Maximal Covariance Group of Wigner Transforms and Pseudo-Differential\n Operators

We show that the linear symplectic and anti-symplectic transformations form\nthe maximal covariance group for both the Wigner transform and Weyl operators.\nThe proof is based on a new result from symplectic geometry which characterizes\nsymplectic and anti-symplectic matrices, and which allows us, in addition, to\nrefine a classical result on the preservation of symplectic capacities of\nellipsoids.\n