The orbit method for the Virasoro algebra
Tuan Anh Pham
TL;DR
This work develops an orbit-method framework for the Witt algebra and its Virasoro extension by constructing a strong Dixmier-type correspondence between Poisson-primitive structures on symmetric algebras and primitive ideals of universal enveloping algebras. Central to the approach are local functions on W (and their Virasoro pullbacks), which yield canonical local representations whose annihilators are primitive (often completely prime) ideals, enabling a Dixmier map from P.Prim S(g) to Prim U(g). The authors introduce master ring-homomorphisms Ψ and their Poisson counterparts Φ that relate infinite-dimensional enveloping algebras to finite-dimensional solvable subquotients, providing a bridge to the finite-dimensional orbit method and rendering a large class of primitive ideals accessible via a Weyl-algebra reduction. They also prove new structural results, including non-surjectivity in certain cases to first Weyl algebra, and establish a precise connection between pseudo-orbits of local functions and coadjoint orbits of the finite-dimensional reductions, culminating in a robust strong Dixmier map for W, W_{≥-1}, and Vir. These findings extend the orbit-method paradigm to a countable-dimensional setting with explicit constructions, offering a framework for understanding primitive ideals and their geometric origins in infinite-dimensional Lie algebras with potential applications in representation theory and mathematical physics.
Abstract
Let $W = \mathbb{C}[t, t^{-1}]\partial_t$ be the Witt algebra of algebraic vector fields on $\mathbb{C}^\times$ and let $V\!ir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. In 2023, Petukhov and Sierra showed that Poisson primitive ideals of $\mathrm{S}(W)$ and $\mathrm{S}(V\!ir)$ can be constructed from elements of $W^*$ and $V\!ir^*$ of a particular form, called local functions. In this paper, we show how to use a local function on $W$ or $V\!ir$ to construct a representation of the Lie algebra. We further show that the annihilators of these representations are new completely prime primitive ideals of $\mathrm{U}(W)$ and $\mathrm{U}(V\!ir)$. We use this to define a Dixmier map from the Poisson primitive spectrum of $\mathrm{S}(V\!ir)$, respectively $\mathrm{S}(W)$, to the primitive spectrum of $\mathrm{U}(V\!ir)$, respectively $\mathrm{U}(W)$, successfully extending the orbit method from finite-dimensional solvable Lie algebras to our countable-dimensional setting. Our method involves new ring homomorphisms from $\mathrm{U}(W)$ to the tensor product of a localized Weyl algebra and the enveloping algebra of a finite-dimensional solvable subquotient of $W$. We further show that the kernels of these homomorphisms are intersections of the primitive ideals constructed from natural subsets of $W^*$. As a corollary, we disprove the conjecture that any primitive ideal of $\mathrm{U}(W)$ is the kernel of some map from $\mathrm{U}(W)$ to the first Weyl algebra.