Finite group actions on reductive groups and buildings and tamely-ramified descent in Bruhat-Tits theory

TL;DR

This paper introduces a new approach to tamely-ramified descent in Bruhat-Tits theory, enhancing understanding of reductive groups and their actions on buildings, building on Rousseau's foundational work.

Contribution

It presents a novel method for tamely-ramified descent in Bruhat-Tits theory, expanding the theoretical framework for reductive group actions on buildings.

Findings

New approach to tamely-ramified descent

Improved understanding of group actions on buildings

Extension of Rousseau's foundational work

Abstract

The purpose of the paper is to give a new approach to tamely-ramified descent in Bruhat-Tits theory. This descent was first studied by Guy Rousseau in his thesis.

Equations

$$\mathcal{O}[\mathcal{G}]=\{f\in K[G]|f(\mathcal{G}(\mathcal{O}))\subset \mathcal{O}\}.$$

$$\mathcal{O}[{\mathcal{H}}_{\Omega }]=\{f\in K[H]|f(H(K{)}^{\Omega })\subset \mathcal{O}\}.$$

$$\mathcal{O}[\mathcal{G}]=\{f\in K[G]|f(\mathcal{G})\subset \mathcal{O}\}.$$

$$\mathcal{O}[{\mathcal{G}}^{\circ }]=\{f\in K[G]|f(\mathcal{P})\subset \mathcal{O}\}.$$

$$\mathrm{For}f\in {\underline{\mathrm{Hom}}}_{\mathrm{Spec}(\mathcal{O})\text{textendash}\mathrm{gr}}(\mathcal{S},\mathcal{H}),\theta \in \Theta ,\mathrm{and}s\in \mathcal{S}(C),(\theta \cdot f)(s)=\theta (f({\theta }^{-1}(s))),$$

$$\chi (\phi (z))=-\omega (\chi (z))\mathrm{for}\mathrm{all}\chi \in {\mathrm{X}}_K^{\ast }(Z)(\hookrightarrow {\mathrm{X}}_K^{\ast }(S)),$$

$${\mathrm{Hom}}_K({\mathrm{GL}}_1,T)={\mathrm{X}}_{\ast }(T)\to {\mathrm{X}}_{\ast }(nT{n}^{-1})={\mathrm{Hom}}_K({\mathrm{GL}}_1,nT{n}^{-1}),$$

$$\chi (f(z))=-\omega (\chi (z))\mathrm{for}\mathrm{all}\chi \in {\mathrm{X}}_K^{\ast }(T).$$

$$\mathcal{O}[{\mathcal{G}}_{z_j}^{\circ }]=\{f\in K[G]|f({\mathcal{G}}_{F_{j-1}}^{\circ }(\mathcal{O}))\subset \mathcal{O}\}=\mathcal{O}[{\mathcal{G}}_{F_{j-1}}^{\circ }],$$

$$R⇝\{h\in {\mathcal{H}}_y(R)|\underset{t\to 0}{\lim }\lambda (t)h\lambda (t{)}^{-1}=1\},$$

$$\mathcal{C}={\mathcal{C}}_0,{\mathcal{C}}_1,\dots ,{\mathcal{C}}_m={\mathcal{C}}^{\prime }$$

Full text

By Gopal Prasad

Dedicated to Guy Rousseau

Introduction. This paper is a sequel to our recent paper [P2]. We will assume familiarity with that paper and will freely use notions and results given in it.

Let $\mathcal{O}$ be a discretely valued Henselian local ring with valuation $\omega$. Let $\mathfrak{m}$ be the maximal ideal of $\mathcal{O}$ and $K$ the field of fractions of $\mathcal{O}$. We will assume throughout that the residue field $\kappa$ of $\mathcal{O}$ is separably closed. Let $\widehat{\mathcal{O}}$ denote the completion of $\mathcal{O}$ with respect to the valuation $\omega$ and $\widehat{K}$ the completion of $K$. For any $\mathcal{O}$-scheme $\mathscr{X}$, $\mathscr{X}(\mathcal{O})$ and $\mathscr{X}(\widehat{\mathcal{O}})$ will always be assumed to carry the Hausdorff-topology induced from the metric-space topology on $\mathcal{O}$ and $\widehat{\mathcal{O}}$ respectively. It is known that if $\mathscr{X}$ is smooth, then $\mathscr{X}(\mathcal{O})$ is dense in $\mathscr{X}(\widehat{\mathcal{O}})$, [GGM, Prop. 3.5.2]. Similarly, for any $K$-variety ${\mathcal{X}}$, ${\mathcal{X}}(K)$ and ${\mathcal{X}}(\widehat{K})$ will be assumed to carry the Hausdorff-topology induced from the metric-space topology on $K$ and $\widehat{K}$ respectively. In case ${\mathcal{X}}$ is a smooth $K$-variety, ${\mathcal{X}}(K)$ is dense in ${\mathcal{X}}(\widehat{K})$, [GGM, Prop. 3.5.2].

Throughout this paper $H$ will denote a connected reductive $K$-group and $\mathscr{D}(H)$ its derived subgroup. In this introduction, and beginning with §2 everywhere, we will assume that Bruhat-Tits theory is available for $H$ over $K$ [P2, 1.9, 1.10]. It is known (cf. [P2, 3.11, 1.11]) that then Bruhat-Tits theory is also available over $K$ for the centralizer of any $K$-split torus in $H$ and for the derived subgroup of such centralizers. Thus there is an affine building called the Bruhat-Tits building of $H(K)$, that is a polysimplicial complex given with a metric, and $H(K)$ acts on it by polysimplicial isometries. This building is also the Bruhat-Tits building of $\mathscr{D}(H)(K)$ and we will denote it by ${\mathcal{B}}(\mathscr{D}(H)/K)$.

Let $\mathfrak{Z}$ be the maximal $K$-split torus in the center of $H$. Let $V(\mathfrak{Z})={\mathbb{R}}\otimes_{{\mathbb{Z}}}{\rm{Hom}}_{K}({\rm{GL}}_{1},\mathfrak{Z})$. Then there is a natural action of $H(K)$ on this Euclidean space by translations, with $\mathscr{D}(H)(K)$, as well as every bounded subgroup of $H(K)$, acting trivially (cf. [P2, 1.9]). The enlarged Bruhat-Tits building ${\mathcal{B}}(H/K)$ of $H(K)$ is the product $V(\mathfrak{Z})\times{\mathcal{B}}(\mathscr{D}(H)/K)$. The apartments of this building, as well as that of ${\mathcal{B}}(\mathscr{D}(H)/K)$, are in bijective correspondence with maximal $K$-split tori of $H$. Given a maximal $K$-split torus $T$ of $H$, the corresponding apartment of ${\mathcal{B}}(H/K)$ is an affine space under $V(T):={\mathbb{R}}\otimes_{{\mathbb{Z}}}{\rm{Hom}}_{K}({\rm{GL}}_{1},T)$.

Given a nonempty bounded subset $\Omega$ of an apartment of ${\mathcal{B}}(H/K)$, or of ${\mathcal{B}}(\mathscr{D}(H)/K)$, there is a smooth affine $\mathcal{O}$-group scheme ${\mathscr{H}}_{\Omega}$ with generic fiber $H$, associated to $\Omega$, such that ${\mathscr{H}}_{\Omega}(\mathcal{O})$ is the subgroup $H(K)^{\Omega}$ of $H(K)$ consisting of elements that fix $\Omega$ and $V(\mathfrak{Z})$ pointwise. The neutral component ${\mathscr{H}}^{\circ}_{\Omega}$ of ${\mathscr{H}}_{\Omega}$ is an open affine $\mathcal{O}$-subgroup scheme of the latter; it is by definition the union of the generic fiber $H$ of ${\mathscr{H}}_{\Omega}$ and the identity component of its special fiber. The group scheme ${\mathscr{H}}^{\circ}_{\Omega}$ is called the Bruhat-Tits group scheme associated to $\Omega$. It has the properties described in [P2, 1.9 -1.10]. The special fiber of ${\mathscr{H}}^{\circ}_{\Omega}$ will be denoted by $\overline{\mathscr{H}}^{\circ}_{\Omega}$.

Let $\Theta$ be a finite group of automorphisms of $H$. We assume that the order of $\Theta$ is not divisible by the characteristic of the residue field $\kappa$. Let $G=(H^{\Theta})^{\circ}$. This group is also reductive, see [Ri, Prop. 10.1.5] or [PY1, Thm. 2.1]. The goal of this paper is to show that Bruhat-Tits theory is available for $G$ over $K$, and the enlarged Bruhat-Tits building of $G(K)$ can be identified with the subspace ${\mathcal{B}}(H/K)^{\Theta}$ of ${\mathcal{B}}(H/K)$ consisting of points fixed under $\Theta$ (see §3). These results have been inspired by the main theorem of [PY1], which implies that if the residue field $\kappa$ is algebraically closed (then every reductive $K$-group is quasi-split [P2, 1.7], so Bruhat-Tits theory is available for any such group over $K$), the enlarged Bruhat-Tits building of $G(K)$ is indeed ${\mathcal{B}}(H/K)^{\Theta}$.

In §4, we will use the above results to obtain “tamely-ramified descent”: (1) We will show that if a connected reductive $K$-group $G$ is quasi-split over a finite tamely-ramified extension $L$ of $K$, then it is quasi-split over $K$ (Theorem 4.4); this result has been proved by Philippe Gille in [Gi] by an entirely different method. (2) The enlarged Bruhat-Tits building ${\mathcal{B}}(G/K)$ of $G(K)$ can be identified with the subspace of points of the enlarged Bruhat-Tits building of $G(L)$ that are fixed under the action of the Galois group ${\rm{Gal}}(L/K)$. This latter result was proved by Guy Rousseau in his unpublished thesis [Rou, Prop. 5.1.1]. It is a pleasure to dedicate this paper to him for his important contributions to Bruhat-Tits theory.

Acknowledgements. I thank Brian Conrad, Bas Edixhoven and Philippe Gille for their helpful comments. I thank the referee for carefully reading the paper and for her/his detailed comments and suggestions which helped to improve the exposition. I was partially supported by NSF-grant DMS-1401380.

For a $K$-split torus $S$, let ${\rm X}_{\ast}(S)={\rm{Hom}}({\rm{GL}}_{1},S)$ and $V(S):={\mathbb{R}}\otimes_{{\mathbb{Z}}}{\rm X}_{\ast}(S)$. Then for a maximal $K$-split torus $T$ of $H$, the apartment $A(T)$ of ${\mathcal{B}}(H/K)$ corresponding to $T$ is an affine space under $V(T)$.

1. Passage to completion

We begin by proving the following well-known result.

Proposition 1.1. $K$-rank $H$ = $\widehat{K}$-rank $H$.

Proof. Let $T$ be a maximal $K$-split torus of $H$ and $Z$ be its centralizer in $H$. Let $Z_{\rm a}$ be the maximal $K$-anisotropic connected normal subgroup of $Z$. Then

$\widehat{K}$-rank $H=\widehat{K}$-rank $Z=\dim(T)+$ $\widehat{K}$-rank $Z_{\rm a}=$ $K$-rank H+$$\widehat{K}-rank $Z_{\rm a}$ .

So to prove the proposition, it suffices to show that $Z_{\rm a}$ is anisotropic over $\widehat{K}$. But according to Theorem 1.1 of [P2], $Z_{\rm a}$ is anisotropic over $\widehat{K}$ if and only if $Z_{\rm a}(\widehat{K})$ is bounded. The same theorem implies that $Z_{\rm a}(K)$ is bounded. As $Z_{\rm a}(K)$ is dense in $Z_{\rm a}(\widehat{K})$, we see that $Z_{\rm a}(\widehat{K})$ is bounded. $\Box$

Proposition 1.2. Bruhat-Tits theory for $H$ is available over $K$ if and only if it is available over $\widehat{K}$. Moreover, if Bruhat-Tits theory for $H$ is available over $K$, then the enlarged Bruhat-Tits buildings of $H(K)$ and $H(\widehat{K})$ are equal.

It was shown by Guy Rousseau in his thesis that the enlarged Bruhat-Tits buildings of $H(K)$ and $H(\widehat{K})$ coincide [Rou, Prop. 2.3.5]. Moreover, every apartment in the building of $H(K)$ is also an apartment in the building of $H(\widehat{K})$; however, the latter may have many more apartments.

Proof. We assume first that Bruhat-Tits theory is available for $H$ over $K$ and let ${\mathcal{B}}(\mathscr{D}(H)/K)$ and ${\mathcal{B}}(H/K)$ respectively denote the Bruhat-Tits building and the enlarged Bruhat-Tits building of $H(K)$. We begin by showing that the actions of $H(K)$ on ${\mathcal{B}}(\mathscr{D}(H)/K)$ and ${\mathcal{B}}(H/K)$ extend to actions of $H(\widehat{K})$ by isometries. For this purpose, we recall that $H(K)$ is dense in $H(\widehat{K})$ and the isotropy at any point of the two buildings is a bounded open subgroup of $H(K)$. Now let $\{h_{i}\}$ be a sequence in $H(K)$ which converges to a point $\widehat{h}\in H(\widehat{K})$, then given any open subgroup of $H(K)$, for all large $i$ and $j$, $h_{i}^{-1}h_{j}$ lies in this open subgroup. Thus for any point $x$ of ${\mathcal{B}}(\mathscr{D}(H)/K)$ or ${\mathcal{B}}(H/K)$, the sequence $h_{i}\cdot x$ is eventually constant, i.e., there exists a positive integer $n$ such that $h_{i}\cdot x=h_{n}\cdot x$ for all $i\geqslant n$. We define $\widehat{h}\cdot x=h_{n}\cdot x$. This gives well-defined actions of $H(\widehat{K})$ on ${\mathcal{B}}(\mathscr{D}(H)/K)$ and ${\mathcal{B}}(H/K)$ by isometries.

For a nonempty bounded subset $\Omega$ of an apartment of ${\mathcal{B}}(H/K)$, or of ${\mathcal{B}}(\mathscr{D}(H)/K)$, let ${\mathscr{H}}_{\Omega}$ and ${\mathscr{H}}^{\circ}_{\Omega}$ be the smooth affine $\mathcal{O}$-group schemes as in the Introduction. Then as ${\mathscr{H}}_{\Omega}(\widehat{\mathcal{O}})$ is a closed and open subgroup of $H(\widehat{K})$ containing ${\mathscr{H}}_{\Omega}(\mathcal{O})$ as a dense subgroup, we see that ${\mathscr{H}}_{\Omega}(\widehat{\mathcal{O}})$ equals the subgroup $H(\widehat{K})^{\Omega}$ of $H(\widehat{K})$ consisting of elements that fix $\Omega$, as well as $V(\mathfrak{Z})$, pointwise.

Let $T$ be a maximal $K$-split torus of $H$, then by Proposition 1.1, $T_{\widehat{K}}$ is a maximal $\widehat{K}$-split torus of $H_{\widehat{K}}$. Let $A$ be the apartment of ${\mathcal{B}}(\mathscr{D}(H)/K)$, or of ${\mathcal{B}}(H/K)$, corresponding to $T$. Then every maximal $\widehat{K}$-split torus of $H_{\widehat{K}}$ is of the form $\widehat{h}T_{\widehat{K}}\widehat{h}^{-1}$ for an $\widehat{h}\in H(\widehat{K})$, and we define the corresponding apartment to be $\widehat{h}\cdot A$. We now declare ${\mathcal{B}}(\mathscr{D}(H)/K)$ (resp. ${\mathcal{B}}(H/K)$) to be the Bruhat-Tits building (resp. the enlarged Bruhat-Tits building) of $H(\widehat{K})$ with these apartments.

