  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.
