A CONJECTURE FOR THE COMPUTATION OF THE D ECOMPOSITION MATRICES OF THE HECKE ALGEBRAS OF TYPE-A AT ROOTS OF UNITY

We exhibit a connection behaving rite combinatorics of the Hecke algeb ras as of type A at roots of unity, and the combinatorics of the cryst al basis of the basic representation of the affine Lie algebras of typ e A((1)). In particular, we conjecture that the decomposition matrices of the algebras H-m ((n) root 1) are easily obtained from the global lower crystal basis of the basic representation ($) over cap sl(n).