Categories O for Dynkin Borel Subalgebras of Root-Reductive Lie Algebras

TL;DR

This paper extends the classical category O to certain infinite-dimensional Lie algebras, focusing on special Borel subalgebras called Dynkin Borel subalgebras, and explores their structure, block decomposition, and reciprocity properties.

Contribution

It introduces and studies extended categories O for infinite-dimensional Lie algebras, focusing on Dynkin Borel subalgebras, and establishes analogues of key properties like BGG reciprocity.

Findings

Block decomposition results carry over from classical category O.

Analogues of BGG reciprocity are established for truncated categories.

Differences include the absence of some injective hulls in the new categories.

Abstract

The purpose of my Ph.D. research is to define and study an analogue of the classical Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}$ for the Lie algebra $\mathfrak{g}$, where $\mathfrak{g}$ is one of the finitary, infinite-dimensional Lie algebras $\mathfrak{gl}_\infty(\mathbb{K})$, $\mathfrak{sl}_\infty(\mathbb{K})$, $\mathfrak{so}_\infty(\mathbb{K})$, and $\mathfrak{sp}_\infty(\mathbb{K})$. Here, $\mathbb{K}$ is an algebraically closed field of characteristic $0$. We call these categories "extended categories $\mathcal{O}$" and use the notation $\bar{\mathcal{O}}$. While the categories $\bar{\mathcal{O}}$ are defined for all splitting Borel subalgebras of $\mathfrak{g}$, this research focuses on the categories $\bar{\mathcal{O}}$ for very special Borel subalgebras of $\mathfrak{g}$ which we call Dynkin Borel subalgebras. Some results concerning block decomposition and Kazhdan-Lusztig multiplicities carry over from usual categories $\mathcal{O}$ to our categories $\bar{\mathcal{O}}$. There are differences which we shall explore in detail, such as the lack of some injective hulls. In this connection, we study truncated categories $\bar{\mathcal{O}}$ and are able to establish an analogue of BGG reciprocity in the categories $\bar{\mathcal{O}}$.