Let $G$ be a Lie group and $B$ 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 $BS(s_1,...s_n)$ 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 $X_w \subset G/B$?