Hilbert series of nearly holomorphic sections on generalized flag manifolds

Let X=G/P be a complex flag manifold and E→X be a G-homogeneous holomorphic vector bundle. Fix a U-invariant Kähler metric on X with U⊆G maximal compact. We study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in E coincides with the space of U-finite smooth sections in E. The degree of nearly holomorphic sections defines a U-invariant filtration on this space. Using sheaf cohomology, we determine in suitable cases the corresponding Hilbert series. It turns out that this is given in terms of Lusztig's q-analog of Kostant's weight multiplicity formula, and hence gives a new representation theoretic interpretation of these formulas.