  The quantum cat map is a model for a quantum system with underlying chaotic
dynamics. In this paper we study the matrix elements of smooth observables in
this model, when taking arithmetic symmetries into account. We give explicit
formulas for the matrix elements as certain exponential sums. With these
formulas we can show that there are sequences of eigenfunctions for which the
matrix elements decay significantly slower then was previously conjectured. We
also prove a limiting distribution for the fluctuation of the normalized matrix
elements around their average.
