  We consider the Weyl quantization on a flat non-standard symplectic vector
space. We focus mainly on the properties of the Wigner functions defined
therein. In particular we show that the sets of Wigner functions on distinct
symplectic spaces are different but have non-empty intersections. This extends
previous results to arbitrary dimension and arbitrary (constant) symplectic
structure. As a by-product we introduce and prove several concepts and results
on non-standard symplectic spaces which generalize those on the standard
symplectic space, namely the symplectic spectrum, Williamson's theorem and
Narcowich-Wigner spectra. We also show how Wigner functions on non-standard
symplectic spaces behave under the action of an arbitrary linear coordinate
transformation.
