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Let $G$ be a connected semisimple algebraic group with Lie algebra $g$ and$P$ a parabolic subgroup of $G$ with $Lie(P)=p$. The parabolic contraction of$g$ is the semi-direct product of $p$ and a $p$-module $g/p$ regarded as anabelian ideal. We are interested in the polynomial invariants of the adjointand coadjoint representations of $q$. In the adjoint case the algebra ofinvariants is easy to describe and turns out to be a graded polynomial algebra.The coadjoint case is more complicated. Here we found a connection betweensymmetric invariants of $q$ and symmetric invariants of centralisers$g_e\subset g$, where $e$ is a Richardson element with polarisation $p$. Usingthis connection and results of Panyushev, Premet, and Yakimova (seearxiv:0610049), we prove that the algebra of symmetric invariants of $q$ isfree for all parabolics in types $A$ and $C$ and some parabolics in type $B$.The technique also applies to minimal parabolics in all types. For a Borelsubalgebra, one gets a contraction of $g$ recently introduced by E.Feigin(arXiv:1007.0646 and arXiv:1101.1898) and studied from invariant-theoreticpoint of view in our previous paper (arxiv:1107.0702).

Parabolic contractions of semisimple Lie algebras and their invariants