A BGG type resolution of holomorphic Verma modules

For a Hermitian symmetric space X = G/K of nan-compact type let theta denot e the Cartan involution of the semisimple Lie group G with respect to the m aximal compact subgroup K of G, and let q denote a theta-stable parabolic s ubalgebra of the complexified Lie algebra g of G with corresponding Levi su bgroup L of G. Given a finite-dimensional irreducible L module F-L we find Bernstein-Gelfand-Gelfand type resolutions of the induced (g, L boolean AND K) module U(g) X-U(q) F-L and its Hermitian dual, the produced module Hom( U((q) over bar)), (U(g), F-L)(L boolean AND K-finite), where U(.) is the un iversal enveloping algebra functor and (q) over bar is the complex conjugat e of q. The results coupled with a Grothendick spectral sequence provide fo r application to certain (g, K) modules obtained by cohomological parabolic induction, and they extend results obtained initially by Stanke.