The Orbit Method for Poisson Orders

TL;DR

This paper develops a geometric and categorical framework for understanding the primitive spectrum of Poisson orders, linking symplectic leaves and cores with module categories, and introduces tools like the Poisson enveloping algebra.

Contribution

It introduces a stratification of the primitive spectrum into symplectic cores and establishes a homeomorphism with annihilators of simple modules, generalizing Poisson Dixmier–Moeglin theory.

Findings

Homeomorphism between primitive spectrum and symplectic cores

PBW theorem for Poisson enveloping algebra

Characterization of annihilators of simple Poisson modules

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.