### 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