Take the 2-minute tour ×  Are there natural generalizations of
the Drinfeld symmetric space? For $\mathbb{K}$, a non-
Archimedean local field, the Drinfeld symmetric space can be
defined as the complement of all $\mathbb{K}$-rational
hyperplanes in $\mathbb{P}^r_{\overline{\mathbb{K}}}$. One can
generalize it in the following direction: let $X$ be some
variety defined over $\mathbb{K}$ and $L$ some line bundle on
$X$ defined over $\mathbb{K}$; consider the complement of the
union of the zero loci of sections $L$ defined over
$\mathbb{K}$. An example of this would be to consider $X=G/H$, a
homogeneous space and $L$, a line bundle defined by a
representation of $G$. Even more concretely, one can consider a
flag variety or a Grassmannian and the polarization induced by
its Plucker embedding.  This seems like a natural thing to
consider. Moreover, the analogs of the Drinfeld's symmetric
space's connections to Bruhat-Tits theory and to degenerations
of $\mathbb{P}^r$ might be interesting.  Has any such thing
appear