Let $A$ be an apartment of ${\mathcal{B}}(\mathscr{D}(H)/K)$, or of ${\mathcal{B}}(H/K)$, corresponding to a maximal $K$-split torus $T$ of $H$ and $\widehat{h}\in H(\widehat{K})$. Given a nonempty bounded subset $\widehat{\Omega}$ of $\widehat{A}:=\widehat{h}\cdot A$, the subset $\Omega:=\widehat{h}^{-1}\cdot{\widehat{\Omega}}$ is contained in $A$. The closed and open subgroup $\widehat{h}H(\widehat{K})^{\Omega}\widehat{h}^{-1}=\widehat{h}{\mathscr{H}}_{\Omega}(\widehat{\mathcal{O}})\widehat{h}^{-1}$ of $H(\widehat{K})$ is the subgroup $H(\widehat{K})^{\widehat{\Omega}}$ consisting of elements that fix $\widehat{\Omega}$, as well as $V(\mathfrak{Z})$, pointwise. Now as $H(K)$ is dense in $H(\widehat{K})$ and $H(\widehat{K})^{\widehat{\Omega}}$ is an open subgroup, $H(\widehat{K})=H(\widehat{K})^{\widehat{\Omega}}\cdot H(K)$, so $\widehat{h}=h^{\prime}\cdot h$, with $h^{\prime}\in H(\widehat{K})^{\widehat{\Omega}}$ and $h\in H(K)$. Thus the apartment $\widehat{A}=\widehat{h}\cdot A=h^{\prime}\cdot hA$, and $hA$ is an apartment of the Bruhat-Tits building of $H(K)$ (or of the enlarged building of $H(K)$). As $h^{\prime}\in H(\widehat{K})^{\widehat{\Omega}}$, the apartment $hA$ contains $\widehat{\Omega}$. This shows that any bounded subset $\widehat{\Omega}$ of an apartment of the Bruhat-Tits building of $H(\widehat{K})$ (or of the enlarged building of $H(\widehat{K})$) is contained in an apartment of the Bruhat-Tits building of $H(K)$ (or of the enlarged building of $H(K)$). We define the $\widehat{\mathcal{O}}$-group schemes ${\mathscr{H}}_{\widehat{\Omega}}$ and ${\mathscr{H}}^{\circ}_{\widehat{\Omega}}$ associated to $\widehat{\Omega}$ to be the group schemes obtained from the corresponding $\mathcal{O}$-group schemes (given by considering $\widehat{\Omega}$ to be a nonempty bounded subset of an apartment of the building, or of the enlarged building, of $H(K)$) by extension of scalars $\mathcal{O}\hookrightarrow\widehat{\mathcal{O}}$.

Let us assume now that Bruhat-Tits theory is available for $H$ over $\widehat{K}$. Then Bruhat-Tits theory is also available for $\mathscr{D}(H)$ over $\widehat{K}$ [P2, 1.11]. The action of $H(\widehat{K})$ on its building ${\mathcal{B}}(\mathscr{D}(H)/\widehat{K})$ restricts to an action of $H(K)$ by isometries. Let $T$ be a maximal $K$-split torus of $G$ and $A$ be the apartment of ${\mathcal{B}}(\mathscr{D}(H)/\widehat{K})$ corresponding to $T_{\widehat{K}}$. We consider the polysimplicial complex ${\mathcal{B}}(\mathscr{D}(H)/\widehat{K})$, with apartments $h\cdot A$, $h\in H(K)$, as the building of $H(K)$ and denote it by ${\mathcal{B}}(\mathscr{D}(H)/K)$. The enlarged Bruhat-Tits building of $H(K)$ is by definition the product $V(\mathfrak{Z})\times{\mathcal{B}}(\mathscr{D}(H)/K)$.

Let $\widehat{\Omega}$ be a nonempty bounded subset of the apartment ${\widehat{A}}=\widehat{h}\cdot A$, $\widehat{h}\in H(\widehat{K})$, of the building ${\mathcal{B}}(\mathscr{D}(H)/\widehat{K})$, or of the enlarged building ${\mathcal{B}}(H/\widehat{K})$. As $H(K)$ is dense in $H(\widehat{K})$, the intersection ${\mathscr{H}}_{\widehat{\Omega}}(\widehat{\mathcal{O}})\widehat{h}\cap H(K)$ is nonempty. For any $h$ in this intersection, $\widehat{\Omega}$ is contained in the apartment $h\cdot A$ of ${\mathcal{B}}(\mathscr{D}(H)/K)$ (or of ${\mathcal{B}}(H/K)$). This implies, in particular, that any two facets lie on an apartment of ${\mathcal{B}}(\mathscr{D}(H)/K)$. We now note that the $\widehat{\mathcal{O}}$-group schemes ${\mathscr{H}}_{\widehat{\Omega}}$ and ${\mathscr{H}}^{\circ}_{\widehat{\Omega}}$ admit unique descents to smooth affine $\mathcal{O}$-group schemes with generic fiber $H$, [BLR, Prop. D.4(b) in §6.1]; the affine rings of these descents are $K[H]\cap\widehat{\mathcal{O}}[{\mathscr{H}}_{\widehat{\Omega}}]$ and $K[H]\cap\widehat{\mathcal{O}}[{\mathscr{H}}^{\circ}_{\widehat{\Omega}}]$ respectively. $\Box$

2. Fixed points in ${\mathcal{B}}(H/K)$ under a finite automorphism group $\Theta$ of $H$

We will henceforth assume that Bruhat-Tits theory is available for $H$ over $K$.

2.1. Let $G$ be a smooth affine $K$-group and ${\mathscr{G}}$ be a smooth affine $\mathcal{O}$-group scheme with generic fiber $G$. According to [BrT2, 1.7.1-1.7.2] ${\mathscr{G}}$ is “étoffé” and hence by (ET) of [BrT2, 1.7.1] its affine ring has the following description:

[TABLE]

Let $\Omega$ be a nonempty bounded subset of an apartment of ${\mathcal{B}}(\mathscr{D}(H)/K)$, or of ${\mathcal{B}}(H/K)$. As the $\mathcal{O}$-group scheme ${\mathscr{H}}_{\Omega}$ is smooth and affine and its generic fiber is $H$, the affine ring of ${\mathscr{H}}_{\Omega}$ has thus the following description:

[TABLE]

Proposition 2.2. Let $\Omega$ be a nonempty bounded subset of an apartment of ${\mathcal{B}}(H/K)$, or of ${\mathcal{B}}(\mathscr{D}(H)/K)$. Let ${\mathscr{H}}_{\Omega}$ and ${\mathscr{H}}^{\circ}_{\Omega}$ be as above. Let $G$ be a smooth connected $K$-subgroup of $H$ and ${\mathscr{G}}$ be a smooth affine $\mathcal{O}$-group scheme with generic fiber $G$ and connected special fiber. Assume that a subgroup ${\mathcal{G}}$ of  ${\mathscr{G}}(\mathcal{O})$ of finite index fixes $\Omega$ pointwise $($so, ${\mathcal{G}}\subset H(K)^{\Omega})$. Then there is a $\mathcal{O}$-group scheme homomorphism $\varphi:\,{\mathscr{G}}\rightarrow{\mathscr{H}}^{\circ}_{\Omega}$ that is the natural inclusion $G\hookrightarrow H$ on the generic fibers. So the subgroup ${\mathscr{G}}(\mathcal{O})$ of $G(K)$ is contained in ${\mathscr{H}}^{\circ}_{\Omega}(\mathcal{O})$ and hence it fixes $\Omega$ pointwise. If $F$ is a facet of ${\mathcal{B}}(\mathscr{D}(H)/K)$ that meets $\Omega$, then ${\mathscr{G}}(\mathcal{O})$ fixes $F$ pointwise.

*Let $S$ be a $K$-split torus of $H$ and ${\mathscr{S}}$ the split $\mathcal{O}$-torus with generic fiber $S$. If a subgroup of the maximal bounded subgroup ${\mathscr{S}}(\mathcal{O})$ of $S(K)$ of finite index fixes $\Omega$ pointwise, then there is a maximal $K$-split torus $T$ of $H$ containing $S$ such that $\Omega$ is contained in the apartment corresponding to $T$. *

Proof. Since the fibers of the smooth affine group scheme ${\mathscr{G}}$ are connected and the residue field $\kappa$ is separably closed, the subgroup ${\mathcal{G}}$ is Zariski-dense in $G$, and its image in ${\mathscr{G}}(\kappa)$ is Zariski-dense in the spacial fiber of ${\mathscr{G}}$. Using this observation, we easily see that the affine ring $\mathcal{O}[{\mathscr{G}}]\,(\subset K[G])$ of ${\mathscr{G}}$ has the following description (cf.  [BrT2, 1.7.2]):

[TABLE]

This description of $\mathcal{O}[{\mathscr{G}}]$ implies at once that the inclusion ${\mathcal{G}}\hookrightarrow H(K)^{\Omega}$ induces a $\mathcal{O}$-group scheme homomorphism $\varphi:\,{\mathscr{G}}\rightarrow{\mathscr{H}}_{\Omega}$ that is the natural inclusion $G\hookrightarrow H$ on the generic fibers (cf. [BrT2, Prop. 1.7.6]). Since ${\mathscr{G}}$ has connected fibers, the homomorphism $\varphi$ factors through ${\mathscr{H}}^{\circ}_{\Omega}$.

Any facet $F$ of ${\mathcal{B}}(\mathscr{D}(H)/K)$ that meets $\Omega$ is stable under ${\mathscr{G}}(\mathcal{O})\,(\subset H(K))$, so a subgroup of ${\mathscr{G}}(\mathcal{O})$ of finite index fixes it pointwise. Now applying the result of the preceding paragraph, for $F$ in place of $\Omega$, we see that there is a $\mathcal{O}$-group scheme homomorphism ${\mathscr{G}}\rightarrow{\mathscr{H}}^{\circ}_{F}$ that is the natural inclusion $G\hookrightarrow H$ on the generic fibers and hence ${\mathscr{G}}(\mathcal{O})$ fixes $F$ pointwise.

Now we will prove the last assertion of the proposition. It follows from what we have shown above that there is a $\mathcal{O}$-group scheme homomorphism $\iota:{\mathscr{S}}\rightarrow{\mathscr{H}}^{\circ}_{\Omega}$ that is the natural inclusion $S\hookrightarrow H$ on the generic fibers ($\iota$ is actually a closed immersion, see [SGA3II, Exp. IX, Thm. 6.8 and Cor. 2.5] and also the proof of Lemma 4.1 in [PY2]). Applying [P2, Prop. 2.1(i)] to the centralizer of $\iota({\mathscr{S}})$ (in ${\mathscr{H}}^{\circ}_{\Omega}$) in place of ${\mathscr{G}}$, and $\mathcal{O}$ in place of $\mathfrak{o}$, we see that there is a closed $\mathcal{O}$-torus ${\mathscr{T}}$ of ${\mathscr{H}}^{\circ}_{\Omega}$ that commutes with $\iota({\mathscr{S}})$ and whose generic fiber $T$ is a maximal $K$-split torus of $H$. The torus $T$ clearly contains $S$, and [P2, Prop. 2.2(ii)] implies that $\Omega$ is contained in the apartment corresponding to $T$. $\Box$

The following is a simple consequence of the preceding proposition.

Corollary 2.3. Let $G$, $S$, ${\mathscr{G}}$, and ${\mathscr{S}}$ be as in the preceding proposition. Then the set of points of ${\mathcal{B}}(\mathscr{D}(H)/K)$ that are fixed under ${\mathscr{G}}(\mathcal{O})$ is the union of facets pointwise fixed under ${\mathscr{G}}(\mathcal{O})$. The set of points of the enlarged building ${\mathcal{B}}(H/K)$ that are fixed under a finite-index subgroup $\mathcal{S}$ of the maximal bounded subgroup $S(K)_{b}\,(={\mathscr{S}}(\mathcal{O}))$ of $S(K)$ is the enlarged Bruhat-Tits building ${\mathcal{B}}(Z_{H}(S)/K)$ of the centralizer $Z_{H}(S)(K)$ of $S$ in $H(K)$.

2.4. Let $\Theta$ be a finite group of automorphisms of the reductive $K$-group $H$. There is a natural action of $\Theta$ on ${\mathcal{B}}(H/K)$, as well as on ${\mathcal{B}}(\mathscr{D}(H)/K)$, by isometries such that for all $h$ in $H(K)$, $x$ in ${\mathcal{B}}(H/K)$ or ${\mathcal{B}}(\mathscr{D}(H)/K)$, and $\theta$ in $\Theta$, we have $\theta(h\cdot x)=\theta(h)\cdot\theta(x)$. The action of $\Theta$ on ${\mathcal{B}}(\mathscr{D}(H)/K)$ is polysimplicial.

In the following we will assume that the characteristic $p$ of the residue field $\kappa$ does not divide the order of $\Theta$. Then $G:=(H^{\Theta})^{\circ}$ is a reductive group, see [Ri, Prop. 10.1.5] or [PY1, Thm. 2.1]. We will prove that Bruhat-Tits theory is available for $G$ over $K$ and the enlarged Bruhat-Tits building of $G(K)$, as a metric space, can be identified with the closed subspace ${\mathcal{B}}(H/K)^{\Theta}$ of points of ${\mathcal{B}}(H/K)$ fixed under $\Theta$.

For a subset $X$ of a set given with an action of $\Theta$, $X^{\Theta}$ will denote the subset of points of $X$ that are fixed under $\Theta$. We will denote ${\mathcal{B}}(\mathscr{D}(H)/K)^{\Theta}$ by $\mathscr{B}$ in the sequel.

If a facet of ${\mathcal{B}}(\mathscr{D}(H)/K)$ is stable under the action of $\Theta$, then its barycenter is fixed under $\Theta$. Conversely, if a facet $F$ of ${\mathcal{B}}(\mathscr{D}(H)/K)$ contains a point $x$ fixed under $\Theta$, then being the unique facet containing $x$, $F$ is stable under the action of $\Theta$.

2.5. (i) Bruhat-Tits group schemes ${\mathscr{G}}^{\circ}_{\Omega}$. Let $\Omega$ be a nonempty $\Theta$-stable bounded subset of an apartment of ${\mathcal{B}}(\mathscr{D}(H)/K)$, or of ${\mathcal{B}}(H/K)$. Then ${\mathscr{H}}_{\Omega}(\mathcal{O})\,(=H(K)^{\Omega})$ is stable under the action of $\Theta$ on $H(K)$, so the affine ring $\mathcal{O}[{\mathscr{H}}_{\Omega}]$ is stable under the action of $\Theta$ on $K[H]$. This implies that $\Theta$ acts on the group scheme ${\mathscr{H}}_{\Omega}$ by $\mathcal{O}$-group scheme automorphisms. The neutral component ${\mathscr{H}}^{\circ}_{\Omega}$ of ${\mathscr{H}}_{\Omega}$ is of course stable under this action. Define the functor ${\mathscr{H}}_{\Omega}^{\Theta}$ of $\Theta$-fixed points that associates to a commutative $\mathcal{O}$-algebra $C$ the subgroup ${\mathscr{H}}_{\Omega}(C)^{\Theta}$ of ${\mathscr{H}}_{\Omega}(C)$ consisting of elements fixed under $\Theta$. The functor ${\mathscr{H}}_{\Omega}^{\Theta}$ is represented by a closed smooth $\mathcal{O}$-subgroup scheme of ${\mathscr{H}}_{\Omega}$ (see Propositions 3.1 and 3.4 of [E], or Proposition A.8.10 of [CGP]); we will denote this closed smooth $\mathcal{O}$-subgroup scheme also by ${\mathscr{H}}_{\Omega}^{\Theta}$. Its generic fiber is $H^{\Theta}$, and so the identity component of the generic fiber is $G$. The neutral component ${({\mathscr{H}}_{\Omega}^{\Theta})}^{\circ}$ of ${\mathscr{H}}_{\Omega}^{\Theta}$ is by definition the union of the identity components of its generic and special fibers; it is an open (so smooth) affine $\mathcal{O}$-subgroup scheme [PY2, §3.5] with generic fiber $G$. The index of the subgroup ${({\mathscr{H}}_{\Omega}^{\Theta})}^{\circ}(\mathcal{O})$ in ${\mathscr{H}}_{\Omega}^{\Theta}(\mathcal{O})$ is known to be finite [EGA IV3, Cor. 15.6.5]. It is obvious that ${({\mathscr{H}}_{\Omega}^{\Theta})}^{\circ}=(({\mathscr{H}}^{\circ}_{\Omega})^{\Theta})^{\circ}$. We will denote $({\mathscr{H}}_{\Omega}^{\Theta})^{\circ}$ by ${\mathscr{G}}^{\circ}_{\Omega}$ in the sequel and call it the Bruhat-Tits $\mathcal{O}$-group scheme associated to $G$ and $\Omega$. As ${\mathscr{G}}^{\circ}_{\Omega}(\mathcal{O})\subset{\mathscr{H}}_{\Omega}(\mathcal{O})$, ${\mathscr{G}}^{\circ}_{\Omega}(\mathcal{O})$ fixes $\Omega$ pointwise.

