Surprising properties of centralisers in classical Lie algebras

Abstract

Let $g$ be a classical Lie algebra, i.e., either $g{l}_n$, $s{p}_n$, or $s{o}_n$ and let $e\in g$ be a nilpotent element. We study various properties of centralisers $g_e$. The first four sections deal with rather elementary questions, like the centre of $g_e$, commuting varieties associated with $g_e$, or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on $g_e^{\ast }$ and symmetric invariants of $g_e$.

Final version, to appear in Ann. Inst. Fourier (Grenoble), v. 59 (2009)