Projective functors on a regular integral block

Assume the Axiom of Choice, inherited from the published block theory (Translation functors by tensoring and projection, The Axiom of Choice).

Fix a regular integral block ${\mathcal{O}}_{\lambda }$ as in The regular integral block and its antidominant weight, with inclusion ${\mathrm{incl}}_{\lambda }:{\mathcal{O}}_{\lambda }\to \mathcal{O}$ and exact projection ${\mathrm{pr}}_{\lambda }:\mathcal{O}\to {\mathcal{O}}_{\lambda }$ of Translation functors by tensoring and projection, so that ${\mathrm{pr}}_{\lambda }\circ {\mathrm{incl}}_{\lambda }=\mathrm{id}$; the block is a direct summand of $\mathcal{O}$ and contains the projective covers $P(w\cdot \lambda )$ of Category O has enough projectives. For a finite-dimensional $\mathfrak{h}$-semisimple $\mathfrak{g}$-module $E$ define the endofunctor $F_E:={\mathrm{pr}}_{\lambda }\circ (E\otimes -)\circ {\mathrm{incl}}_{\lambda }\text{\hspace{0.17em}⁣}:{\mathcal{O}}_{\lambda }⟶{\mathcal{O}}_{\lambda }\mathrm{.}$

Basic properties. The functor $F_E$ is well defined and additive: $E\otimes -$ is an exact endofunctor of $\mathcal{O}$ and maps $\mathcal{O}$ into itself (Finite-dimensional tensoring preserves O), and ${\mathrm{pr}}_{\lambda }$ is exact. Hence $F_E$ is exact as a composite of exact functors. It preserves projective objects: if $P\in {\mathcal{O}}_{\lambda }$ is projective, then $E\otimes P$ is projective in $\mathcal{O}$ (Finite-dimensional tensoring preserves projectives in category O), and ${\mathrm{pr}}_{\lambda }$ preserves projectives (Exact projections onto linkage blocks preserve projectives). The zero module $E=0$ gives the zero functor, and $E=\mathbb{C}$ the identity functor, since the one-dimensional trivial module acts by the identity.

Projective functors. A projective functor on ${\mathcal{O}}_{\lambda }$ is an endofunctor of ${\mathcal{O}}_{\lambda }$ isomorphic to a direct summand of some $F_E$, $E$ finite dimensional and $\mathfrak{h}$-semisimple. Write ${\mathcal{F}}_{\lambda }$ for the full subcategory of the endofunctor category of ${\mathcal{O}}_{\lambda }$ whose objects are the projective functors, with arbitrary natural transformations as morphisms; it is additive with the pointwise biproducts of Additive category, and "direct summand" means natural maps $i:F\to F_E$, $p:F_E\to F$ with $pi=1_F$; then $ip$ is an idempotent natural transformation.

Closure properties. The class of projective functors is closed under finite direct sums, composition, direct summands and passage to adjoints:

1. $F_{E\oplus E^{\mathrm{\prime }}}\cong F_E\oplus F_{E^{\mathrm{\prime }}}$, since tensoring with a direct sum distributes over direct sums and ${\mathrm{pr}}_{\lambda }$ is additive.
2. $F_E\circ F_{E^{\mathrm{\prime }}}$ is a natural direct summand of $F_{E\otimes E^{\mathrm{\prime }}}$: the natural block decomposition of $E^{\mathrm{\prime }}\otimes X$ splits off its $\lambda$-component before applying $E\otimes -$ and the final projection. Its complementary blocks can return to ${\mathcal{O}}_{\lambda }$, so an isomorphism with the entire $F_{E\otimes E^{\mathrm{\prime }}}$ is not asserted.
3. $F_{E^{\ast }}$ is both a left and a right adjoint of $F_E$: this is Translation functors are exact and biadjoint applied to the two generalized central characters of the block, since $F_{E^{\ast }}={\mathrm{pr}}_{\lambda }\circ (E^{\ast }\otimes -)\circ {\mathrm{incl}}_{\lambda }$ is the translation functor $T_{{\chi }_{\lambda },E^{\ast },{\chi }_{\lambda }}$ and similarly for $F_E$.
4. A direct summand of an exact additive functor is exact, and a direct summand of a functor preserving projective objects again preserves projective objects: apply the splitting to the image of each object. Hence every projective functor is exact and preserves projective objects, and composition of summand splittings makes a composite of projective functors a summand of the composite of their ambient tensor functors. Clause 2 therefore proves closure under composition; clause 1 proves closure under sums. For an adjoint, take the mate of the summand idempotent under clause 3; the triangle identities carry its square to itself, and its image is the corresponding adjoint summand of $F_{E^{\ast }}$.

This is the definition of Bernstein-Gelfand [BGe, Section 3.1] and of [Soe90, Section 1.2]; unlike the functors $T_{\lambda }^{\mu }$ of Translation functors by tensoring and projection, projective functors are not required to come from a single finite-dimensional tensor functor, only from a direct summand of one.