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Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel. (English) Zbl 1052.33014

Dunkl operators on ${ℝ}^{d}$ generalize the classical partial derivatives; they generate a commutative algebra of differential-difference operators. Due to the work of Dunkl, Opdam, de Jeu, Rösler and others, it is well-known that there exist associated Dunkl kernels and Dunkl transforms (generalizing the classical exponentials and Fourier tranform) as well as heat semigroups and Hermite polynomials.

In the paper under review, Fock spaces and the chaos transform associated with Dunkl operators are introduced. In particular, relations to Hermite functions and commutation relations between Dunkl and multiplication operators are studied (while these relations are partially known). These commutation relations in particular lead to Weyl commutation relations.

##### MSC:
 33C99 Hypergeometric functions 39A70 Difference operators 46E99 Linear function spaces and their duals 47B99 Special classes of linear operators