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On a theorem of Ramanan

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Department of Mathematics, Nagoya University
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Let G be a simply connected Lie group and P a parabolic subgroup without simple factor. A finite dimensional irreducible representation of P defines a homogeneous vector bundle E over the homogeneous space G/P. Ramanan [2] proved that, if the second Betti number b2 of G/P is 1, the inequality in Definition (2.3) holds provided F is locally free. Since the notion of the H-stability was not established at that time, it was inevitable to assume that b2 = 1 and F is locally free. In this paper, pushing Ramanan’s idea through, we prove that E is H-stable for any ample line bundle H. Our proof as well as Ramanan’s depends on the Borel-Weil theorem. If we recall that the Borel-Weil theorem fails in characteristic p > 0, it is interesting to ask whether our theorem remains true in characteristic p > 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 69 , February 1978 , pp. 131 - 138
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1978

References

[1] Kostant, B. Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. Vol. 74, 1961.Google Scholar
[2] Ramanan, S. Holomorphic vector bundles on homogeneous spaces, Topology vol. 5, 1966.Google Scholar