#### Milestone Year

### 1988

#### Classification of Primitive Ideals

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.