The symmetric invariants of centralizers and Slodowy grading II

2016/04/05 by Charbonnel, Jean-Yves, Moreau, Anne
#13A50 #14L24 #17B20 #17B35 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1604.01274

Abstract

Let \mathfrakg be a finite-dimensional simple Lie algebra of rank ℓ over an algebraically closed field \Bbbk of characteristic zero, and let (e,h,f) be an \mathfraksl₂-triple of g. Denote by \mathfrakg^e the centralizer of e in \mathfrakg and by \rm S(\mathfrakg^e)^\mathfrakg^e the algebra of symmetric invariants of \mathfrakg^e. We say that e is good if the nullvariety of some ℓ homogenous elements of \rm S(\mathfrakg^e)^\mathfrakg^e in (\mathfrakg^e)^* has codimension ℓ. If e is good then \rm S(\mathfrakg^e)^\mathfrakg^e is a polynomial algebra. In this paper, we prove that the converse of the main result of arXiv:1309.6993 is true. Namely, we prove that e is good if and only if for some homogenous generating sequence q₁,…,q_ℓ, the initial homogenous components of their restrictions to e+\mathfrakg^f are algebraically independent over \Bbbk.

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