Vogan’s talks concentrated on describing the so-called “orbit method” or “orbit philosophy”, which posits a bijection for Lie groups G between
This is described as a “method” or “philosophy” rather than a theorem because it doesn’t always work, and remains poorly understood in some cases, while at the same time having shown itself to be a powerful source of inspiration in representation theory.
It is probably best understood as an expression of the deep relationship between quantum mechanics and representation theory, and the surprising power of the notion of “quantization” of a classical mechanical system. In the Hamiltonian formalism, a classical mechanical system with symmetry group G corresponds to what a mathematician would call a symplectic manifold with an action of the group G preserving the symplectic structure. “Geometric quantization” is supposed to associate in some natural way a quantum mechanical system with symmetry group G to this symplectic manifold with G-action, with the Hilbert space of the quantum system providing a unitary representation of the group G. The representation is expected to be irreducible just when the group G acts transitively on the symplectic manifold. One can show that symplectic manifolds with transitive G action correspond to orbits of G on (Lie G)*, the dual space to the Lie algebra of G, with G acting by the dual of the adjoint action. So it is these “co-adjoint orbits” that provide geometrical versions of classical mechanical systems with G symmetry, and the orbit philosophy says that we should be able to quantize them to get irreducible unitary representations, and any irreducible unitary representation should come from this construction.
That such a “quantization” exists is perhaps surprising. To a quantum system one expects to be able to associate a classical system by taking Planck’s constant to zero, but there is no good reason to expect that there should be a natural way of “quantizing” a classical system and getting a unique quantum system. Remarkably, we are able to do this for many classes of symplectic manifolds. For nilpotent groups like the Heisenberg group, that the orbit method works is a theorem, and this can be extended to solvable groups. What remains to be understood is what happens for reductive groups.
Already for the simplest case here, compact Lie groups, the situation is very interesting. Here co-adjoint orbits are things like flag manifolds, and the Borel-Weil-Bott theorem says that if an integrality condition is satisfied one gets the expected irreducible representations, sometimes in higher cohomology spaces. One can take “geometric quantization” here to be essentially “integration in K-theory”, realizing representations using solutions to the Dirac equation. Recently Freed-Hopkins-Teleman gave a beautiful construction that gives the inverse map, associating an orbit to a given representation.
For non-compact real forms of complex reductive groups, like SL(2,R), the situation is much trickier, with the unitary representations infinite dimensional. Vogan’s lectures were designed to lead up to and explain the still poorly understood problem of how to associate representations to nilpotent orbits of such groups. At the end of his slides, he gives two references one can consult to find out more about this.
Finally, there is a good graduate level textbook about the orbit method, Kirillov’s Lectures on the Orbit Method. For more about the orbit method philosophy, its history and current state, a good source to consult is Vogan’s review of this book in the Bulletin of the AMS.