The geometric meaning of Zhelobenko operators

Abstract

Let \( \mathfrak{g} \) be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, \( \mathfrak{b} \) a Borel subalgebra of \( \mathfrak{g} \), \( \mathfrak{h}\subset \mathfrak{b} \) the Cartan sublagebra, and N ⊂ G the unipotent subgroup corresponding to the nilradical \( \mathfrak{n}\subset \mathfrak{b} \). We show that the explicit formula for the extremal projection operator for \( \mathfrak{g} \) obtained by Asherova, Smirnov, and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence \( N\times \mathfrak{h}\to \mathfrak{b} \) given by the restriction of the adjoint action. Simple geometric proofs of formulas for the “classical” counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.