On symmetric invariants of centralisers in reductive Lie algebras

2006/10/01 by D. Panyushev, Panyushev, D., A. Premet +3
Mathematics · #13A50 #14L24 #17B45 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:13A50 #msc:14L24 #msc:17B45

paper · pdf · doi:10.48550/arxiv.math/0610049

49 pages, 2 figures

arxiv created 2006/12/13 · arxiv updated 2009/12/01

Abstract

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

Related

• The symmetric invariants of the centralizers and Slodowy grading
• The symmetric invariants of centralizers and Slodowy grading II
• Surprising properties of centralisers in classical Lie algebras
• Invariants of Centralisers in Positive Characteristic
• The centralisers of nilpotent elements in classical Lie algebras