Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by Humphreys). If $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{g}$ contained in $\mathfrak{b}$, then there exists a unique indecomposable module $D(\lambda)$, where $\lambda\in\mathfrak{h}^*$, such that the weight space of $D(\lambda)$ with weight $\lambda$ has dimension $1$, and $D(\lambda)$ has a finite filtration with standard subquotients as well as a finite filtration with costandard subquotients. The construction of $D(\lambda)$ is discussed in the same book.

However, when I read Ringel's definition of tilting modules, I noticed some discrepancies. For example, a tilting module $D(\lambda)$ does not need to have projective dimension $1$ in $\mathcal{O}$ (this is the first condition in the link). Plus, the enveloping algebra $U(\mathfrak{g})$ is not in $\mathcal{O}$, so it is not the kernel of any epimorphism from a direct sum of finitely many copies of $D(\lambda)$ to another direct sum of finitely many copies of $D(\lambda)$, violating the third condition in the link. (I am not sure anyhow if the third condition makes any sense. After all, $\mathcal{O}$ is not the whole category of $U(\mathfrak{g})$-modules.) The only condition $D(\lambda)$ satisfies is $\operatorname{Ext}_{\mathcal{O}}^1\big(D(\lambda),D(\lambda)\big)=0$ (i.e., the second condition of the link).

The first condition can be fixed by using the definition of generalized tilting modules, but this definition may be not-so-helpful for other abelian categories. However, there doesn't seem to be a fix for the third condition. There is still no exact sequence $0\to U(\mathfrak{g})\to T_0\to T_1\to T_2 \to \ldots \to T_n \to 0$, where each $T_i$ is a finite direct sum of $D(\lambda)$.

Hence, I would like to ask why are $D(\lambda)$ called

anyway? Is there a version of tilting theory that works with $D(\lambda)$? Do we have something like Ringel duality here? Is there a tilting theory for abelian categories in general?tilting modules

For tilting objects in a general abelian category, I am aware of the work by Colpi and Fuller in 2007, but it requires that the category contains the direct sums of arbitrary, possibly infinite, numbers of copies of some objects. I think this requirement is too restrictive.