On the integrability of a representation of sl (2, R)

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12 Citations (Scopus)

Abstract

The Dunkl operators involve a multiplicity function k as parameter [C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989) 167-183]. For positive real values of this function, we consider on the Schwartz space S (RN) a representation ωk of sl (2, R) defined in terms of the Dunkl-Laplacian operator. By means of a beautiful theorem due to E. Nelson, we prove that ωk exponentiates to a unique unitary representation of the universal covering group of SL (2, R). The representation theory is used to derive an identity of Bochner type for the Dunkl transform.

Original languageEnglish
Pages (from-to)249-264
Number of pages16
JournalJournal of Functional Analysis
Volume250
Issue number2
DOIs
Publication statusPublished - Sept 15 2007
Externally publishedYes

Keywords

  • Bochner identity
  • Dunkl operators
  • Dunkl transform
  • Integrable representations
  • Schrödinger model

ASJC Scopus subject areas

  • Analysis

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