zbMATH — the first resource for mathematics

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].

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