(ii) Parahoric subgroups of $G(K)$. For $x\in\mathscr{B}$, ${\mathscr{G}}^{\circ}_{\{x\}}$ and $P_{x}:={\mathscr{G}}^{\circ}_{\{x\}}(\mathcal{O})$ will respectively be called the Bruhat-Tits parahoric $\mathcal{O}$-group scheme and the parahoric subgroup of $G(K)$ associated with the point $x$. The parahoric subgroups of $G(K)$ are by definition the subgroups $P_{x}$ for $x\in\mathscr{B}$. For a given parahoric subgroup $P_{x}$, the associated Bruhat-Tits parahoric $\mathcal{O}$-group scheme is ${\mathscr{G}}^{\circ}_{\{x\}}$.

(iii) Let $P$ be a parahoric subgroup of $G(K)$, ${\mathscr{G}}^{\circ}$ the associated Bruhat-Tits parahoric $\mathcal{O}$-group scheme, $\overline{\mathscr{G}}^{\circ}$ the special fiber of ${\mathscr{G}}^{\circ}$, and $\mathcal{P}$ be a subgroup of $P$ of finite index. Then the image of $\mathcal{P}$ in $\overline{\mathscr{G}}^{\circ}(\kappa)$ is Zariski-dense in the connected group $\overline{\mathscr{G}}^{\circ}$, so the affine ring of ${\mathscr{G}}^{\circ}$ is:

[TABLE]

Thus the subgroup $\mathcal{P}$ “determines” the group scheme ${\mathscr{G}}^{\circ}$, and hence $P$ is the unique parahoric subgroup of $G(K)$ containing $\mathcal{P}$ as a subgroup of finite index.

(iv) Let $P$ and ${\mathscr{G}}^{\circ}$ be as in the preceding paragraph. Let $\Omega$ be a nonempty $\Theta$-stable bounded subset of an apartment of ${\mathcal{B}}(\mathscr{D}(H)/K)$, or of ${\mathcal{B}}(H/K)$, and ${\mathscr{G}}^{\circ}_{\Omega}$ be as in (i). We assume that $\Omega$ is fixed pointwise by $P$. Then the inclusion of $P$ in $H(K)^{\Omega}\,(={\mathscr{H}}_{\Omega}(\mathcal{O}))$ gives a $\mathcal{O}$-group scheme homomorphism ${\mathscr{G}}^{\circ}\rightarrow{\mathscr{H}}^{\circ}_{\Omega}$ (Proposition 2.2). This homomorphism obviously factors through ${\mathscr{G}}^{\circ}_{\Omega}$ to give a $\mathcal{O}$-group scheme homomorphism ${\mathscr{G}}^{\circ}\rightarrow{\mathscr{G}}^{\circ}_{\Omega}$ that is the identity on the generic fiber $G$.

Suppose $x,\,y$ are two points of $\mathscr{B}$ that are fixed by $P$, and $[xy]$ is the geodesic joining $x$ and $y$. Then $P$ fixes every point $z$ of $[xy]$. Let ${\mathscr{G}}^{\circ}_{[xy]}$ be as in (i) for $\Omega=[xy]$. There are $\mathcal{O}$-group scheme homomorphisms ${\mathscr{G}}^{\circ}\rightarrow{\mathscr{G}}^{\circ}_{[xy]}$ and ${\mathscr{G}}^{\circ}\rightarrow{\mathscr{G}}^{\circ}_{\{z\}}$ that are the identity on the generic fiber $G$.

We will show that $\mathscr{B}$ is an affine building with apartments described in 2.16. We begin with the following proposition which has been suggested by Proposition 1.1 of [PY1]. The proof given here is different from that in [PY1].

Proposition 2.6. Let $\Omega$ be a nonempty $\Theta$-stable bounded subset of an apartment of ${\mathcal{B}}(\mathscr{D}(H)/K)$. Let ${\mathscr{H}}:={\mathscr{H}}^{\circ}_{\Omega}$ be the Bruhat-Tits smooth affine $\mathcal{O}$-group scheme with generic fiber $H$, and connected special fiber $\overline{\mathscr{H}}$, associated to $\Omega$. Let $\overline{\mathscr{H}}^{\,\rm pred}:=\overline{\mathscr{H}}/{{\mathscr{R}}_{u,\kappa}}(\overline{\mathscr{H}})$ be the maximal pseudo-reductive quotient of $\overline{\mathscr{H}}$. Then there exist closed $\mathcal{O}$-tori ${\mathscr{S}}\subset{\mathscr{T}}$ in ${\mathscr{H}}$ such that

(i)* the generic fiber $T$ of ${\mathscr{T}}$ is a maximal $K$-split torus of $H$ and $\Omega$ is contained in the apartment corresponding to $T$;*

(ii)* ${\mathscr{S}}$, and therefore its generic fiber $S$, is stable under $\Theta$ and the special fiber of ${\mathscr{S}}$ maps onto the central torus of $\overline{\mathscr{H}}^{\,\rm pred}$. *

We first prove the following lemma in which $\kappa$ is any field of characteristic $p\geqslant 0$.

Lemma 2.7. Let $\mathcal{H}$ be a smooth connected affine algebraic $\kappa$-group given with an action by a finite group $\Theta$ and $\mathcal{U}$ be a $\Theta$-stable smooth connected unipotent normal $\kappa$-subgroup of $\mathcal{H}$. We assume that $p$ does not divide the order of $\Theta$. Let $\overline{\mathcal{S}}$ be a $\Theta$-stable $\kappa$-torus of $\overline{\mathcal{H}}:=\mathcal{H}/\mathcal{U}$. Then there exists a $\Theta$-stable $\kappa$-torus $\mathcal{S}$ in $\mathcal{H}$ that maps isomorphically onto $\overline{\mathcal{S}}$. In particular, there exists a $\Theta$-stable $\kappa$-torus in $\mathcal{H}$ that maps isomorphically onto the central torus of $\overline{\mathcal{H}}$.

Proof. Let $\mathcal{T}$ be a $\kappa$-torus of $\mathcal{H}$ that maps isomorphically onto $\overline{\mathcal{S}}\,(\subset\overline{\mathcal{H}})$. Considering the $\Theta$-stable solvable subgroup $\mathcal{T}\,\mathcal{U}$; using conjugacy under $\mathcal{U}(\kappa)$ of maximal $\kappa$-tori of this solvable group [Bo, Thm. 19.2], we see that for $\theta\in\Theta$, $\theta(\mathcal{T})=u(\theta)^{-1}\mathcal{T}u(\theta)$ for some $u(\theta)\in\mathcal{U}(\kappa)$. Let $\mathcal{U}(\kappa)=:\mathcal{U}_{0}\supset\mathcal{U}_{1}\supset\mathcal{U}_{2}\cdots\supset\mathcal{U}_{n}=\{1\}$ be the descending central series of the nilpotent group $\mathcal{U}(\kappa)$. Each subgroup $\mathcal{U}_{i}$ is $\Theta$-stable and $\mathcal{U}_{i}/\mathcal{U}_{i+1}$ is a commutative $p$-group if $p\neq 0$, and a ${\mathbb{Q}}$-vector space if $p=0$. Now let $i\leqslant n$, be the largest integer such that there exists a $\kappa$-torus $\mathcal{S}$ in $\mathcal{T}\,\mathcal{U}$ that maps onto $\overline{\mathcal{S}}$, and for every $\theta\in\Theta$, there is a $u(\theta)\in\mathcal{U}_{i}$ such that $\theta(\mathcal{S})=u(\theta)^{-1}\mathcal{S}u(\theta)$. Let $\mathcal{N}_{i}$ be the normalizer of $\mathcal{S}$ in $\mathcal{U}_{i}$. Then, for $\theta\in\Theta$, $\theta(\mathcal{N}_{i})=u(\theta)^{-1}\mathcal{N}_{i}u(\theta)$ and hence as $\mathcal{U}_{i}/\mathcal{U}_{i+1}$ is commutative, we see that $\theta(\mathcal{N}_{i}\mathcal{U}_{i+1})=\mathcal{N}_{i}\mathcal{U}_{i+1}$, i.e., $\mathcal{N}_{i}\mathcal{U}_{i+1}$ is $\Theta$-stable. It is easy to see that $\theta\mapsto u(\theta)$ mod $(\mathcal{N}_{i}\mathcal{U}_{i+1})$ is a $1$-cocycle on $\Theta$ with values in $\mathcal{U}_{i}/\mathcal{N}_{i}\mathcal{U}_{i+1}$. But ${\rm{H}}^{1}(\Theta,\,\mathcal{U}_{i}/\mathcal{N}_{i}\mathcal{U}_{i+1})$ is trivial since the finite group $\Theta$ is of order prime to $p$ if $p\neq 0$, and $\mathcal{U}_{i}/\mathcal{N}_{i}\mathcal{U}_{i+1}$ is divisible if $p=0$. So there exits a $u\in\mathcal{U}_{i}$ such that for all $\theta\in\Theta$, $u^{-1}u(\theta)\theta(u)$ lies in $\mathcal{N}_{i}\mathcal{U}_{i+1}$. Now let $\mathcal{S}^{\prime}=u^{-1}\mathcal{S}u$. Then the normalizer of $\mathcal{S}^{\prime}$ in $\mathcal{U}_{i}$ is $u^{-1}\mathcal{N}_{i}u$ and again as $\mathcal{U}_{i}/\mathcal{U}_{i+1}$ is commutative, $u^{-1}\mathcal{N}_{i}u\cdot\mathcal{U}_{i+1}=\mathcal{N}_{i}\mathcal{U}_{i+1}$. For $\theta\in\Theta$, we choose $u^{\prime}(\theta)\in\mathcal{U}_{i+1}$ such that $u^{-1}u(\theta)\theta(u)\in u^{-1}\mathcal{N}_{i}u\cdot u^{\prime}(\theta)$. Then $\theta(\mathcal{S}^{\prime})=u^{\prime}(\theta)^{-1}\mathcal{S}^{\prime}u^{\prime}(\theta)$ for all $\theta\in\Theta$. This contradicts the maximality of $i$ unless $i=n$. $\Box$

We will now prove Proposition 2.6. There is a natural action of $\Theta$ on ${\mathscr{H}}$ by $\mathcal{O}$-group scheme automorphisms (2.5(i)). This action induces an action of $\Theta$ on $\overline{\mathscr{H}}$, as well as on $\overline{\mathscr{H}}^{\rm{pred}}$, by $\kappa$-group automorphisms. According to the preceding lemma, there exists a $\Theta$-stable $\kappa$-torus ${\overline{\mathscr{S}}}$ in $\overline{\mathscr{H}}$ that maps isomorphically onto the central torus of $\overline{\mathscr{H}}^{\rm{pred}}$. We will denote the natural inclusion ${\overline{\mathscr{S}}}\hookrightarrow\overline{\mathscr{H}}$ by $\overline{\iota}$. The character group ${\rm{X}}^{\ast}({\overline{\mathscr{S}}})$ of ${\overline{\mathscr{S}}}$ is a free abelian group on which $\Theta$ acts by automorphisms. Let ${\mathscr{S}}$ be the split $\mathcal{O}$-torus with affine ring the group ring $\mathcal{O}[{\rm{X}}^{\ast}({\overline{\mathscr{S}}})]$. The action of $\Theta$ on ${\rm{X}}^{\ast}({\overline{\mathscr{S}}})$ induces an action (of $\Theta$) on ${\mathscr{S}}$ by $\mathcal{O}$-group scheme automorphisms. According to [SGA3II, Exp. XI, 4.2] the homomorphism functor ${\underline{\rm{Hom}}}_{{\rm{Spec}}(\mathcal{O}){{\textendash}\rm{gr}}}({\mathscr{S}},{\mathscr{H}})$ is representable by a smooth $\mathcal{O}$-scheme $\mathscr{X}$. There is a natural action of $\Theta$ on $\mathscr{X}$ induced by its action on ${\underline{\rm{Hom}}}_{{\rm{Spec}}(\mathcal{O}){{\textendash}\rm{gr}}}({\mathscr{S}},{\mathscr{H}})$ described as follows.

[TABLE]

for any commutative $\mathcal{O}$-algebra $C$. By Proposition A.8.10 of [CGP] the subscheme $\mathscr{X}^{\Theta}$ of $\Theta$-fixed points in $\mathscr{X}$ is a closed smooth $\mathcal{O}$-subscheme of $\mathscr{X}$. The inclusion $\overline{\iota}:\,{\overline{\mathscr{S}}}\hookrightarrow\overline{\mathscr{H}}$ clearly lies in $\mathscr{X}^{\Theta}(\kappa)$. Now since $\mathcal{O}$ is Henselian, the natural map $\mathscr{X}^{\Theta}(\mathcal{O})\rightarrow\mathscr{X}^{\Theta}(\kappa)$ is surjective [EGA IV4, 18.5.17] and hence there is a $\mathcal{O}$-group scheme homomorphism $\iota:\,{\mathscr{S}}\rightarrow{\mathscr{H}}$ that is $\Theta$-equivariant and lies over $\overline{\iota}$. As $\overline{\iota}$ is a closed immersion, using [SGA3II, Exp. IX, 2.5 and 6.6] we see that $\iota$ is a closed immersion. We identify ${\mathscr{S}}$ with a $\Theta$-stable closed $\mathcal{O}$-torus of ${\mathscr{H}}$ in terms of $\iota$.

We now consider the centralizer $\mathscr{C}$ of ${\mathscr{S}}$ in ${\mathscr{H}}$ which by [CGP, Prop. A.8.10(2)] is a smooth affine $\mathcal{O}$-subgroup scheme of ${\mathscr{H}}$. Its special fiber clearly contains a maximal torus ${\overline{\mathscr{T}}}$ of $\overline{\mathscr{H}}$. Hence, using Proposition 2.1(i) of [P2] we see that $\mathscr{C}$ contains a closed $\mathcal{O}$-torus ${\mathscr{T}}$ whose special fiber is ${\overline{\mathscr{T}}}$. This ${\mathscr{T}}$ is $\mathcal{O}$-split since its special fiber ${\overline{\mathscr{T}}}$ is split over the separably closed field $\kappa$; ${\mathscr{T}}$ clearly contains ${\mathscr{S}}$, and its generic fiber $T$ is a maximal $K$-split torus of $G$ since ${\overline{\mathscr{T}}}$ is a maximal $\kappa$-torus of $\overline{\mathscr{H}}$. Proposition 2.2(ii) of [P2] implies that the apartment of ${\mathcal{B}}(\mathscr{D}(H)/K)$ corresponding to $T$ contains $\Omega$. $\Box$

2.8. A reduction. Let $Z$ be the central torus of $H$, then $H=Z\cdot\mathscr{D}(H)$. Hence, $G=(H^{\Theta})^{\circ}=(Z^{\Theta})^{\circ}\cdot(\mathscr{D}(H)^{\Theta})^{\circ}$. As the enlarged Bruhat-Tits building ${\mathcal{B}}(H/K)$ of $H(K)$ is by definition $V(\mathfrak{Z})\times{\mathcal{B}}(\mathscr{D}(H)/K)$, ${\mathcal{B}}(H/K)^{\Theta}=V(\mathfrak{Z})^{\Theta}\times{\mathcal{B}}(\mathscr{D}(H)/K)^{\Theta}$. So to prove that ${\mathcal{B}}(H/K)^{\Theta}$, as a metric space, is the enlarged Bruhat-Tits building of $G(K)$, it would suffice to show that as a metric space, ${\mathcal{B}}(\mathscr{D}(H)/K)^{\Theta}$ is the enlarged Bruhat-Tits building of $(\mathscr{D}(H)^{\Theta})^{\circ}$. In view of this, we may (and do) replace $H$ by $\mathscr{D}(H)$ and assume in the sequel that $H$ is semi-simple.

