Schubert duality for $SL(n,\mathbb{R})$-flag domains

This paper is concerned with the study of spaces of naturally defined cycles associated to $SL(n,\mathbb{R})$-flag domains. These are compact complex submanifolds in open orbits of real semisimple Lie groups in flag domains of their complexification. It is known that there are optimal Schubert varieties which intersect the cycles transversally in finitely many points and in particular determine them in homology. Here we give a precise description of these Schubert varieties in terms of certain subsets of the Weyl group and compute their total number. Furthermore, we give an explicit description of the points of intersection in terms of flags and their number.

Keywords

cycle spaces, flag domains, Schubert varieties, transversality

2010 Mathematics Subject Classification

14M15

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Supported by SPP 1388 of the DFG.

Received 13 March 2016

Published 17 July 2017