There is a natural duality between orbits γ of a real form G of a complex semisimple group G C on a homogeneous rational manifold Z = G C /P and those κ of the complexification K C 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 partially supported by Schwerpunkt "Global methods in complex geometry" and SFB-237 of the Deutsche Forschungsgemeinschaft.† Supported by a stipend of the Deutsche Akademische Austauschdienst.
A necessary and sufficient condition is proved for a generalized steepest descent approximation to converge to the zeros of m-accretive operators. Related results deal with the convergence of the scheme to fixed points of pseudocontractive maps.
The orbits of a real form G of a complex semisimple Lie group GC and those of the complexification KC of its maximal compact subgroup K acting on Z=GC/Q, a homogeneous, algebraic, GC-manifold, are finite. Consequently, there is an open G-orbit. Lower-dimensional orbits are on the boundary of the open orbit with the lowest dimensional one being closed. Induced action on the parameter space of certain compact geometric objects (cycles) related to the manifold in question has been characterized using duality relations between G- and KC-orbits in the case of an open G-orbit and more recently lower-dimensional G-orbits. We show that the parameter space associated with the unique closed G-orbit in Z agrees with that of the other orbits characterized as a certain explicitly defined universal domain.
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