Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas

We give a complete description of the graded multiplicity space which appears in the Feigin-Loktev fusion product of graded Kirillov-Reshetikhin modules for all simple Lie algebras. This construction is used to obtain an upper bound formula for the fusion coefficients in these cases. The formula generalizes the case of g = A_r of our previous paper, where the multiplicities are generalized Kostka polynomials. In the case of other Lie algebras, the formula is the fermionic side of the X = M conjecture. In the cases where the Kirillov-Reshetikhin conjecture, regarding the decomposition formula for tensor products of KR-modules, has been proven in its original, restricted form, our result provides a proof of the conjectures of Feigin and Loktev regarding the fusion product multiplicities.