A survey on Weyl calculus for representations of nilpotent Lie groups

Abstract

We survey some aspects of the pseudo-differential Weyl calculus for irreducible unitary representations of nilpotent Lie groups, ranging from the classical ideas to recently obtained results. The classical Weyl-H\"ormander calculus is recovered for the Schr\"odinger representation of the Heisenberg group. Our discussion concerns various extensions of this classical situation to arbitrary nilpotent Lie groups and to some infinite-dimensional Lie groups that allow us to handle the magnetic pseudo-differential calculus.