Isospectral deformations of the Lagrangian Grassmannians

We study the special Lagrangian Grassmannian SU(n)/SO(n), with n >= 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 >= 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 R-6; in fact, this space is isometric to the reduced space of SU(4)/SO(4).