Equivariant Deformation Quantization for the Cotangent Bundle of a Flag Manifold

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Let be a (generalized) flag manifold of a non-compact real semisimple Lie group, where and have complexifications X and G. We investigate the problem of constructing a graded star product on Pol(T*) which corresponds to a -equivariant quantization of symbols into smooth differential operators acting on half-densities on. We show that any solution is algebraic in that it restricts to a G-equivariant graded star product star on the algebraic part R of Pol(T*). We construct, when R is generated by the momentum functions μˣ for G, a preferred choice of star where μˣ⋆ϕ has the form μˣϕ+{μˣ,ϕ}t+Λˣ(ϕ)t². Here Λˣ are operators on R which are not differential in the known examples and so μˣ⋆ϕ is not local in ϕ. R acquires an invariant positive definite inner product compatible with its grading. The completion of R is a new Fock space type model of the unitary representation of G on L² half-densities on X.