  We look at the expectation values for quantized linear symplectic maps on the
multidimensional torus and their distribution in the semiclassical limit. We
construct super-scars that are stable under the arithmetic symmetries of the
system and localize around invariant manifolds. We show that these super-scars
exist only when there are isotropic rational subspaces, invariant under the
linear map. In the case where there are no such scars, we compute the variance
of the fluctuations of the matrix elements for the desymmetrized system, and
present a conjecture for their limiting distributions.
