A Deodhar type stratification on the double flag variety

Details A Deodhar type stratification on the double flag variety A Deodhar type stratification on the double flag variety

2006-07-16

arxiv.org/abs/math/0607374v2

Transform. Groups 12 (2007), no. 4, 769--785.

Mathematics

Symplectic Geometry

21 pages

Scientific paper

10.1007/s00031-007-0061-8

We describe a stratification on the double flag variety $G/B^+\times G/B^-$ of a complex semisimple algebraic group $G$ analogous to the Deodhar stratification on the flag variety $G/B^+$, which is a refinement of the stratification into orbits both for $B^+\times B^-$ 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 $G/B^+\times G/B^-$, which would generalize results of Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double Schubert cells.

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