  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 an
abelian ideal. We are interested in the polynomial invariants of the adjoint
and coadjoint representations of $q$. In the adjoint case the algebra of
invariants is easy to describe and turns out to be a graded polynomial algebra.
The coadjoint case is more complicated. Here we found a connection between
symmetric invariants of $q$ and symmetric invariants of centralisers
$g_e\subset g$, where $e$ is a Richardson element with polarisation $p$. Using
this connection and results of Panyushev, Premet, and Yakimova (see
arxiv:0610049), we prove that the algebra of symmetric invariants of $q$ is
free 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 Borel
subalgebra, one gets a contraction of $g$ recently introduced by E.Feigin
(arXiv:1007.0646 and arXiv:1101.1898) and studied from invariant-theoretic
point of view in our previous paper (arxiv:1107.0702).
