C. F. Dunkl and E. M. Opdam

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameterized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the 'rational Cherednik algebra', and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups $G(m, p, N)$, the set of singular parameters in the parameterized family of these structures is described explicitly, using the theory of non-symmetric Jack polynomials.

2000 Mathematical Subject Classification: 20F55 (primary), 52C35, 05E05, 33C08 (secondary).

Keywords: complex reflection group, Dunkl operator, rational Cherednik algebra.

E-mail: cfd5z@virginia.edu and opdam@science.uva.nl