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Zbl 1140.44001
Gasqui, Jacques; Goldschmidt, Hubert
Isospectral deformations of the Lagrangian Grassmannians. (English)
[J] Ann. Inst. Fourier 57, No. 7, 2143-2182 (2007). ISSN 0373-0956; ISSN 1777-5310/e

Let $G=SU(n)$ and $K=SO(n)$. We denote by $K_S$ the subgroup of $G$ defined by $K_S=\{e^{i\pi k/n}B\mid k\in{\Bbb Z}$, $B\in \text{SO}(n)\}$. 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.
[Takeshi Kawazoe (Yokohama)]

MSC 2000:: *44A12 Radon transform
53C35 Symmetric spaces (differential geometry)
58A10 Differential forms
58J53 Isospectrality

Keywords: symmetric space; special Lagrangian Grassmannian; reduced Lagrangian Grassmannian; isospectral deformation; Guillemin condition

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