Jacques Gasqui; Hubert Goldschmidt
Isospectral deformations of the Lagrangian Grassmannians
(Déformations isospectrales des grassmanniennes Lagrangiennes)
Annales de l'institut Fourier, 57 no. 7 (2007), p. 2143-2182, doi: 10.5802/aif.2329
Article PDF | Reviews MR 2394538 | Zbl 1140.44001
Class. Math.: 44A12, 53C35, 58A10, 58J53
Keywords: Symmetric space, special Lagrangian Grassmannian, reduced Lagrangian Grassmannian, Radon transform, infinitesimal isospectral deformation, symmetric form, Guillemin condition
Résumé - Abstract
We study the special Lagrangian Grassmannian $SU(n)/SO(n)$, with $n\ge 3$, 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 $\ge 2$, 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 $\mathbb{R}^6$; in fact, this space is isometric to the reduced space of $SU(4)/SO(4)$.
Bibliography
[2] J. Gasqui & H. Goldschmidt, “Infinitesimal isospectral deformations of the Grassmannian of $3$-planes in $\mathbb{R}^6$”, Mém. Soc. Math. Fr. (N.S.) (2007) no. 109 Zbl 1152.53040
[3] Victor Guillemin, Some microlocal aspects of analysis on compact symmetric spaces, Seminar on Microlocal Analysis, Princeton Univ. Press, 1979, p. 79–111 MR 560313 | Zbl 0425.58020
[4] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, 1978 MR 514561 | Zbl 0451.53038
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