Parabolic contractions of semisimple Lie algebras and their invariants
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- 作者：Dmitri I. Panyushev (1)
Oksana S. Yakimova (2) - 关键词：Algebra of invariants ; Coadjoint representation ; Contraction ; Richardson orbit ; 13A50 ; 14L30 ; 17B08 ; 17B45 ; 22E46
- 刊名：Selecta Mathematica, New Series
- 出版年：2013
- 出版时间：August 2013
- 年：2013
- 卷：19
- 期：3
- 页码：699-717
- 全文大小：369KB
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Oksana S. Yakimova (2)
1. Institute for Information Transmission Problems, Russian Academy of Sciences, B. Karetnyi per. 19, 127994, Moscow, Russia
2. Mathematisches Institut, Friedrich-Schiller-Universit?t Jena, Jena, Deutschland
文摘
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 313:343-91, 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 18:513-37, 2012) and studied from invariant-theoretic point of view in our previous paper (Panyushev and Yakimova in Ann Inst Fourier 62(6):2053-068, 2012).