Kirillov orbit theory

The Kirillov orbit theory or the method of orbits establishes a correspondence between the set of unitary equivalence classes of irreducible representations of a Lie group and the orbits of the action of G on the dual of its Lie algebra $\mathfrak{g}^*$ . These orbits are also called coadjoint orbits.

For example if G is a connected, simply connected nilpotent Lie group , the equivalence classes of irreducible unitary representations of G are parametrized by the orbits of the action G on $\mathfrak{g}^*$ . The theory was developed by Alexandre Kirillov originally for nilpotent groups and further by Bertram Kostant, Louis Auslander and others for solvable groups.