What question did this study set out to answer?

This work aims to show the stability of the principal H-bundle associated with a parabolic subgroup of a semisimple group.

March 29, 2026Open Access

On the Principal Bundles over a Flag Manifold

HAH. Azad IBI. Biswas

Key Points

This work aims to show the stability of the principal H-bundle associated with a parabolic subgroup of a semisimple group.
Defined a principal P-bundle using the quotient map from G to G/P.
Employed an irreducible homomorphism from P to a complex reductive group H.
Described the Harder-Narasimhan reduction process for the G-bundle over G/P.
Demonstrated that the principal H-bundle over G/P is stable.
Characterized the Harder-Narasimhan reduction of the G-bundle, employing the Levi factor L(P).

Abstract

Let P be a parabolic subgroup of a semisimple simply connected linear algebraic group G over C and an irreducible homomorphism from P to a complex reductive group H .We show that the associated principal H -bundle over G/P , associated for 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 .

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