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

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite dimensional, complex, semisimple Lie algebra and let <italic>V</italic> be a finite dimensional, irreducible <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <italic>V</italic> obtained by H. Garland and J. Lepowsky are identical. Let <italic>G</italic> 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 <italic>G</italic> , which has the same cohomology as the de Rham complex.