Real characters for Demazure modules of rank two affine Lie algebras
Let \({\mathfrak g}\) be a Kac-Moody algebra of type \(A_1^{(1)}\) or \(A_2^{(2)} \). Given a dominant weight \(\lambda\), let \(V(\lambda)\) denote the irreducible \({\mathfrak g}\)-module with heighest weight \(\lambda\). Given an element \(w\) of the Weyl group, let \(E_w (\lambda)\) denote the Demazure module associated to \(w\), namely, \(E_w (\lambda)\) is the \({\mathfrak b}\)-module generated by the weight space \(V_{w (\lambda)}\) (of dimension one), \({\mathfrak b}\) being the Borel subalgebra of \({\mathfrak g}\). For \(n\in \mathbb Z\), let \(E_w (\lambda)_n = \bigoplus_{\mu\in \lambda- n\alpha_0 +\mathbb Z \delta} E_w (\lambda)_\mu\), where \(\alpha_0, \delta\) have the usual meaning. In this paper, the author computes \(\chi_n (E_w (\lambda)): =\sum_{n\in \mathbb Z} \dim E_w (\lambda)_nq^n\) in an explicit way. Unlike the other known character formulae, namely the Kac-Weyl character formula and Demazure character formula, which involve massive cancellations, the formula given in this paper is positive-termed.
scientific article; zbMATH DE number 941276