  Let X=G/P be a complex flag manifold and E->X be a G-homogeneous holomorphic
vector bundle. Fix a U-invariant Kaehler metric on X with U in 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 to
be given in terms of Lusztig's q-analog of Kostant's weight multiplicity
formula.
