Cohomogeneity-Three HyperKähler Metrics on Nilpotent Orbits

2004/06/01 by Martin Villumsen, Martin Dalgaard Villumsen, Villumsen, Martin
Mathematics · Physics and Astronomy · #17B20 #32M15 #53C25 #57S25 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Waves and Solitons #math.DG #msc:17B20 #msc:32M15 #msc:53C25 #msc:57S25

paper · pdf · doi:10.48550/arxiv.math/0406009

arxiv created 2004/06/01 · openalex publication_date 2004/06/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let O be a nilpotent orbit in g^C where G is a compact, simple group and g=Lie(G). It is known that O carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov-Kostant-Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when O is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKähler metrics with G-invariant Kähler potentials if and only if g is sp(3), su(6), so(7), so(12) or e₇ and otherwise is the unique G-invariant hyperKähler metric with G-invariant Kähler potential.

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