@inbook{941788741b3642e3aa6530a7f87b3eb3,

title = "Notes on homogeneous vector bundles over complex flag manifolds",

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{\textquoteright}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.",

keywords = "METIS-210104",

author = "Sergei Igonine",

year = "2002",

doi = "10.1007/978-94-010-0575-3_11",

language = "English",

isbn = "1-4020-0792-2",

series = "NATO science series",

publisher = "Kluwer Academic Publishers",

pages = "245--254",

editor = "V.A. Malyshev and A.M. Vershik",

booktitle = "Asymptotic Combinatorics with Application to Mathematical Physics",

address = "Netherlands",

}