  We study the category of Z^l-graded modules with finite-dimensional graded
pieces for certain Z+^l-graded Lie algebras. We also consider certain Serre
subcategories with finitely many isomorphism classes of simple objects. We
construct projective resolutions for the simple modules in these categories and
compute the Ext groups between simple modules. We show that the projective
covers of the simple modules in these Serre subcategories can be regarded as
multigraded generalizations of Kirillov-Reshetikhin modules and give a
recursive formula for computing their graded characters.
