On decomposition of Bott-Samelson sheaves

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