Analysis of the affine transformations of the time-frequency plane

We consider two aspects of the action of the extended metaplectic representation of the group G of affine, measure and orientation preserving maps of the time-frequency plane on L2 functions on the line. On the one hand, we list, up to equivalence, all possible reproducing formulas that arise by restricting the representation to connected Lie subgroups of G. On the other hand, we describe, in terms of Weyl calculus, the commutative von Neumann algebras generated by restriction to one-parameter subgroups.