Categories $\mathcal{O}$ for Root-Reductive Lie Algebras: II. Translation Functors and Tilting Modules

This is the second paper of a series of papers on a version of categories $\mathcal{O}$ for root-reductive Lie algebras. Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. For some pairs of blocks $\overline{\mathcal{O}}[\lambda]$ and $\overline{\mathcal{O}}[\mu]$, the subcategories whose objects have finite length are equivalence via functors obtained by the direct limits of translation functors. Tilting objects can also be defined in $\overline{\mathcal{O}}$. There are also universal tilting objects $D(\lambda)$ in parallel to the finite-dimensional cases.