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 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 $G ={\rm GL}(n,{\mathbb C})$, 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.