, Volume 112, Issue 4, pp 433-440
Date: 15 Oct 2003

Cycles in G-orbits in G -flag manifolds

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Abstract.

There is a natural duality between orbits γ of a real form G of a complex semisimple group G on a homogeneous rational manifold Z=G /P and those κ of the complexification K of any of its maximal compact subgroups K: (γ,κ) is a dual pair if γ∩κ is a K-orbit. The cycle space C(γ) is defined to be the connected component containing the identity of the interior of {g:g(κ)∩γ is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C(γ) is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C(γ) is a certain explicitly computable universal domain.

Research of the first author partially supported by Schwerpunkt ‘‘Global methods in complex geometry’’ and SFB-237 of the Deutsche Forschungsgemeinschaft.
The second author was supported by a stipend of the Deutsche Akademische Austauschdienst.