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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?

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    $\begingroup$ 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 $\nu-\eta$? $\endgroup$ – Jeremy Rickard May 10 at 13:37
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Jantzen first defined translation functors around 1977, using a definite finite dimensional module, in 2.10 of his Habilitationsschrift here. Note that at around the same time, Gregg Zuckerman was also considering tensor products with finite dimensional modules in the Lie group representation framework here. The ideas in both constructions are similar but independent, and the use of a finite dimensional representation is quite specific.

Chapter 7 of my 2008 AMS textbook (with corrections!) follows the same pattern. But note that the Jantzen idea of crossing walls does require a choice of weight in the wall involved: see 7.15 in my book. This seems to need input from Jantzen's student Soergel, which still leaves an open question as to methodology: Is there a simpler method of wall-crossing?

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