Schubert varieties and cycle spaces. (English) Zbl 1048.32005

Authors’ abstract: Let \(G_0\) be a real semisimple Lie group. It acts naturally on every complex flag manifold \(z= G/Q\) of its complexification. Given an Iwasawa decomposition \(G_0= K_0 A_0 N_0\), a \(G_0\)-orbit \(\gamma\subset Z\), and the dual \(K\)-orbit \(\kappa \subset Z\), Schubert varieties are studied and a theory of Schubert slices for arbitrary \(G_0\)-orbits is developed. For this, certain geometric properties of dual pairs \((\gamma,\kappa)\) are underlined. Canonical complex analytic slices contained in a given \(G_0\)-orbit which are transversal to the dual \(K_0\)-orbit \(\gamma\cap \kappa\) are constructed and analyzed. Associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains. In particular, it is shown that the linear cycle space \(\Omega_W(D)\) is a Stein domain that contains the universally defined Iwasawa domain \(\Omega_I\). This is one of the main ingredients in the proof that \(\Omega_W(D)= \Omega_{AG}\) for all but a few Hermitian exceptions. In the Hermitian case, \(\Omega_W(D)\) is concretely described in terms of the associated bounded symmetric domain.


32E10 Stein spaces
22E30 Analysis on real and complex Lie groups
14M15 Grassmannians, Schubert varieties, flag manifolds
32M10 Homogeneous complex manifolds
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