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Dunkl operators: Theory and applications. (English) Zbl 1029.43001
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 ${T}_{\xi }$ which are modifications of the partial derivatives ${\partial }_{\xi }$ on ${ℝ}^{d}$ and which depend on some fixed root system $R$ on ${ℝ}^{d}$ and additional multiplicity parameters $k\ge 0$. The ${T}_{\xi }$ generate a commutative algebra of differential-reflection operators; the joint eigenfunctions lead to the Dunkl kernel $K\left(x,y\right)$ $\left(x,y\in V\right)$ which forms a generalization of the exponential function ${e}^{〈x,y〉}$. Motivated by the case $d=1$, the symmetrizations of $K$ with respect to the reflection group $W$ generated by $R$ 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 $V=V\left(k,R\right)$ with ${T}_{\xi }V=V{\partial }_{\xi }$. Moreover, this operator is positive by a result of the author, i.e., there exists a positive integral representation of $V$, which may be seen in the case of spherical functions as a Harish-Chandra formula. This result has some consequences for strong estimations for $K$.

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 ${ℝ}^{d}$. 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.

##### MSC:
 43-02 Research monographs (abstract harmonic analysis) 33C52 Orthogonal polynomials and functions associated with root systems 33C80 Connections of hypergeometric functions with groups and algebras 82C22 Interacting particle systems 44A20 Integral transforms of special functions 44A35 Convolution (integral transforms)