On the Orbit Method for the Lie Algebra of Vector Fields on a Curve

Let be the Lie algebra of vector fields on an affine smooth curve Σ. Our goal is to establish an orbit method for . Since is infinite-dimensional, we face some technical problems. Without having groups acting on , we try nevertheless to define the notion of “orbits.” So, we focus our attention to a subspace *fof *. This subspace consists of the “finite-dimensional orbits.” To almost all λ in *fit corresponds a simple induced representation of whose annihilator is a primitive ideal. We conjecture that this ideal has a finite Gelfand–Kirillov codimension. What we are actually looking for is a bijection similar to Dixmierʹs bijection (in the finite-dimensional case) between the “orbits” of *fand certain primitive ideals of the enveloping algebra of .