The Symmetric Invariants of Centralizers and Slodowy Grading II

Abstract

Let 𝔤 be a finite-dimensional simple Lie algebra of rank ℓ over an algebraically closed field 𝕜 of characteristic zero. We identify 𝔤 with 𝔤^∗ through the Killing form of 𝔤. Let (e, h, f) be an 𝔰𝔩₂-triple of 𝔤. Denote by 𝔤^e the centralizer of e in 𝔤 and by $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ the algebra of symmetric invariants of 𝔤^e. We say that e is good if the nullvariety of some ℓ homogeneous elements of $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ in (𝔤^e)^∗ has codimension ℓ. In our previous work (Charbonnel and Moreau. Math. Zeitsch. 282, n° 1-2, 273–339 2016), we showed that if e is good then $\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of Charbonnel and Moreau (Math. Zeitsch. 282, n° 1-2, 273–339 2016) is true. Namely, we prove that e is good if and only if for some homogeneous generating sequence q ₁, … , q _l of S(𝔤)^𝔤, the initial homogeneous components of their restrictions to e + 𝔤^f are algebraically independent over 𝕜.