The category $\mathcal{O}$ for a general Coxeter system

TL;DR

This paper explores the structure of the category al for general Coxeter systems, defining key functors and establishing their fundamental properties, thereby extending classical results to a broader algebraic context.

Contribution

It introduces a formulation of al for general Coxeter systems and proves fundamental properties of associated functors, generalizing classical results in representation theory.

Findings

Defined translation, Zuckerman, and twisting functors for al

Proved duality of Zuckerman functor

Generalized Verma's homomorphism results

Abstract

We study the category $\mathcal{O}$ for a general Coxeter system using a formulation of Fiebig. The translation functors, the Zuckerman functors and the twisting functors are defined. We prove the fundamental properties of these functors, the duality of Zuckerman functor and generalization of Verma's result about homomorphisms between Verma modules.