The geometric meaning of Zhelobenko operators

TL;DR

This paper explores the geometric interpretation of Zhelobenko operators and extremal projection operators in complex semisimple Lie algebras, linking algebraic formulas to birational geometric equivalences and providing simple geometric proofs.

Contribution

It establishes a geometric framework for understanding Zhelobenko operators and extremal projection operators through birational equivalences and offers geometric proofs for their formulas.

Findings

Connection between algebraic formulas and birational geometric equivalences.

Geometric proofs for classical extremal projection and Zhelobenko operators.

Explicit formula relating N×h to b via the adjoint action.

Abstract

Let g be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, b a Borel subalgebra of g, h the Cartan sublagebra contained in b and N the unipotent subgroup of G corresponding to the nilradical n of b. We show that the explicit formula for the extremal projection operator for g obtained by Asherova, Smirnov and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence N\times h -> 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.