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First, let me fix some notation. Let $\mathfrak{g}\_1$ be a semisimple Lie algebra and let $\mathfrak{p}$ be its parabolic subalgebra which induces the grading $\mathfrak{g} = {\mathfrak{g}}\_{-} \oplus {\mathfrak{g}}\_{0} \oplus {\mathfrak{g}}\_{+}$, where $\mathfrak{g}\_{0}$ is the Levi part of $\mathfrak{p}$ and $\mathfrak{g}\_+$ its nilpotent radical. Let $\lambda$ be a weight of an irreducible finite dimensional $\mathfrak{p}$-representation $F_\lambda$ and let $V_\lambda$ be the generalized Verma module $\mathfrak{U_g}\otimes_{\mathfrak{U_p}} F_\lambda$.

There is an article1, which suggest that the invariant operators between associated bundles over $G/P$ are given by obstruction in cohomology $H^k(\mathfrak{g}\_-,V_\lambda)$, at least in the 1-graded case.

Is this result known and if yes is there a more mathematical treatment?

What is known about the cohomological groups $H^k(\mathfrak{g}\_-,V_\lambda)$? (Results of Collingwood & Boe for the 1-graded case are referenced in the mentioned article, but they are 25 years old.)

  1. The article is "Unfolded form of conformal equations in M dimensions and o(M+2)-modules" by O.V. Shaynkman, I.Yu. Tipunin and M.A. Vasiliev and it was published in Rev. Math. Phys.
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@r0b0t: It's good etiquette, I think, to spell out at least part of the metadata of any cited article. Not just it might save the odd click, but also helps to give some more context to the question. In this case, the article is "Unfolded form of conformal equations in M dimensions and o(M+2)-modules" by O.V. Shaynkman, I.Yu. Tipunin and M.A. Vasiliev. –  José Figueroa-O'Farrill Jun 21 '11 at 20:24
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OK. Thank you. I will edit the question accordingly. –  Vít Tuček Jun 22 '11 at 10:39

1 Answer 1

In: A. Tolpygo, Estimation of the cohomology of Verma modules, Russ. Math. Surv. 48 (1993), N1, 193-194, some estimations on the dimension of cohomology in question are given. In the later paper: Lie algebra cohomology and generating functions, Homology Homotopy Appl. 6 (2004), 59-85 http://www.intlpress.com/HHA/v6/n1/a6/ , Tolpygo elaborates on the technique used in the former shorter note, but in the context of finite-dimensional modules. But probably something could be inferred from the latter paper also in the case of Verma modules.

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