Tensor product decompositions for cohomologies of Bott-Samelson varieties

Let $T$ be a maximal torus of a semisimple complex algebraic group, $\mathrm{BS}(s)$ be the Bott-Samelson variety for a sequence of simple reflections $s$ and $\mathrm{BS}(s)^T$ be the set of $T$-fixed points of $\mathrm{BS}(s)$. We prove the tensor product decompositions for the image of the restriction $H^\bullet_T(\mathrm{BS}(s),k)\to H_T^\bullet(X,k)$, where $X\subset\mathrm{BS}(s)^T$ is defined by some special not overlapping equations $\gamma_i\gamma_{i+1}\cdots\gamma_j=w_{i,j}$ with right-hand sides belonging to the Weyl group.