Isospectral deformations of the Lagrangian Grassmannians
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2143-2182.

We study the special Lagrangian Grassmannian SU(n)/SO(n), with n3, and its reduced space, the reduced Lagrangian Grassmannian X. The latter is an irreducible symmetric space of rank n-1 and is the quotient of the Grassmannian SU(n)/SO(n) under the action of a cyclic group of isometries of order n. The main result of this paper asserts that the symmetric space X possesses non-trivial infinitesimal isospectral deformations. Thus we obtain the first example of an irreducible symmetric space of arbitrary rank 2, which is both reduced and non-infinitesimally rigid. Our result may be viewed as a generalization of the construction which we had given previously for the reduced Grassmannian of 3-planes in 6 ; in fact, this space is isometric to the reduced space of SU(4)/SO(4).

Nous étudions la grassmannienne lagrangienne spéciale SU(n)/ SO(n), avec n3, et son espace réduit X, qui est l’espace symétrique irréductible de rang n-1 quotient de SU(n)/SO(n) par l’action d’un groupe cyclique d’isometries d’ordre n. Notre résultat principal est la construction de déformations infinitésimales isospectrales non triviales de X. Nous obtenons ainsi les premiers exemples en rang quelconque 2 d’espaces symétriques irréductibles réduits et non infinitésimalement rigides. Notre résultat peut être vu comme une généralisation de la construction que nous avions donnée dans un précédent papier pour la grassmannienne réduite des 3-plans de 6 , espace qui est en fait isométrique à l’espace réduit de SU(4)/SO(4).

DOI: 10.5802/aif.2329
Classification: 44A12,  53C35,  58A10,  58J53
Keywords: Symmetric space, special Lagrangian Grassmannian, reduced Lagrangian Grassmannian, Radon transform, infinitesimal isospectral deformation, symmetric form, Guillemin condition
Gasqui, Jacques 1; Goldschmidt, Hubert 2

1 Université Joseph Fourier Institut Fourier 100 rue des Maths BP 74 38402 Saint-Martin d’Hères (France)
2 Columbia University Department of Mathematics MC 4406 2990 Broadway New York, NY 10027 (USA)
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Gasqui, Jacques; Goldschmidt, Hubert. Isospectral deformations of the Lagrangian Grassmannians. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2143-2182. doi : 10.5802/aif.2329. https://aif.centre-mersenne.org/articles/10.5802/aif.2329/

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[2] Gasqui, J.; Goldschmidt, H. Infinitesimal isospectral deformations of the Grassmannian of 3-planes in  6 , Mém. Soc. Math. Fr. (N.S.) (2007) no. 109, pp. vi+92 | Numdam | Zbl

[3] Guillemin, Victor Some microlocal aspects of analysis on compact symmetric spaces, Seminar on Microlocal Analysis (Ann. of Math. Stud.), Volume 93, Princeton Univ. Press, Princeton, N.J., 1979, pp. 79-111 | MR | Zbl

[4] Helgason, S. Differential geometry, Lie groups, and symmetric spaces, Academic Press, Orlando, FL, 1978 | MR | Zbl

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