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

Salem Ben Saïd

doi:10.1016/j.jfa.2007.04.022

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

Salem Ben Saïd

Research output: Contribution to journal › Article › peer-review

11 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 (R^N) a representation ω_k 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 ω_k 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 language	English
Pages (from-to)	249-264
Number of pages	16
Journal	Journal of Functional Analysis
Volume	250
Issue number	2
DOIs	https://doi.org/10.1016/j.jfa.2007.04.022
Publication status	Published - Sept 15 2007
Externally published	Yes

Keywords

Bochner identity
Dunkl operators
Dunkl transform
Integrable representations
Schrödinger model

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2007.04.022

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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.",

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AB - 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.

KW - Bochner identity

KW - Dunkl operators

KW - Dunkl transform

KW - Integrable representations

KW - Schrödinger model

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

Abstract

Keywords

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