Tensoring with infinite-dimensional modules in $\scr O_0$

Abstract

We show that the principal block $\scr O_0$ of the BGG category $\scr O$ for a semisimple Lie algebra $\germ g$ acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category $\scr O$. We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on $\scr O_0$. Furthermore, all this generalises to parabolic subcategories of $\scr O_0$. As an example, we present some explicit computations for the algebra $\germ{sl}_3$.