A COMBINATORIAL STUDY OF AFFINE SCHUBERT VARIETIES IN THE AFFINE GRASSMANNIAN

Abstract

Let ${\overline{\mathrm{X}}}_{\uplambda}$ be the closure of the I-orbit ${\overline{\mathrm{X}}}_{\uplambda}$ in the affine Grassmanian Gr of a simple algebraic group G of adjoint type, where I is the Iwahori subgroup and λ is a coweight of G. We find a simple algorithm which describes the set Ψ(λ) of all I-orbits in ${\overline{\mathrm{X}}}_{\uplambda}$ in terms of coweights. We introduce R-operators (associated to positive roots) on the coweight lattice of G, which exactly describe the closure relation of I-orbits. These operators satisfy Braid relations generically on the coweight lattice. We also establish a duality between the set Ψ(λ) and the weight system of the level one affine Demazure module of ${}^L\tilde{\mathfrak{g}}$ indexed by λ, where ${}^L\tilde{\mathfrak{g}}$ is the affine Kac–Moody algebra dual to the affine Kac–Moody Lie algebra $\tilde{\mathfrak{g}}$ associated to the Lie algebra $\mathfrak{g}$ of G.