The complexification Sp(2 n, C) of the symplectic group is defined by the same relation, but allowing the matrices A, B, C and D to be complex. The subsemigroup of group elements that take the Siegel upper half plane into itself has a natural double cover. The representations of Mp(2 n, R) on L (R) and F n extend naturally to a representation of this semigroup by contraction operators defined by kernels, which generalise the one-dimensional case (taking determinants where necessary). The action of Mp(2 n, R) on coherent states applies equally well to operators in this larger semigroup.