Equivariant Deformation Quantization for the Cotangent Bundle of a Flag Manifold

Abstract

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 star on the algebraic part R of $Pol(T^*\XR)$. We construct, when R is generated by the momentum functions $\mu^x$ for G, a preferred choice of star where $\mu^x\star\phi$ has the form $\mu^x\phi+\half\{\mu^x,\phi\}t+\Lambda^x(\phi)t^2$. Here $\Lambda^x$ are operators on R which are not differential in the known examples and so $\mu^x\star\phi$ is not local in $\phi$. 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 $L^2$ half-densities on X.