Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas

Abstract

In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups. The main tools in this paper are the resolutions of standard modules given in [ <bold>21</bold> ] and a generalization to the case of Kac-Moody Lie algebras of Jantzen’s character sum formula for a quotient of two Verma modules (one of the main results of this article). Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).