Injectives have costandard filtrations

$D(P(\lambda ))$ is injective by [F1]. If $0=X_0\subseteq X_1\subseteq \cdots \subseteq X_n=P(\lambda )$ is a Verma flag with factors $\mathrm{\Delta }({\mu }_i)=X_i\mathrm{/}{X}_{i-1}$, then applying the exact contravariant functor $D$ to the defining sequences $0\to X_{i-1}\to X_i\to \mathrm{\Delta }({\mu }_i)\to 0$ gives exact sequences $0\to \mathrm{\nabla }({\mu }_i)\to D(X_i)\to D(X_{i-1})\to 0$; by induction on $i$, a finite costandard flag of $D(X_{i-1})$ concatenated with the subobject $\mathrm{\nabla }({\mu }_i)$ gives a finite costandard flag of $D(X_i)$, because extensions of objects with finite costandard flags again have finite costandard flags. For $i=n$ this gives a finite costandard flag of $I(\lambda )=D(P(\lambda ))$ with the factors $\mathrm{\nabla }({\mu }_i)$, hence $(I(\lambda ):\mathrm{\nabla }(\mu ))=(P(\lambda ):\mathrm{\Delta }(\mu ))=[\mathrm{\Delta }(\mu ):L(\lambda )]$ by [F2].

Let $I\in \mathcal{O}$ be injective. By [F3] it has finite length and $I=I_1\oplus \cdots \oplus I_n$ with each $I_j$ indecomposable. Applying the exact contravariant involution $D$ gives $D(I)={⨁}_{j}D(I_j)$ with each $D(I_j)$ an indecomposable projective: $D$ is an equivalence, so it preserves indecomposability and exchanges projectives with injectives. Each $D(I_j)$ is therefore a projective cover of its simple head $L({\mu }_j)$ by [F2], hence $D(I_j)\cong P({\mu }_j)$ by uniqueness of projective covers and $I_j\cong D(D(I_j))\cong D(P({\mu }_j))=I({\mu }_j)$.

By step 1.1 each $I({\mu }_j)$ has a finite costandard flag, and a finite direct sum of objects with finite costandard flags again has one, by concatenating flags along the summands; hence every injective object $I\cong {⨁}_{j}I({\mu }_j)$ has a finite costandard flag, and the bijection between indecomposable projectives and indecomposable injectives is induced by $D$.