Deformation Quantization of Coadjoint Orbits

A method for the deformation quantization of coadjoint orbits of semisimple Lie groups is proposed. It is based on the algebraic structure of the orbit. Its relation to geometric quantization and differentiable deformations is explored. Let G be a complex Lie group of dimension n and GR a real form of G. Let G and GR be their respective Lie algebras with Lie bracket [ , ]. As it is well known, G R has a Poisson structure, {f1, f2}(λ) =< [(df1)λ, (df2)λ], λ >, f1, f2 ∈ C (G R), λ ∈ G ∗ R. (1) Choosing a basis {X1, . . .Xn} of GR and its dual, {ξ , . . . ξ}, the Poisson bracket can be written as {f1, f2}(x) = c k ijλk ∂f1 ∂xi ∂f2 ∂xj , x = n