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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 [4] Helgason, S., Differential geometry, Lie groups, and symmetric spaces, (1978), Academic Press, Orlando, FL · Zbl 0451.53038
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