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Line bundles on a symmetric space and invariant analysis. (Fibrés en droites sur un espace symétrique et analyse invariante.) (French) Zbl 0820.43007

Let \(G/H\) be a real symmetric space, \(\chi : H \to \mathbb{C}^ \times\) a character and \(L_ \chi = G \times_ H \mathbb{C}\) the associated line bundle over \(G/H\). Further let \({\mathfrak g} = {\mathfrak h} + {\mathfrak s}\) be the corresponding infinitesimal decomposition of the Lie algebra \(\mathfrak g\) of \(G\) and \(\text{Exp} : {\mathfrak s} \to G/H\) the usual exponential map of \(G\) composed with the quotient map. Then Exp is a local diffeomorphism. It can be used to transfer distributions on \(L_ \chi\), i.e., elements in the dual of the space of compactly supported sections of \(L_ \chi\), to distributions on \(\mathfrak s\). The author provides a universal function \(e_ \chi : {\mathfrak s} \times {\mathfrak s} \to \mathbb{C}\) which allows one to write an explicit formula for the convolution of two \(H\)-invariant distributions on \(L_ \chi\) in the transferred picture. A detailed study of the transfer mechanism yields sufficient conditions on \(\chi\) for the commutativity of the convolution. Similarly it is possible to give conditions for the commutativity of the algebra of invariant differential operators \({\mathbf D}(L_ \chi)\) acting on the sections of \(L_ \chi\).

MSC:

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