A combinatorial study of affine Schubert varieties in affine Grassmannian

Let $\overline{\mathtt{X}}_\lambda$ be the closure of the $\mathtt{I}$-orbit $\mathtt{X}_\lambda$ in the affine Grassmanian $\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\mathtt{I}$ is the Iwahori group and $\lambda$ is a coweight of $G$. We find a simple algorithm which describes the set $\Psi(\lambda)$ of all $\mathtt{I}$-orbits in $\overline{\mathtt{X}}_\lambda$ 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 $\mathtt{I}$-orbits. These operators satisfy Braid relations generically on the coweight lattice. We also establish a duality between the set $\Psi(\lambda)$ and the weight system of the level one affine Demazure module $\hat{\mathscr{D}}_\lambda$ of $^L\tilde{\mathfrak{g}}$ indexed by $\lambda$, 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 $G$.