Associated to a simple root of a finite-dimensional complex semisimple Lie
algebra, there are several endofunctors (defined by Arkhipov, Enright, Frenkel,
Irving, Jantzen, Joseph, Mathieu, Vogan and Zuckerman) on the BGG category
$\mathcal{O}_0$. We study their relations, compute cohomologies of their
derived functors and describe the monoid generated by Arkhipov’s and Joseph’s
functors and the monoid generated by Irving’s functors. It turns out that the
endomorphism rings of all elements in these monoids are isomorphic. We prove
that the functors give rise to an action of the singular braid monoid on the
bounded derived category of $\mathcal{O}_0$. We also use Arkhipov’s, Joseph’s
and Irving’s functors to produce new generalized tilting modules.