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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 appeared in the literature?

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One keyphrase to search - Rapoport-Zink period domains.. – David Ben-Zvi Apr 1 '11 at 3:41
Let me add a reference to David's (roght to the point) remark: There is a book by Rapoport and Zink "Period Spaces for p-divisible groups", where Drinfeld symmetric spaces are generalized to other classical groups. By now there is wealth of literature studying and using Rapoport-Zink spaces. – Torsten Wedhorn Apr 1 '11 at 9:20

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