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

Xu Shen, Ruishen Zhao

Abstract

Let $G$ be a connected reductive group over a $p$-adic local field $F$. Rémy-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 Rémy-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 $μ=(1^d, 0^{n-d})$ 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.

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

Abstract

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

Paper Structure

This paper contains 19 sections, 29 theorems, 217 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bruhat-Tits buildings and $p$-adic flag varieties
  3. Embeddings of Bruhat-Tits buildings into Berkovich flag varieties
  4. Berkovich compactifications of Bruhat-Tits buildings
  5. Bruhat-Tits buildings and $p$-adic period domains
  6. Berkovich spaces and adic spaces
  7. Newton and Harder-Narasimhan stratifications of $p$-adic flag varieties
  8. Embeddings of Bruhat-Tits buildings into $p$-adic period domains
  9. Comparison of boundaries
  10. Functorialities of Newton stratifications
  11. Strongly regular elements in the Kottwitz set
  12. Boundaries of Berkovich compactifications and non basic Newton strata
  13. Retraction maps for $\mathrm{GL}_n$
  14. $p$-adic period domain and the stable locus
  15. The retraction map for a maximal torus
  16. ...and 4 more sections

Key Result

Theorem 1.1

The continuous map $\theta$ factors through the open Newton stratum $\mathcal{F}\ell(G,\mu)^{b_0}$, i.e. $\theta(\mathcal{B}(G,F))\subset \mathcal{F}\ell(G,\mu)^{b_0}$.

Figures (1)

  • Figure 1: Example

Theorems & Definitions (60)

  • Theorem 1.1: Theorem \ref{['thm BT vs p-adic period']}
  • Theorem 1.2: Theorem \ref{['thm boundaries']}
  • Theorem 1.3: Theorem \ref{['thm retract continuous']}, Theorem \ref{['retract']}, Proposition \ref{['prop retract Drinfeld']}
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5: rtw2010 Theorem 4.1, Propositions 4.5, 4.6
  • Proposition 2.6
  • proof
  • ...and 50 more