Graded limits of minimal affinizations and beyond: the multiplicity free case for type $E_6$

We obtain a graded character formula for certain graded modules for the current algebra over a simple Lie algebra of type $E_6$. For certain values of their highest weight, these modules were conjectured to be isomorphic to the classical limit of the corresponding minimal affinizations of the associated quantum group. We prove that this is the case under further restrictions on the highest weight. Under another set of conditions on the highest weight, Chari and Greenstein have recently proved that they are projective objects of a full subcategory of the category of graded modules for the current algebra. Our formula applies to all of these projective modules.