On the inverse Abel transforms for certain Riemannian symmetric spaces of rank 2
It is well known that the Abel transform and its inverse play an important role in harmonic analysis on noncompact Riemannian symmetric spaces. Up to now, we know the explicit expressions for this transform and its inverse only in some cases: the complex symmetric spaces [\textit{F. Rouvière}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10, 263-290 (1983; Zbl 0527.53006)], the spaces of rank one [\textit{N. Lohoué} and \textit{Th. Rychener}, Comment. Math. Helv. 57, 445-468 (1982; Zbl 0505.53022); \textit{F. Rouvière}, op. cit.], the first kind of classical bounded symmetric domain \(\text{SU} (p,q)/S(U(p) \times U(q))\), the symmetric spaces \(\text{SU}^*(6)/\text{Sp}(3)\) and \(e_{6(-26)}/f_4\). Using shift operators and other tools, \textit{A. Hba} obtained in his thesis [Sur l'inversion de la transformation d'Abel et les fonctions sphériques, Univ. Nice, 1988] the inverse Abel transform for the symmetric space \(\text{Sp} (2,n)/\text{Sp} (2) \times \text{Sp} (n)\). At the end of this thesis Hba suggested that one can use the shift operator \(D_2 (m) D_3(m)\) (see section 1 for the precise definition) to deduce the spherical functions on \(\text{SO} (2,4k)/\text{SO} (2) \times \text{SO} (4k)\), thus to express the inverse Abel transforms for these spaces by integral-differential operators. Motivated by the idea of \textit{R. J. Beerends} [Compos. Math. 66, 145-197 (1988; Zbl 0661.43006)] and \textit{A. Hba} [C. R. Acad. Sci., Paris, Sér. I 305, 77-80 (1987; Zbl 0616.43011)], we consider in this paper the inverse Abel transforms for the symmetric spaces \(\text{SO} (2,n)/\text{SO}(2) \times \text{SO} (n)\). Our results include Hba's assertion as a special case.
scientific article; zbMATH DE number 931999
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