator with . Moreover, this operator is positive by a result of the author, i.e., there exists a positive integral representation of , which may be seen in the case of spherical functions as a Harish-Chandra formula. This result has some consequences for strong estimations for .
Motivated by the connection with quantum CMS-models, there is a Dunkl-type Laplace operator which is, similar to the classical case, the generator of a Dunkl-type heat subgroup on . The connection with quantum CMS-models also forms the motivation to introduce Dunkl-type Hamiltonians and the associated generalized Hermite polynomials.
Finally, some asymptotic results for the Dunkl kernel inside of Weyl chambers are discussed.