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

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 \(G ={\rm GL}(n,{\mathbb C})\), this reduction is the Harder–Narasimhan filtration of the vector bundle associated to \(E_G\) by the standard representation of \({\rm GL}(n,{\mathbb C})\). The reduction of \(E_G\) in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and \(E_P\subset E_G\) 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 \({\mathfrak u}\) the Lie algebra of the unipotent radical of P, the second condition says that for any irreducible P-module V occurring in \({\mathfrak u}/[{\mathfrak u} , {\mathfrak u}]\), the associated vector bundle \(E_P\times_P V\) is of positive degree; here \({\mathfrak u}/[{\mathfrak u} , {\mathfrak u}]\) 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.