### Abstract

Generalizing the Harder-Narasimhan filtration of a vector bundle it is shown that a principal G-bundle E_{G} over a compact Kähler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where G = GL(n, ℂ), this reduction is the Harder-Narasimhan filtration of the vector bundle associated to E_{G} by the standard representation of GL(n, ℂ). The reduction of E_{G} in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and E_{P} ⊂ E_{G} the canonical reduction, then the first condition says that the principal Lbundle obtained by extending the structure group of the P-bundle E_{P} using the natural projection of P to L is semistable. Denoting by u the Lie algebra of the uhipotent radical of P, the second condition says that for any irreducible P-module V occurring in u/[u, u], the associated vector bundle E_{P × P} V is of positive degree; here u/[u, u] is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character χ of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to E_{P} for χ is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here.

Original language | English |
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Pages (from-to) | 693-712 |

Number of pages | 20 |

Journal | Mathematische Annalen |

Volume | 323 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2002 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Annalen*,

*323*(4), 693-712. https://doi.org/10.1007/s002080200322

**Harder-Narasimhan reduction for principal bundles over a compact Kähler manifold : Harder-Narasimhan reduction.** / Anchouche, Boudjemaa; Azad, Hassan; Biswas, Indranil.

Research output: Contribution to journal › Article

*Mathematische Annalen*, vol. 323, no. 4, pp. 693-712. https://doi.org/10.1007/s002080200322

}

TY - JOUR

T1 - Harder-Narasimhan reduction for principal bundles over a compact Kähler manifold

T2 - Harder-Narasimhan reduction

AU - Anchouche, Boudjemaa

AU - Azad, Hassan

AU - Biswas, Indranil

PY - 2002

Y1 - 2002

N2 - Generalizing the Harder-Narasimhan filtration of a vector bundle it is shown that a principal G-bundle EG over a compact Kähler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where G = GL(n, ℂ), this reduction is the Harder-Narasimhan filtration of the vector bundle associated to EG by the standard representation of GL(n, ℂ). The reduction of EG in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and EP ⊂ EG the canonical reduction, then the first condition says that the principal Lbundle obtained by extending the structure group of the P-bundle EP using the natural projection of P to L is semistable. Denoting by u the Lie algebra of the uhipotent radical of P, the second condition says that for any irreducible P-module V occurring in u/[u, u], the associated vector bundle EP × P V is of positive degree; here u/[u, u] is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character χ of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to EP for χ is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here.

AB - Generalizing the Harder-Narasimhan filtration of a vector bundle it is shown that a principal G-bundle EG over a compact Kähler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where G = GL(n, ℂ), this reduction is the Harder-Narasimhan filtration of the vector bundle associated to EG by the standard representation of GL(n, ℂ). The reduction of EG in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and EP ⊂ EG the canonical reduction, then the first condition says that the principal Lbundle obtained by extending the structure group of the P-bundle EP using the natural projection of P to L is semistable. Denoting by u the Lie algebra of the uhipotent radical of P, the second condition says that for any irreducible P-module V occurring in u/[u, u], the associated vector bundle EP × P V is of positive degree; here u/[u, u] is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character χ of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to EP for χ is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here.

UR - http://www.scopus.com/inward/record.url?scp=0036983016&partnerID=8YFLogxK

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U2 - 10.1007/s002080200322

DO - 10.1007/s002080200322

M3 - Article

VL - 323

SP - 693

EP - 712

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 4

ER -