2.9. We introduce the following partial order “$\prec$” on the set of nonempty subsets of ${\mathcal{B}}(H/K)$: Given two nonempty subsets $\Omega$ and $\Omega^{\prime}$, $\Omega^{\prime}\prec\Omega$ if the closure $\overline{\Omega}$ of $\Omega$ contains $\Omega^{\prime}$. If $F$ and $F^{\prime}$ are facets of ${\mathcal{B}}(H/K)$, with $F^{\prime}\prec F$, or equivalently, ${\mathscr{H}}^{\circ}_{F}(\mathcal{O})\subset{\mathscr{H}}^{\circ}_{F^{\prime}}(\mathcal{O})$, we say that $F^{\prime}$ is a face of $F$. In a collection $\mathcal{C}$ of facets, thus a facet is maximal if it is not a proper face of any facet belonging to $\mathcal{C}$, and a facet is minimal if no proper face of it belongs to $\mathcal{C}$.

Now let $X$ be a convex subset of ${\mathcal{B}}(H/K)$ and $\mathcal{C}$ be the set of facets of ${\mathcal{B}}(H/K)$, or facets lying in a given apartment $A$, that meet $X$. Then the following assertions are easy to prove (see Proposition 9.2.5 of [BrT1]): (1) All maximal facets in $\mathcal{C}$ are of equal dimension and a facet $F\in\mathcal{C}$ is maximal if and only if $\dim(F\cap X)$ is maximal. (2) Let $F$ be a facet lying in an apartment $A$. Assume that $F$ is maximal among the facets of $A$ that meet $X$, and let $A_{F}$ be the affine subspace of $A$ spanned by $F$. Then every facet of $A$ that meets $X$ is contained in $A_{F}$ and $A\cap X$ is contained in the affine subspace of $A$ spanned by $F\cap X$.

The subset $\mathscr{B}\,(={\mathcal{B}}(H/K)^{\Theta})$ of ${\mathcal{B}}(H/K)$ is closed and convex. Hence the assertions of the preceding paragraph hold for $\mathscr{B}$ in place of $X$.

2.10. Let $x,\,y\in\mathscr{B}\,(={\mathcal{B}}(H/K)^{\Theta})$. Let $F$ be a facet of ${\mathcal{B}}(H/K)$ which contains $x$ in its closure and is maximal among the facets that meet $\mathscr{B}$, and let $\Omega=F\cup\{y\}$. Let $S\subset T$ be a pair of $K$-split tori with properties (i) and (ii) of Proposition 2.6. Let $S_{G}$ and $T_{G}$ be the maximal subtori of $S$ and $T$ respectively contained in $G$. Let $A$ be the apartment of ${\mathcal{B}}(H/K)$ corresponding to $T$. Then $A$ contains $y$ and the closure of $F$, and so it also contains $x$. Moreover, $A$ is an affine space under $V(T)$, the affine subspace $V(S)+x$ of $A$ contains $F$ and is spanned by it. The affine subspaces $V(S_{G})+x\subset V(T_{G})+x$ of $A$ are clearly contained in $\mathscr{B}$. As $V(S)^{\Theta}=V(S_{G})$ and $F\subset V(S)+x$, we see that $F^{\Theta}=F\cap\mathscr{B}$ is contained in $V(S_{G})+x$. But since the facet $F$ is maximal among the facets that meet $\mathscr{B}$, $A^{\Theta}\,(=A\cap\mathscr{B})$ is contained in the affine subspace of $A$ spanned by $F^{\Theta}$. Therefore, $A^{\Theta}=V(S_{G})+x$. This implies that $V(S_{G})+x=V(T_{G})+x$ and hence $S_{G}=T_{G}$. We will now show that $S_{G}$ is a maximal $K$-split torus of $G$.

Let $S^{\prime}$ be a maximal $K$-split torus of $G$ containing $S_{G}$. The centralizer $Z_{H}(S^{\prime})$ of $S^{\prime}$ in $H$ is stable under $\Theta$. The enlarged Bruhat-Tits building ${\mathcal{B}}(Z_{H}(S^{\prime})/K)$ of $Z_{H}(S^{\prime})(K)$ is identified with the union of apartments of ${\mathcal{B}}(H/K)$ corresponding to maximal $K$-split tori of $H$ that contain $S^{\prime}$, cf. [P2, 3.11]. Let $z$ be a point of ${\mathcal{B}}(Z_{H}(S^{\prime})/K)^{\Theta}$ and $T^{\prime}\,(\supset S^{\prime})$ be a maximal $K$-split torus of $H$ such that the corresponding apartment $A^{\prime}$ of ${\mathcal{B}}(Z_{H}(S^{\prime})/K)$ contains $z$. Then $A^{\prime}=V(T^{\prime})+z$ and hence ${A^{\prime}}^{\Theta}=A^{\prime}\cap\mathscr{B}=V(T^{\prime})^{\Theta}+z=V(S^{\prime})+z$ is an affine subspace of $A^{\prime}$ of dimension $\dim(S^{\prime})$. Let $F^{\prime}$ be a facet of $A^{\prime}$ that contains the point $z$ in its closure and is maximal among the facets of $A^{\prime}$ meeting $\mathscr{B}$. Then ${A^{\prime}}^{\Theta}$ is contained in the affine subspace of $A^{\prime}$ spanned by ${F^{\prime}}^{\Theta}$, so $\dim({F^{\prime}}^{\Theta})=\dim(S^{\prime})\geqslant\dim(S_{G})$. But $\dim(F^{\Theta})=\dim(S_{G})\geqslant\dim({F^{\prime}}^{\Theta})$. This implies that $\dim(S_{G})=\dim(S^{\prime})$ and hence $S^{\prime}=S_{G}$. Thus we have shown that $S_{G}$ is a maximal $K$-split torus of $G$.

Now let $x$ be an arbitrary point of $\mathscr{B}$ and take $y=x$. Let $S_{G}$ be as above and $C$ be the maximal $K$-split central torus of $G$. Then $C$ is contained in $S_{G}$ and hence $Z_{H}(S_{G})$ is contained in the centralizer $Z_{H}(C)$ of $C$ in $H$. So ${\mathcal{B}}(Z_{H}(S_{G})/K)$ is contained in ${\mathcal{B}}(Z_{H}(C)/K)$. In particular, $x\in{\mathcal{B}}(Z_{H}(C)/K)$, which implies that $\mathscr{B}\subset{\mathcal{B}}(Z_{H}(C)/K)$.

Thus we have established the following proposition:

Proposition 2.11. (i) $\mathscr{B}\subset{\mathcal{B}}(Z_{H}(C)/K)$.

(ii) Given points $x,\,y\in\mathscr{B}$, there exists a maximal $K$-split torus $S_{G}$ of $G$, and a maximal $K$-split torus $T$ of $H$ containing $S_{G}$, and hence contained in $Z_{H}(S_{G})$, such that the apartment $A\,(\subset{\mathcal{B}}(Z_{H}(S_{G})/K))$ corresponding to $T$ contains $x$ and $y$. Moreover, $A^{\Theta}=A\cap\mathscr{B}$ is the affine subspace $V(S_{G})+x$ of $A$ of dimension $\dim(S_{G})$.

We will now derive the following proposition which will give us apartments in the enlarged Bruhat-Tits building of $G(K)$. In the sequel, we will use $S$, instead of $S_{G}$, to denote a maximal $K$-split torus of $G$. As $M:=Z_{H}(S)$ is stable under $\Theta$, the enlarged Bruhat-Tits building ${\mathcal{B}}(M/K)$ of $M(K)$ contains a $\Theta$-fixed point.

Proposition 2.12. Let $S$ be a maximal $K$-split torus of $G$ and let $T$ be a maximal $K$-split torus of $H$ containing $S$ such that the apartment $A$ of ${\mathcal{B}}(H/K)$ corresponding to $T$ contains a $\Theta$-fixed point $x$. Then ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}=V(S)+x=A^{\Theta}$. So ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ is an affine space under the ${\mathbb{R}}$-vector space $V(S)$.

Proof. Let $Z$ be the maximal $K$-split central torus and $Z_{H}(S)^{\prime}$ be the derived subgroup of $Z_{H}(S)$. Then $Z$, $Z_{H}(S)$ and $Z_{H}(S)^{\prime}$ are stable under $\Theta$; $G^{\prime}:={({Z_{H}(S)^{\prime}}^{\Theta})}^{\circ}$ is anisotropic over $K$ since $S$ is a maximal $K$-split torus of $(Z_{H}(S)^{\Theta})^{\circ}\,(\subset G)$. Now applying Proposition 2.11(ii) to $Z_{H}(S)^{\prime}$ in place of $H$, we see that the Bruhat-Tits building ${\mathcal{B}}(Z_{H}(S)^{\prime}/K)$ of $Z_{H}(S)^{\prime}(K)$ contains only one point fixed under $\Theta$. For if $y,z\in{\mathcal{B}}(Z_{H}(S)^{\prime}/K)^{\Theta}$, then there is an apartment $A^{\prime}$ of ${\mathcal{B}}(Z_{H}(S)^{\prime}/K)$ containing these points and ${A^{\prime}}^{\Theta}$ is an affine subspace of $A^{\prime}$ of dimension 0 as $G^{\prime}$ is anisotropic over $K$. Therefore, $y=z$. This proves that ${\mathcal{B}}(Z_{H}(S)^{\prime}/K)^{\Theta}$ consists of a single point. Hence, ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}=V(Z)^{\Theta}+x=V(S)+x$, and so it is an affine space under $V(S)$. $\Box$

2.13. Another reduction. Let $C$ be the maximal $K$-split central torus of $G$. Then ${\mathcal{B}}(H/K)^{\Theta}=\mathscr{B}\subset{\mathcal{B}}(Z_{H}(C)/K)$ (Proposition 2.11(i)). Let $H^{\prime}$ be the derived subgroup of $Z_{H}(C)$; $H^{\prime}$ is a connected semi-simple subgroup of $H$ stable under the group $\Theta$ of automorphisms of $H$; ${({H^{\prime}}^{\Theta})}^{\circ}\,(\subset G)$ contains the derived subgroup of $G$ and its central torus is $K$-anisotropic. To prove that ${\mathcal{B}}(H/K)^{\Theta}$ is the enlarged Bruhat-Tits building of $G(K)$, we may (and will) assume, after replacing $H$ with $H^{\prime}$, that $H$ is semi-simple and the central torus of $G$ is $K$-anisotropic (cf. [P2, 3.11, 1.11]) and show that ${\mathcal{B}}(H/K)^{\Theta}$ is an affine building.

2.14. Let $S$ be a maximal $K$-split torus of $G$. Let $N:=N_{G}(S)$ and $Z:=Z_{G}(S)$ respectively be the normalizer and the centralizer of $S$ in $G$. As $N$ (in fact, the normalizer $N_{H}(S)$ of $S$ in $H$) normalizes the centralizer $Z_{H}(S)$ of $S$ in $H$, there is a natural action of $N(K)$ on ${\mathcal{B}}(Z_{H}(S)/K)$ and $N(K)$ stabilizes ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ under this action. For $n\in N(K)$, the action of $n$ carries an apartment $A$ of ${\mathcal{B}}(Z_{H}(S)/K)$ to the apartment $n\cdot A$ by an affine transformation.

Now let $T$ be a maximal $K$-split torus of $Z_{H}(S)$ such that the corresponding apartment $A:=A_{T}$ of ${\mathcal{B}}(Z_{H}(S)/K)$ contains a $\Theta$-fixed point $x$. According to the previous proposition, ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}=V(S)+x=A^{\Theta}$. So we can view ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ as an affine space under $V(S)$. We will now show, using the proof of the lemma in 1.6 of [PY1], that ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ has the properties required of an apartment corresponding to the maximal $K$-split torus $S$ in the Bruhat-Tits building of $G(K)$ if such a building exists. We need to check the following three conditions.

A1: The action of $N(K)$ on ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}=A^{\Theta}$ is by affine transformations and the maximal bounded subgroup $Z(K)_{b}$ of $Z(K)$ acts trivially.

Let ${\rm Aff}(A^{\Theta})$ be the group of affine automorphisms of $A^{\Theta}$ and $\varphi:N(K)\rightarrow{\rm Aff}(A^{\Theta})$ be the action map.

A2: The group $Z(K)$ acts by translations, and the action is characterized by the following formula: for $z\in Z(K)$,

[TABLE]

here we regard the translation $\varphi(z)$ as an element of $V(S)$.

A3: For $g\in{\rm{Aff}}(A^{\Theta})$, denote by $dg\in{\rm{GL}}(V(S))$ the derivative of $g$. Then the map $N(K)\rightarrow{\rm{GL}}(V(S))$, $n\mapsto d\varphi(n)$, is induced from the action of $N(K)$ on ${{\rm X}_{\ast}}(S)$ $($i.e., it is the Weyl group action$)$.

Moreover, as the central torus of $G$ is $K$-anisotropic, these three conditions determine the affine structure on ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ uniquely; see [T, 1.2].

Proposition 2.15. Conditions A1, A2 and A3 hold.

Proof. The action of $n\in N(K)$ on ${\mathcal{B}}(Z_{H}(S)/K)$ carries the apartment $A=A_{T}$ via an affine isomorphism $f(n)\,:\,A\rightarrow A_{nTn^{-1}}$ to the apartment $A_{nTn^{-1}}$ corresponding to the torus $nTn^{-1}$ containing $S$. As $(A_{nTn^{-1}})^{\Theta}={\mathcal{B}}(Z_{H}(S)/K)^{\Theta}=A^{\Theta}$, we see that $f(n)$ keeps $A^{\Theta}$ stable and so $\varphi(n):=f(n)|_{A^{\Theta}}$ is an affine automorphism of $A^{\Theta}$.

The derivative $df(n):V(T)\rightarrow V(nTn^{-1})$ is induced from the map

[TABLE]

$\lambda\mapsto{\rm{Int}}\,n\cdot\lambda$, where ${\rm{Int}}\,n$ is the inner automorphism of $H$ determined by $n\in N(K)\subset H(K)$. So, the restriction $d\varphi(n):V(S)\rightarrow V(S)$ is induced from the homomorphism ${\rm X}_{\ast}(S)\rightarrow{\rm X}_{\ast}(S)$, $\lambda\mapsto{\rm{Int}}\,n\cdot\lambda$. This proves A3.

Condition A3 implies that $d\varphi$ is trivial on $Z(K)$. Therefore, $Z(K)$ acts by translations. The action of the bounded subgroup $Z(K)_{b}$ on $A^{\Theta}$ admits a fixed point by the fixed point theorem of Bruhat-Tits. Therefore, $Z(K)_{b}$ acts by the trivial translation. This proves A1.

Since the image of $S(K)$ in $Z(K)/Z(K)_{b}\simeq{\mathbb{Z}}^{\dim(S)}$ is a subgroup of finite index, to prove the formula in A2, it suffices to prove it for $z\in S(K)$. But for $z\in S(K)$, $zTz^{-1}=T$, and $f(z)$ is a translation of the apartment $A$ ($\varphi(z)$ is regarded as an element of $V(T)$) which satisfies (see 1.9 of [P2]):

[TABLE]

This implies the formula in A2, since the restriction map ${\rm X}_{K}^{\ast}(T)\rightarrow{\rm X}_{K}^{\ast}(S)$ is surjective and the image of the restriction map ${\rm X}_{K}^{\ast}(Z)\rightarrow{\rm X}_{K}^{\ast}(S)$ is of finite index in ${\rm X}_{K}^{\ast}(S)$. $\Box$

2.16. Apartments of $\mathscr{B}$. By definition, the apartments of $\mathscr{B}$ are the affine spaces ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ under the ${\mathbb{R}}$-vector space $V(S)$ (of dimension $=K$-rank $G$) for maximal $K$-split tori $S$ of $G$. For any apartment $A$ of ${\mathcal{B}}(Z_{H}(S)/K)$ that contains a $\Theta$-fixed point, ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}=A^{\Theta}$ (Proposition 2.11(ii)). The subgroup $N_{G}(S)(K)$ of $G(K)$ acts by affine transformations on the apartment ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ and $Z_{G}(S)(K)$ acts on it by translations (Proposition 2.15). Conjugacy of maximal $K$-split tori of $G$ under $G(K)$ implies that this group acts transitively on the set of apartments of $\mathscr{B}$.

Propositions 2.11(ii) and 2.12 imply the following proposition at once:

Proposition 2.17. Given any two points of $\mathscr{B}$, there is a maximal $K$-split torus $S$ of $G$ such that the corresponding apartment of $\mathscr{B}$ contains these two points.

