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 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 × 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. (2000): 32L05, 53C05
Mathematics Subject Classification
The symmetry classification problem for wave equation on sphere is considered. Symmetry algebra is found and a classification of its subalgebras, up to conjugacy, is obtained. Similarity reductions are performed for each class, and some examples of exact invariant solutions are given.
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