Tensoring with infinite-dimensional modules in \(\mathcal {O}_0\) (Q605330)

Let \(\mathcal O_0\) be the principal block of the BGG category \(\mathcal O\) for a finite-dimensional complex semisimple Lie algebra \(\mathfrak g\) [\textit{I. N. Bernstein, I. M. Gelfand} and \textit{S. I. Gelfand}, Funct. Anal. Appl. 10, 87--92 (1976; Zbl 0353.18013)] and let \(\downarrow_0\) denote the projection onto \(\mathcal O_0\). The purpose of this paper is the study of some functors from \(\mathcal O_0\) to the category of endofunctors on \(\mathcal O_0\), obtained via tensoring with \(\mathfrak g\)-modules in \(\mathcal O_0\) which are not necessarily finite-dimensional. The exact functor \(G\) is defined as follows: (i) For \(M\in\mathcal O_0\), the endofunctor \(G_M\) on \(\mathcal O_0\) is given by \(G_MN = (M\otimes N)\!\!\downarrow_0\) on objects and by \(G_M\varphi = (\text{Id}_M\otimes\varphi)\!\!\downarrow_0: G_MK\rightarrow G_ML\) on morphisms \(\varphi: K\rightarrow L\), where \[ (\text{Id}_M\otimes\varphi)\!\!\downarrow_0 = \pi_{G_ML}\circ(\text{Id}_M\otimes\varphi)\circ\iota_{G_MK}, \] with \(\pi_{G_ML}\!:\! M\otimes L\rightarrow (M\otimes L)\!\!\downarrow_0\) and \(\iota_{G_MK}\!:\! (M\otimes K)\!\!\downarrow_0\rightarrow M\otimes K\) being, respectively, the natural projection and inclusion. (ii) For a morphism \(\varphi: M\rightarrow N\) in \(\mathcal O_0\), the corresponding natural transformation \(G_\varphi: G_M\rightarrow G_N\) is given by \(G_\varphi K =(\varphi\otimes\text{Id}_K)\!\!\downarrow_0: G_MK\rightarrow G_NK\) \((K\in\mathcal O_0)\). Then the right exact contravariant functor \(F\) and the left exact contravariant functor \(H\) are defined by taking, respectively, the left and the right adjoints \(F_M\) and \(H_M\) of \(G_M\) for all \(M\in\mathcal O_0\). The author demonstrates the faithfulness of the functors \(F, G, H\) and determines some of their properties. In particular, he proves that, for any \(M\in\mathcal O_0\), the endofunctors \(F_M, G_M\), and \(H_M\) preserve the additive subcategories of projective, tilting, and injective \(\mathfrak g\)-modules, respectively. He also shows that the functor \(G\) gives rise to natural comonad and monad structures on \(\mathcal O_0\) and that all the results on the functors \(F, G, H\) generalize to parabolic subcategories of \(\mathcal O_0\).