A categorification of the Lusztig--Vogan module

TL;DR

This paper develops two new categorifications of the Lusztig--Vogan module for real reductive groups, linking geometric and algebraic approaches through derived categories and bimodule modules.

Contribution

It introduces two novel categorifications of the Lusztig--Vogan module, connecting geometric and algebraic frameworks via equivariant hypercohomology.

Findings

Constructed a categorification using semisimple complexes in an equivariant derived category.

Developed a second categorification as a module category over Soergel bimodules.

Established a relationship between the two categorifications through equivariant hypercohomology.

Abstract

We construct two categorifications of the Lusztig--Vogan module associated to a real reductive algebraic group. The first categorification is given by semisimple complexes in an equivariant derived category, and the second is constructed as a module category over Soergel bimodules. Our categorifications are related by taking equivariant hypercohomology.