Orbit duality for flag manifolds
This article gives a geometric presentation of the ``Matsuki duality for orbits on real and complex flag manifolds'', using the ``moment norm technique''. Let \(G\) be a complex semi-simple Lie group, \(G_{0}\), \(G_{u}\) the real forms corresponding to a Cartan involution \(\theta\) and a complex conjugation \(\sigma\) commuting with \(\theta\), \(K_{0}=G_{0}\cap G_{u}\). Let \(G_{0}^{\prime }\) such that \(G_{0}^{0}\subset G_{0}'\subset G_{0}\), \(K_{0}^{\prime }=G_{0}'\cap K_{0}\), \(K=\left( K_{0}\right) _{\mathbb{C}}\), \(K'\) such that \(K^{0}\subset K'\subset K\) and \(K'\cap K_{0}=K_{0}'\) (\(H^{0}=\) identity component of \(H\)). Let \(X=G/Q\) a flag variety, where \(Q\) is any parabolic subgroup of \(G\). The Matsuki duality is a one-to-one correspondence between \(G_{0}'\)-orbits and \(K'\)-orbits in \(X\). To establish this duality, the authors define a function \(f^{+}:X\rightarrow \mathbf{R}\) as the square norm of the moment map for an appropriate Kähler structure on \(X\) and maps \(\pi^{\pm}:X\rightarrow\mathcal{C}\) on the critical set \(\mathcal{C}\) of \(f^{+}\). Inverse images under \(\pi^{+}\) (\(\pi^{-}\)) of \(K_{0}'\)-orbits in \(\mathcal{C}\) are the \(G_{0}'\) (\(K^{\prime }\))-orbits in \(X\). A real version of Matsuki duality is also given. The paper gives detailed proofs but no example is worked out.
scientific article; zbMATH DE number 1837468