Notes on homogeneous vector bundles over complex flag manifolds

Sergei Igonine

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    Abstract

    Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset ∑ ⊂ ∏ of simple roots of G, and let E ϕ be a homogeneous vector bundle over the flag manifold M = G/P corresponding to a linear representationϕ of P. Using Bott’s theorem, we obtain sufficient conditions on ϕ in terms of the combinatorial structure of ∑ ⊂ ∏ for cohomology groups H q(M,ε ϕ ) to be zero, where ε ϕ is the sheaf of holomorphic sections of E ϕ . In particular, we define two numbers d(P), ℓ(P) ∈ ℕ such that for any ϕ obtained by natural operations from a representation φ~ of dimension less than d(P) one has H q(M,ε ϕ ) = 0 for 0 < q < ℓ(P). Applying this result to H 1(M,εϕϕ), we see that the vector bundle E ϕ, is rigid.
    Original languageEnglish
    Title of host publicationAsymptotic Combinatorics with Application to Mathematical Physics
    EditorsV.A. Malyshev, A.M. Vershik
    Place of PublicationDordrecht
    PublisherKluwer Academic Publishers
    Pages245-254
    Number of pages10
    ISBN (Print)1-4020-0792-2
    DOIs
    Publication statusPublished - 2002

    Publication series

    NameNATO science series
    Volume77

    Keywords

    • METIS-210104

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