Coherent state realizations of su(n+1) on the n-torus

We obtain a new family of coherent state representations of SU(n+1), in which the coherent states are Wigner functions over a subgroup of SU(n+1). For representations of SU(n+1) of the type (λ, 0, 0,...), the basis functions are simple products of n exponential. The corresponding coherent state representations of the algebra su(n+1) are also obtained, and provide a polar decomposition of su(n+1) for any n+1. The su(n+1) modules thus obtained are useful in understanding contractions of su(n+1) and su(n+1)-phase states of quantum optics.