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Let X=G/P be a complex flag manifold and E->X be a G-homogeneous holomorphicvector bundle. Fix a U-invariant Kaehler metric on X with U in G maximalcompact. We study the sheaf of nearly holomorphic sections and show that thespace of global nearly holomorphic sections in E coincides with the space ofU-finite smooth sections in E. The degree of nearly holomorphic sectionsdefines a U-invariant filtration on this space. Using sheaf cohomology, wedetermine in suitable cases the corresponding Hilbert series. It turns out tobe given in terms of Lusztig's q-analog of Kostant's weight multiplicityformula.

Hilbert series of nearly holomorphic sections on generalized flag manifolds