The Orbit Method for the Jacobi Group

g∗ −→ g, λ 7→ Xλ characterized by λ(Y ) =< Xλ, Y >, Y ∈ g. Therefore the coadjoint G-orbits in g∗ may be identified with adjoint G-orbits in g. The philosophy of the orbit method says that we may attach the irreducible unitary representations of G to the coadjoint orbits in g∗. Historically the orbit method that was first initiated by A.A. Kirillov (cf. [K]) early in the 1960s in a real nilpotent Lie group worked beautifully. Thereafter the orbit method was extended nicely to a solvable Lie group of type I by Auslander and Kostant (cf. [A-K]). Their proof was based on the existence of complex polarizations satisfying a positivity condition. Unfortunately Kirillov’s work fails to be generalized in some ways to the case of compact Lie groups or semisimple Lie groups. Relatively simple groups like SL(2,R) have irreducible unitary representations that do not correspond to any symplectic homogeneous space. Conversely, P. Torasso [T] found that the double cover of SL(3,R) has a homogeneous symplectic manifold corresponding to no unitary representations. The orbit method for reductive Lie groups is a kind of a philosophy but not a theorem. Many large families of orbits correspond in comprehensible ways to unitary representations, and provide a clear geometric picture of these representations. The coadjoint orbits for a reductive Lie group are classified into three kinds of orbits, namely, hyperbolic, elliptic and nilpotent ones. The hyperbolic orbits are related to the unitary representations obtained by the parabolic induction and on the other hand, the elliptic ones are related to the unitary representations obtained by the cohomological induction. However, we still have no idea of attaching unitary representations to nilpotent orbits. It is known that there are only finitely many nilpotent orbits. In a certain case, some nilpotent orbits are corresponded to the so-called unipotent representations. For instance, a minimal nilpotent orbit is attached to a minimal representation. In fact, the notion of unipotent representations is not still well defined. The investigation of unipotent representations is now under way.