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On symmetric invariants of centralisers in reductive Lie algebras

D. Panyushev·A. Premet·O. Yakimova
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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 $g$.

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BibTeXRIS

D. Panyushev, A. Premet, O. Yakimova. 2006-12-13. On symmetric invariants of centralisers in reductive Lie algebras. https://arxiv.org/abs/math/0610049

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