ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 33C80, 44A15, 33C50 Rösler, Margit, Technische Universität München, Germany Positivity of Dunkl’s intertwining operator Dunkl operators are parametrized differential-difference operators which are associated with a finite reflection group on ℝ^{N} and 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. We report on the proof of a conjecture of Dunkl, which states that for non-negative parameter values, this intertwining operator allows a positive integral representation on certain algebras of analytic fuctions. Our result in particular implies that the generalized exponential kernel of the Dunkl transform is positive-definite. Reference: M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J. , to appear.