How to regularize singular vectors and kill the dynamical Weyl group (Q1826882)

Let \(\mathfrak{g}\) be a simple Lie algebra, \(M(\lambda)\) denote the Verma \(\mathfrak{g}\)-module with highest weight \(\lambda\), and \(F\) be a finite-dimensional \(\mathfrak{g}\)-module. In the paper under review the authors study the singularities of certain intertwining operators \(\Phi_{\lambda}^{v}\in \text{Hom}_{\mathfrak{g}}(M(\lambda+\mu), M(\lambda)\otimes F)\), where \(v\in F\) is a vector of weight \(\mu\). The main objective of the present paper is to regularize the family \(\Phi_{\lambda}^{v}\), that is to construct a new family, \(\widetilde{\Phi}_{\lambda}^{v}\), of intertwining operators holomorphically depending on \(\lambda\) and \(v\). To do this the authors introduce the notion of a regularizing operator on \(V\), collecting the axioms along the way that are necessary to do the job described above. It is not immediate that a regularizing operator exists, and, in fact, the authors only show that such operators exist for the algebras \(\mathfrak{sl}_2\) and \(\mathfrak{sl}_3\). In the case when a regularizing operator exists, it is shown to be unique in some sense. Finally, it is shown that any regularizing operator conjugates the dynamical Weyl group operators to constant operators.