For an additive category $\mathcal C$ there is the notion of a Serre functor on $\mathcal C$, i.e. a an autoequivalence $S$ of $C$ such that there exist isomorphisms
$$Hom(A, S(B)) \cong Hom(B, A)^\*$$
natural in $A,B \in \mathcal C$.


If $\mathcal D$ is a full subcategory of $\mathcal C$ that is preserved by $S$, when is the restriction of $S$ a Serre functor of $\mathcal D$?


The particular case I am wondering about is the following: In the paper [1], the authors compute the Serre functor S of $D^b(\mathcal O\_\lambda)$ of the derived category of an integral block of the BGG category $\mathcal O$. Let $\mathcal O^{\mathfrak p}\_\lambda$ be the corresponding block of the parabolic category $\mathcal O$. They authors state that $D^b(\mathcal O^{\mathfrak p}\_\lambda)$ is a full triangulated subcategory of $D^b(\mathcal O\_\lambda)$ that is preserved by $S$. They then prove that the restriction of $S$ shifted by a non-zero integer $n$ is a Serre functor of $D^b(\mathcal O^{\mathfrak p}\_\lambda)$.


Doesn't this imply that for complexes $A, B \in D^b(\mathcal O^{\mathfrak p}\_\lambda)$ the above equality of Hom-spaces holds both for $S$ and $S[n]$?


[1] Volodymyr Mazorchuk and Catharina Stroppel. “Projective-injective modules,
Serre functors and symmetric algebras”. *Journal für die reine und angewandte Mathematik* 616 (2008), pp. 131–165.



Serre functor of a subcategory (in particular parabolic category O)
It's not true that $D^b(\mathcal{O}^{\mathfrak{p}})$ is a full subcategory of $D^b(\mathcal{O})$. Think about the case $\mathfrak{g}=\mathfrak{p}=\mathfrak{sl}\_2$. The category $\mathcal{O}^{\mathfrak{p}}$ is finite-dimensional modules and thus semi-simple, whereas in $D^b(\mathcal{O})$, we have that $\mathrm{Ext}^2(\mathbb{C},\mathbb{C})\cong \mathbb{C}$. That's why the restriction of the Serre functor is not itself a Serre functor; it's a minor miracle that a shift of it is.


