Astérisque N° 220/1994 (Broché)

Consider an indecomposable finite root system R. Associate to R two families of objects : 1.) For any odd integer p > 1 (prime to 3 if R is of type G2) let Up be the quantized enveloping algebra at a p-th root of unity. We take here Lusztig's version. 2.) For any prime p let Gp be the semisimple connected and simply connected algebraic group over an algebraically closed field of characteristic p. Restrict to the case where p is greater than the Coxeter number h of R.
Consider the block of the trivial one dimensional module for Up resp. for Gp. The simple modules in this block are indexed by certain elements in the affine Weyl group Wa of R. Suppose that L. is the simple module indexed by w and that Vw is the Weyl module with head Lw. We show that there are integers dw, x, independent of p such that in the Up case (resp. in the Gp case) dw, x is equal to the multiplicity of Lx, as a composition factor of Vw for all p > h (resp.
for all p >> 0). This implies : If the Lusztig conjecture holds in the quantum case, then it holds for p >> 0 in the prime characteristic case.