Proposition 2.18. Let $\mathscr{A}$ be an apartment of $\mathscr{B}$. Then there is a unique maximal $K$-split torus $S$ of $G$ such that $\mathscr{A}={\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$. So the stabilizer of $\mathscr{A}$ in $G(K)$ is $N_{G}(S)(K)$.

Proof. We fix a maximal $K$-split torus $S$ of $G$ such that $\mathscr{A}={\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$. We will show that $S$ is uniquely determined by $\mathscr{A}$. For this purpose, we observe that the subgroup $N_{G}(S)(K)$ of $G(K)$ acts on $\mathscr{A}$ and the maximal bounded subgroup $Z_{G}(S)(K)_{b}$ of $Z_{G}(S)(K)$ acts trivially (Proposition 2.15). So the subgroup $\mathcal{Z}$ of $G(K)$ consisting of elements that fix $\mathscr{A}$ pointwise is a bounded subgroup of $G(K)$, normalized by $N_{G}(S)(K)$, and it contains $Z_{G}(S)(K)_{b}$. Now, using the Bruhat decomposition of $G(K)$ with respect to $S$, we see that every bounded subgroup of $G(K)$ that is normalized by $N_{G}(S)(K)$ is a normal subgroup of the latter. Hence the identity component of the Zariski-closure of ${\mathcal{Z}}$ is $Z_{G}(S)$. As $S$ is the unique maximal $K$-split torus of $G$ contained in $Z_{G}(S)$, both the assertions follow. $\Box$

2.19. The affine Weyl group of $G$. Let $G(K)^{+}$ denote the (normal) subgroup of $G(K)$ generated by $K$-rational elements of the unipotent radicals of parabolic $K$-subgroups of $G$. Let $S$ be a maximal $K$-split torus of $G$, $N$ and $Z$ respectively be the normalizer and centralizer of $S$ in $G$. Let $N(K)^{+}:=N(K)\cap G(K)^{+}$. Then $N(K)^{+}$ maps onto the Weyl group $W:=N(K)/Z(K)$ of $G$ (this can be seen using, for example, [CGP, Prop. C.2.24(i)]).

Let $\mathscr{A}$ be the apartment of $\mathscr{B}$ corresponding to $S$. As in 2.14, let $\varphi:N(K)\rightarrow{\rm{Aff}}(\mathscr{A})$ be the action map, then the affine Weyl group $W_{\rm{aff}}$ of $G/K$ is by definition the subgroup $\varphi(N(K)^{+})$ of ${\rm{Aff}}(\mathscr{A})$.

3. Bruhat-Tits theory for $G$ over $K$

3.1. As before, we will continue to assume in this section that $H$ is semi-simple and the central torus of $G$ is $K$-anisotropic (see 2.13). For a $\Theta$-stable nonempty bounded subset $\Omega$ of an apartment of ${\mathcal{B}}(H/K)$, let ${\mathscr{G}}^{\circ}_{\Omega}$ be the Bruhat-Tits group scheme as defined in 2.5(i). We will denote the special fiber of ${\mathscr{G}}^{\circ}_{\Omega}$ by $\overline{\mathscr{G}}^{\circ}_{\Omega}$ in the sequel.

Given a point $x\in\mathscr{B}$, for simplicity we will denote ${\mathscr{G}}^{\circ}_{\{x\}}$, ${\mathscr{H}}_{\{x\}}$, ${\mathscr{H}}^{\circ}_{\{x\}}$ and ${\mathscr{H}}^{\Theta}_{\{x\}}$ by ${\mathscr{G}}^{\circ}_{x}$, ${\mathscr{H}}_{x}$, ${\mathscr{H}}^{\circ}_{x}$ and ${\mathscr{H}}^{\Theta}_{x}$ respectively, and the special fibers of these group schemes will be denoted by ${\overline{\mathscr{G}}}^{\circ}_{x}$, ${\overline{\mathscr{H}}}_{x}$, ${\overline{\mathscr{H}}}^{\circ}_{x}$ and ${\overline{\mathscr{H}}}^{\Theta}_{x}$ respectively. The subgroup of $H(K)$ (resp. $G(K)$) consisting of elements that fix $x$ will be denoted by $H(K)^{x}$ (resp. $G(K)^{x}$). The subgroup ${\mathscr{G}}^{\circ}_{x}(\mathcal{O})\,(\subset G(K)^{x})$ is of finite index in $G(K)^{x}$.

3.2. Let $\Omega^{\prime}\prec\Omega$ be nonempty $\Theta$-stable bounded subsets of an apartment of ${\mathcal{B}}(H/K)$. The $\mathcal{O}$-group scheme homomorphism ${\mathscr{H}}_{\Omega}\rightarrow{\mathscr{H}}_{\Omega^{\prime}}$ of [P2, 1.10] restricts to a homomorphism $\rho_{\Omega^{\prime},\Omega}:{\mathscr{H}}^{\circ}_{\Omega}\rightarrow{\mathscr{H}}^{\circ}_{\Omega^{\prime}}$, and by [E, Prop. 3.5], or [CGP, Prop. A.8.10(2)], it induces a $\mathcal{O}$-group scheme homomorphism ${\mathscr{H}}_{\Omega}^{\Theta}\rightarrow{\mathscr{H}}_{\Omega^{\prime}}^{\Theta}$. The last homomorphism gives a $\mathcal{O}$-group scheme homomorphism $\rho^{G}_{\Omega^{\prime},\Omega}:({\mathscr{H}}^{\Theta}_{\Omega})^{\circ}={\mathscr{G}}^{\circ}_{\Omega}\rightarrow{\mathscr{G}}^{\circ}_{\Omega^{\prime}}=({\mathscr{H}}_{\Omega^{\prime}}^{\Theta})^{\circ}$ that is the identity homomorphism on the generic fiber $G$.

3.3. Let $\mathscr{A}$ be the apartment of $\mathscr{B}$ corresponding to a maximal $K$-split torus $S$ of $G$ and $\Omega$ be a nonempty bounded subset of $\mathscr{A}$. The apartment $\mathscr{A}$ is contained in an apartment $A$ of ${\mathcal{B}}(H/K)$ that corresponds to a maximal $K$-split torus $T$ of $H$ containing $S$ and $\mathscr{A}=A\cap\mathscr{B}=A^{\Theta}$ (2.16). So $\Omega$ is a bounded subset of $A$. The group scheme ${\mathscr{H}}_{\Omega}$ contains a closed split $\mathcal{O}$-torus ${\mathscr{T}}$ with generic fiber $T$, see [P2, 1.9]. Let ${\mathscr{S}}$ be the closed $\mathcal{O}$-subtorus of ${\mathscr{T}}$ whose generic fiber is $S$ (${\mathscr{S}}$ is the schematic closure of $S$ in ${\mathscr{T}}$). The automorphism group $\Theta$ of ${\mathscr{H}}_{\Omega}$ acts trivially on the $\mathcal{O}$-torus ${\mathscr{S}}$ (since $S\subset G\subset H^{\Theta}$) and hence ${\mathscr{S}}$ is contained in ${\mathscr{G}}^{\circ}_{\Omega}$. The special fiber ${\overline{\mathscr{S}}}$ of ${\mathscr{S}}$ is a maximal torus of $\overline{\mathscr{G}}^{\circ}_{\Omega}$ since $S$ is a maximal $K$-split torus of $G$.

For $x\in\mathscr{B}$, let $P_{x}:={\mathscr{G}}^{\circ}_{x}(\mathcal{O})$ be the parahoric subgroup of $G(K)$ associated with the point $x$. Let $S$ be a maximal $K$-split torus of $G$ such that $x$ lies in the apartment $\mathscr{A}$ of $\mathscr{B}$ corresponding to $S$. Then the group scheme ${\mathscr{G}}^{\circ}_{x}$ contains a closed split $\mathcal{O}$-torus ${\mathscr{S}}$ whose generic fiber is $S$.

Proposition 3.4. Let $\mathscr{A}$ and $\mathscr{A}^{\prime}$ be apartments of $\mathscr{B}$ and $\Omega$ a nonempty bounded subset of $\mathscr{A}\cap\mathscr{A}^{\prime}$. Then there exists an element $g\in{\mathscr{G}}^{\circ}_{\Omega}(\mathcal{O})$ that maps $\mathscr{A}$ onto $\mathscr{A}^{\prime}$. Any such element fixes $\Omega$ pointwise.

Proof. We will use Proposition 2.1(ii) of [P2], with $\mathcal{O}$ in place of $\mathfrak{o}$, and denote ${\mathscr{G}}^{\circ}_{\Omega}$ by ${\mathscr{G}}$, and its special fiber by $\overline{\mathscr{G}}$, in this proof. Let $S$ and $S^{\prime}$ be the maximal $K$-split tori of $G$ corresponding to the apartments $\mathscr{A}$ and $\mathscr{A}^{\prime}$ respectively and ${\mathscr{S}}$ and ${\mathscr{S}}^{\prime}$ be the closed $\mathcal{O}$-tori of ${\mathscr{G}}$ with generic fibers $S$ and $S^{\prime}$ respectively. The special fibers ${\overline{\mathscr{S}}}$ and ${\overline{\mathscr{S}}}^{\prime}$ of ${\mathscr{S}}$ and ${\mathscr{S}}^{\prime}$ are maximal split tori of $\overline{\mathscr{G}}$, and hence according to a result of Borel and Tits there is an element $\overline{g}$ of $\overline{\mathscr{G}}(\kappa)$ which conjugates ${\overline{\mathscr{S}}}$ onto ${\overline{\mathscr{S}}}^{\prime}$ [CGP, Thm. C.2.3]. Now [P2, Prop. 2.1(ii)] implies that there exists a $g\in{\mathscr{G}}(\mathcal{O})$ lying over $\overline{g}$ that conjugates ${\mathscr{S}}$ onto ${\mathscr{S}}^{\prime}$. This element fixes $\Omega$ pointwise and conjugates $S$ onto $S^{\prime}$ and hence maps $\mathscr{A}$ onto $\mathscr{A}^{\prime}$. $\Box$

3.5. Polysimplicial structure on $\mathscr{B}$. Let $P$ be a parahoric subgroup of $G(K)$ and ${\mathscr{G}}^{\circ}$ be the Bruhat-Tits parahoric $\mathcal{O}$-group scheme associated with $P$ (2.5(ii)). Let ${\mathcal{B}}(H/K)^{P}$ denote the set of points of ${\mathcal{B}}(H/K)$ fixed by $P$. According to Corollary 2.3, ${\mathcal{B}}(H/K)^{P}$ is the union of facets pointwise fixed by $P$. Let $\overline{\mathcal{F}}_{P}:={\mathcal{B}}(H/K)^{P}\cap\mathscr{B}$. This closed convex subset is by definition the closed facet of $\mathscr{B}$ associated with the parahoric subgroup $P$. The $\mathcal{O}$-group scheme ${\mathscr{G}}^{\circ}$ contains a closed split $\mathcal{O}$-torus ${\mathscr{S}}$ whose generic fiber $S$ is a maximal $K$-split torus of $G$ (3.3). The subgroup ${\mathscr{S}}(\mathcal{O})$ (of $S(K)$) is the maximal bounded subgroup of $S(K)$ [PY2, 3.6] and it is contained in $P\,(={\mathscr{G}}^{\circ}(\mathcal{O}))$, so, according to Corollary 2.3, ${\mathcal{B}}(H/K)^{P}$ is contained in the enlarged building ${\mathcal{B}}(Z_{H}(S)/K)$ of $Z_{H}(S)(K)$. This implies that the closed facet $\overline{\mathcal{F}}_{P}$ is contained in the apartment $\mathscr{A}:={\mathcal{B}}(Z_{H}(S)/K)^{\Theta}\,(={\mathcal{B}}(Z_{H}(S)/K)\cap\mathscr{B})$ of $\mathscr{B}$ corresponding to the maximal $K$-split torus $S$ of $G$.

Given another parahoric subgroup subgroup $Q$ of $G(K)$, if ${\overline{\mathcal{F}}}_{Q}={\overline{\mathcal{F}}}_{P}$, then $Q=P$. (To see this, we choose points $x,\,y\in\mathscr{B}$ such that ${\mathscr{G}}^{\circ}_{x}(\mathcal{O})=P$ and ${\mathscr{G}}^{\circ}_{y}(\mathcal{O})=Q$. Then $y\in\overline{\mathcal{F}}_{Q}=\overline{\mathcal{F}}_{P}$. So $P$ fixes $y$. Now using 2.5(iv) we see that $P\subset Q$. We similarly see that $Q\subset P$.) Hence if $Q\supsetneq P$, then ${\overline{\mathcal{F}}}_{Q}$ is properly contained in ${\overline{\mathcal{F}}}_{P}$. Let $\mathcal{F}_{P}$ be the subset of points of $\overline{\mathcal{F}}_{P}$ that are not fixed by any parahoric subgroup of $G(K)$ larger than $P$. Then $\mathcal{F}_{P}=\overline{\mathcal{F}}_{P}-\bigcup_{Q\supsetneq P}\overline{\mathcal{F}}_{Q}$. By definition, $\mathcal{F}_{P}$ is the facet of $\mathscr{B}$ associated with the parahoric subgroup $P$ of $G(K)$, and as $P$ varies over the set of parahoric subgroups of $G(K)$, these are are all the facets of $\mathscr{B}$. We will show below (Propositions 3.8 and 3.10) that $\mathcal{F}_{P}$ is convex and bounded.

In the following two lemmas (3.6 and 3.7), $\kappa$ is any field of characteristic $p\geqslant 0$. We will use the notation introduced in [CGP, §2.1].

Lemma 3.6. Let $\mathcal{H}$ be a smooth connected affine algebraic $\kappa$-group and $\mathcal{Q}$ be a pseudo-parabolic $\kappa$-subgroup of $\mathcal{H}$. Let $\mathcal{S}$ be a $\kappa$-torus of $\mathcal{Q}$ whose image in the maximal pseudo-reductive quotient $\mathcal{M}:=\mathcal{Q}/{\mathscr{R}}_{u,\kappa}(\mathcal{Q})$ of $\mathcal{Q}$ contains the maximal central torus of $\mathcal{M}$. Then any $1$-parameter subgroup $\lambda:{\rm{GL}}_{1}\rightarrow\mathcal{H}$ such that $\mathcal{Q}=P_{\mathcal{H}}(\lambda){\mathscr{R}}_{u,\kappa}(\mathcal{H})$ has a conjugate under ${\mathscr{R}}_{u,\kappa}(\mathcal{Q})(\kappa)$ with image in $\mathcal{S}$.

