Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits
Annales de l'Institut Fourier, Volume 59 (2009) no. 1, p. 167-197

In this paper, we construct a hyperkähler structure on the complexification 𝒪 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 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.

Dans cet article, nous construisons une métrique hyperkählerienne sur l’orbite complexifiée 𝒪 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 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.

DOI : https://doi.org/10.5802/aif.2428
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
@article{AIF_2009__59_1_167_0,
     author = {Tumpach, Alice Barbara},
     title = {Infinite-dimensional hyperk\"ahler manifolds associated with Hermitian-symmetric affine coadjoint orbits},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     pages = {167-197},
     doi = {10.5802/aif.2428},
     mrnumber = {2514863},
     zbl = {1170.58002},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_1_167_0}
}
Tumpach, Alice Barbara. Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 167-197. doi : 10.5802/aif.2428. http://www.numdam.org/item/AIF_2009__59_1_167_0/

[1] Andruchow, E.; Larotonda, G. Hopf-Rinow Theorem in the Sato Grassmannian, to appear in J. Funct. Analysis | MR 2442079 | Zbl 1160.22010 | Zbl pre05365129

[2] Andruchow, E.; Larotonda, G. Nonpositively curved metric in the positive cone of a finite von Neumann algebra, preprint | MR 2254561 | Zbl 1098.53033

[3] Arvanitoyeorgos, A. An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, American Math. Society, Providence, R.I., Student Math. Library no. 22 (2003) | MR 2011126 | Zbl 1045.53001

[4] Balachandran, V. K. Simple L * -algebras of classical type, Math. Ann., Tome 180 (1969), pp. 205-219 | Article | MR 243362 | Zbl 0159.42203

[5] Beltiţă, D.; Ratiu, T.; Tumpach, A. B. The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits, J. Funct. Anal., Tome 247 (2007), pp. 138-168 | Article | MR 2319757 | Zbl 1120.22007

[6] Besse, A. L. Einstein manifolds, Springer, Folge 3, Ergebnisse der Mathematik und ihrere Grenzgebiete, Tome 10 (1987) | MR 867684 | Zbl 0613.53001

[7] Biquard, O. Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes, Math. Ann., Tome 304 (1996) no. 2, pp. 253-276 | Article | MR 1371766 | Zbl 0843.53027

[8] Biquard, O.; Gauduchon, P. Hyperkähler metrics on cotangent bundles of Hermitian Symmetric spaces, Geometry and Physics, Lect. notes Pure Appl. Math. Serie 184, Marcel Dekker (1996), pp. 287-298 | MR 1423175 | Zbl 0879.53051

[9] Biquard, O.; Gauduchon, P. La métrique hyperkählérienne des orbites coadjointes de type symétrique d’un groupe de Lie complexe semi-simple, C. R. Acad. Sci. Paris, série I, Tome 323 (1996), pp. 1259-1264 | MR 1428547 | Zbl 0866.58007

[10] Biquard, O.; Gauduchon, P. Géométrie hyperkählérienne des espaces hermitiens symétriques complexifiés, Séminaire de théorie spectrale et géométrie, Grenoble, Tome 16 (1998), pp. 127-173 | Numdam | MR 1666451 | Zbl 0943.53029

[11] Cheeger, J.; Ebin, D. G. Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam (1975) | MR 458335 | Zbl 0309.53035

[12] Ekeland, I. The Hopf-Rinow Theorem in infinite dimension, J. Differential Geometry, Tome 13 (1978), pp. 287-301 | MR 540948 | Zbl 0393.58004

[13] De La Harpe, P. Classification des L * -algèbres semi-simples réelles séparables, C.R. Acad. Sci. Paris, Ser. A, Tome 272 (1971), pp. 1559-1561 | MR 282218 | Zbl 0215.48501

[14] Helgason, S. Differential Geometry and Symmetric Spaces, Academic Press, New York (1962) | MR 145455 | Zbl 0111.18101

[15] Hitchin, N. J. The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Tome 55 (1987) no. 1, pp. 59-126 | Article | MR 887284 | Zbl 0634.53045

[16] Kaup, W. Algebraic characterization of symmetric complex Banach manifolds, Math. Ann., Tome 228 (1977), pp. 39-64 | Article | MR 454091 | Zbl 0335.58005

[17] Kaup, W. Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension I, II, Math. Ann., Tome 257 (1981), pp. 463-486 (262, (1983), p. 57–75.) | Article | MR 639580 | Zbl 0482.32010

