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.

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