Abstract
We study singular hyperkähler quotients of the cotangent bundle of a complex semisimple Lie group as stratified spaces whose strata are hyperkähler. We focus on one particular case where the stratification satisfies the frontier condition and the partial order on the set of strata can be described explicitly by Lie theoretic data.
Highlights
Let G be a complex reductive group, or equivalently, the complexification KC of a compact connected Lie group K
Kronheimer [K] showed that the cotangent bundle T ∗G can be endowed with a hyperkahler structure
The action of K × K on G by left and right multiplications lifts to an action on T ∗G which preserves the hyperkahler structure and has a hyperkahler moment map, i.e., a moment map for each of the three Kahler forms [DS1, Lem. 2]
Summary
Let G be a complex reductive group, or equivalently, the complexification KC of a compact connected Lie group K. We will describe a coarser stratification which keeps the property that the strata are disjoint unions of copies of spaces D(gΨ)top In this case, the set of strata is in bijection with the set of root subsystems modulo the Weyl group action, or equivalently, the set of conjugacy classes of regular semisimple subalgebras of g. The set of strata is in bijection with the set of root subsystems modulo the Weyl group action, or equivalently, the set of conjugacy classes of regular semisimple subalgebras of g To obtain these results, we will first prove a Kempf–Ness type theorem for the hyperkahler quotient of T ∗G by any closed subgroup H of K × K.
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