Infinitesimal isospectral deformations of symmetric spaces of compact type
Tuesday May 6, 2008
02:30PM - 03:30PM
Speakers:
Hubert Goldschmidt
Abstract:
Infinitesimal isospectral deformations of symmetric spaces of
compact type
Let (X, g) be an irreducible symmetric space of compact type. According
to a result of Guillemin, the infinitesimal deformation corresponding to an
isospectral deformation of the metric g belongs to the kernel of a certain
Radon transform acting on the symmetric 2-forms on X. This is the motivation
for defining the space I(X) of infinitesimal isospectral deformations
of X as a subspace of the kernel of this Radon transform. If I(X) vanishes,
an isospectral deformation of the metric g is trivial to first-order.
We shall give an overview of our joint work with Jacques Gasqui concerning
the space I(X):
1) A necessary condition for the vanishing of I(X) is that it be reduced,
i.e., it is not the cover of another symmetric space.
2) Let K be a division algebra over R. If X is the reduced space of the
Grassmannian of m-planes in Km+n, with m 6= n, the space I(X) vanishes.
This was known when X is a projective space which is not equal a sphere;
using work of Duistermaat–Guillemin, this leads to spectral rigidity results
for the projective space.
3) If X is the reduced space of the special Lagrangian Grassmannian
SU(n)/SO(n), or of the unitary group SU(n), or of the symmetric space
SU(2n)/Sp(n), with n 3, the space I(X) does not vanish and we give
explicit constructions of non-trivial infinitesimal deformations.
4) We determine the space I(X) when X is the reduced space of the
symmetric space SU(3)/SO(3) or of the unitary group SU(3).
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