Cayley transforms and orbit structure in complex flag manifolds
type: Item
instance of: scholarly article
title: Cayley transforms and orbit structure in complex flag manifolds
zbMATH Open document ID: 0887.22007
DOI: 10.1007/BF01234542
published in: Transformation Groups
publication date: 1998-04-28T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 22E10
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 1112645
zbMATH Keywords: Complex flag manifolds
zbMATH Keywords: Cayley transforms
zbMATH Keywords: semisimple symmetric spaces
zbMATH Keywords: orbit structure
zbMATH Keywords: real semisimple Lie group
author: Joseph A. Wolf
author: R. Zierau
reviewed by: Edward G. Dunne
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1112645