Veselov, A. P.; Chalykh, O. A. Explicit formulae for spherical functions on symmetric spaces of type AII. (English. Russian original) Zbl 0760.43004 Funct. Anal. Appl. 26, No. 1, 59-60 (1992); translation from Funkts. Anal. Prilozh. 26, No. 1, 74-76 (1992). In this note there is proposed an explicit formula for a spherical function \(\varphi_ \lambda\) for the noncompact type symmetric space \(SU^*(2n)/Sp(n)\): \[ \varphi_ \lambda(g)=b(\lambda)\prod_{i<j}sh^{-2}(x_ i-x_ j)\sum_{w\in S_ n}\text{sgn}(w)\cdot\psi(\sqrt{-1} w\lambda,x), \] where \(\lambda\in\mathbb{R}^ n\), \(b(\lambda)\) is an explicitly written factor, \(g=\exp a\), \(a=\text{diag}(x_ 1,\dots,x_ n,x_ 1,\dots,x_ n)\), \(\sum x_ i=0\). The function \(\psi(k,x)\) is obtained by applying a differential operator \(D\) (its explicit expression is given) to the function \(\exp\sum k_ i x_ i\). Moreover the authors give an explicit inversion formula of the Abel transform for the symmetric space in question. Here they are based on some results of R. Beerends. Reviewer: V.F.Molchanov (Freiberg) Cited in 7 Documents MSC: 43A85 Harmonic analysis on homogeneous spaces 43A90 Harmonic analysis and spherical functions 22E46 Semisimple Lie groups and their representations Keywords:spherical function; symmetric space; inversion formula; Abel transform PDFBibTeX XMLCite \textit{A. P. Veselov} and \textit{O. A. Chalykh}, Funct. Anal. Appl. 26, No. 1, 59--60 (1992; Zbl 0760.43004); translation from Funkts. Anal. Prilozh. 26, No. 1, 74--76 (1992) Full Text: DOI References: [1] I. M. Gelfand, Sov. Math. Dokl.,70 (1950). [2] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York (1978). · Zbl 0451.53038 [3] S. Helgason, Groups and Geometric Analysis, Academic Press, Orlando (1984). · Zbl 0543.58001 [4] L. Vretare, SIAM J. Math. Anal.,15, 805-833 (1984). · Zbl 0549.43006 · doi:10.1137/0515062 [5] R. J. Beerends, Compositio Math.,66, 145-197 (1988). [6] O. A. Chalykh and A. P. V. Veselov, Commun. Math. Phys.,126, 597-611 (1990). · Zbl 0746.47025 · doi:10.1007/BF02125702 [7] E. Opdam, Compositio Math.,67, 21-49 (1988). [8] M. A. Olshanetsky and A. M. Perelomov, Phys. Rep.,94, 313-404 (1983). · doi:10.1016/0370-1573(83)90018-2 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.