Relative version of Weyl-Kac character formula (Q911698)

Let \({\mathfrak g}\) be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra H spanned by \(h_ 1,...,h_ l\). Assume that \(\lambda \in H^*\) is an integral weight such that \(\lambda +\rho\) is dominant. Let \(W_{\lambda}\) be the subgroup of the Weyl group generated by the simple reflections \(\{s_ i:\) \(\lambda (h_ i)\geq 0\}\). For \(w\in W_{\lambda}\), the Verma module M(w.\(\lambda)\) is a submodule of the Verma module M(\(\lambda)\). Let K(\(\lambda)\) be the sum of all the M(w.\(\lambda)\), \(e\neq w\in W_{\lambda}\). The author proves that the character of the quotient module is given by the formula \(\sum_{w\in W_{\lambda}}(-1)^{\ell (w)}ch M(w.\lambda).\) This is the relative version of the Weyl-Kac character formula.