Cayley transforms and orbit structure in complex flag manifolds
The authors describe the orbit structure for a real semisimple Lie group acting on a complex flag manifold in the following setting. The real group \(G_0\) is the real form of the complex group \(G\). The flag manifold \(Z=G/Q\) is such that some open \(G_0\) orbit \(D_0\) is a semisimple symmetric space (not necessarily Riemannian). The main result is that the \(G_0\)-orbits in \(Z\) are in one-to-one correspondence with the orbits of a smaller real group \(M_0 \subset G_0\) acting on a compact Hermitian symmetric space \(F \subset Z\), realized as the \(M\)-orbit of an appropriately chosen basepoint \(z_0\) in \(D_0\). (Here, \(M\) is a complex subgroup of \(G\), with \(M_0\) a real form of \(M\).) Moreover, the parametrization (via partial Cayley transforms) reveals the amount of `open-ness' or `closed-ness' of any particular orbit by relating it to \(\text{SL}(2,{\mathbf R})\times\text{SL}(2,{\mathbf R})\times\ldots\times\text{SL}(2,{\mathbf R})\) orbits in a product of Riemann spheres. An open \(G_0\)-orbit in \(Z\) corresponds to an open \(M_0\)-orbit in \(F\), which corresponds to a product of open \(\text{SL}(2,{\mathbf R})\)-orbits in the spheres. The closed \(G_0\)-orbit corresponds to the closed \(M_0\)-orbit, which corresponds to the product of equators. For the other orbits, the partial Cayley transform keeps track of how many upper or lower half-planes and how many equators make up the corresponding orbit in the product of spheres. The authors' main result presents an appealing picture. The generalized flag variety \(Z\) contains a subvariety \(F\) that cuts across all the \(G_0\)-orbits of \(Z\) and whose orbit structure is relatively easy to describe and completely mimics that of \(Z\). Furthermore, the proofs in the current paper are much simpler than the proofs of \textit{A. Korányi} and \textit{J. A. Wolf} [Ann. Math., II. Ser. 81, 265-288 (1965; Zbl 0137.27402); Am. J. Math. 87, 899-939 (1965; Zbl 0137.27403)], which cover the case where \(Z\) is Hermitian symmetric. For a detailed picture of the geometry of \(G_0\)-orbits, including the case where no orbit is symmetric, see \textit{J. A. Wolf}'s \textit{Bulletin} article [Bull. Am. Math. Soc. 75, 1121-1237 (1969; Zbl 0183.50901)].
scientific article; zbMATH DE number 1112645
scientific article