  A SU(2) intertwiner with N legs can be interpreted as the quantum state of a
convex polyhedron with N faces (when working in 3d). We show that the
intertwiner Hilbert space carries a representation of the non-compact group
SO*(2N). This group can be viewed as the subgroup of the symplectic group
Sp(4N,R) which preserves the SU(2) invariance. We construct the associated
Perelomov coherent states and discuss the notion of semi-classical limit, which
is more subtle that we could expect. Our work completes the work by Freidel and
Livine which focused on the U(N) subgroup of SO*(2N).
