A survey on Weyl calculus for representations of nilpotent Lie groups

TL;DR

This survey reviews Weyl calculus for nilpotent Lie group representations, covering classical and recent developments, including extensions to infinite-dimensional groups and magnetic pseudo-differential calculus.

Contribution

It provides a comprehensive overview of Weyl calculus extensions for nilpotent Lie groups, connecting classical and modern approaches, and exploring infinite-dimensional cases.

Findings

Classical Weyl-H"ormander calculus is recovered for the Schr"odinger representation of the Heisenberg group.

Extensions to arbitrary nilpotent Lie groups are discussed.

Inclusion of infinite-dimensional Lie groups and magnetic pseudo-differential calculus.

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.