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Fonctions généralisées sur une algèbre de Lie semi-simple réelle. (Generalized functions on a real semisimple Lie algebra). (French) Zbl 0647.22009
Summary: Let \({\mathfrak g}\) be a real semisimple Lie algebra and G a connected Lie group with Lie algebra \({\mathfrak g}\). We construct in a suitable neighbourhood of 0 in \({\mathfrak g}^ a \)G-invariant generalized function, which is an eigenfunction for all G-invariant constant coefficients differential operators on \({\mathfrak g}\). We show on examples how this function occurs in inversion formulas for symmetric spaces of type \(G_ c/G\). We hope that this result extends to all symmetric spaces of this type.

MSC:
22E60 Lie algebras of Lie groups
53C35 Differential geometry of symmetric spaces
22E30 Analysis on real and complex Lie groups
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