Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety. 

Let $w$ be an element of the Weyl group $W$ with a reduced expression 
$w = s_1 \cdots s_n$. Let $X_w$ be the corresponding Schubert variety in the flag variety, and $BS(s_1,...s_n)$ be the corresponding Bott-Samelson resolution. 

There is a birational map $f: BS(s_1,s_2,...,s_n) \rightarrow X_{s_1s_2...s_n} \subset G/B$. What can we say about the (derived) pushforward of the structure sheaf $\mathcal{O}$ on $BS(s_1,s_2,...,s_n)$? Is that the convolution product of some 
$G-$equivariant coherent sheaves on $X_w \subset G/B$?