Geometric Height on Flag Varieties in Positive Characteristic

Let $k$ be an algebraically closed field of characteristic $p\neq 0$. Let $G$ be a connected reductive group over $k$, $P \subseteq G$ be a parabolic subgroup and $\lambda: P \longrightarrow \mathbb G_m$ be a strictly anti-dominant character. Let $C$ be a projective smooth curve over $k$ with function field $K=k(C)$ and $F$ be a principal $G$-bundle on $C$. Then $F/P \longrightarrow C$ is a flag bundle and $\mathcal{L}_\lambda=F \times_P k_\lambda$ on $F/P$ is a relatively ample line bundle. We compute the height filtration and successive minima of the height function $h_{\mathcal{L}_\lambda}: X(\overline{K}) \longrightarrow \mathbb{R}$ over the flag variety $X=(F/P)_K$.