Mini-course on Joseph's proof of the Annihilation Theorem

In August-September 2025, we held a minicourse on Joseph's proof of the Annihilation Theorem.

Dates:  

Aug. 18, 21 

Sept. 14, 19, 21, 26

Here are the notes.

Prof. Anthony Joseph (photo - courtesy of Weizmann Institute of Science)

Title: Joseph's proof of Annihilation Theorem

Abstract: The aim of this mini-course is to discuss Joseph’s proof of the following theorem: the annihilator of Verma module in the enveloping algebra of a semisimple Lie algebra is generated by its intersection with  the centre of the enveloping algebra. This result leads to Bernstein-Gelfand equivalence of categories and Duflo theorem: primitive spectrum of enveloping algebra consists of the annihilators of simple modules of highest weight. The classical proof of this theorem contains some complicated algebro-geometric arguments, so the generalization to Kac-Moody case and quantum case required another approach.

Joseph’s approach is based on the fact that PRV determinants and Shapovalov’s form have the same set of linear factors.

Joseph's proof is presented in the following papers:

• A. Joseph, Sur l'annulateur d'un module de Verma, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514 (1998).

• A. Joseph, G. Letzter, Verma modules annihilators and quantized enveloping algebras, Ann. Ec. Norm. Sup. (1995).