Good grading polytopes

Let be a finite‐dimensional semisimple Lie algebra over ℂ and e ∈ 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 . 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.