In mathematics, the Dixmier mapping describes the space Prim(U(g)) of primitive ideals of the universal enveloping algebra U(g) of a finite-dimensional solvable Lie algebra g over an algebraically closed field of characteristic 0 in terms of coadjoint orbits.
The Dixmier map is closely related to the orbit method, which relates the irreducible representations of a nilpotent Lie group to its coadjoint orbits.
Suppose that g is a completely solvable Lie algebra, and f is an element of the dual g*.
A polarization of g at f is a subspace h of maximal dimension subject to the condition that f vanishes on [h,h], that is also a subalgebra.
The Dixmier map I is defined by letting I(f) be the kernel of the twisted induced representation Ind~(f|h,g) for a polarization h.