Chapter 5 - The Orbit Method and Unitary Representations for Reductive Lie Groups

Abstract

The orbit method pioneered by Kirillov and Kostant seeks to construct irreducible unitary representations by analogy with quantization procedures in mechanics. A classical physical system may sometimes be modeled by a configuration space M, a manifold the points of which represent the possible positions of the bodies in the system. The corresponding phase space is the cotangent bundle T'M, the points of which represent the possible states (positions and momenta) of the bodies in the system. In this setting, a classical observable is a function on T*M. The quantum mechanical analog of this system is based on the Hilbert space L2(M) of square-integrable half-densities on M. A quantum mechanical observable is an operator T on L2(M); the scalar product (Tv, v) is the expected value of the observable on the state v. Some connection between classical and quantum observables is provided by a symbol calculus; an interesting quantum observable is often a differential operator on M, and the corresponding classical observable is related to the symbol of the differential operator.