In the study of semisimple Lie groups, lattices appear all over the place. In the theory of elliptic functions and modular forms, (equivalence classes of) lattices correspond to elliptic curves and to points on the modular space. I believe furthermore that it's the case that lattices corresponding to elliptic curves with extra automorphisms (equivalently, elements of the upper half plane with nontrivial stabilizer in the modular group in the one-dimensional case) tend to be the ones corresponding to root systems, e.g. $\mathfrak{sl}_3$. I don't know much about higher dimensional modular forms, but I imagine that this generalizes to higher dimensional semisimple Lie algebras. Is there a deeper connection here? Do holomorphic functions invariant under root lattices always have interesting properties? Are both part of a general theory? If this is well-known, where might I find a reference?

EDIT: Also, the Weyl chambers remind me of the fundamental domains, which appear so often in elliptic functions/modular forms.