Good grading polytopes

Abstract

Let 𝔤 be a finite-dimensional semisimple Lie algebra over ℂ and e ∈ 𝔤 a nilpotent element. Elashvili and Kac have recently classified all good ℤ-gradings for e. We instead consider good ℝ-gradings, which are naturally parameterized by an open convex polytope in a Euclidean space arising from the reductive part of the centralizer of e in 𝔤. As an application, we prove that the isomorphism type of the finite W-algebra attached to a good ℝ-grading for e is independent of the particular choice of good grading.