dorsal/arxiv
View SchemaPositivity of Dunkl's intertwining operator
| Authors | Margit Rösler |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9710029 |
| URL | https://arxiv.org/abs/q-alg/9710029 |
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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"date_created": "2026-03-02T18:01:28.825000Z",
"date_modified": "2026-03-02T18:01:28.825000Z",
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"abstract": "For a finite reflection group on $\\b R^N,$ the associated Dunkl operators are\nparametrized first-order differential-difference operators which generalize the\nusual partial derivatives. They generate a commutative algebra which is - under\nweak assumptions - intertwined with the algebra of partial differential\noperators by a unique linear and homogeneous isomorphism on polynomials. In\nthis paper it is shown that for non-negative parameter values, this\nintertwining operator is positivity-preserving on polynomials and allows a\npositive integral representation on certain algebras of analytic functions.\nThis result in particular implies that the generalized exponential kernel of\nthe Dunkl transform is positive-definite.",
"arxiv_id": "q-alg/9710029",
"authors": [
"Margit R\u00f6sler"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Positivity of Dunkl\u0027s intertwining operator",
"url": "https://arxiv.org/abs/q-alg/9710029"
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