Weight spaces and root spaces of Kac-Moody algebras
Let \({\mathfrak g} (A)\) be the Kac-Moody algebra associated to a symmetric generalized Cartan matrix. The purpose of the paper under review is to give an explicit realization of the weight spaces of an integrable highest weight \({\mathfrak g} (A)\)-module \(L(\lambda)\) and the weight spaces of the universal enveloping algebra \(U({\mathfrak n}_ -)\) in terms of certain spaces of rational functions. The approach is based on an embedding of \(L(\lambda)\) in the (shifted) Fock space \(V_ \lambda\) constructed from the root lattice of \({\mathfrak g} (A)\) [see \textit{R. E. Borcherds}, Proc. Natl. Acad. Sci. USA 84, 3068-3071 (1986; Zbl 0613.17012)]. It is difficult to characterize the weight subspace \(L(\lambda)_{\lambda - \ell_ 1 \alpha_ 1 - \cdots - \ell_ N \alpha_ N}\) within \(V_ \lambda\), and the main results are obtained by passing to the matrix coefficients \(x \mapsto \langle e^ \lambda\), \(\prod Y(e^{\alpha_ j}\), \(z_{j,i})x \rangle\), \(1 \leq j \leq N\), \(l \leq i \leq \ell_ j\), for vertex operators, and characterizing the weight subspace in terms of this map.
scientific article; zbMATH DE number 645115