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Journal of Lie Theory 14 (2004), No. 2, 569--581
Copyright Heldermann Verlag 2004

On the Principal Bundles over a Flag Manifold

Hassan Azad
Dept. of Mathematical Sciences, King Fahd University, Dhahran 31261, Saudi Arabia,

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


Let $P$ be a parabolic subgroup of a semisimple simply connected linear algebraic group $G$ over $\mathbb C$ and $\rho$ an irreducible homomorphism from $P$ to a complex reductive group $H$. We show that the associated principal $H$--bundle over $G/P$, associated for $\rho$ to the principal $P$--bundle defined by the quotient map $G\, \longrightarrow\, G/P$, is stable. We describe the Harder--Narasimhan reduction of the $G$--bundle over $G/P$ obtained using the composition $P\, \longrightarrow\, L(P)\, \longrightarrow\, G$, where $L(P)$ is the Levi factor of $P$.

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