Some homological properties of the category $\mathcal{O}$, II

TL;DR

This paper proves that Lusztig's a-function determines the projective dimensions of key modules in category O, linking homological properties with representation theory and categorification.

Contribution

It establishes a general proof connecting Lusztig's a-function to projective dimensions in category O and explores derived category representations and categorification of Weyl group actions.

Findings

Lusztig's a-function describes projective dimensions of tilting and injective modules.

Simple modules under projective functors can be represented by linear complexes of tilting modules.

Categorification decomposes the regular Weyl group representation into cell modules.

Abstract

We show, in full generality, that Lusztig's $\mathbf{a}$-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category $\mathcal{O}$, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of tilting modules. These complexes, in turn, can be interpreted as the images of simple modules under projective functors in the Koszul dual of the category $\mathcal{O}$. Finally, we describe the dominant projective module and also projective-injective modules in some subcategories of $\mathcal{O}$ and show how one can use categorification to decompose the regular representation of the Weyl group into a direct sum of cell modules, extending the results known for the symmetric group (type $A$).