Metaplectic operators on {$C^n$}

The metaplectic representation describes a class of automorphism of the Heisenberg group H = H(G), defined for a locally compact abelian group G. For $G = R^d$, H is the usual Heisenberg group. For the case when G is the finite cyclic group $Z^n$, only partial constructions are known. Here we present new results for this case and we obtain an explicit construction of the metaplectic operators on $C^n$. We also include applications to Gabor frames.