Line bundles on $G$-Bott-Samelson-Demazure-Hansen varieties

Let $G$ be a semi-simple simply connected algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G\mathrm{.}$ Let $W$ be the Weyl group of $G$ with respect to $T$. For an arbitrary sequence $w=(s_{i_1},s_{i_2},\dots ,s_{i_r})$ of simple reflections in $W,$ let $Z_w$ be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to $w\mathrm{.}$ Let $\widetilde{Z_w}:=G{\times }^B{Z}_w$ denote the fibre bundle over $G\mathrm{/}B$ with the fibre over $B\mathrm{/}B$ is $Z_w\mathrm{.}$ In this article, we give necessary and sufficient conditions for the varieties $Z_w$ and $\widetilde{Z_w}$ to be Fano (weak-Fano). We show that a line bundle on $Z_w$ is globally generated if and only if it is nef. We show that Picard group $\text{Pic}(\widetilde{Z_w})$ is free abelian and we construct a $\mathcal{O}(1)$-basis. We characterize the nef, globally generated, ample and very ample line bundles on $\widetilde{Z_w}$ in terms of the $\mathcal{O}(1)$-basis.