Resumen de Parabolic contractions of semisimple Lie algebras and their invariants
Dmitri I. Panyushev, Oksana Yakimova
Let GG be a connected semisimple algebraic group with Lie algebra gg and PP a parabolic subgroup of GG with LieP=pLieP=p . The parabolic contraction qq of gg is the semi-direct product of pp and a pp -module g/pg/p regarded as an abelian ideal. We are interested in the polynomial invariants of the adjoint and coadjoint representations of qq . 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 qq and symmetric invariants of centralisers ge⊂gge⊂g , where e∈ge∈g is a Richardson element with polarisation pp . Using this connection and results of Panyushev et al. (J Algebra 313:343–391, 2007), we prove that the algebra of symmetric invariants of qq is free for all parabolic subalgebras in types AA and CC and some parabolics in type BB . This technique also applies to the minimal parabolic subalgebras in all types. For p=bp=b , a Borel subalgebra of gg , one gets a contraction of gg 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).
