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

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 $\bar{\mathcal{O}}$.