Kirillov orbit method

Summary

The orbit method relates the irreducible unitary representations of a Lie group $G$ to the coadjoint orbits. These are the orbits of $G$ in the dual $\mathfrak g^*$ of its Lie algebra. Kirillov showed that for connected simply connected nilpotent Lie groups this relation is an exact bijection, built by inducing characters from polarizing subgroups. For other groups it is a guiding principle rather than a theorem. It works for exponential solvable groups and gives good heuristics for compact and reductive groups. It ties representation theory to symplectic geometry, since each coadjoint orbit carries a natural symplectic form.