Polynomiality for the Poisson centre of truncated maximal parabolic subalgebras

TL;DR

This paper proves that the Poisson centre of certain truncated maximal parabolic subalgebras in specific Lie algebras is a polynomial algebra, using algebraic slices and improved bounds, extending known results.

Contribution

It establishes polynomiality of the Poisson centre for new cases of truncated maximal parabolic subalgebras in Lie algebras of types B, D, and E_6.

Findings

Poisson centre is polynomial in these cases.

Constructed algebraic slices for new cases.

Provided improved bounds for the Poisson centre.

Abstract

We show that the Poisson centre of truncated maximal parabolic subalgebras of a simple Lie algebra of type B, D and E_6 is a polynomial algebra. In roughly half of the cases the polynomiality of the Poisson centre was already known by a completely different method. For the rest of the cases, our approach is to construct an algebraic slice in the sense of Kostant given by an adapted pair and the computation of an improved upper bound for the Poisson centre.