We study and give a complete classification of good Zgradings of all simple finite-dimensional Lie algebras. This problem arose in the quantum Hamiltonian reduction for affine Lie algebras.Lemma 1.1. Let e ∈ g 2 , e = 0. Then there exists h ∈ g 0 and f ∈ g −1 such that {e, h, f } form an sℓ 2 -triple, i.e., [h, e] Proof. By the Jacobson-Morozov theorem (see [J]), there exist h, f ∈ g such that {e, h, f } is an sℓ 2 -triple. We write h = j∈Z h j , f = j∈Z f j according to the given Z-grading of g. Then [h 0 , e] = 2e and [e, g] ∋ h 0 (since [e, f −2 ] = h 0 ). Therefore, by Morozov's lemma (see [J]), there exists f ′ such that {e, h, f ′ } is an sℓ 2 -triple. But then {e, h 0 , f ′ 0 } is an sℓ 2 -triple. The following lemma is well-known [C] (and easy to prove).Lemma 1.2. Let e be a non-zero nilpotent element of g and let s = {e, h, f } be an sℓ 2 -triple. Then g s (the centralizer of s in g ) is a maximal reductive subalgebra of g e . Theorem 1.1. Let (1.1) be a good Z-grading and e ∈ g 2 a good element. Let H ∈ g be the element defining the Z-grading (i.e., g j = {a ∈ g|[H, a] = ja}), and let s = {e, h, f } be an sℓ 2 -triple given by Lemma 1.1. Then z := H − h lies in the center of g s .
