In the sixties the very ambitious Dixmier-Gabriel programme of
classifying primitive ideals was presented and soon became of
consuming interest. Duflo made the first breakthrough on the
difficult case pertaining to a reductive Lie algebra.
In 1980-1988 at Weizmann, a numerical invariant, the Goldie rank, assigned to each of infinitely many primitive ideals, was found quite amazingly to be given by a finite family of polynomials which we called Goldie rank polynomials. This gave a classification of the primitive spectrum as a set.
Determining the scale factors in the Goldie rank polynomials, a problem still of active current interest, involves constructing primitive ideals of Goldie rank one. An early example of the latter was the Joseph ideal. Further in conjunction with a researcher from Wuppertal, the topology of the primitive spectrum was studied through the notion of a sheet borrowed from geometry. A significant result was that (in the semisimple case) there are only finitely-many Goldie rank one sheets. These have yet to be fully classified.
Goldie rank polynomials are a representational theory analogue of the famous Springer representation of the Weyl group constructed through etale cohomology. A link between these two theories came through Joseph polynomials. In much greater generality these give cohomology classes for algebraic varieties with a torus action.
WIS work on Goldie rank polynomials was also one of the origins of Kazhdan-Lusztig theory, the latter having now had far-reaching implications in representation theory.
The classification of primitive ideals for quantum groups was
settled at Weizmann in 1993.
Further a quantum analogue of the all-important Beilinson-Bernstein equivalence of categories was given. These involved algebras, generally distinct, but all specializing to the same (polynomial) algebra.
In 1998 their primitive spectrum was also determined at Weizmann involving a hitherto unused aspect of Bruhat order on the Weyl group. The fact that these spectra were rather different was a striking example of a purely quantum phenomenon.