Extremal projectors in the semi-classical case (Q1374008)

Let \({\mathfrak g}\) be a complex semisimple Lie algebra, and let \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}\). Suppose \(R({\mathfrak h})\) is a field of rational functions on \({\mathfrak h}^*\). Consider the algebra \(U'({\mathfrak g})=U({\mathfrak g})\otimes_{S({\mathfrak h})} R({\mathfrak h})\) and the generic Verma module \(V={U'({\mathfrak g}) \over U'({\mathfrak g}){\mathfrak n}}\), where \({\mathfrak n}=\bigoplus_{\gamma\in \Delta^+} {\mathfrak g}_\gamma\) and \(\Delta^+\) is a set of positive roots. Then \(V={\mathfrak n}^- V\oplus R({\mathfrak h})1_+\), where \(1_+=1 +U'({\mathfrak g}){\mathfrak n}\), and hence a projector \(p\) onto \(R({\mathfrak h})1_+\) are defined. \(p\) is called the extremal projector. It was shown by D. P. Zhelobenko that \(p\) factorizes into elementary projectors of special kind. The author derives the same results for the symmetric algebra (the so-called semi-classical case). In the appendix a factorization for the extremal projector of the Virasoro algebra in the semi-classical case is derived.