Orbit method

Roger Howe found a version of the orbit method that applies to p-adic Lie groups.

[1] David Vogan proposed that the orbit method should serve as a unifying principle in the description of the unitary duals of real reductive Lie groups.

In this way the orbit method may be viewed as a precise mathematical manifestation of a vague physical principle of quantization.

This point of view has been significantly advanced by Kostant in his theory of geometric quantization of coadjoint orbits.

At its simplest, it states that a character of a Lie group may be given by the Fourier transform of the Dirac delta function supported on the coadjoint orbits, weighted by the square-root of the Jacobian of the exponential map, denoted by

If G is a compact semisimple Lie group with a Cartan subalgebra h then its coadjoint orbits are closed and each of them intersects the positive Weyl chamber h*+ in a single point.