Weight Spaces and Root Spaces of Kac-Moody Algebras

Let g(A) be the Kac-Moody algebra associated to a generalized symmetric Cartan matrix A, and g(A) = n_⊕ h ⊕ n_+ be its triangular decomposition. The purpose of this paper is to give an explicit realization of the weight spaces of an integrable highest weight module of g(A) and the weight spaces of the universal enveloping algebra U(n_) in terms of certain function spaces. We also discuss a similar construction for the root spaces. Our approach is based on a vertex operator construction of Kac-Moody algebras and representations by Borcherds [Bo} and uses the results on matrix coefficient of vertex operators given by Frenkel et al. [FLM). We state here our main results for the rank two Kac-Moody algebras; see Theorems 4.1 and 4.2 for the general case.