  We obtain a graded character formula for certain graded modules for the
current algebra over a simple Lie algebra of type E6. 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.
