Infinitesimal isospectral deformations of symmetric spaces: quotients of the special unitary group (Q612272)

The authors pursue their study of the infinitesimal deformations of the symmetric spaces of compact type, that are both irreducible and reduced. Motivated by a result of \textit{V. Guillemin} [Ann. Math. Stud. 93, 79--111 (1979; Zbl 0425.58020)], the authors introduce the space \(I(X)\) of infinitesimal isospectral deformations of a Riemannian symmetric space \((X; g)\) of compact type. The authors show that the reduced spaces of the unitary group \(\mathrm{SU} (n)\) and the symmetric space \(\mathrm{SU} (2n)=\mathrm{Sp}(n)\) with \(n\geq 3\) possesses non-trivial infinitesimal isospectral deformations. For the reduced space \(X\) of the unitary group \(\mathrm{SU} (n)\); they also prove a related result: in all degrees \(p\geq 2\), there exist symmetric \(p\)-forms on \(X\) which satisfy the Guillemin condition and are not symmetrized covariant derivatives of symmetric \((p - 1)\)-forms, unless \(n = p = 3\). Here, one says that a symmetric \(p\)-form \(u\) on a symmetric space \((X, g)\) satisfies the Guillemin condition if, for every maximal flat totally geodesic torus \(Z\) contained in \(X\) and for all parallel vector \(\zeta\) on \(Z\), the integral \[ \int_{Z} u(\zeta,\dots,\zeta) dZ \] vanishes, where \(dZ\) is the Riemannian measure of \(Z\).