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.

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U2 - 10.1007/s002080200322

DO - 10.1007/s002080200322

M3 - Article

AN - SCOPUS:0036983016

VL - 323

SP - 693

EP - 712

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 4

ER -