Abstract
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.
Highlights
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
Let G be a noncompact semisimple Lie group which is embedded in its complexification GC and let Q be a parabolic subgroup in the sense that it contains a Borel subgroup of GC, Z = GC/Q is a compact, homogeneous, algebraic, rational GC-flag manifold
Since G is semisimple, it decomposes as a product of almost simple factors G = G1 × G2 × ⋅ ⋅ ⋅ × Gk in the sense that the map ΠiGi → G is surjective with finite kernel
Summary
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. 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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