On the Orbit Method and the Homomorphism of Harish-Chandra

Let be a complex semisimple Lie algebra with adjoint group G. Let U( ) be the enveloping algebra of . Following an idea of J. Dixmier, (Sur la méthode des orbites, in “Proceedings de la conférence: Non commutative Harmonic analysis, Marseille–Luminy, 1978,” Lecture Notes in Mathematics, Vol. 728, Springer-Verlag, New York/Berlin) we construct a bijection from the set of regular coadjoint orbits onto the set of minimal primitive ideals in U( ), without using the notion of “polarization” (see J. Dixmier, Algèbres Enveloppantes, Gauthier-Villars, Paris, 1974). This is a partial answer to the problem posed by J. Dixmier in the former reference.