On a theorem of Ramanan

Abstract

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.