Proof. Let $\lambda:{\rm{GL}}_{1}\rightarrow\mathcal{H}$ be a $1$-parameter subgroup such that $\mathcal{Q}=P_{\mathcal{H}}(\lambda){\mathscr{R}}_{u,\kappa}(\mathcal{H})$. The image $\mathcal{T}$ of $\lambda$ is contained in $\mathcal{Q}$ and it maps into the central torus of $\mathcal{M}$. Therefore, $\mathcal{T}$ is contained in the solvable subgroup $\mathcal{S}{\mathscr{R}}_{u,\kappa}(\mathcal{Q})$ of $\mathcal{Q}$. Note that as $\mathcal{S}$ is commutative, the derived subgroup of $\mathcal{S}{\mathscr{R}}_{u,\kappa}(\mathcal{Q})$ is contained in ${\mathscr{R}}_{u,\kappa}(\mathcal{Q})$, so the maximal $\kappa$-tori of $\mathcal{S}{\mathscr{R}}_{u,\kappa}(\mathcal{Q})$ are conjugate to each other under ${\mathscr{R}}_{u,\kappa}(\mathcal{Q})(\kappa)$ [Bo, Thm. 19.2]. Hence, there is a $u\in{\mathscr{R}}_{u,\kappa}(\mathcal{Q})(\kappa)$ such that $u\mathcal{T}u^{-1}\subset\mathcal{S}$. Then the image of the $1$-parameter subgroup $\mu:{\rm{GL}}_{1}\rightarrow\mathcal{S}$, defined as $\mu(t)=u\lambda(t)u^{-1}$, is contained in $\mathcal{S}$. $\Box$

Lemma 3.7. Let $\mathcal{H}$ be a smooth connected affine algebraic $\kappa$-group given with an action by a finite group $\Theta$. We assume that $p$ does not divide the order of $\Theta$. Let ${\mathcal{G}}={(\mathcal{H}^{\Theta})}^{\circ}$. Then

(i)* ${\mathscr{R}}_{u,\kappa}({\mathcal{G}})=({\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H}))^{\circ}={({\mathscr{R}}_{u,\kappa}(\mathcal{H})^{\Theta})}^{\circ}$; moreover, ${\mathcal{G}}/({\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H}))$ is pseudo-reductive, and if $\kappa$ is perfect then ${\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H})={\mathscr{R}}_{u,\kappa}({\mathcal{G}})$.*

(ii)* Given a $\Theta$-stable pseudo-parabolic $\kappa$-subgroup $\mathcal{Q}$ of $\mathcal{H}$, $\mathcal{P}:={\mathcal{G}}\cap\mathcal{Q}$ is a pseudo-parabolic $\kappa$-subgroup of  ${\mathcal{G}}$, so $\mathcal{P}$ is connected and it equals ${(\mathcal{Q}^{\Theta})}^{\circ}$.*

(iii)* Conversely, given a pseudo-parabolic $\kappa$-subgroup $\mathcal{P}$ of ${\mathcal{G}}$, and a maximal $\kappa$-torus $\mathcal{S}\subset\mathcal{P}$, there is a $\Theta$-stable pseudo-parabolic $\kappa$-subgroup $\mathcal{Q}$ of $\mathcal{H}$, $\mathcal{Q}$ containing the centralizer $Z_{\mathcal{H}}(\mathcal{S})$ of $\mathcal{S}$ in $\mathcal{H}$, such that $\mathcal{P}={\mathcal{G}}\cap\mathcal{Q}={(\mathcal{Q}^{\Theta})}^{\circ}$.*

Proof. The first assertion of (i) immediately follows from [CGP, Prop. A.8.14(2)]. Now we observe that as ${\mathscr{R}}_{u,\kappa}({\mathcal{G}})=({\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H}))^{\circ}$, the quotient $({\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H}))/{\mathscr{R}}_{u,\kappa}({\mathcal{G}})$ is a finite étale (unipotent) normal subgroup of ${\mathcal{G}}/{\mathscr{R}}_{u,\kappa}({\mathcal{G}})$ so it is central. Thus the kernel of the quotient map $\pi:{\mathcal{G}}/{\mathscr{R}}_{u,\kappa}({\mathcal{G}})\rightarrow{\mathcal{G}}/({\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H}))$ is an étale unipotent central subgroup. Hence, ${\mathcal{G}}/({\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H}))$ is pseudo-reductive as ${\mathcal{G}}/{\mathscr{R}}_{u,\kappa}({\mathcal{G}})$ is. Moreover, if $\kappa$ is perfect then every pseudo-reductive $\kappa$-group is reductive and such a group does not contain a nontrivial étale unipotent normal subgroup. This implies that if $\kappa$ is perfect, then ${\mathscr{R}}_{u,\kappa}({\mathcal{G}})={\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H})$.

Since ${\mathscr{R}}_{u,\kappa}({\mathcal{G}})\subset{\mathcal{G}}\cap{\mathscr{R}}_{u,\kappa}(\mathcal{H})\subset{\mathcal{G}}\cap\mathcal{Q}$, to prove (ii), we can replace $\mathcal{H}$ by its pseudo-reductive quotient $\mathcal{H}/{\mathscr{R}}_{u,\kappa}(\mathcal{H})$ and assume that $\mathcal{H}$ is pseudo-reductive. Then ${\mathcal{G}}$ is also pseudo-reductive by (i). The $\kappa$-unipotent radical ${\mathscr{R}}_{u,\kappa}(\mathcal{Q})$ of $\mathcal{Q}$ is $\Theta$-stable. Let $\mathcal{S}$ be a $\Theta$-stable $\kappa$-torus in $\mathcal{Q}$ that maps isomorphically onto the maximal central torus of the pseudo-reductive quotient $\overline{\mathcal{Q}}:=\mathcal{Q}/{\mathscr{R}}_{u,\kappa}(\mathcal{Q})$ (Lemma 2.7). By Lemma 3.6, there exists a 1-parameter subgroup $\lambda:{\rm{GL}}_{1}\rightarrow\mathcal{S}$ such that $\mathcal{Q}=P_{\mathcal{H}}(\lambda)$. Let $\mu=\sum_{\theta\in\Theta}\theta\cdot\lambda$. Then $\mu$ is invariant under $\Theta$ and so it is a 1-parameter subgroup of ${\mathcal{G}}$. We will now show that $\mathcal{Q}=P_{\mathcal{H}}(\mu)$. Let $\Phi$ (resp. $\Psi$) be the set of weights in the Lie algebra of $\mathcal{Q}$ (resp. $P_{\mathcal{H}}(\mu)$) with respect to the adjoint action of $\mathcal{S}$. Then since $\mathcal{Q}$, $P_{\mathcal{H}}(\mu)$ and $\mathcal{S}$ are $\Theta$-stable, the subsets $\Phi$ and $\Psi$ (of ${\rm{X}}(\mathcal{S})$) are stable under the action of $\Theta$ on ${\rm{X}}(\mathcal{S})$. Hence, for all $a\in\Phi$, as $\langle a,\lambda\rangle\geqslant 0$, we conclude that $\langle a,\mu\rangle\geqslant 0$. Therefore, $\Phi\subset\Psi$. On the other hand, for $b\in\Psi$, $\langle b,\mu\rangle\geqslant 0$. If $b\,(\in\Psi)$ does not belong to $\Phi$, then for $\theta\in\Theta$, $\theta\cdot b\notin\Phi$, so for all $\theta\in\Theta$, $\langle\theta\cdot b,\lambda\rangle<0$, which implies that $\langle b,\mu\rangle<0$. This is a contradiction. Therefore, $\Phi=\Psi$ and so $\mathcal{Q}=P_{\mathcal{H}}(\mu)$. Now observe that ${(\mathcal{Q}^{\Theta})}^{\circ}\subset{\mathcal{G}}\cap\mathcal{Q}\subset\mathcal{Q}^{\Theta}$. As $\mathcal{Q}^{\Theta}$ is a smooth subgroup ([E, Prop. 3.4] or [CGP, Prop. A.8.10(2)]), ${\mathcal{G}}\cap\mathcal{Q}$ is a smooth $\kappa$-subgroup, and since it contains the pseudo-parabolic $\kappa$-subgroup $P_{{\mathcal{G}}}(\mu)$ of ${\mathcal{G}}$, it is a pseudo-parabolic $\kappa$-subgroup of ${\mathcal{G}}$ [CGP, Prop. 3.5.8], hence in particular it is connected. Therefore, ${\mathcal{G}}\cap\mathcal{Q}={(\mathcal{Q}^{\Theta})}^{\circ}$.

Now we will prove (iii). Let $\lambda:{\rm{GL}}_{1}\rightarrow\mathcal{S}$ be a $1$-parameter subgroup such that $\mathcal{P}=P_{{\mathcal{G}}}(\lambda){\mathscr{R}}_{u,\kappa}({\mathcal{G}})$. Then $\mathcal{Q}:=P_{\mathcal{H}}(\lambda){\mathscr{R}}_{u,\kappa}(\mathcal{H})$ is a pseudo-parabolic $\kappa$-subgroup of $\mathcal{H}$ that is $\Theta$-stable (since $\lambda$ is $\Theta$-invariant) and it contains $\mathcal{P}$ as well as $Z_{\mathcal{H}}(\mathcal{S})$. According to (ii), ${\mathcal{G}}\cap\mathcal{Q}={(\mathcal{Q}^{\Theta})}^{\circ}$ is a pseudo-parabolic $\kappa$-subgroup of ${\mathcal{G}}$ containing $\mathcal{P}$. The Lie algebras of $\mathcal{P}$ and ${(\mathcal{Q}^{\Theta})}^{\circ}$ are clearly equal. This implies that $\mathcal{P}={\mathcal{G}}\cap\mathcal{Q}={(\mathcal{Q}^{\Theta})}^{\circ}$ and we have proved (iii). $\Box$

Proposition 3.8. Let $P$ be a parahoric subgroup of $G(K)$ and $\mathcal{F}_{P}$ and $\overline{\mathcal{F}}_{P}$ be as in $3.5$.

(i) Given $x\in\mathcal{F}_{P}$ and $y\in\overline{\mathcal{F}}_{P}$, for every point $z$ of the geodesic $[xy]$, except possibly for $z=y$, ${\mathscr{G}}^{\circ}_{z}(\mathcal{O})=P$.

(ii) Let $F$ be a facet of ${\mathcal{B}}(H/K)$ that meets $\overline{\mathcal{F}}_{P}$ and is maximal among such facets. Then ${\mathscr{G}}^{\circ}_{F}(\mathcal{O})=P$. Thus $F\cap\mathscr{B}\subset\mathcal{F}_{P}$.

The first assertion of this proposition implies that $\mathcal{F}_{P}$ is convex. The second assertion implies that $\mathcal{F}_{P}$ is an open-dense subset of $\overline{\mathcal{F}}_{P}$, hence the closure of $\mathcal{F}_{P}$ is $\overline{\mathcal{F}}_{P}$.

Proof. To prove the first assertion, let $[xy]$ be the geodesic joining $x$ and $y$. Let $F_{0},\,F_{1},\ldots,\,F_{n}$ be the facets of ${\mathcal{B}}(H/K)$ containing a segment of positive length of the geodesic $[xy]$ (so each $F_{i}$ is $\Theta$-stable and is fixed pointwise by $P$, hence $P\subset{\mathscr{G}}^{\circ}_{F_{i}}(\mathcal{O})$, cf. 2.5(iv)). Then $[xy]\subset\bigcup_{i}\overline{F}_{i}$. We assume the facets $\{F_{i}\}$ indexed so that $x$ lies in $\overline{F}_{0}$, $y$ lies in $\overline{F}_{n}$, and for each $i<n$, $\overline{F}_{i}\cap\overline{F}_{i+1}$ is nonempty. Let $z_{0}=x$. For every positive integer $i\,(\leqslant n)$, $\overline{F}_{i-1}\cap\overline{F}_{i}$ contains a unique point of $[xy]$; we will denote this point by $z_{i}$.

To prove the second assertion of the proposition along with the first, we take $x$ to be a point of $\mathscr{B}$ such that ${\mathscr{G}}^{\circ}_{x}(\mathcal{O})=P$ (so $x\in\mathcal{F}_{P}$) and take $y$ to be any point of $F\cap\mathscr{B}$. Let $[xy]$, and for $i\leqslant n$, $F_{i}$ and $z_{i}$ be as in the preceding paragraph. Then $F_{n}=F$.

Since $x\in{\overline{F}}_{0}$, there is a $\mathcal{O}$-group scheme homomorphism ${\mathscr{G}}^{\circ}_{F_{0}}\rightarrow{\mathscr{G}}^{\circ}_{x}$ that is the identity on the generic fiber $G$. Thus, ${\mathscr{G}}^{\circ}_{F_{0}}(\mathcal{O})\subset P$. But $P\subset{\mathscr{G}}^{\circ}_{F_{0}}(\mathcal{O})$, so ${\mathscr{G}}^{\circ}_{z_{0}}(\mathcal{O})={\mathscr{G}}^{\circ}_{F_{0}}(\mathcal{O})=P$. Let $j\,(\leqslant n)$ be a positive integer such that for all $i<j$, ${\mathscr{G}}^{\circ}_{z_{i}}(\mathcal{O})={\mathscr{G}}^{\circ}_{F_{i}}(\mathcal{O})=P$. The inclusion of $\{z_{j}\}$ in $\overline{F}_{j-1}\cap\overline{F}_{j}$ gives rise to $\mathcal{O}$-group scheme homomorphisms ${\mathscr{H}}_{F_{j-1}}\stackrel{{\scriptstyle\sigma_{j}}}{{\longrightarrow}}{\mathscr{H}}_{z_{j}}\stackrel{{\scriptstyle\rho_{j}}}{{\longleftarrow}}{\mathscr{H}}_{F_{j}}$ that are the identity on the generic fiber $H$. The images of the induced homomorphisms $\overline{\mathscr{H}}^{\circ}_{F_{j-1}}\stackrel{{\scriptstyle{\overline{\sigma}}_{j}}}{{\longrightarrow}}\overline{\mathscr{H}}^{\circ}_{z_{j}}\stackrel{{\scriptstyle{\overline{\rho}}_{j}}}{{\longleftarrow}}\overline{\mathscr{H}}^{\circ}_{F_{j}}$ are pseudo-parabolic $\kappa_{s}$-subgroups of $\overline{\mathscr{H}}^{\circ}_{z_{j}}$ ([P2, 1.10(2)]). We conclude by Lie algebra consideration that ${\overline{\sigma}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j-1}})={({\overline{\sigma}}_{j}(\overline{\mathscr{H}}^{\circ}_{F_{j-1}})^{\Theta})}^{\circ}$ and ${\overline{\rho}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j}})={({\overline{\rho}}_{j}(\overline{\mathscr{H}}^{\circ}_{F_{j}})^{\Theta})}^{\circ}$, and Lemma 3.7(ii) implies that both of these subgroups are pseudo-parabolic subgroups of $\overline{\mathscr{G}}^{\circ}_{z_{j}}$. As ${\mathscr{G}}^{\circ}_{F_{j-1}}(\mathcal{O})=P$, whereas, $P\subset{\mathscr{G}}^{\circ}_{F_{j}}(\mathcal{O})\,(\subset{\mathscr{G}}^{\circ}_{z_{j}}(\mathcal{O}))$, we see that ${\overline{\sigma}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j-1}})$ is contained in ${\overline{\rho}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j}})$. Let $\overline{Q}$ and $\overline{Q}^{\prime}$ respectively be the images of  ${\overline{\sigma}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j-1}})$ and ${\overline{\rho}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j}})$ in the maximal pseudo-reductive quotient $\overline{G}^{\,\rm pred}_{z_{j}}:=\overline{\mathscr{G}}^{\circ}_{z_{j}}/{\mathscr{R}}_{u,\kappa_{s}}(\overline{\mathscr{G}}^{\circ}_{z_{j}})$ of $\overline{\mathscr{G}}^{\circ}_{z_{j}}$. Then $\overline{Q}\subset\overline{Q}^{\prime}$, and both of them are pseudo-parabolic subgroups of $\overline{G}^{\,\rm pred}_{z_{j}}$.

Now let $S$ be a maximal $K$-split torus of $G$ such that the apartment of $\mathscr{B}$ corresponding to $S$ contains the geodesic $[xy]$ and let $v\in V(S)$ so that $v+x=y$. Then for all sufficiently small positive real number $\epsilon$, $-\epsilon v+z_{j}\in F_{j-1}$ and $\epsilon v+z_{j}\in F_{j}$. Using [P2, 1.10(3)] we infer that the images of the pseudo-parabolic subgroups ${\overline{\sigma}}_{j}(\overline{\mathscr{H}}^{\circ}_{F_{j-1}})$ and ${\overline{\rho}}_{j}(\overline{\mathscr{H}}^{\circ}_{F_{j}})$ (of $\overline{\mathscr{H}}^{\circ}_{z_{j}}$) in the maximal pseudo-reductive quotient $\overline{H}^{\,\rm pred}_{z_{j}}:=\overline{\mathscr{H}}^{\circ}_{z_{j}}/{\mathscr{R}}_{u,\kappa_{s}}(\overline{\mathscr{H}}^{\circ}_{z_{j}})$ of $\overline{\mathscr{H}}^{\circ}_{z_{j}}$ are opposite pseudo-parabolic subgroups. Therefore, the image $\mathcal{H}$ of  ${\overline{\sigma}}_{j}(\overline{\mathscr{H}}^{\circ}_{F_{j-1}})\cap{\overline{\rho}}_{j}(\overline{\mathscr{H}}^{\circ}_{F_{j}})$ in $\overline{H}^{\,\rm pred}_{z_{j}}$ is pseudo-reductive. Proposition A.8.14 (2) of [CGP] implies then that ${(\mathcal{H}^{\Theta})}^{\circ}$ is pseudo-reductive. It is obvious that under the natural homomorphism $\pi:\overline{G}^{\,\rm pred}_{z_{j}}\rightarrow\overline{H}^{\,\rm pred}_{z_{j}}$, the image of $\overline{Q}=\overline{Q}\cap\overline{Q}^{\prime}$ is ${(\mathcal{H}^{\Theta})}^{\circ}$. As the kernel of the homomorphism $\pi$ is a finite (étale unipotent) subgroup (Lemma 3.7(i)), and ${(\mathcal{H}^{\Theta})}^{\circ}$ is pseudo-reductive, we see that $\overline{Q}$ is a pseudo-reductive subgroup of $\overline{G}^{\,\rm pred}_{z_{j}}$. But since $\overline{Q}$ is a pseudo-parabolic subgroup of the latter, we must have $\overline{Q}=\overline{G}^{\,\rm pred}_{z_{j}}$, and hence, $\overline{Q}^{\prime}=\overline{G}^{\,\rm pred}_{z_{j}}$. So, ${{\overline{\sigma}}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j-1}})=\overline{\mathscr{G}}^{\circ}_{z_{j}}={{\overline{\rho}}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j}})$.

