Weyl Groups and Abelian Varieties

Let G be a finite group. For each integral representation $\rho$ of G we consider $\rho-$decomposable principally polarized abelian varieties; that is, principally polarized abelian varieties (X,H) with $\rho(G)-$action, of dimension equal to the degree of $\rho$, which admit a decomposition of the lattice for X into two G-invariant sublattices isotropic with respect to $\Im H$, with one of the sublattices $\mathbb{Z}G-$isomorphic to $\rho$. We give a construction for $\rho-$decomposable principally polarized abelian varieties, and show that each of them is isomorphic to a product of elliptic curves. Conversely, if $\rho$ is absolutely irreducible, we show that each $\rho-$decomposable p.p.a.v. is (isomorphic to) one of those constructed above, thereby characterizing them. In the case of irreducible, reduced root systems, we consider the natural representation of its associated Weyl group, apply the preceding general construction, and characterize completely the associated families of principally polarized abelian varieties, which correspond to modular curves.