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Isospectral deformations of the Lagrangian Grassmannians. (English) Zbl 1140.44001
Let \(G=SU(n)\) and \(K=SO(n)\). We denote by \(K_S\) the subgroup of \(G\) defined by \(K_S=\{e^{i\pi k/n}B\mid k\in{\mathbb Z}\), \(B\in \text{SO}(n)\}\). Then \(Y=G/K_S\) is a quotient of \(X=G/K\) and is an irreducible symmetric space of rank \(n-1\), which is called the reduced Lagrangian Grassmannian. Let \(I(X)\) denote the space of infinitesimal isospectral deformation of \(X\), consisting of all symmetric 2-forms \(h\) satisfying the Guillemin condition and \(\operatorname{div}\;h=0\).
The authors obtain explicitly a subspace of \(I(X)\) isomorphic to the infinite-dimensional space of real valued functions on \(X\) orthogonal to a finite dimensional space. Then, by using an induced relation between \(I(X)\) and \(I(Y)\), they prove that \(I(Y)\) does not vanish, or \(Y\) is not rigid in the sense of Guillemin. This is the first example of an irreducible symmetric space of arbitrary rank \(\geq2\), which is reduced and non rigid.

MSC:
44A12 Radon transform
53C35 Differential geometry of symmetric spaces
58A10 Differential forms in global analysis
58J53 Isospectrality
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References:
[1] Gasqui, J.; Goldschmidt, H., Radon transforms and the rigidity of the Grassmannians, 156, (2004), Princeton University Press, Princeton, NJ, Oxford · Zbl 1051.44003
[2] Gasqui, J.; Goldschmidt, H., Infinitesimal isospectral deformations of the Grassmannian of \(3\)-planes in \(\mathbb{R}^6,\) Mém. Soc. Math. Fr. (N.S.), 109, vi+92 pp., (2007) · Zbl 1152.53040
[3] Guillemin, Victor, Seminar on Microlocal Analysis, 93, Some microlocal aspects of analysis on compact symmetric spaces, 79-111, (1979), Princeton Univ. Press, Princeton, N.J. · Zbl 0425.58020
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