Kirillov orbit theory

In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of the group on the dual space of its Lie algebra. The theory was introduced by Kirillov (1961, 1962) for nilpotent groups and later extended by Bertram Kostant, Louis Auslander, Lajos Pukánszky and others to the case of solvable groups. Roger Howe found a version of the orbit method that applies to p-adic Lie groups. 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.

One of the key observations of Kirillov was that coadjoint orbits of a Lie group G have natural structure of symplectic manifolds whose symplectic structure is invariant under G. If an orbit is the phase space of a G-invariant classical mechanical system then the corresponding quantum mechanical system ought to be described via an irreducible unitary representation of G. Geometric invariants of the orbit translate into algebraic invariants of the corresponding representation. In this way the orbit method may be viewed as a precise mathematical manifestation of a vague physical principle of quantization. In the case of a nilpotent group G the correspondence involves all orbits, but for a general G additional restrictions on the orbit are necessary (polarizability, integrality, Pukanszky condition). This point of view has been significantly advanced by Kostant in his theory of geometric quantization of coadjoint orbits.

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