A COMBINATORIAL STUDY OF AFFINE SCHUBERT VARIETIES IN THE AFFINE GRASSMANNIAN

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