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  "body": "\n## Summary\n\nThe orbit method relates the irreducible unitary representations of a Lie group $G$ to the coadjoint orbits.\nThese are the orbits of $G$ in the dual $\\mathfrak g^*$ of its Lie algebra.\nKirillov showed that for connected simply connected nilpotent Lie groups this relation is an exact bijection, built by inducing characters from polarizing subgroups.\nFor other groups it is a guiding principle rather than a theorem.\nIt works for exponential solvable groups and gives good heuristics for compact and reductive groups.\nIt ties representation theory to symplectic geometry, since each coadjoint orbit carries a natural symplectic form.\n\n## Explanations\n\n### L1 human\n\nThe dual of the Lie algebra is a vector space on which the group acts, and it breaks into orbits.\nThe orbit method says: each orbit should give one irreducible representation, and each irreducible representation should come from one orbit.\nFor the Heisenberg group, the orbits are parallel planes and single points.\nThe planes give the representations of quantum mechanics, one for each value of Planck's constant, and the points give the one-dimensional characters.\nIn this way the method is a mathematical version of passing from classical to quantum mechanics.\n",
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