Harder-Narasimhan reduction for principal bundles over a compact Kahler manifold

Abstract

Generalizing the Harder-Narasimhan filtration of a vector bundle it is shown that a principal G-bundle 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 , this reduction is the Harder-Narasimhan filtration of the vector bundle associated to by the standard representation of . The reduction of in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and the canonical reduction, then the first condition says that the principal L-bundle obtained by extending the structure group of the P-bundle using the natural projection of P to L is semistable. Denoting by the Lie algebra of the unipotent radical of P, the second condition says that for any irreducible P-module V occurring in , the associated vector bundle is of positive degree; here is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character 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 for is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here.