Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules

Let $Z_{\bf i}$ (resp., $X(w)$) be the Bott-Samelson variety (resp., the Schubert variety), and $\mathcal{L}_{\bf m}$ (resp., $\mathcal{L}_\lambda$) a line bundle on $Z_{\bf i}$ (resp., on $X(w)$). We can think of $H^0(Z_{\bf i}, \mathcal{L}_{\bf m})$ as a generalization of $H^0(X(w), \mathcal{L}_\lambda)$. We extend the string polytope for the Demazure module $H^0(X(w), \mathcal{L}_\lambda)^\ast$ to $H^0(Z_{\bf i}, \mathcal{L}_{\bf m})^\ast$, and prove that it is identical to the Newton-Okounkov body of $Z_{\bf i}$ with respect to a specific valuation. As applications of this result, we show that these are indeed polytopes, and also construct a basis of $H^0(Z_{\bf i}, \mathcal{L}_{\bf m})$, which can be thought of as a perfect basis.