Some remarks on periodic gradings

Let $\mathfrak{q}$ be a finite-dimensional Lie algebra, $\vartheta \in Aut(\mathfrak{q})$ a finite order automorphism, and ${\mathfrak{q}}_0$ the subalgebra of fixed points of $\vartheta$. Using $\vartheta$ one can construct a pencil $\mathcal{P}$ of compatible Poisson brackets on $S(\mathfrak{q})$, and thereby a `large’ Poisson-commutative subalgebra $Z(\mathfrak{q},\vartheta )$ consisting of ${\mathfrak{q}}_0$-invariants in $S(\mathfrak{q})$. We study one particular bracket $\{,{\}}_{\mathrm{\infty }}\in \mathcal{P}$ and the related Poisson centre ${\mathcal{Z}}_{\mathrm{\infty }}$. It is shown that ${\mathcal{Z}}_{\mathrm{\infty }}$ is a polynomial ring, if $\mathfrak{q}$ is reductive.