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Journal of Lie Theory 17 (2007), No. 3, 669--684
Copyright Heldermann Verlag 2007

On the Principal Bundles over a Flag Manifold: II

Hassan Azad
Dept. of Mathematics, University of Management Sciences, Lahore, Pakistan

Indranil Biswas
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India


[Part I of this article has been published in J. Lie Theory 14 (2004) 569--581.] Let $G$ be a connected semisimple linear algebraic group defined over an algebraically closed field $k$ and $P\subset G$, $P\ne G$, a reduced parabolic subgroup that does not contain any simple factor of $G$. Let $\rho : P\longrightarrow H$ be a homomorphism, where $H$ is a connected reductive linear algebraic group defined over $k$, with the property that the image $\rho(P)$ is not contained in any proper parabolic subgroup of $H$. We prove that the principal $H$-bundle $G\times^P H$ over $G/P$ constructed using $\rho$ is stable with respect to any polarization on $G/P$. When the characteristic of $k$ is positive, the principal $H$-bundle $G\times^P H$ is shown to be strongly stable with respect to any polarization on $G/P$.

Keywords: Homogeneous space, principal bundle, Frobenius, stability.

MSC: 14M15, 14F05

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