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Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds. (English) Zbl 1043.22007

The duality property of orbits on flag manifolds is studied and applications to the SL(\(n,R\)), SO(\(p,q\)), SL(\(n,C\)), SO(\(n,C\)), or Sp(\(n,C\)) cases are presented. One considers a connected complex semisimple Lie group \(G_{C}\) and denotes by \(G_{R}\) a connected real form of \(G_{C}\). Then, the complexification \(K_{C}\) in \(G_{C}\) of a maximal compact subgroup \(K\) of \(G_{R}\) is constructed. A previous result established by one of the authors [T. Matsuki, Hiroshima Math. J. 18, 59–67 (1988; Zbl 0652.53035)] is that the \(K_{C}\)-orbits \(S\) and \(G_{R}\)-orbits \(S^{\prime}\) on a complex flag manifold can be put in a one-to-one \(S\leftrightarrow S^{\prime}\) duality correspondence by the condition that \(S\cap S^{\prime}\) is nonempty and compact.
In this paper it is shown that it is possible to replace \(K_{C}\) by some conjugate \(xK_{C}x^{-1}\) so that the duality correspondence is preserved. The sets \(C(S)\) of such \(x\) for various orbits \(S\) and their relations with each other are investigated. It is proven that for classical groups the intersection \(C=\bigcap_{S}C(S)\) equals \(\widetilde{D_{0}}Z\) where \(D_{0}=\widetilde{D_{0}}/K_{C}\) is the universal domain in \(G_{C}/K_{C}\) and \(Z\) is the center of \(G_{C}\). As a corollary, it is established that \(D_{0}\) is a Stein extension for classical groups. Moreover, it is conjectured that \(C(S)_{0}=\widetilde{D_{0}}\) for a generic \(S\), where \(C(S)_{0}\) is the connected component of \(C(S)\) containing the identity. Applications to the non-compact and complex classical Lie groups are presented in detail.

MSC:

22E46 Semisimple Lie groups and their representations
53C35 Differential geometry of symmetric spaces

Citations:

Zbl 0652.53035
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