Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle

[摘要] A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space E , and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.

[关键词] Dunkl operators;quantum principal bundle;quantum connection;quantum curvature;Coxeter groups