A Dixmier map for the Lie algebra of vector fields on a curve

The aim of my talk is to propose a Dixmier map for the Lie algebra $\mathfrak{g}$ of vector fields on an affine smooth curve. To achieve this, we will focus on a particular subspace $\mathfrak{g}_f^*$ of $\mathfrak{g}^*$, whose elements will define a simple induced representation. These representations will lead to primitive ideals of the enveloping algebra $\mathcal{U}(\mathfrak{g})$ of $\mathfrak{g}$. Ultimately, we are seeking a bijection, similar to Dixmier’s in the finite-dimensional case, between the orbits of $\mathfrak{g}_f^*$ and certain primitive ideals of the enveloping algebra $\mathcal{U}(\mathfrak{g})$.