What does translation through the wall correspond to under Beilinson Bernstein localization?

More precisely I am interested in the following:

There is a well known equivalence between the principal block of category $\mathcal O$ and perverse sheaves on the flag manifold, constructible along $B$ orbits:

$$\mathcal O_0 \cong \mathcal P_{(B)}(G/B)$$

Now for a singular integral weight $\lambda$ one can consider the translation through the wall functor $$ \theta_\lambda:\mathcal O_0 \rightarrow \mathcal O_\lambda \rightarrow \mathcal O_0$$

What does it correspond to under the above equivalence? My naive guess/hope would be, that it is given by convolution with the sheaf corresponding to $\theta_\lambda (L_e)$ where $L_e$ is the antidominant simple. Is this correct? If so, is there a geometric way to construct this sheaf?

PS: I am aware that there are descritions of the translation functors using slightly more elaborate version of localization, for example in this paper by Beilinson Ginzburg. However I would prefer to keep the above setup.

`$\lambda$`

(thought of as lying in just the`$s$`

-wall of the fundamental Weyl chamber). In Jantzen's original treatment of translation functors it was hard to understand this independence directly. I think the standard proof relies on the study of projective functors by S. Gelfand and J. Bernstein. – Jim Humphreys Dec 21 '12 at 1:17`$\lambda$`

doesn't matter. (But the original setting for translation functors is very elementary, so going this far already increases the sophistication.) Soergel's deeper methods seem essential, however, when you also consider non-integral weights (and introduce an "integral" Weyl subgroup): for instance, it's nontrivial to show that everything just depends on that small Weyl group. – Jim Humphreys Dec 21 '12 at 14:25