# Well-definedness of translation functor

Fix $$\eta, \nu \in \mathfrak{h}^*$$ and write $$pr_\eta$$ and $$pr_\nu$$ for the natural projections of the category $$\mathcal{O}$$ onto $$\mathcal{O}_{\chi_\eta}$$ and $$\mathcal{O}_{\chi_\nu}$$. If $$L$$ is finite dimensional and $$M \in \mathcal{O}$$, then $$M \mapsto pr_ \nu \left(L \otimes (pr_\eta M)\right)$$ followed by inclusion into $$\mathcal{O}$$ defines an exact functor $$\mathcal{O} \to \mathcal{O}$$. Its restriction to $$\mathcal{O}_{\chi_\eta}$$ (without the inclusion) relates the two subcategories $$\mathcal{O}_{\chi_\eta}$$ and $$\mathcal{O}_{\chi_\nu}$$. Write $$T^\nu_\eta$$ for the resulting functor on $$\mathcal{O}$$ (or on $$\mathcal{O}_{\chi_\eta}$$). We call $$T^\nu_\eta$$ a translation functor.

My question: The definition of a translation functor $$T^\nu_\eta$$ depends on the choice of a finite dimensional module $$L$$, how to show it is well-defined?

• I think you’re missing conditions on $L$, otherwise the functor you describe is clearly not independent of $L$. I think $L$ should be a highest weight module of highest weight $\nu-\eta$? – Jeremy Rickard May 10 at 13:37