A geometric proof of Duflo's Theorem

TL;DR

This paper provides a geometric proof of Duflo's theorem by utilizing the Beilinson-Bernstein localisation mechanism for sheaves of twisted differential operators on schemes, offering new insights into primitive ideals of Lie algebra enveloping algebras.

Contribution

It introduces a geometric approach to prove Duflo's theorem, connecting localisation techniques with primitive ideal classification.

Findings

New geometric proof of Duflo's theorem

Enhanced understanding of primitive ideals in Lie algebra theory

Application of localisation mechanisms to algebraic structures

Abstract

Let $R$ be a commutative ring: we explain the Beilinson-Bernstein localisation mechanism for sheaves of homogeneous twisted differential operators defined over a smooth, separated, locally of finite type $R$-scheme. As an application, we give a new proof of Duflo's theorem characterising the primitive ideals of the enveloping algebra of a complex semisimple Lie algebra.