2013/09/26 by Jean-Yves Charbonnel, Anne Moreau, Charbonnel, Jean-Yves +1
Mathematics · #13A50 #14L24 #17B20 #17B35 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:13A50 #msc:14L24 #msc:17B20 #msc:17B35
paper · pdf · doi:10.48550/arxiv.1309.6993
59 pages. Final version, to appear in Math. Zeitschrift
arxiv created 2015/09/27 · arxiv updated 2015/09/29
Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by ge the centralizer of e in g and by S(ge)ge the algebra of symmetric invariants of ge. We say that e is good if the nullvariety of some r homogeneous elements of S(ge)ge in the dual of ge has codimension r. If e is good then S(ge)ge is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(ge)ge, the initial homogeneous component of their restrictions to e+gf are algebraically independent, with (e,h,f) an sl2-triple of g, then e is good. As applications, we obtain new examples of nilpotent elements that verify the above polynomiality condition, in in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type D7.