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Fonctions généralisées sphériques sur \(G_{{\mathbb{C}}}/G_{{\mathbb{R}}}\). (Generalized spherical functions on \(G_{{\mathbb{C}}}/G_{{\mathbb{R}}})\). (French) Zbl 0714.43013
A spherical function on a symmetric space G/H is an H-invariant generalized function which is an eigenfunction of the invariant differential operators. The author studies the case of a semisimple complex group G, H being a real form of G. There is an analogy between the analysis of such a symmetric space, and the analysis of the real semisimple group H. In particular the author proves that the spherical functions are locally integrable. The spherical functions are determined by their restrictions to the Cartan subspaces, and these restrictions are solutions of a differential system with constant coefficients.
In the second part of the paper one assumes that the Lie algebra \({\mathfrak h}\) of H has a compact Cartan subalgebra \({\mathfrak t}\), then i\({\mathfrak t}\) is a split Cartan subspace for G/H. In this setting the author constructs a family of spherical functions on G/H. Let Exp be the map Exp: \({\mathfrak h}\to G/H\) defined by \(Exp(X)=\exp (iX)H\), and let J(X) be its Jacobian. One starts with an H-invariant generalized function \(\Theta\) on \({\mathfrak h}\) which is an eigenfunction of the Ad(H)-invariant differential operators on \({\mathfrak h}\) with constant coefficients. Then one defines a new generalized function \({\tilde \Theta}\) on \({\mathfrak h}\) as a series indexed by the set \(\{\) \(Y|\) \(Exp(Y)=Exp(X)\}\). Then the generalized function \({\bar \Theta}\) on G/H given by \({\bar \Theta}\)(Exp X)\(=J(X)^{-1/2}{\tilde \Theta}(X)\), is a spherical function.
Reviewer: J.Faraut

MSC:
43A90 Harmonic analysis and spherical functions
53C35 Differential geometry of symmetric spaces
33C55 Spherical harmonics
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