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.