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. More precisely, it is a homeomorphism from the space of orbits g*/G of the dual g* of g (with the Zariski topology) under the action of the adjoint group G to Prim(U(g)) (with the Jacobson topology). The Dixmier map is closely related to the orbit method, which relates the irreducible representations of a nilpotent Lie group to its coadjoint orbits. Dixmier ( 1963 ) introduced the Dixmier map for nilpotent Lie algebras and then in (Dixmier 1966 ) extended it to solvable ones. Dixmier (1996 , chapter 6) describes the Dixmier mapping in detail.
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.