Positivity of Dunkl’s Intertwining Operator

For a finite reflection group on R , the associated Dunkl operators are parametrized firstorder differential-difference operators which generalize the usual partial derivatives. They generate a commutative algebra which is – under weak assumptions – intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials. In this paper it is shown that for non-negative parameter values, this intertwining operator is positivity-preserving on polynomials and allows a positive integral representation on certain algebras of analytic functions. This result in particular implies that the generalized exponential kernel of the Dunkl transform is positive-definite. 1991 AMS Subject Classification: Primary: 33C80; Secondary: 44A15, 33C50.