The impact of Goldie’s theorem on primitive ideal theory
Seminar Room 1, Newton Institute
In 1958 Alfred Goldie published a seemingly rather abstract theorem stating that many important rings admit a calculation of fractions. This was soon realized to be a deep and fundamental result, particularly leading to a numerical invariant known as Goldie rank. Through Duflo’s theorem one may parameterize the primitive spectrum of an enveloping algebra in the semisimple case by the dual of the Cartan. Then astonishingly, Goldie rank is given through a family of polynomials. Moreover these polynomials have some remarkable properties. For example they form a basis of a multiplicity free representation of the Weyl group. One thereby obtains a quite unattended connection with the Springer theory relating Weyl group representations to the geometry of nilpotent orbits.
The polynomials that define Goldie rank are determined up to a scale factor by an explicit formula involving the Kazhdan-Lusztig polynomials and were even a motivation for the precise definition of the latter. These scale factors can be largely determined by finding the locus of Goldie rank one, a problem which has remained open for some thirty years.
Another related question is to describe the Goldie rank one sheets (of which there are just finitely many, by virtue of a positivity property of Goldie rank polynomials coming from geometry) and in particular to determine their topology. Combined with the Gelfand-Kirillov conjecture, primitive quotients of enveloping algebras are described (in principle) as matrix rings over differential operators linked to symplectic structure, exactly like Dirac’s relativistic quantum mechanical equation. Thus primitive ideal theory is intimately related to Quantization. In this lecture we review the main results and open problems of the theory.