Abstract
We carry out a detailed study of \(\Xi ^+\), a distinguished \(G\)-invariant Stein domain in the complexification of an irreducible Hermitian symmetric space \(G/K\). The domain \(\Xi ^+\) contains the crown domain \(\Xi \) and is naturally diffeomorphic to the anti-holomorphic tangent bundle of \(G/K\). The unipotent parametrization of \(\Xi ^+\) introduced in Krötz and Opdam (GAFA Geom Funct Anal 18:1326–1421, 2008) and Krötz (Invent Math 172:277–288, 2008) suggests that \(\Xi ^+\) also admits the structure of a twisted bundle \(G\times _K {\mathcal N}^+\), with fiber a nilpotent cone \({\mathcal N}^+\). Here we give a complete proof of this fact and use it to describe the \(G\)-orbit structure of \(\Xi ^+\) via the \(K\)-orbit structure of \({\mathcal N}^+\). In the tube case, we also single out a Stein, \(G\)-invariant domain contained in \(\Xi ^+ {\setminus } \Xi \) which is relevant in the classification of envelopes of holomorphy of invariant subdomains of \(\Xi ^+\).
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Acknowledgments
We are grateful to the referee for his accurate comments and for suggesting an argument which simplified the proofs of Lemmas 3.1 and 6.4.
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Geatti, L., Iannuzzi, A. Orbit structure of a distinguished Stein invariant domain in the complexification of a Hermitian symmetric space. Math. Z. 278, 769–793 (2014). https://doi.org/10.1007/s00209-014-1333-3
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DOI: https://doi.org/10.1007/s00209-014-1333-3