## Duke Mathematical Journal

### Schubert varieties and cycle spaces

#### Abstract

Let G0 be a real semisimple Lie group. It acts naturally on every complex flag manifold z=G/Q of its complexification. Given an Iwasawa decomposition G0=K0A0N0, a G0-orbit γ⊂Z, and the dual K-orbit κ⊂Z, Schubert varieties are studied and a theory of Schubert slices for arbitrary G0-orbits is developed. For this, certain geometric properties of dual pairs (γ,κ) are underlined. Canonical complex analytic slices contained in a given G0-orbit γ which are transversal to the dual K0-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 ΩW(D) is a Stein domain that contains the universally defined Iwasawa domain ΩI. This is one of the main ingredients in the proof that ΩW(D)=ΩAG for all but a few Hermitian exceptions. In the Hermitian case, ΩW(D) is concretely described in terms of the associated bounded symmetric domain.

#### Article information

Source
Duke Math. J. Volume 120, Number 2 (2003), 229-249.

Dates
First available in Project Euclid: 16 April 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1082138583

Digital Object Identifier
doi:10.1215/S0012-7094-03-12021-9

Mathematical Reviews number (MathSciNet)
MR2019975

Zentralblatt MATH identifier
1048.32005

#### Citation

Huckleberry, Alan T.; Wolf, Joseph A. Schubert varieties and cycle spaces. Duke Math. J. 120 (2003), no. 2, 229--249. doi:10.1215/S0012-7094-03-12021-9. http://projecteuclid.org/euclid.dmj/1082138583.