A result on Ext over Kac-Moody algebras (Q1283762)

Let \(\mathfrak{g}\) be a (not necessarily symmetrizable) Kac-Moody algebra over a field of characteristic zero and let \(W\) be the corresponding Weyl group. Fix a dominant integral weight, \(\lambda\), and \(x,y\in W\) such that \(x\geq y\) with respect to the Bruhat order. Set \(n=l(x)-l(y)\). Denote by \(C(\lambda)\) the category of all weight \(\mathfrak{g}\)-modules, whose weights are less or equal than \(\lambda\). For any weight \(\mu\) let \(M(\mu)\) denote the Verma module with highest weight \(\mu\) and \(L(\mu)\) denote its unique simple quotient. The author proves the following result: the dimension of \(\text{Ext}_{C(\lambda)}^n (M(x\cdot\lambda), L(y\cdot\lambda))\) equals one.