Orbit method: Difference between revisions

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== Relation with symplectic geometry ==

One of the key observations of Kirillov was that coadjoint orbits of a Lie group ''G'' have natural structure of [[symplectic manifold]]s whose symplectic structure is invariant under ''G''. If an orbit is the [[phase space]] of a ''G''-invariant [[Hamiltonian mechanics|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, PukanszkyPukánszky condition). This point of view has been significantly advanced by Kostant in his theory of [[geometric quantization]] of coadjoint orbits.

== Kirillov character formula ==