Journal of Lie Theory

no. 3
On the Principal Bundles over a Flag Manifold: II
Hassan Azad1; Indranil Biswas2
1Dept. of Mathematics, University of Management Sciences, Lahore, Pakistan
2School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
Journal of Lie Theory, Volume 17 (2007) no. 3, pp. 669-684
[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 $G/P$.
DOI: 10.5802/jolt.469
Classification: 14M15, 14F05
Keywords: Homogeneous space, principal bundle, Frobenius, stability

Hassan Azad  1 ; Indranil Biswas  2

1 Dept. of Mathematics, University of Management Sciences, Lahore, Pakistan
2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India 
Hassan Azad; Indranil Biswas. On the Principal Bundles over a Flag Manifold: II. Journal of Lie Theory, Volume 17 (2007) no. 3, pp. 669-684. doi: 10.5802/jolt.469
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