Centralizers of distinguished nilpotent pairs and related problems

In this paper, by establishing an explicit and combinatorial description of the centralizer of a distinguished nilpotent pair in a classical simple Lie algebra, we solve in the classical case Panyushev's Conjecture which says that distinguished nilpotent pairs are wonderful, and the classification problem on almost principal nilpotent pairs. More precisely, we show that disinguished nilpotent pairs are wonderful in types A, B and C, but they are not always wonderful in type D. Also, as the corollary of the classification of almost principal nilpotent pairs, we have that almost principal nilpotent pairs do not exist in the simply-laced case and that the centralizer of an almost principal nilpotent pair in a classical simple Lie algebra is always abelian.Comment: 25 page

Semi-direct products of Lie algebras and their invariants

The goal of this paper is to extend the standard invariant-theoretic design, well-developed in the reductive case, to the setting of representation of certain non-reductive groups. This concerns the following notions and results: the existence of generic stabilisers and generic isotropy groups for finite-dimensional representations; structure of the fields and algebras of invariants; quotient morphisms and structure of their fibres. One of the main tools for obtaining non-reductive Lie algebras is the semi-direct product construction. We observe that there are surprisingly many non-reductive Lie algebras whose adjoint representation has a polynomial algebra of invariants. We extend results of Takiff, Geoffriau, Rais-Tauvel, and Levasseur-Stafford concerning Takiff Lie algebras to a wider class of semi-direct products. This includes $Z_2$-contractions of simple Lie algebras and generalised Takiff algebras.Comment: 49 pages, title changed, section 11 is shortened, numerous minor corrections; accepted version, to appear in Publ. RIMS 43(2007

On divisible weighted Dynkin diagrams and reachable elements

Let D(e) denote the weighted Dynkin diagram of a nilpotent element $e$ in complex simple Lie algebra \g. We say that D(e) is divisible if D(e)/2 is again a weighted Dynkin diagram. (That is, a necessary condition for divisibility is that $e$ is even.) The corresponding pair of nilpotent orbits is said to be friendly. In this note, we classify the friendly pairs and describe some of their properties. We also observe that any subalgebra sl(3) in \g determines a friendly pair. Such pairs are called A2-pairs. It turns out that the centraliser of the lower orbit in an A2-pair has some remarkable properties. Let $Gx$ be such an orbit and $h$ a characteristic of $x$. Then $h$ determines the Z-grading of the centraliser $z=z(x)$. We prove that $z$ is generated by the Levi subalgebra $z(0)$ and two elements in $z(1)$. In particular, (1) the nilpotent radical of $z$ is generated by $z(1)$ and (2) $x\in [z,z]$. The nilpotent elements having the last property are called reachable.Comment: 17 pages; v2 minor corrrections; final version, to appear in Transformation Groups (2010