BGG category $\mathcal{O}$ and $\mathbb{Z}$-graded representation theory

TL;DR

This paper introduces the $bZ$-graded representation theory of the BGG category $bO$ for complex semisimple Lie algebras, focusing on combinatorial functors and graded duality and translation functors.

Contribution

It provides foundational definitions and insights into the $bZ$-graded structures within the BGG category $bO$, emphasizing Soergel's combinatorial $bV$ functor.

Findings

Development of $bZ$-graded duality functors

Introduction of $bZ$-graded translation functors

Clarification of the role of Soergel's $bV$ functor

Abstract

We give an introduction to the $\mathbb{Z}$-graded representation theory of the BGG category $\mathcal{O}$ of a complex semisimple Lie algebras, with an emphasis on Soergel's combinatorial $\mathbb{V}$ functor, definitions of $\mathbb{Z}$-graded duality functors and definitions of $\mathbb{Z}$-graded translation functors.