[18] Kovalev, A. G. Nahm’s equation and complex adjoint orbits, Quart. J. Math., Tome 47 (1993), pp. 41-58 | Article | Zbl 0852.53033

[19] Kronheimer, P. B. A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group, J. London Math. Soc. (2), Tome 42 (1990), pp. 193-208 | Article | MR 1083440 | Zbl 0721.22006

[20] Kronheimer, P. B. Instantons and the geometry of the nilpotent variety, J. Differential Geometry, Tome 32 (1990), pp. 473-490 | MR 1072915 | Zbl 0725.58007

[21] Larotonda, G. Geodesic Convexity, Symmetric Spaces and Hilbert-Schmidt Operators, Buenos Aires, Argentina, Universidad Nacional de General Sarmiento (2005) (Ph. D. Thesis)

[22] Mostow, G. D. Some new decomposition theorems for semi-simple groups, Mem. Amer. Math. Soc. (1955) no. 14, pp. 31-54 | MR 69829 | Zbl 0064.25901

[23] Neeb, K. -H. A Cartan-Hadamard theorem for Banach-Finsler manifolds, Geom. Dedicata, Tome 95 (2002), pp. 115-156 | Article | MR 1950888 | Zbl 1027.58003

[24] Neeb, K. -H. Highest weight representations and infinite-dimensional Kähler manifolds, Recent advanceds in Lie theory (Vigo, 2000), Tome 25 (2002), pp. 367-392 (Res. Exp. Math., Heldermann, Lemgo) | MR 1937991 | Zbl 1020.22008

[25] Neeb, K. -H. Infinite-dimensional groups and their representations, Lie theory, Progr. Math., Tome 228 (2004), pp. 213-328 (Birkhäuser Boston, Boston, MA) | MR 2042690 | Zbl 1076.22016

[26] O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York (1983) | MR 719023 | Zbl 0531.53051

[27] Pressley, A.; Segal, G. Loop Groups, Clarendon Press, Oxford Mathematical Monographs. Oxford (UK) Tome viii (1988) | MR 900587 | Zbl 0638.22009

[28] Schue, J. R. Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc., Tome 95 (1960), pp. 69-80 | Article | MR 117575 | Zbl 0093.30601

[29] Schue, J. R. Cartan decompositions for L * -algebras, Trans. Amer. Math. Soc., Tome 98 (1961), pp. 334-349 | MR 133408 | Zbl 0099.10205

[30] Tumpach, A. B. On the classification of affine Hermitian-symmetric coadjoint orbits of L * -groups, to appear in Forum Mathematicum

[31] Tumpach, A. B. Variétés kählériennes et hyperkählériennes de dimension infinie, École Polytechnique, Palaiseau, France (2005) (Ph. D. Thesis)

[32] Tumpach, A. B. Mostow Decomposition Theorem for a L * -group and Applications to affine coadjoint orbits and stable manifolds, preprint arXiv:math-ph/0605039 (2006)

[33] Tumpach, A. B. Hyperkähler structures and infinite-dimensional Grassmannians, J. Funct. Anal., Tome 243 (2007), pp. 158-206 | Article | MR 2291435 | Zbl 1124.58004

[34] Unsain, I. Classification of the simple real separable L * -algebras, J. Diff. Geom., Tome 7 (1972), pp. 423-451 | MR 325721 | Zbl 0279.46044

[35] Wolf, J. A. On the classification of Hermitian Symmetric Spaces, J. Math. Mech., Tome 13 (1964), pp. 489-496 | MR 160850 | Zbl 0245.32011

[36] Wolf, J. A. Fine structure of Hermitian Symmetric Spaces, Symmetric Spaces, short Courses presented at Washington Univ., pure appl. Math., Tome 8 (1972), pp. 271-357 | MR 404716 | Zbl 0257.32014

[37] Wolf, J. A. Spaces of Constant Curvature, Department of Mathematics, University of California, Berkeley, Calif., Second edition (1972) | MR 343213 | Zbl 0281.53034

[38] Wurzbacher, T. Fermionic Second Quantization and the Geometry of the Restricted Grassmannian, Birkhäuser Verlag, Basel, in Infinite-Dimensional Kähler Manifolds, DMV Seminar, Band, Tome 31 (2001) | MR 1853244 | Zbl 1058.53064