On decomposition of Bott-Samelson sheaves

We give an exact algorithm to calculate (under some GKM-restriction) the matrix describing the embedding $\B(\s)_x\subset\B(\s)^x$, where the first module is the costalk and the second one is the stalk at $x$ of a Bott-Samelson module (sheaf) $\B(\s)$. This allows us to calculate the first few terms of the decomposition of $\B(\s)$ into a sum of indecomposable modules (sheaves) and to calculate the characters of Braden--MacPherson sheaves in some previously unknown cases.