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


Author: Alvany Rocha-Caridi
Journal: Trans. Amer. Math. Soc. 262 (1980), 335-366
MSC: Primary 17B10; Secondary 22E47
DOI: https://doi.org/10.1090/S0002-9947-1980-0586721-0
MathSciNet review: 586721
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Abstract: Let $\mathcal {g}$ be a finite dimensional, complex, semisimple Lie algebra and let V be a finite dimensional, irreducible $\mathcal {g}$-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.


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