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Bases of T-equivariant cohomology of Bott-Samelson varieties

We construct combinatorial bases of the $T$-equivariant ($T$ is the maximal torus) cohomology $H^\bullet_T(Σ,k)$ of the Bott-Samelson variety $Σ$ under some mild restrictions on the field of coefficients $k$. This bases allow us to prove the surjectivity of the restrictions $H^\bullet_T(Σ,k)\to H^\bullet_T(π^{-1}(x),k)$ and $H^\bullet_T(Σ,k)\to H^\bullet_T(Σ\setminusπ^{-1}(x),k)$, where $π:Σ\to G/B$ is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of $H^\bullet_T(Σ,k)$ and restricting them to $π^{-1}(x)$ or $Σ\setminusπ^{-1}(x)$ respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image $π_*{\underline k}_Σ$. This algorithm avoids division by 2, which allows us to reestablish 2-torsion for parity sheaves in Braden's example.