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

1980; American Mathematical Society; Volume: 262; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1980-0586721-0

1088-6850

Alvany Rocha-Caridi,

Spectral Theory in Mathematical Physics

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.