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Nahm’s equations and complex adjoint orbits. (English) Zbl 0852.53033
The author obtains a generalization of the results of Kronheimer about the existence of hyper-Kähler structures on some adjoint orbits of a complex semisimple Lie group. He proves the following Theorem. (1) Let \(G/H\) be an adjoint orbit of a compact semisimple Lie group \(G\) and \(G^c\), \(H^c\) the complexifications of \(G\), \(H\). Then the manifold \(G^c/H^c\) has a family of complete hyper-Kähler metrics. It is parametrised by triples \((a_1,a_2, a_3)\) consisting of elements from a Cartan subalgebra \(t\) of \(h = \text{Lie }H\) with stabilizer \(H\). The complex manifold \(G^c/H^c\) is isomorphic to the adjoint orbit \(\text{Ad }G^c(a_2 + ia_3)\) if \(H^c\) is the stabilizer of \(a_2 + ai_3\). (2) Let \(b^c \in g^c\) be a nilpotent element which commutes with \(a_2 + ia_3\). Then the complex adjoint orbit \(\text{Ad }G^c(a_2 + ia_3 + b^c)\) has a family of hyper-Kähler metrics parametrised by elements \(a_1 \in t\) such that the centralizers of the pair \((a_2,a_3)\) and the triple \((a_1, a_2, a_3)\) coincide.

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
22E46 Semisimple Lie groups and their representations
22E60 Lie algebras of Lie groups
81T13 Yang-Mills and other gauge theories in quantum field theory
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