Tilting modules in category O and sheaves on moment graphs

TL;DR

This paper links tilting modules in category O to sheaves on moment graphs, providing new character formulas and insights into filtrations via geometric and combinatorial methods.

Contribution

It introduces a geometric description of tilting modules as sheaves on moment graphs and connects them to Braden-MacPherson sheaves, offering new proofs and character formulas.

Findings

Tilting modules correspond to specific sheaves on moment graphs.

Character formulas for tilting modules are derived.

Provides a Koszul dual proof of semisimplicity of Jantzen filtration subquotients.

Abstract

We describe tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of category O. We prove that they map to Braden-MacPherson sheaves constructed along the reversed Bruhat order under Fiebig's localization functor. By this means, we get character formulas for tilting modules and explain how Soergel's result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration.