Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits

Tumpach, Alice Barbara

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 167-197.

Dans cet article, nous construisons une métrique hyperkählerienne sur l’orbite complexifiée ${𝒪}^{ℂ}$ de toute orbite coadjointe affine hermitienne symétrique $𝒪$ d’un $L^{*}$-groupe semi-simple de type compact, qui est compatible avec la forme symplectique complexe de Kirillov-Kostant-Souriau et qui se restreint en la structure kählérienne de $𝒪$. Grâce à une identification pertinente de l’orbite complexifiée ${𝒪}^{ℂ}$ avec l’espace cotangent $T𝒪$ de l’orbite de type compact $𝒪$ induite par le théorème de décomposition de Mostow, nous en déduisons l’existence d’une structure hyperkählérienne sur $T𝒪$ compatible avec la forme symplectique complexe de Liouville et dont la restriction à la section nulle est la structure kählérienne de $𝒪$. Des formules explicites de la métriques en termes de l’orbite complexifiée et de l’espace cotangent sont données. Comme cas particulier, nous retrouvons la famille à un paramètre de structures hyperkählériennes sur une complexification naturelle de la grassmannienne restreinte et sur l’espace cotangent de la grassmannienne restreinte précédemment obtenue par l’auteur via une réduction hyperkählérienne.

In this paper, we construct a hyperkähler structure on the complexification ${𝒪}^{ℂ}$ of any Hermitian symmetric affine coadjoint orbit $𝒪$ of a semi-simple $L^{*}$-group of compact type, which is compatible with the complex symplectic form of Kirillov-Kostant-Souriau and restricts to the Kähler structure of $𝒪$. By a relevant identification of the complex orbit ${𝒪}^{ℂ}$ with the cotangent space $T𝒪$ of $𝒪$ induced by Mostow’s decomposition theorem, this leads to the existence of a hyperkähler structure on $T𝒪$ compatible with Liouville’s complex symplectic form and whose restriction to the zero section is the Kähler structure of $𝒪$. Explicit formulas of the metric in terms of the complex orbit and of the cotangent space are given. As a particular case, we obtain the one-parameter family of hyperkähler structures on a natural complexification of the restricted Grassmannian and on the cotangent space of the restricted Grassmannian previously constructed by the author via a hyperkähler reduction.

Classification : 17B65, 22E65, 58B25, 81R10, 46T05, 53D05
Keywords: Infinite-dimensional hyperkähler manifolds, affine coadjoint orbit, Hermitian-symmetric spaces, hyperkähler reduction, cotangent space, strongly orthogonal roots, $L^*$-algebra, restricted Grassmannian
Mot clés : variétés hyperkählériennes de dimension infinie, orbites coadjointes affines, espaces hermitien-symétriques, réduction hyperkählérienne, espace cotangent, racines fortement orthogonales, $L^*$-algèbres, Grassmannienne restreinte

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