2008.4: On symmetric invariants of centralisers in reductive Lie algebras

2008.4: D. Panyushev, A. Premet and O. Yakimova (2007) On symmetric invariants of centralisers in reductive Lie algebras. Journal of Algebra, 313 (Special issue celebrating the 70th birthday of E. B. Vinberg). pp. 343-391. ISSN 0021-8693

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DOI: 10.1016/j.jalgebra.2006.12.026

Abstract

Let g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let q be the centraliser of e in g. In this paper we study the algebra S(q)^q of symmetric invariants of q. We prove that if g is of type A or C, then S(q)^q is always a graded polynomial algebra in l variables, and we show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type A we prove that the invariant algebra S(q)q is freely generated by a regular sequence in S(q) and describe the tangent cone at e to the nilpotent variety of g.

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