Geometric Height on Flag Varieties in Positive Characteristic

Let $k$ be an algebraically closed field of characteristic $p\mathrm{\ne }0$. Let $G$ be a connected reductive group over $k$, $P\subseteq G$ be a parabolic subgroup and $\lambda :P⟶{\mathbb{G}}_m$ 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 $F\mathrm{/}P⟶C$ is a flag bundle and ${\mathcal{L}}_{\lambda }=F{\times }_P{k}_{\lambda }$ on $F\mathrm{/}P$ is a relatively ample line bundle. We compute the height filtration and successive minima of the height function $h_{{\mathcal{L}}_{\lambda }}:X(\bar{K})⟶\mathbb{R}$ over the flag variety $X=(F\mathrm{/}P{)}_K$.