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Generalized gradients and Poisson transforms. (English) Zbl 0989.22018
Bourguignon, Jean Pierre (ed.) et al., Global analysis and harmonic analysis. Papers from the conference, Marseille-Luminy, France, May 1999. Paris: Société Mathématique de France. Sémin. Congr. 4, 235-249 (2000).
Denote by \(C^\infty({\mathbb E})\) the smooth sections of a homogeneous vector bundle over the real flag manifold \(G/P\), where \(G\) is a semi-simple Lie group, \(P\) a parabolic subgroup, and \(\mathbb E\) induced by a representation \(E\) of \(P\), i.e. \({\mathbb E}=G\times_PE\). The author constructs a large class of first-order differential operators on smooth sections \(D: C^\infty({\mathbb E})\to C^\infty({\mathbb F})\) which is \(G\)-invariant between two such bundles. These operators are called generalized gradients. It is shown that these gradients can in some sense be extended to the Riemannian symmetric space \(G/K\) (\(K\) is a maximal compact subgroup of \(G\)) in a canonical way, which is consistent with natural vector-valued Poisson transforms from \(C^\infty({\mathbb E})\) to sections of bundles over \(G/K\).
For the entire collection see [Zbl 0973.00041].

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