Root vector localization and minimal nilpotent finite W-algebras

Let \(g\) be any finite-dimensional complex simple Lie algebra, let h^ be its dual Coxeter number, and let \(e\) be its highest-root vector which is a minimal nilpotent element. In this paper, using a result of Bavula on invariants of commuting locally nilpotent derivations, we show that \[ U(g)_f D^+h^-2⊗ D_1⊗ U(g,e), \] where the first factor is a polynomial Weyl algebra of rank h⁻2, the second factor is a Laurent Weyl algebra of rank one, and U(g,e) is the finite W-algebra associated with g and e. As an application, we prove that if M is a simple infinite-dimensional g-module with the minimal GK dimension on which $f-1$ acts locally nilpotently, then M P⊗ V for some simple D⁺h⁻2⊗ D₁-module P and some finite-dimensional simple U(g,e)-module V. We also give a new proof of the Skryabine quivalence and Petukhov's birational decomposition of the quotient division ring of U(g).