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Summary: A DEODHAR TYPE STRATIFICATION
ON THE DOUBLE FLAG VARIETY
BEN WEBSTER AND MILEN YAKIMOV
Abstract. We describe a stratification on the double flag variety G/B+ ×
G/B- of a complex semisimple algebraic group G analogous to the Deodhar
stratification on the flag variety G/B+, which is a refinement of the stratifi-
cation into orbits both for B+ × B- and for the diagonal action of G, just as
Deodhar's stratification refines the orbits of B+ and B-.
We give a coordinate system on each stratum, and show that all strata are
coisotropic subvarieties. Also, we discuss possible connections to the positive
and cluster geometry of G/B+ × G/B-, which would generalize results of
Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double
Schubert cells.
1. Introduction
Let G be a complex semisimple algebraic group with a fixed pair of dual Borel
subgroups Bą
and let T = B+
B-
be the corresponding maximal torus of G.
The Weyl group of the pair (G, T ) will denoted by W and its identity element by
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