On Weyl calculus in infinitely many variables

We outline an abstract approach to the pseudo-differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of representations associated with infinite-dimensional coadjoint orbits. We illustrate the approach by the case of infinite-dimensional Heisenberg groups. The classical Weyl-H\"ormander calculus is recovered for the Schr\"odinger representations of the finite-dimensional Heisenberg groups.