A surjectivity theorem for rigid highest weight modules
Let \({\mathfrak g}\) be a complex semisimple Lie algebra. Fix a Cartan subalgebra \({\mathfrak h}\) and a set of simple roots \(B\subset {\mathfrak h}^*\). Let \(L=L(\mu):\) \(\mu\in {\mathfrak h}^*\) be the simple highest weight \({\mathfrak g}\)-module with highest weight \(\mu\)-\(\rho\). \(End_{{\mathbb{C}}}(L)\) is a \({\mathcal U}({\mathfrak g})\)-\({\mathcal U}({\mathfrak g})\)- bimodule, it admits an action of \({\mathfrak g}\) via the diagonal map j. Let A (resp. F) be the subring of \(End_{{\mathbb{C}}}(L)\) of locally j(\({\mathfrak n^-})\)-nilpotent (resp. j(\({\mathfrak g})\)-finite) elements. There is an injection map \(F\hookrightarrow A\). The main result of the article is that this injection is surjective iff L is rigid, i.e. if it is not a submodule of an induced \({\mathfrak g}\)-module \(M={\mathcal U}({\mathfrak g})\otimes_{{\mathcal U}({\mathfrak p})}K\), here \({\mathfrak p}\) is any proper parabolic subalgebra of \({\mathfrak g}\) and K is any simple \({\mathfrak p}\)- module. This result leads to a simple proof of the surjectivity theorem of Levasseur, Smith and Stafford: For certain choices of \(\mu\), there exists a bijective map of \({\mathcal U}({\mathfrak g})/Ann(L(\mu))\) onto the algebra of all differential operators on the associated variety \({\mathcal V}(\mu)\) of the \({\mathfrak g}\)-module L(\(\mu)\). In addition to this, the main theorem gives an information on the Goldie skew field, associated to a primitive quotient of \({\mathcal U}({\mathfrak g})\) and could be used in studying completely prime primitive ideals of \({\mathcal U}({\mathfrak g})\) (see Sect. 8 of the paper).
scientific article; zbMATH DE number 4073226
scientific article