Unipotent Representations Attached to Principal Nilpotent Orbits

The Orbit Method is a conjectural correspondence between co-adjoint orbits of a Lie group G and its irreducible unitary representations. When G is nilpotent or simply connected and solvable, this correspondence is perfect and complete. But when G is reductive, serious problems arise. The worst of these problems have to do with the nilpotent orbits of G. As of yet, there is no general method for attaching unitary representations to nilpotent orbits. In this talk, I will attach a finite set of irreducible unitary representations to the principal nilpotent orbits. I will deduce formulas for the K-types and associated varieties of the representations in question.