Geometric interpretation of translation through the wall 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.
 A: I am guessing the following is well known to you/not what your are looking for, but nonetheless:
Let $s$ be a simple reflection, $P_s$ the corresponding minimal parabolic, $\pi_s\colon G/B\to G/P_s$ the projection. Translation across the $s$-wall `corresponds' to $\pi_s^*\pi_{s*}$. I use quotation marks because as stated this is clearly not true (translation across the wall is t-exact, $\pi_s^*\pi_{s*}$ is certainly not). However, $\pi_s^*\pi_{s*}$ does correspond to translation across the wall under Koszul duality. This is also the same as convolving with the $IC$-complex corresponding to $s$.
Morally (as you point out), reflection across the wall should correspond to convolving with the corresponding tilting. But there is an annoying issue here: tiltings are not $B$-equivariant. Similar problem occurs if instead of convolution using equivariant derived categories you try to use the standard Fourier-Mukai formalism and try to use an object on $G/B\times G/B$ as a kernel. However, there is a fix that comes at some technical expense. Namely, Bezrukavnikov and Yun's free monodromic sheaves http://arxiv.org/abs/1101.1253. The idea actually goes back to the paper of Beilinson and Ginzburg that you cite (look at Section 5).
