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

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.