The Poisson spectrum of the symmetric algebra of the Virasoro algebra

Let $W = \mathbb{C}[t,t^{-1}]\partial_t$ be the Witt algebra of algebraic vector fields on $\mathbb{C}^\times$ and let $Vir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In this paper, we study the Poisson ideal structure of the symmetric algebras of $Vir$ and $W$, as well as several related Lie algebras. We classify prime Poisson ideals and Poisson primitive ideals of $S(Vir)$ and $S(W)$. In particular, we show that the only functions in $W^*$ which vanish on a nontrivial Poisson ideal (that is, the only maximal ideals of $S(W)$ with a nontrivial Poisson core) are given by linear combinations of derivatives at a finite set of points; we call such functions local. Given a local function $\chi\in W^*$, we construct the associated Poisson primitive ideal through computing the algebraic symplectic leaf of $\chi$, which gives a notion of coadjoint orbit in our setting. As an application, we prove a structure theorem for subalgebras of $Vir$ of finite codimension and show in particular that any such subalgebra of $Vir$ contains the central element $z$, substantially generalising a result of Ondrus and Wiesner on subalgebras of codimension 1. As a consequence, we deduce that $S(Vir)/(z-\lambda)$ is Poisson simple if and only if $\lambda \neq 0$.