The geometric meaning of Zhelobenko operators (Q367143)

Let \(\mathfrak{g}\) be a complex finite dimensional semisimple Lie algebra with a fixed Borel subalgebra \(\mathfrak{b}\) whose radical is \(\mathfrak{n}\). Extremal projectors are projectors on the space of \(\mathfrak{n}\)-invariants in certain \(\mathfrak{g}\)-modules on which \(\mathfrak{n}\) acts locally finitely (for example, in Verma modules). They admit a generalization called Zhelobenko operators. In this paper the author considers Zhelobenko operators for a universal Verma module and on the space of functions on \(\mathfrak{b}\). Via a geometric interpretation the author observes that the classical formula for both extremal projectors and Zhelobenko operators can be obtained by elementary geometric methods. The formula turns out to be related to the existence of a birational equivalence \(N\times\mathfrak{h}\to\mathfrak{b}\) (here \(\mathfrak{h} \subset \mathfrak{b}\) is the Cartan subalgebra) given by restriction of the adjoint action.