Positivity of Dunkl's intertwining operator

1997/10/24 by Margit Rösler, Rösler, Margit · 5 citations
Mathematics · #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.QA #msc:33C50 #msc:33C80 #msc:44A15 #q-alg

paper · pdf · doi:10.48550/arxiv.q-alg/9710029

18 pages, LaTeX2e; some minor corrections made

arxiv created 1997/12/11 · arxiv updated 2009/11/30

Abstract

For a finite reflection group on \b R^N, the associated Dunkl operators are parametrized first-order 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.

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