Abstract
Let \(G\) be a connected semisimple algebraic group with Lie algebra \(\mathfrak{g }\) and \(P\) a parabolic subgroup of \(G\) with \(\mathrm{Lie\, }P=\mathfrak{p }\). The parabolic contraction \(\mathfrak{q }\) of \(\mathfrak{g }\) is the semi-direct product of \(\mathfrak{p }\) and a \(\mathfrak{p }\)-module \(\mathfrak{g }/\mathfrak{p }\) regarded as an abelian ideal. We are interested in the polynomial invariants of the adjoint and coadjoint representations of \(\mathfrak{q }\). In the adjoint case, the algebra of invariants is easily described and it turns out to be a graded polynomial algebra. The coadjoint case is more complicated. Here we found a connection between symmetric invariants of \(\mathfrak{q }\) and symmetric invariants of centralisers \(\mathfrak{g }_e\subset \mathfrak{g }\), where \(e\in \mathfrak{g }\) is a Richardson element with polarisation \(\mathfrak{p }\). Using this connection and results of Panyushev et al. (J Algebra 313:343–391, 2007), we prove that the algebra of symmetric invariants of \(\mathfrak{q }\) is free for all parabolic subalgebras in types \(\mathbf A\) and \(\mathbf C\) and some parabolics in type \(\mathbf B\). This technique also applies to the minimal parabolic subalgebras in all types. For \(\mathfrak{p }=\mathfrak{b }\), a Borel subalgebra of \(\mathfrak{g }\), one gets a contraction of \(\mathfrak{g }\) recently introduced by Feigin (Selecta Math 18:513–537, 2012) and studied from invariant-theoretic point of view in our previous paper (Panyushev and Yakimova in Ann Inst Fourier 62(6):2053–2068, 2012).
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Panyushev, D.I., Yakimova, O.S. Parabolic contractions of semisimple Lie algebras and their invariants. Sel. Math. New Ser. 19, 699–717 (2013). https://doi.org/10.1007/s00029-013-0122-x
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DOI: https://doi.org/10.1007/s00029-013-0122-x