  Let $k$ be an algebraically closed field of characteristic zero. Let $C/k$ be
a projective smooth curve with function field $K=k(C)$ and $G/k$ be a connected
reductive group. Let $F$ be a principal $G$-bundle on $C$. Let $P \subseteq G$
be a parabolic subgroup and $\lambda: P \longrightarrow G$ be a strictly
anti-dominant character. 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.
  Let $h_{\mathcal{L}_\lambda}$ be the height function on $X=(F/P)_K$ induced
by $\mathcal{L}_\lambda$. We compute its height filtration and successive
minima: the height filtration is given by a Bruhat decomposition and successive
minima are given by slopes.
  An interesting application is that, in this case, Zhang's inequality on
successive minima can be enhanced into an equality. This is proved from the
convex geometry point of view.
  Let $f \in N^1(F/P)$ be the numerical class of a vertical fiber. We also
compute the augmented base loci $\mathrm{B}_+(\mathcal{L}_\lambda-tf)$ for any
$t \in \mathbb{R}$ and it turns out that they are almost the same as the height
filtration, which accords with the philosophy in Arakelov geometry that the
positivity of the line bundle $\mathcal{L}_\lambda$ is equivalent to the
positivity of its induced height function $h_{\mathcal{L}_\lambda}$. As a
corollary, we compute the $k$-th movable cones of flag bundles over curves for
all $k$.
