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A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group. (English) Zbl 0721.22006
Let G be a compact Lie group. The author studies the orbits of the complexified group \(G^ c\) in the dual space of its own Lie algebra. His main result is that in the case of regular semisimple orbits (i.e. those of the form \(C^ c/T^ c\), where \(T^ c\) is a maximal complex torus), these manifolds have a quaternionic, or hyper-Kählerian structure (i.e. they possess 3 anti-commuting complex structures I, J, K and a Riemannian metric g which is Kähler with respect to all three).

MSC:
22E46 Semisimple Lie groups and their representations
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
22E60 Lie algebras of Lie groups
81T13 Yang-Mills and other gauge theories in quantum field theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B99 Lie algebras and Lie superalgebras
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