Slices for maximal parabolic subalgebras of a semisimple Lie algebra

TL;DR

This paper constructs a slice for the coadjoint action of maximal truncated parabolic subalgebras in simple Lie algebras, extending Kostant's theorem by using an adapted pair instead of a principal sl_2-triple.

Contribution

It introduces a new method to construct slices for the coadjoint action of these subalgebras, generalizing previous results and connecting to the polynomial nature of the Poisson center.

Findings

Constructed a slice for the coadjoint action of p

Extended Kostant's theorem to maximal truncated parabolic subalgebras

Linked the slice construction to the polynomial structure of the Poisson center

Abstract

Let p be a maximal truncated parabolic subalgebra of a simple Lie Algebra. It was shown in many cases that the Poisson centre Y(p) is a polynomial algebra. We construct a slice for the coadjoint action of p, thus extending a theorem of Kostant. The role of the principal sl_2-triple is played by an adapted pair.