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A survey on Weyl calculus for representations of nilpotent Lie groups

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-Hörmander calculus is recovered for the Schrödinger 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.