Let \({\mathfrak g}\) be a complex solvable Lie algebra with enveloping algebra \(U({\mathfrak g})\) and algebraic adjoint group \(G\). The (factorized) Dixmier map \(I\) is a correspondence between coadjoint \(G\)-orbits and primitive ideals in \(U({\mathfrak g})\) (all of which are completely prime). It is known to be a continuous bijection and in fact a piecewise homeomorphism [W. Borho, P. Gabriel and R. Rentschler, Primideale in Einhüllenden auflösbarer Lie-algebren, Lect. Notes Math. 357 (1973; Zbl 0293.17005)].
The present article shows that it is a homeomorphism on all of \({\mathfrak g}^*/{\mathfrak G}\); this was previously known for nilpotent \({\mathfrak g}\) [N. Conze-Berline, J. Algebra 34, 444-450 (1975; Zbl 0308.17006)].
The first step of the proof is to use standard generic flatness techniques from commutative algebra to replace \({\mathfrak g}\) by a solvable restricted Lie algebra in characteristic \(p\). Next the author introduces an “orbital correspondence”, which can be defined only in prime characteristic, to pass from one stratum of \({\mathfrak g}^*\) on which \(I\) is known to be bicontinuous to another. Finally, he appeals to a classification theorem of B. Ju. Weisfeiler and V. G. Kac [Funct. Anal. Appl. 5, 111-117 (1971); translation from Funkts. Anal. Prilozh. 5, No. 2, 28-36 (1971; Zbl 0237.17003)] to show that certain induced ideals are primitive and certain subsets of primitive ideals are dense in the Jacobson topology.
{Reviewer’s remark: The use of modular methods to prove results in characteristic 0 has been an important theme in much of the author’s recent work}.