Since the natural homomorphism ${\mathscr{G}}^{\circ}_{F_{j-1}}(\mathcal{O})\rightarrow\overline{\mathscr{G}}^{\circ}_{F_{j-1}}(\kappa)$ is surjective (as $\mathcal{O}$ is henselian and ${\mathscr{G}}^{\circ}_{F_{j-1}}$ is smooth, [EGA IV4, 18.5.17]), and ${{\overline{\sigma}}}_{j}(\overline{\mathscr{G}}^{\circ}_{F_{j-1}})=\overline{\mathscr{G}}^{\circ}_{z_{j}}$, the image of ${\mathscr{G}}^{\circ}_{F_{j-1}}(\mathcal{O})\,(\subset{\mathscr{G}}^{\circ}_{z_{j}}(\mathcal{O}))$ in $\overline{\mathscr{G}}^{\circ}_{z_{j}}(\kappa)$ is Zariski-dense in $\overline{\mathscr{G}}^{\circ}_{z_{j}}$. From this we see that

[TABLE]

cf. [BrT2, 1.7.2] and 2.1. Therefore, $\sigma_{j}|_{{\mathscr{G}}^{\circ}_{F_{j-1}}}:{\mathscr{G}}^{\circ}_{F_{j-1}}\rightarrow{\mathscr{G}}^{\circ}_{z_{j}}$ is a $\mathcal{O}$-group scheme isomorphism. We similarly see that $\rho_{j}|_{{\mathscr{G}}^{\circ}_{F_{j}}}:{\mathscr{G}}^{\circ}_{F_{j}}\rightarrow{\mathscr{G}}^{\circ}_{z_{j}}$ is a $\mathcal{O}$-group scheme isomorphism. Now since ${\mathscr{G}}^{\circ}_{F_{j-1}}(\mathcal{O})=P$, we conclude that $P={\mathscr{G}}^{\circ}_{z_{j}}(\mathcal{O})={\mathscr{G}}^{\circ}_{F_{j}}(\mathcal{O})$. By induction it follows that $P={\mathscr{G}}^{\circ}_{z_{i}}(\mathcal{O})={\mathscr{G}}^{\circ}_{F_{i}}(\mathcal{O})$ for all $i\leqslant n$. In particular, for all $z\in[xy]$, except possibly for $z=y$, ${\mathscr{G}}^{\circ}_{z}(\mathcal{O})=P$, and ${\mathscr{G}}^{\circ}_{F_{n}}(\mathcal{O})=P$. $\Box$

For parahoric subgroups $P$ and $Q$ of $G(K)$, if $\mathcal{F}_{P}\cap\mathcal{F}_{Q}$ is nonempty, then for any $z$ in this intersection, $P={\mathscr{G}}^{\circ}_{z}(\mathcal{O})=Q$ (Proposition 3.8(i)). Thus every point of $\mathscr{B}$ is contained in a unique facet.

For a parahoric subgroup $Q$ of $G(K)$ containing $P$, obviously, $\mathcal{F}_{Q}\subset\overline{\mathcal{F}}_{Q}\subset\overline{\mathcal{F}}_{P}$, thus $\mathcal{F}_{Q}\prec\mathcal{F}_{P}$ and hence $\mathcal{F}_{P}$ is a maximal facet if and only if $P$ is a minimal parahoric subgroup of $G(K)$. The maximal facets of $\mathscr{B}$ are called the chambers of $\mathscr{B}$. It is easily seen using the observations contained in 2.9 that all the chambers are of equal dimension. We say that a facet $\mathcal{F}^{\prime}$ of $\mathscr{B}$ is a face of a facet $\mathcal{F}$ if  $\mathcal{F}^{\prime}\prec\mathcal{F}$, i.e., if  $\mathcal{F}^{\prime}$ is contained in the closure of  $\mathcal{F}$.

We will use the following simple lemma in the proof of the next proposition.

Lemma 3.9. Let $S$ be a maximal $K$-split torus of $G$, $\mathscr{A}$ the corresponding apartment of $\mathscr{B}$, and $\mathcal{C}$ be a noncompact closed convex subset of $\mathscr{A}$. Then for any point $x\in\mathcal{C}$, there is an infinite ray originating at $x$ and contained in $\mathcal{C}$.

Proof. Recall that $\mathscr{A}$ is an affine space under the vector space $V(S)={\mathbb{R}}\otimes_{{\mathbb{Z}}}{\rm{X}}_{\ast}(S)$. We identify $\mathscr{A}$ with $V(S)$ using translations by elements in the latter, with $x$ identified with the origin [math], and use a positive definite inner product on $V(S)$ to get a norm on $\mathscr{A}$. With this identification, $\mathcal{C}$ is a closed convex subset of $V(S)$ containing [math]. Since $\mathcal{C}$ is noncompact, there exist unit vectors $v_{i}\in V(S)$, $i\geqslant 1$, and positive real numbers $s_{i}\rightarrow\infty$ such that $s_{i}v_{i}$ lies in $\mathcal{C}$. After replacing $\{v_{i}\}$ by a subsequence, we may (and do) assume that the sequence $\{v_{i}\}$ converges to a unit vector $v$. We will now show that for every nonnegative real number $t$, $tv$ lies in $\mathcal{C}$, this will prove the lemma. To see that $tv$ lies in $\mathcal{C}$, it suffices to observe that for a given $t$, the sequence $\{tv_{i}\}$ converges to $tv$, and for all sufficiently large $i$ (so that $s_{i}\geqslant t$), $tv_{i}$ lies in $\mathcal{C}$. $\Box$

Proposition 3.10. For any parahoric subgroup $P$ of $G(K)$, the associated closed facet ${\overline{\mathcal{F}}}_{P}$ of $\mathscr{B}$, and so also the associated facet $\mathcal{F}_{P}\,(\subset{\overline{\mathcal{F}}}_{P})$, is bounded.

Proof. Let $S$ be a maximal $K$-split torus of $G$ such that the corresponding apartment of $\mathscr{B}$ contains $\overline{\mathcal{F}}_{P}$ (3.5). Assume, if possible, that ${\overline{\mathcal{F}}}_{P}$ is noncompact and fix a point $x$ of $\mathcal{F}_{P}$. Then, according to the preceding lemma, there is an infinite ray $\mathcal{R}:=\{tv+x\,|\,t\in{\mathbb{R}}_{\geqslant 0}\}$, for some $v\in V(S)$, originating at $x$ and contained in $\overline{\mathcal{F}}_{P}$. It is obvious from Proposition 3.8(i) that this ray is actually contained in $\mathcal{F}_{P}$. Hence, for every point $z\in\mathcal{R}$, ${\mathscr{G}}^{\circ}_{z}(\mathcal{O})=P$.

As the central torus of $G$ has been assumed to be $K$-anisotropic, there is a nondivisible root $a$ of $G$, with respect to $S$, such that $\langle a,\,v\rangle>0$. Let $S_{a}$ be the identity component of the kernel of $a$ and $G_{a}$ (resp. $H_{a}$) be the derived subgroup of the centralizer of $S_{a}$ in $G$ (resp. $H$). Fix $t\in{\mathbb{R}}_{\geqslant 0}$, and let $y=tv+x\in\mathcal{R}$. Let ${\mathscr{S}}$ be the closed $1$-dimensional $\mathcal{O}$-split torus of ${\mathscr{G}}^{\circ}_{y}$ whose generic fiber is the maximal $K$-split torus of $G_{a}$ contained in $S$ and let $\lambda:{\rm{GL}}_{1}\rightarrow{\mathscr{S}}\,(\hookrightarrow{\mathscr{G}}^{\circ}_{y}\hookrightarrow{\mathscr{H}}_{y})$ be the $\mathcal{O}$-isomorphism such that $\langle a,\lambda\rangle>0$. Let $c=\langle a,v\rangle/\langle a,\lambda\rangle$. Then $\langle a,v-c\lambda\rangle=0$.

Let $\mathscr{U}_{y}$ be the $\mathcal{O}$-subgroup scheme of ${\mathscr{H}}_{y}$ representing the functor

[TABLE]

cf. [CGP, Lemma 2.1.5]. Using the last assertion of 2.1.8(3), and the first assertion of 2.1.8(4), of [CGP] (with $k$, which is an an arbitrary commutative ring in these assertions, replaced by $\mathcal{O}$, and $G$ replaced by ${\mathscr{H}}_{y}$), we see that $\mathscr{U}_{y}$ is a closed smooth unipotent $\mathcal{O}$-subgroup scheme of ${\mathscr{H}}_{y}$ with connected fibers; the generic fiber of $\mathscr{U}_{y}$ is $U_{H}(\lambda)$, where $U_{H}(\lambda)$ is as in [CGP, Lemma 2.1.5] with $G$ replaced by $H$. We consider the smooth closed $\mathcal{O}$-subgroup scheme $\mathscr{U}_{y}^{\Theta}$ of $\mathscr{U}_{y}$. As $\mathscr{U}_{y}^{\Theta}$ is clearly normalized by ${\mathscr{S}}$, it has connected fibers, and hence it is contained in $({\mathscr{H}}_{y}^{\Theta})^{\circ}={\mathscr{G}}_{y}^{\circ}$. The generic fiber of $\mathscr{U}_{y}^{\Theta}$ is $U_{H}(\lambda)^{\Theta}$ that contains the root group $U_{a}\,(=U_{G_{a}}(\lambda))$ of $G$ corresponding to the root $a$.

As $\bigcup_{z\in\mathcal{R}}\mathscr{U}_{z}(\mathcal{O})\supset U_{H_{a}}(\lambda)(K)\supset U_{a}(K)$, we see that $\bigcup_{z\in\mathcal{R}}\mathscr{U}^{\Theta}_{z}(\mathcal{O})\supset U_{a}(K)$. Now since ${\mathscr{G}}^{\circ}_{z}\supset\mathscr{U}^{\Theta}_{z}$, we conclude that $\bigcup_{z\in\mathcal{R}}{\mathscr{G}}^{\circ}_{z}(\mathcal{O})\supset U_{a}(K)$. But for all $z\in\mathcal{R}$, ${\mathscr{G}}^{\circ}_{z}(\mathcal{O})=P$, so the parahoric subgroup $P$ contains the unbounded subgroup $U_{a}(K)$. This is a contradiction. $\Box$

Proposition 3.10 implies that each closed facet of $\mathscr{B}$ is a compact polyhedron. Considering the facets lying on the boundary of a maximal closed facet of $\mathscr{B}$, we see that $\mathscr{B}$ contains facets of every dimension $\leqslant K$-rank $G$.

3.11. Let $P$ be a parahoric subgroup of $G(K)$ and $\mathcal{F}:=\mathcal{F}_{P}$ be the facet of $\mathscr{B}$ associated to $P$ in 3.5. Then for any $x\in\mathcal{F}$, since $P\subset{\mathscr{G}}^{\circ}_{\mathcal{F}}(\mathcal{O})\subset{\mathscr{G}}^{\circ}_{x}(\mathcal{O})=P$  (2.5(iv)), ${\mathscr{G}}^{\circ}_{\mathcal{F}}(\mathcal{O})=P$ and hence the natural $\mathcal{O}$-group scheme homomorphism ${\mathscr{G}}^{\circ}_{\mathcal{F}}\rightarrow{\mathscr{G}}^{\circ}_{x}$ is an isomorphism. In particular, for any facet $F$ of ${\mathcal{B}}(H/K)$ that meets $\mathcal{F}$, ${\mathscr{G}}^{\circ}_{\mathcal{F}}={\mathscr{G}}^{\circ}_{F}$.

Proposition 3.12. Let $\mathcal{F}$ be a facet of $\mathscr{B}$. Then the $\kappa$-unipotent radical ${\mathscr{R}}_{u,\kappa}(\overline{\mathscr{G}}^{\circ}_{\mathcal{F}})$ of $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$ equals ${(\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}\cap{\mathscr{R}}_{u,\kappa}(\overline{\mathscr{H}}^{\circ}_{\mathcal{F}}))}^{\circ}$.

Let $\mathcal{F}$ and $\mathcal{F}^{\prime}$ be two facets of $\mathscr{B}$, with $\mathcal{F}^{\prime}\prec\mathcal{F}$. Then$:$

(i)* The kernel of the induced homomorphism ${\overline{\rho}}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}:\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}\rightarrow\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$ between the special fibers is a smooth unipotent $\kappa$-subgroup of  $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$ and the image $\mathfrak{p}(\mathcal{F}^{\prime}/\mathcal{F})$ is a pseudo-parabolic $\kappa$-subgroup of $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$.*

(ii)* If $F$ and $F^{\prime}$ are facets of ${\mathcal{B}}(H/K)$, $F^{\prime}\prec F$, that meet $\mathcal{F}$ and $\mathcal{F}^{\prime}$ respectively, then $\mathfrak{p}(\mathcal{F}^{\prime}/\mathcal{F})=(\overline{\mathscr{Q}}^{\Theta})^{\circ}$, where $\overline{\mathscr{Q}}$ is the image of ${\overline{\rho}}_{F^{\prime},F}:\overline{\mathscr{H}}^{\circ}_{F}\rightarrow\overline{\mathscr{H}}^{\circ}_{F^{\prime}}$.*

(iii)* The inverse image of the subgroup $\mathfrak{p}(\mathcal{F}^{\prime}/\mathcal{F})(\kappa)$ of  $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}(\kappa)$, under the natural surjective homomorphism ${\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}(\mathcal{O})\rightarrow\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}(\kappa)$, is $\rho^{G}_{\mathcal{F}^{\prime},\mathcal{F}}({\mathscr{G}}^{\circ}_{\mathcal{F}}(\mathcal{O}))\,(\subset{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}(\mathcal{O}))$.*

Given a pseudo-parabolic $\kappa$-subgroup $\overline{\mathscr{P}}$ of $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$, there is a facet $\mathcal{F}$ of $\mathscr{B}$ with $\mathcal{F}^{\prime}\prec\mathcal{F}$ such that the image of the homomorphism $\overline{\rho}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}:\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}\rightarrow\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$ equals $\overline{\mathscr{P}}$.

Proof. The first assertion of the proposition follows immediately from Lemma 3.7(i).

