Elementary divisors of the Shapovalov form on the basic representation of Kac–Moody algebras

We calculate the invariant factors of the standard lattice VZ inside the basic representation of untwisted Kac–Moody algebras of ADE-type. This, in turn, gives the invariant factors of Cartan matrices for symmetric groups in prime characteristic as well as for the Iwahori–Hecke algebra at roots of unity.