Translation functors by tensoring and projection

Assume the Axiom of Choice (The Axiom of Choice). Work in category $\mathcal{O}$ with the conventions of the preceding definitions. For a weight $\lambda$ write ${\chi }_{\lambda }$ for the generalized central character obtained from $\lambda$, so that ${\chi }_{\lambda }={\chi }_{\mu }$ if and only if $\mu \in W\cdot \lambda$ (Central characters are dot-Weyl orbits), and let $\mathcal{O}={⨁}_{\chi }{\mathcal{O}}_{\chi }$ be the central-character decomposition of Generalized central-character decomposition of O with inclusions ${\mathrm{incl}}_{\chi }$ and exact projections ${\mathrm{pr}}_{\chi }$, so that ${\mathrm{pr}}_{\chi }\circ {\mathrm{incl}}_{\chi }=\mathrm{id}$.

For a finite-dimensional $\mathfrak{h}$-semisimple $\mathfrak{g}$-module $E$ (Weight and weight space) and two generalized central characters $\chi ,{\chi }^{\mathrm{\prime }}$, set $T_{\chi ,E,{\chi }^{\mathrm{\prime }}}:={\mathrm{pr}}_{{\chi }^{\mathrm{\prime }}}\circ (E\otimes -)\circ {\mathrm{incl}}_{\chi }\text{\hspace{0.17em}⁣}:{\mathcal{O}}_{\chi }⟶{\mathcal{O}}_{{\chi }^{\mathrm{\prime }}}\mathrm{.}$ This is well defined because $E\otimes -$ is an exact endofunctor of $\mathcal{O}$ (Finite-dimensional tensoring preserves O) and the projections and inclusions are exact.

For weights $\lambda ,\mu$ with $\mu -\lambda$ integral (Dot-Weyl facets and single-wall translation data), let $\nu$ be the unique dominant weight in the linear Weyl orbit $W(\mu -\lambda )$, which exists and is unique by Finite Weyl closed chambers and stabilizers and is integral; let $L(\nu )$ be the finite-dimensional simple module of highest weight $\nu$ (Finite-dimensional simple modules are classified by dominant highest weights). Define $T_{\lambda }^{\mu }:=T_{{\chi }_{\lambda },L(\nu ),{\chi }_{\mu }}\text{\hspace{0.17em}⁣}:{\mathcal{O}}_{{\chi }_{\lambda }}⟶{\mathcal{O}}_{{\chi }_{\mu }},T_{\mu }^{\lambda }:=T_{{\chi }_{\mu },L(\nu {)}^{\ast },{\chi }_{\lambda }}\text{\hspace{0.17em}⁣}:{\mathcal{O}}_{{\chi }_{\mu }}⟶{\mathcal{O}}_{{\chi }_{\lambda }},$ where $L(\nu {)}^{\ast }$ is the ordinary linear dual, a finite-dimensional $\mathfrak{h}$-semisimple simple module isomorphic to $L(-w_0\nu )$ by Highest weight of the dual representation.

The labels $\lambda$ and $\mu$ denote actual weights and not $\rho$-shifted parameters. The central-character subcategories used here are those of the published decomposition; a central-character summand can contain several linkage blocks of Central-character summands refine into linkage blocks, while for an indecomposable central-character summand the present functors are translation between that summand and its target.