Journal of Lie Theory Volume 17 (2007) 669–684 c  2007 Heldermann Verlag On the Principal Bundles over a Flag Manifold, II Hassan Azad and Indranil Biswas Communicated by E. B. Vinberg Abstract. Let G be a connected semisimple linear algebraic group defined over an algebraically closed field k and P  G a reduced parabolic subgroup that does not contain any simple factor of G . Let ρ : P −→ H be a homomor- phism, where H is a connected reductive linear algebraic group defined over k , with the property that the image ρ(P ) is not contained in any proper parabolic subgroup of H . We prove that the principal H –bundle G × P H over G/P constructed using ρ is stable with respect to any polarization on G/P . When the characteristic of k is positive, the principal H –bundle G × P H is shown to be strongly stable with respect to any polarization on G/P . Mathematics Subject Classification 2000: 14M15, 14F05. Keywords and phrases: Homogeneous space, principal bundle, Frobenius, stabil- ity. 1. Introduction Let k be an algebraically closed field. Take any connected semisimple linear algebraic group G defined over k . Let P ⊂ G be a (reduced) parabolic subgroup such that the image of P in any simple quotient of G is a proper subgroup. In other words, P does not contain any simple factor of P . The subgroup P being parabolic the quotient G/P is a smooth projective variety. Let H be a connected reductive linear algebraic group defined over the field k . Let ρ : P −→ H be an irreducible homomorphism. This means that the image ρ(P ) is not contained in any proper parabolic subgroup of H . Associated to ρ , we have a principal H –bundle over G/P which can be constructed as follows: Let G × P H be the quotient of G × H for the twisted diagonal action of P whose orbit through any point (g 0 ,h 0 ) ∈ G × H consists of all (g 0 g −1 ,ρ(g)h 0 ), g ∈ P . The composition of the projection G × H −→ G with the quotient map G −→ G/P descends to a projection from G × P H to G/P . This descended projection defines a principal H –bundle over G/P . Let E H denote this principal H –bundle over G/P . We recall that when the characteristic of k is positive, a principal bundle over a smooth polarized projective variety X defined over k , with a reductive ISSN 0949–5932 / $2.50 c  Heldermann Verlag