A Deodhar type stratification on the double flag variety

Abstract

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.

21 pages

References in corpus (5)

• Cluster ensembles, quantization and the dilogarithm
• On the variety of Lagrangian subalgebras
• On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups
• Poisson structures on affine spaces and flag varieties. II. General case
• Connected components of real double Bruhat cells

Cited by in corpus (3)

• On the Spectra of Quantum Groups
• Cluster Structures on Double Bott-Samelson Cells
• Note on the Deodhar decomposition of a double Schubert cell