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 \(\tilde \varphi \) 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.
Partially supported by the RFBR grants 01-01-00709, 00-01-00705.
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Igonin, S. (2002). Notes on Homogeneous Vector Bundles over Complex Flag Manifolds. In: Malyshev, V., Vershik, A. (eds) Asymptotic Combinatorics with Application to Mathematical Physics. NATO Science Series, vol 77. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0575-3_11
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DOI: https://doi.org/10.1007/978-94-010-0575-3_11
Publisher Name: Springer, Dordrecht
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