Projected richardson varieties and affine Schubert varieties

Abstract

Let G be a complex quasi-simple algebraic group and G/P be a partial flag variety. The projections of Richardson varieties from the full flag variety form a stratification of G/P. We show that the closure partial order of projected Richardson varieties agrees with that of a subset of Schubert varieties in the affine flag variety of G. Furthermore, we compare the torus-equivariant cohomology and K-theory classes of these two stratifications by pushing or pulling these classes to the affine Grassmannian. Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian of type A.