On the principal bundles over a flag manifold

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