Crystal graphs of higher level q-deformed Fock spaces, Lusztig a-values and Ariki-Koike algebras

We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\mathrm{\_}q(\widehat{\mathfrak{s}\mathfrak{l}}\mathrm{\_}e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using Lusztig $a$-values.