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 $V\!ir$ Virasoro , unique nontrivial central extension $W$ . In this paper, we study Poisson ideal structure symmetric algebras as well several related Lie algebras. We classify prime ideals primitive $\operatorname {S}(V\!ir)$ {S}(W)$ particular, show that only functions in $W^*$ which vanish a (that is, maximal with core) are given by linear combinations derivatives at finite set points; call such local Given function $\chi \in W^*$ construct associated through computing symplectic leaf $\chi$ gives notion coadjoint orbit our setting. As an application, prove theorem for subalgebras codimension show, any subalgebra contains element $z$ substantially generalising result Ondrus Wiesner one. consequence, deduce {S}(V\!ir)/(z-\zeta )$ is simple if $\zeta \neq ~0$