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The trigonometric Dunkl intertwining operator and its dual associated with the Cherednik operators and the Heckman-Opdam theory. (English) Zbl 1204.33028

Summary: We prove that there exists a unique topological isomorphism \(V_k\) from \({\mathcal E}(\mathbb R^d)\) (the space of \(C^\infty\)-functions on \(\mathbb R^d\)) onto itself which intertwines the Cherednik operators \(T_{j}, j = 1, 2, \dots , d\), and the partial derivatives \(\frac{\partial}{\partial x_j}, j = 1, 2, \dots , d\), called the trigonometric Dunkl intertwining operator (this name has been proposed by G. J. Heckman). To define and study the operator \(V_{k}\) we have introduced first the trigonometric Dunkl dual intertwining operator \(^{t}V_{k}\). The operators \(V_{k}\) and \(^{t}V_{k}\) are the analogue in the Dunkl theory of the Dunkl intertwining operator and its dual [see C. F. Dunkl, Can. J. Math. 43, No. 6, 1213–1227 (1991; Zbl 0827.33010); K. Trim\`che, Integral Transforms Spec. Funct. 12, No. 4, 349–374 (2001; Zbl 1027.47027)].

MSC:

33E30 Other functions coming from differential, difference and integral equations
33C67 Hypergeometric functions associated with root systems
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
43A32 Other transforms and operators of Fourier type
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[1] Cherednik I., Invent. Math. 106 pp 411– (1991) · Zbl 0725.20012 · doi:10.1007/BF01243918
[2] Cherednik I., Int. Math. Res. Not. 15 pp 733– (1997) · Zbl 0882.22016 · doi:10.1155/S1073792897000482
[3] Dunkl C. F., Can. J. Math. 43 pp 1213– (1991) · Zbl 0827.33010 · doi:10.4153/CJM-1991-069-8
[4] Heckman G. J., Compos. Math. 64 pp 329– (1987)
[5] Opdam E. M., Acta Math. 175 pp 75– (1995) · Zbl 0836.43017 · doi:10.1007/BF02392487
[6] Rösler M., Duke Math. J. 98 pp 445– (1999) · Zbl 0947.33013 · doi:10.1215/S0012-7094-99-09813-7
[7] Trimèche K., Integrals Transforms Special Funct. 12 (4) pp 349– (2001) · Zbl 1027.47027 · doi:10.1080/10652460108819358
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