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# Homogeneous vector bundles

Let be a connected complex semisimple Lie group, and a parabolic subgroup. Then is a principal -bundle over the homogeneous manifold . Any holomorphic representation on a complex vector space induces a holomorphic vector bundle over :

where . Then is a holomorphic vector bundle over . A vector bundle over arising in this way is called a homogeneous vector bundle. Let be the corresponding sheaf of holomorphic sections. The homogeneous vector bundle inherits a left -action from the left multiplication in :

for

Thus, all the cohomology groups become -modules.

In [15] Bott proved that if the representation is irreducible, the cohomology groups all vanish except possibly in one single dimension. Moreover, in the nonvanishing dimension , is an irreducible representation of whose highest weight is related to .

This theorem generalizes an earlier theorem of Borel and Weil, who proved it for a positive line bundle.

In Bott's paper one finds a precise way of determining the nonvanishing dimension in terms of the roots and weights of and . Thus, on the one hand, Bott's theorem gives a geometric realization of induced representations, and on the other hand, it provides an extremely useful vanishing criterion for the cohomology of homogeneous vector bundles.

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