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Let      g \mathcal {g}       be a finite dimensional, complex, semisimple Lie algebra and let  V  be a finite dimensional, irreducible      g \mathcal {g}      -module. By computing a certain Lie algebra cohomology we show that the generalized versions of the weak and the strong Bernstein-Gelfand-Gelfand resolutions of  V  obtained by H. Garland and J. Lepowsky are identical. Let  G  be a real, connected, semisimple Lie group with finite center. As an application of the equivalence of the generalized Bernstein-Gelfand-Gelfand resolutions we obtain a complex in terms of the degenerate principal series of  G , which has the same cohomology as the de Rham complex.

Splitting criteria for 𝔤-modules induced from a parabolic and the Berňsteĭn-Gel’fand-Gel’fand resolution of a finite-dimensional, irreducible 𝔤-module