  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 $G$.
