Bruhat-Tits buildings and $p$-adic period domains

TL;DR

This paper explores the embedding of Bruhat-Tits buildings into $p$-adic period domains, showing how these embeddings relate to compactifications, Newton strata, and retraction maps, thus deepening understanding of $p$-adic geometry.

Contribution

It demonstrates that the Rémery-Thuillier-Werner embeddings factor through $p$-adic Hodge-Tate period domains and constructs a retraction map for certain cases, linking buildings with $p$-adic period spaces.

Findings

Embeddings factor through $p$-adic Hodge-Tate period domains.

Boundaries of compactifications relate to non-basic Newton strata.

Constructed a retraction map for $ ext{GL}_n$ cases.

Abstract

Let $G$ be a connected reductive group over a $p$-adic local field $F$. R\'emy-Thuillier-Werner constructed embeddings of the (reduced) Bruhat-Tits building $\mathcal{B}(G,F)$ into the Berkovich spaces associated to suitable flag varieties of $G$, generalizing the work of Berkovich in split case. They defined compactifications of $\mathcal{B}(G,F)$ by taking closure inside these Berkovich flag varieties. We show that, in the setting of a basic local Shimura datum, the R\'emy-Thuillier-Werner embedding factors through the associated $p$-adic Hodge-Tate period domain. Moreover, we compare the boundaries of the Berkovich compactification of $\mathcal{B}(G,F)$ with non basic Newton strata. In the case of $\mathrm{GL}_n$ and the cocharacter $\mu=(1^d, 0^{n-d})$ for an integer $d$ which is coprime to $n$, we further construct a continuous retraction map from the $p$-adic period domain to the building. This reveals new information on these $p$-adic period domains, which share many similarities with the Drinfeld spaces.