Parabolic contractions of semisimple Lie algebras and their invariants

TL;DR

This paper investigates polynomial invariants of parabolic contractions of semisimple Lie algebras, establishing their structure and connections to symmetric invariants of centralisers, with results applicable to various Lie algebra types.

Contribution

It characterizes the algebra of invariants for parabolic contractions, proving it is free in types A and C, and extends previous invariant theory results to these algebraic structures.

Findings

Algebra of invariants is a graded polynomial algebra in the adjoint case.

Proves invariants are free for all parabolics in types A and C.

Extends results to minimal parabolics and Borel subalgebras.

Abstract

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).