The Orbit Method for Poisson Orders

Abstract

A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation $A$ of a Poisson algebra $Z$ should correspond bijectively to the symplectic leaves of $\operatorname{Spec}(Z)$. In this article we consider a Poisson order $A$ over a complex affine Poisson algebra $Z$. We stratify the primitive spectrum $\operatorname{Prim}(A)$ into symplectic cores, which should be thought of as families of non-commutative symplectic leaves. We then introduce a category $A$-$\mathcal{P}$-Mod of $A$-modules adapted to the Poisson structure on $Z$, and we show that when $\operatorname{Spec}(Z)$ is smooth with locally closed symplectic leaves, there is a natural homeomorphism from the spectrum of annihilators of simple objects in $A$-$\mathcal{P}$-Mod to the set of symplectic cores in $\operatorname{Prim}(A)$ with its quotient topology. Several application are given to Poisson representation theory. Our main tool is the Poisson enveloping algebra $A^e$ of a Poisson order $A$, which captures the Poisson representation theory of $A$. For $Z$ regular and affine we prove a PBW theorem for $A^e$ and use this to characterise the annihilators of simple Poisson modules: they coincide with the Poisson weakly locally closed, the Poisson primitive and the Poisson rational ideals. We view this as a generalised weak Poisson Dixmier--Moeglin equivalence.