From Jantzen to Andersen filtration via tilting equivalence (Q2890394)

The main result of this article is to identify certain filtrations on the space of homomorphisms from a projective module \(P\) to a Verma module with that of homomorphisms from a proper Verma module to a tilting module given by \(P\). To make the statement more precise, let \(\mathfrak{g}\) be a semisimple complex Lie algebra and \(\mathfrak{h}\) a Cartan subalgebra of \(\mathfrak{g}\). Given \(\lambda \in \mathfrak{h}^*\), let \(\Delta(\lambda)\) be the Verma module with highest weight \(\lambda\). Observe that, via the deformed Verma module \(\Delta_T(\lambda)\) of \(\Delta(\lambda)\), the Jantzen filtration of \(\Delta(\lambda)\) yields a filtration on \(\Hom_{\mathfrak{g}}(P, \Delta(\lambda))\), where \(P\) is a projective module of the BGG-category \(\mathcal{O}\). On the other hand, in [Represent. Theory 2, 432--448 (1998; Zbl 0964.17018)], \textit{W. Soergel} introduced the tilting functor \(t\), a contravariant self-equivalence of the category of modules with Verma flags. An important fact is that \(t\) sends projective modules to tilting modules and sends \(\Delta(\lambda)\) to \(\Delta(-2\rho-\lambda)\); in particular, the tilting functor \(t\) induces an isomorphism \(\Hom_{\mathfrak{g}}(P, \Delta(\lambda)) \simeq \Hom_{\mathfrak{g}}(\Delta(-2\rho-\lambda), t(P))\), where \(\rho\) is half the sum of the positive roots. The author showed that the induced isomorphism identifies the filtration on \(\Hom_{\mathfrak{g}}(P, \Delta(\lambda))\) induced by the Jantzen filtration on \(\Delta(\lambda)\) with a filtration on \(\Hom_{\mathfrak{g}}(\Delta(-2\rho-\lambda), t(P))\) called the Andersen filtration.