A Deodhar-type stratification on the double flag variety

Abstract

We describe a partition of the double flag variety G/B⁺ × G/B^- of a complex semisimple algebraic group G analogous to the Deodhar partition on the flag variety G/B⁺. This partition is a refinement of the stratification into orbits both for B⁺ × B^- and for the diagonal action of G, just as Deodhar's partition 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.