Equivariant Deformation Quantization for the Cotangent Bundle of a Flag Manifold

Let XR be a (generalized) flag manifold of a non-compact real semisimple Lie group GR, where XR and GR have complexifications X and G. We investigate the problem of constructing a graded star product on Pol(T ∗XR) which corresponds to a GR-equivariant quantization of symbols into smooth differential operators acting on half-densities on XR. We show that any solution is algebraic in that it restricts to a G-equivariant graded star product ⋆ on the algebraic part R of Pol(T ∗XR). We construct, when R is generated by the momentum functions μx for G, a preferred choice of ⋆ where μx ⋆ φ has the form μxφ + 1 2 {μx, φ}t + Λ(φ)t. 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 L half-densities on X .