Moment-norm gradient flow on flag manifolds (Q937826)

An essential tool in the study of flag manifolds in geometry and representation theory is the so-called ``Matsuki duality for orbits on real and complex flag manifolds associated to connected semi-simple Lie groups''. Roughly speaking, Matsuki duality is a one-to-one correspondence between two special orbit spaces of a flag variety \(X=G/Q\), where \(G\) is a connected complex semi-simple Lie group and \(Q\) is a certain kind of parabolic subgroup of \(G\). This duality theorem, although completely analytic in nature, was originally proved by \textit{T. Matsuki} using algebrogeometric methods [cf. J. Math. Soc. Japan 31, 331--357 (1979; Zbl 0396.53025)]. The search for a better understanding of the geometry behind Matsuki duality experienced a decisive breakthrough in 2002, when \textit{R. Bremigan} and \textit{J. Lorch} gave the first general, purely geometric proof of this result [Manuscr. Math. 109, No. 2, 233--261 (2002; Zbl 1019.22013)]. Their proof was based on a subtle analysis of the gradient flow of a certain moment-norm function \(f^+\) on \(X =G/P\) with values in \(\mathbb{R}\). In the paper under review, the author extends this analysis by constructing explicit integral curves for the gradient \(\nabla f^+\), that is, by exhibiting concrete moment-norm gradient flows on such flag manifolds. Although not needed for the geometric proof of Matsuki's duality theorem for orbits on flag manifolds \(G/P\), the explicit description of integral curves for the gradient \(\nabla f^+\) is expected to yield useful additional information about the geometry of the underlying flag manifold. For instance, it is shown that certain integral curves for \(\nabla f^+\) correspond to Cayley transforms for \(G/P\).