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

We provide an algorithm to calculate the invariant factors of the Shapovalov form on the standard $\Z$-lattice inside the basic representation of a Kac-Moody algebra of $ADE$ type, and give explicit formulae in some cases. The techniques developed reduce the problem to finding the invariant factors of a family of bilinear forms on the ring of symmetric functions, having Jack's symmetric functions as an orthonormal basis. These results have applications to the representation theory of Iwahori-Hecke algebras at roots of unity and the modular representation theory of symmetric groups.