The Ext-dual of a Verma module is a Verma module (Q1181450)

Let \(V\) be a split semi-simple Lie algebra of Chevalley type over a field \(k\), \(\ell\) be the rank of \(V\), and \(\Gamma^ +\) be the set of positive roots of \(V\). Let \(U(V)\) be the universal enveloping algebra of \(V\). Let \(\lambda=(\lambda_ 1,\dots,\lambda_ \ell)\in k^ \ell\) be a weight of \(V\), \(M(\lambda)\) be the left Verma module, and \(M'(\lambda)\) be the right Verma module. By constructing a projective resolution of \(M(\lambda)\) the author shows the following: (a) \(\text{Ext}^ n_{U(V)}(M(\lambda),U(V))=0\) unless \(n=\ell+r\), where \(r=\#\Gamma^ +\); (b) \(\text{Ext}^ n_{U(V)}(M(\lambda),U(V))\) is isomorphic to the Verma module \(M'(\lambda-\sum_{\alpha\in\Gamma^ +}\alpha)\), where \(\lambda-\sum_{\alpha\in\Gamma^ +}\alpha\) is the weight \((\lambda_ i-\sum_{\alpha\in\Gamma^ +}\alpha(h_ i))\).