  We propose a new realization, using Harish-Chandra bimodules, of the Serre
functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex
finite dimensional Lie algebra. We further show that our realization carries
over to classical Lie superalgebras in many cases. Along the way we prove that
category $\mathcal{O}$ and its parabolic generalizations for classical Lie
superalgebras are categories with full projective functors. As an application
we prove that in many cases the endomorphism algebra of the basic
projective-injective module in (parabolic) category $\mathcal{O}$ for classical
Lie superalgebras is symmetric. As a special case we obtain that in these cases
the algebras describing blocks of the category of finite dimensional modules
are symmetric. We also compute the latter algebras for the superalgebra
$\mathfrak{q}(2)$.

Representation Theory