From Jantzen to Andersen Filtration via Tilting Equivalence

TL;DR

This paper demonstrates that the tilting equivalence in category O aligns the Jantzen and Andersen filtrations on certain Hom-spaces, revealing a deep connection between these filtrations.

Contribution

It establishes that the tilting equivalence identifies the Jantzen filtration with the Andersen filtration on Hom-spaces in category O.

Findings

The tilting equivalence induces an isomorphism of Hom-spaces.

It shows the Jantzen and Andersen filtrations are compatible under this equivalence.

The result unifies two important filtrations in representation theory.

Abstract

The space of homomorphisms from a projective object to a Verma module in category O inherits an induced filtration from the Jantzen filtration on the Verma module. On the other hand there is the Andersen filtration on the space of homomorphisms from a Verma module to a tilting module as described in [Soe07]. The tilting equivalence from [Soe98] induces an isomorphism of these kinds of Hom-spaces. We will show that this equivalence even identifies both filtrations.