Korean J. Math. Vol. 22 No. 1 (2014) pp.1-28
DOI: https://doi.org/10.11568/kjm.2014.22.1.1

The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman-Opdam theory attached to the root systems of type $B_2$ and $C_2$

Main Article Content

Khalifa Trimeche

Abstract

We consider the hypergeometric translation operator associated to the Cherednik operators and the Heckman-Opdam theory attached to the root system of type B2. We prove in this paper that these operators are positivity preserving and allows positive integral representations. In particular we deduce that the product formulas of the Opdam-Cherednik and the Heckman-Opdam kernels are positive integral transforms, and we obtain best estimates of these kernels.

The method used to obtain the previous results shows that these results are also true in the case of the root system of type $C_2$.



Article Details

References

[1] I. Cherednik. A unification of Knizhnik-Zamslodchnikov equations and Dunkl operators via affine Hecke algebras Invent. Math. 106, (1990) 411-432. Google Scholar

[2] G.J.Heckman and E.M.Opdam. Root systems and hypergeometric functions I. Compos Math. 64, (1987), 329-352. Google Scholar

[3] E.M. Opdam. Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, (1995), 75-121. Google Scholar

[4] W.Rudin. Real and complex analysis. Second Edition, McGraw-Hill, 1974. Google Scholar

[5] B.Schapira. Contribution to the hypergeometric function theory of Heckman and Opdam : sharp estimates, Schwartz spaces, heat kernel. Geom. Funct. Anal. 18, 2008, 222-250. Google Scholar

[6] K. Trim`eche. The trigonometric Dunkl intertwining operator and its dual associated with the Cherednik operators and the Heckman-Opdam theory. Adv. Pure Appl. Math. 1, (2010), 293-323. Google Scholar

[7] K. Trim`eche. Harmonic analysis associated with the Cherednik op- erators and the Heckam-Opdam theory. Adv. Pure Appl. Math. 2, (2011), 23-46. Google Scholar

[8] K. Trim`eche. Hypergeometric convolution structure on Lp-spaces and Applications, for the Heckman-Opdam theory. Preprint. Faculty of Sciences of Tunis. 2012. Google Scholar

[9] K. Trim`eche. Positivity of the hypergeometric translation operators associated with the Cherednik operator, and of the Dunkl translation operators in the one dimentional case. Preprint. Faculty of Sciences of Tunis. 2013. Google Scholar

[10] K. Trim`eche The positivity of the hypergeometric translation oper- ator and of its dual associated with the Cherednik operators and the Heckman-Opdam theory on Rd. Preprint . Faculty of Sciences of Tunis. 2013. Google Scholar

[11] K. Trim`eche The positivity of the transmutation operators associated with the Cherednik operators attached to the root system of type B2 and C2. Preprint. Faculty of Sciences of Tunis. 2014. Google Scholar