Let $R$ be a root lattice of a irreducible root system $\Phi$.
Suppose $W$ is a Weyl group of $\Phi$ and $S$ is a sublattice of $R$ which is $W$-stable and satisfies $|R/S|<\infty$. For example, let $m$ be a positive integer then $mR$ is a $W$-stable sublattice of $R$ and $|R/S| = m^{n}$, rank$R$ = $n$. I don't know how to find all $W$-stable sublattices of a root lattice.
Are there any conclusions or references for such special $S$?
Thank you in advance.
