*(English)*Zbl 1029.43001

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 ${\mathbb{R}}^{d}$ and which depend on some fixed root system $R$ on ${\mathbb{R}}^{d}$ 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 $K(x,y)$ $(x,y\in V)$ which forms a generalization of the exponential function ${e}^{\langle x,y\rangle}$. 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 $V=V(k,R)$ 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 ${\mathbb{R}}^{d}$. 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) |