Description
(Joint work with Valerio Toledano Laredo)
The well known "Dunkl operators" associated to a finite real reflection group constitute a commutative parameter family of deformations of the directional derivatives in Euclidean space. These operators are "differential-reflection" operators. Heckman and Cherednik have defined trigonometric versions of Dunkl's operators. The interest for these
operators lies in their deep ties to Macdonald polynomials and hypergeometric functions, to the Calogero-Moser quantum integrable system, and to the representation theory
of Hecke algebras.
"Hypergeometric shift operators" are tools to study Weyl group symmetric structures and functions in these contexts. In the present talk we present a theorem of existence and uniqueness of ''nonsymmetric shift operators'' for the Dunkl operators themselves. These nonsymmetric
shift operators are differential-reflection operators which "shift"
the parameters of the Dunkl operators by integers by means of a "transmutation relation".