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Distributions sphériques invariantes sur l’espace symétrique semi- simple et son c-dual. (Invariant spherical distributions on a semisimple symmetric space and its c-dual). (French) Zbl 0627.43007
Non-commutative harmonic analysis and Lie groups, Proc. Int. Conf., Marseille-Luminy 1985, Lect. Notes Math. 1243, 283-309 (1987).
[For the entire collection see Zbl 0607.00006.]
The author studies the invariant spherical distribution and proves a Fourier inversion formula for the affine symmetric space GL(n, \({\mathbb{C}})/GL(n, {\mathbb{R}})\). It is the first time that an explicit inversion formula is given for an affine symmetric space G/H of rank \(>1\), for H non-compact, apart from the case where \(G=G_ 1\times G_ 1\), \(G_ 1\) a semisimple Lie group, and H the diagonal of \(G_ 1\times G_ 1.\)
The method is classical. The author generalizes to the case of semisimple symmetric spaces the Harish-Chandra transforms. The Harish-Chandra transform of a function is defined on the union of the Cartan subspaces, and satisfies certain gap relations.
An invariant spherical distribution on G/H is a joint eigendistribution of the invariant differential operators, which is invariant under H. These gap relations and the formulas for the radial parts of these operators permit the construction of invariant spherical distributions.
The Dirac measure at the origin is obtained by differentiating the Harish-Chandra transform associated with one of the Cartan subspaces. Then the inversion formula is obtained by using the gap relations, Euclidean Fourier analysis and Fourier series.
Reviewer: J.Faraut

43A85 Harmonic analysis on homogeneous spaces
53C35 Differential geometry of symmetric spaces