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Abstract : |
Generalizing the Harder-Narasimhan filtration of a vector bundle it is shown that a principal G-bundle E-G over a compact Kahler 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 subset of E-G the canonical reduction, then the first condition says that the principal L-bundle 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 unipotent 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 x (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 chi 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 chi is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here., |