To prove (i), we fix $x\in\mathcal{F}^{\prime}$ and let $F^{\prime}$ be the facet of ${\mathcal{B}}(H/K)$ containing $x$. As the closure of $\mathcal{F}$ contains $x$, there is a facet $F$ of ${\mathcal{B}}(H/K)$ that meets $\mathcal{F}$ and contains $x$ in its closure. Then $F^{\prime}\subset\overline{F}$, i.e., $F^{\prime}\prec F$, and $F$ and $F^{\prime}$ meet $\mathcal{F}$ and $\mathcal{F}^{\prime}$ respectively. Hence, ${\mathscr{G}}^{\circ}_{\mathcal{F}}={\mathscr{G}}^{\circ}_{F}={({\mathscr{H}}_{F}^{\Theta})}^{\circ}$ and ${\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}={\mathscr{G}}^{\circ}_{F^{\prime}}={({\mathscr{H}}_{F^{\prime}}^{\Theta})}^{\circ}$ (3.11). Now we will prove assertions (i) and (ii) together. The kernel $\overline{\mathscr{K}}$ of the homomorphism $\overline{\rho}_{F^{\prime},F}:\overline{\mathscr{H}}_{F}^{\circ}\rightarrow\overline{\mathscr{H}}_{F^{\prime}}^{\circ}$ is a smooth unipotent $\kappa$-subgroup, and the image $\overline{\mathscr{Q}}$ is a pseudo-parabolic $\kappa$-subgroup of $\overline{\mathscr{H}}_{F^{\prime}}^{\circ}$ [P2, 1.10 (1), (2)]. The pseudo-parabolic subgroup $\overline{\mathscr{Q}}$ is clearly $\Theta$-stable as the facets $F$ and $F^{\prime}$ are $\Theta$-stable. The kernel of  ${\overline{\rho}}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}$ is $\overline{\mathscr{K}}\cap\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$, and its image is contained in ${(\overline{\mathscr{Q}}^{\Theta})}^{\circ}$. Therefore, the kernel of ${\overline{\rho}}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}$ contains ${(\overline{\mathscr{K}}^{\Theta})}^{\circ}$ and is contained in $\overline{\mathscr{K}}^{\Theta}$. As $\overline{\mathscr{K}}^{\Theta}$ is a smooth subgroup of $\overline{\mathscr{K}}$, we see that the kernel of ${\overline{\rho}}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}$ is smooth.

Since the image of the Lie algebra homomorphism ${\rm{L}}(\overline{\mathscr{G}}_{\mathcal{F}}^{\circ})\rightarrow{\rm{L}}(\overline{\mathscr{G}}_{\mathcal{F}^{\prime}}^{\circ})$ induced by ${\overline{\rho}}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}$ is ${\rm{L}}(\overline{\mathscr{Q}})^{\Theta}$, the containment $\mathfrak{p}(\mathcal{F}^{\prime}/\mathcal{F})=\overline{\rho}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}(\overline{\mathscr{G}}_{\mathcal{F}}^{\circ})\subset{(\overline{\mathscr{Q}}^{\Theta})}^{\circ}$ is equality. According to Lemma 3.7(ii), ${({\overline{\mathscr{Q}}}^{\Theta})}^{\circ}$ is a pseudo-parabolic $\kappa$-subgroup of $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$.

To prove (iii), let $F^{\prime}\prec F$ be as in the proof of (i) above and $\overline{\mathscr{Q}}$ be the image of  $\overline{\rho}_{F^{\prime},F}:\overline{\mathscr{H}}_{F}^{\circ}\rightarrow\overline{\mathscr{H}}_{F^{\prime}}^{\circ}$. Then, as we saw above, $\overline{\mathscr{Q}}$ is a $\Theta$-stable pseudo-parabolic $\kappa$-subgroup of $\overline{\mathscr{H}}^{\circ}_{F^{\prime}}$ and $\mathfrak{p}(\mathcal{F}^{\prime}/\mathcal{F})=\overline{\mathscr{P}}:={(\overline{\mathscr{Q}}^{\Theta})}^{\circ}$. The inverse image of the subgroup $\overline{\mathscr{Q}}(\kappa)$ of $\overline{\mathscr{H}}^{\circ}_{F^{\prime}}(\kappa)$ under the natural surjective homomorphism ${\mathscr{H}}^{\circ}_{F^{\prime}}(\mathcal{O})\rightarrow\overline{\mathscr{H}}^{\circ}_{F^{\prime}}(\kappa)$ equals $\rho_{F^{\prime},F}({\mathscr{H}}^{\circ}_{F}(\mathcal{O}))\,(\subset{\mathscr{H}}^{\circ}_{F^{\prime}}(\mathcal{O}))$, see [P2, 1.10 (4)]. Let ${\mathscr{G}}_{F}={({\mathscr{H}}^{\circ}_{F})}^{\Theta}$ and ${\mathscr{G}}_{F^{\prime}}={({\mathscr{H}}^{\circ}_{F^{\prime}})}^{\Theta}$. We will denote the $\mathcal{O}$-group scheme homomorphism ${\mathscr{G}}_{F}\rightarrow{\mathscr{G}}_{F^{\prime}}$ induced by $\rho_{F^{\prime},F}$ by $\rho^{\Theta}_{F^{\prime},F}$; the corresponding homomorphism $\overline{\mathscr{G}}_{F}\rightarrow\overline{\mathscr{G}}_{F^{\prime}}$ between the special fibers of  ${\mathscr{G}}_{F}$ and ${\mathscr{G}}_{F^{\prime}}$ will be denoted by $\overline{\rho}^{\Theta}_{F^{\prime},F}$. The neutral components of ${\mathscr{G}}_{F}$ and ${\mathscr{G}}_{F^{\prime}}$ are ${\mathscr{G}}^{\circ}_{\mathcal{F}}$ and ${\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$ respectively (3.11). Let ${\mathscr{G}}^{\natural}_{F}\,(\supset{\mathscr{G}}^{\circ}_{\mathcal{F}})$ be the inverse image of ${\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$ in ${\mathscr{G}}_{F}$ under $\rho^{\Theta}_{F^{\prime},F}$. Since the homomorphism $\rho_{F^{\prime},F}$ is the identity on the generic fiber $H$, we infer that $h\in{\mathscr{H}}^{\circ}_{F}(\mathcal{O})$ is fixed under $\Theta$ if and only if so is $\rho_{F^{\prime},F}(h)$, and as the generic fiber of both ${\mathscr{G}}^{\circ}_{\mathcal{F}}$ and ${\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$ is $G$, the generic fiber of ${\mathscr{G}}^{\natural}_{F}$ is also $G$. It is easily seen now that the inverse image of the subgroup $\mathfrak{p}(\mathcal{F}^{\prime}/\mathcal{F})(\kappa)$ of  $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}(\kappa)$, under the natural surjective homomorphism ${\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}(\mathcal{O})\rightarrow\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}(\kappa)$, is $\rho^{\Theta}_{F^{\prime},F}({\mathscr{G}}^{\natural}_{F}(\mathcal{O}))$. We will presently show that the last group equals $\rho^{G}_{\mathcal{F}^{\prime},\mathcal{F}}({\mathscr{G}}^{\circ}_{\mathcal{F}}(\mathcal{O}))$, this will prove (iii).

${\mathscr{G}}^{\natural}_{F}$ is the union of its generic fiber $G$ and its special fiber $\overline{\mathscr{G}}^{\natural}_{F}$; and the identity component of $\overline{\mathscr{G}}^{\natural}_{\mathcal{F}}$ is clearly $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$. We have shown above that the image $\overline{\mathscr{P}}$ of  $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$ under the homomorphism $\overline{\rho}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}$ is a pseudo-parabolic $\kappa$-subgroup of $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$ and the kernel of this homomorphism is smooth. Hence, as $\kappa$ is separably closed, $\overline{\rho}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}(\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}(\kappa))=\overline{\mathscr{P}}(\kappa)$. So, according to [CGP, Thm. C.2.23], there is a pseudo-parabolic $\kappa$-subgroup $\overline{\mathscr{P}}^{\prime}$ of $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$, that contains $\overline{\mathscr{P}}$, such that $\overline{\rho}^{\Theta}_{F^{\prime},F}(\overline{\mathscr{G}}^{\natural}_{F}(\kappa))=\overline{\mathscr{P}}^{\prime}(\kappa)$. But since $\kappa$ is infinite, $\overline{\mathscr{P}}^{\prime}(\kappa)/\overline{\mathscr{P}}(\kappa)$ is infinite unless $\overline{\mathscr{P}}^{\prime}=\overline{\mathscr{P}}$. So we conclude that $\overline{\mathscr{P}}^{\prime}=\overline{\mathscr{P}}$, and then $\overline{\rho}^{\Theta}_{F^{\prime},F}(\overline{\mathscr{G}}^{\natural}_{F}(\kappa))=\overline{\mathscr{P}}(\kappa)=\overline{\rho}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}(\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}(\kappa))$. Now using this, and the fact that the natural homomorphism ${\mathscr{G}}^{\circ}_{\mathcal{F}}(\mathcal{O})\rightarrow\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}(\kappa)$ is surjective (since $\mathcal{O}$ is henselian and ${\mathscr{G}}^{\circ}_{\mathcal{F}}$ is smooth, [EGA IV4, 18.5.17]) and the kernel of this homomorphism equals the kernel of the natural surjective homomorphism ${\mathscr{G}}^{\natural}_{F}(\mathcal{O})\rightarrow\overline{\mathscr{G}}^{\natural}_{F}(\kappa)$, we see that $\rho^{G}_{\mathcal{F}^{\prime},\mathcal{F}}({\mathscr{G}}^{\circ}_{\mathcal{F}}(\mathcal{O}))=\rho^{\Theta}_{F^{\prime},F}({\mathscr{G}}^{\natural}_{F}(\mathcal{O}))$. This proves (iii).

Finally, to prove the last assertion of the proposition, we fix a facet $F^{\prime}$ of ${\mathcal{B}}(H/K)$ that meets $\mathcal{F}^{\prime}$. Then ${\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}={\mathscr{G}}^{\circ}_{F^{\prime}}$ (3.11). Using Lemma 3.7(iii) for $\kappa$ in place of $k$ and $\overline{\mathscr{H}}^{\circ}_{F^{\prime}}$ in place of $\mathcal{H}$, we find a $\Theta$-stable pseudo-parabolic $\kappa$-subgroup $\overline{\mathscr{Q}}$ of $\overline{\mathscr{H}}^{\circ}_{F^{\prime}}$ such that $\overline{\mathscr{P}}={(\overline{\mathscr{Q}}^{\Theta})}^{\circ}$. Let $(F^{\prime}\prec)\,F$ be the facet of ${\mathcal{B}}(H/K)$ corresponding to the pseudo-parabolic $\kappa$-subgroup $\overline{\mathscr{Q}}$ of $\overline{\mathscr{H}}^{\circ}_{F^{\prime}}$. Then $F$ is stable under $\Theta$-action. As $F^{\prime}\prec F$, there is a natural $\mathcal{O}$-group scheme homomorphism $\rho_{F^{\prime},F}:{\mathscr{H}}^{\circ}_{F}\rightarrow{\mathscr{H}}^{\circ}_{F^{\prime}}$ that restricts to a $\mathcal{O}$-group scheme homomorphism $\rho^{G}_{F^{\prime},F}:{\mathscr{G}}^{\circ}_{F}\rightarrow{\mathscr{G}}^{\circ}_{F^{\prime}}$. Let $\overline{\mathscr{Q}}$ be the image of the former. Then according to (ii), the image of the latter is ${(\overline{\mathscr{Q}}^{\Theta}})^{\circ}=\overline{\mathscr{P}}$. Let $P={\mathscr{G}}^{\circ}_{F}(\mathcal{O})\subset{\mathscr{G}}^{\circ}_{F^{\prime}}(\mathcal{O})=:Q$, and $\mathcal{F}=\mathcal{F}_{P}$. Then $P\subset Q$ are parahoric subgroups of $G(K)$, $\mathcal{F}^{\prime}=\mathcal{F}_{Q}\subset{\overline{\mathcal{F}}}_{Q}\subset{\overline{\mathcal{F}}}_{P}=\overline{\mathcal{F}}$, thus $\mathcal{F}^{\prime}\prec\mathcal{F}$. As $F$ and $F^{\prime}$ meet $\mathcal{F}$ and $\mathcal{F}^{\prime}$ respectively, ${\mathscr{G}}^{\circ}_{\mathcal{F}}={\mathscr{G}}^{\circ}_{F}$ and ${\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}={\mathscr{G}}^{\circ}_{F^{\prime}}$ (3.11), and hence the image of the homomorphism $\overline{\rho}^{G}_{\mathcal{F}^{\prime},\mathcal{F}}:\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}\rightarrow\overline{\mathscr{G}}^{\circ}_{\mathcal{F}^{\prime}}$ equals $\overline{\mathscr{P}}$. $\Box$

Proposition 3.12 and [CGP, Propositions 2.2.10 and 3.5.1] imply the following. (Recall that the residue field $\kappa$ of $K$ has been assumed to be separably closed!)

Corollary 3.13. (i)* A facet $\mathcal{F}$ of $\mathscr{B}$ is a chamber $($=maximal facet$)$ if and only if  $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$ does not contain a proper pseudo-parabolic $\kappa$-subgroup. Equivalently, $\mathcal{F}$ is a chamber if and only if the pseudo-reductive quotient  $\overline{G}^{\,\rm pred}_{\mathcal{F}}:=\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}/{\mathscr{R}}_{u,\kappa}(\overline{\mathscr{G}}^{\circ}_{\mathcal{F}})$ of  $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$ is commutative $($this is the case if and only if  $\overline{G}^{\,\rm pred}_{\mathcal{F}}$ contains a unique maximal $\kappa$-torus, or, equivalently, every torus of this pseudo-reductive group is central $)$. *(ii)The codimension of a facet $\mathcal{F}$ of $\mathscr{B}$ equals the rank of the derived subgroup of the pseudo-split pseudo-reductive quotient  $\overline{G}^{\,\rm pred}_{\mathcal{F}}$. Hence the dimension of $\mathcal{F}$ equals the dimension of the maximal central torus of $\overline{G}^{\,\rm pred}_{\mathcal{F}}$ $($recall that the central torus of $G$ is $K$-anisotropic (2.13), the dimension of $\mathscr{B}$ equals $K$-rank $G$ (2.16), and rank $\overline{G}^{\,\rm pred}_{\mathcal{F}}$= $K$-rank G$$). Thus a point $x$ of $\mathscr{B}$ is a vertex $($of $\mathscr{B})$ if and only if $\overline{G}^{\,\rm pred}_{x}:=\overline{\mathscr{G}}^{\circ}_{x}/{\mathscr{R}}_{u,\kappa}(\overline{\mathscr{G}}^{\circ}_{x})$ does not contain a nontrivial central torus.

We will now establish the following analogues of Propositions 3.5–3.7 of [P2].

Proposition 3.14. Let $\mathscr{A}$ be an apartment of $\mathscr{B}$, and $\mathcal{C}$, $\mathcal{C}^{\prime}$ two chambers in $\mathscr{A}$. Then there is a gallery joining $\mathcal{C}$ and $\mathcal{C}^{\prime}$ in $\mathscr{A}$, i.e., there is a finite sequence

[TABLE]

of chambers in $\mathscr{A}$ such that for $i$ with $1\leqslant i\leqslant m$, $\mathcal{C}_{i-1}$ and $\mathcal{C}_{i}$ share a face of codimension 1.

Proof. Let $\mathscr{A}_{2}$ be the codimension $2$-skelton of $\mathscr{A}$, i.e., the union of all facets in $\mathscr{A}$ of codimension at least $2$. Then $\mathscr{A}_{2}$ is a closed subset of $\mathscr{A}$ of codimension $2$, so $\mathscr{A}-\mathscr{A}_{2}$ is a connected open subset of the affine space $\mathscr{A}$. Hence $\mathscr{A}-\mathscr{A}_{2}$ is arcwise connected. This implies that given points $x\in\mathcal{C}$ and $x^{\prime}\in\mathcal{C}^{\prime}$, there is a piecewise linear curve in $\mathscr{A}-\mathscr{A}_{2}$ joining $x$ and $x^{\prime}$. Now the chambers in $\mathscr{A}$ that meet this curve make a gallery joining $\mathcal{C}$ to $\mathcal{C}^{\prime}$. $\Box$

As the central torus of $G$ is $K$-anisotropic, the dimension of any apartment, or any chamber, in $\mathscr{B}$ is equal to the $K$-rank of $G$. A panel in $\mathscr{B}$ is by definition a facet of codimension $1$.

Proposition 3.15. $\mathscr{B}$* is thick, that is any panel is a face of at least three chambers, and every apartment of $\mathscr{B}$ is thin, that is any panel lying in an apartment is a face of exactly two chambers of the apartment.*

Proof. Let $\mathcal{F}$ be a facet of $\mathscr{B}$ that is not a chamber, and $\mathcal{C}$ be a chamber of which $\mathcal{F}$ is a face. Then there is an $\mathcal{O}$-group scheme homomorphism $\rho^{G}_{\mathcal{F},\mathcal{C}}:{\mathscr{G}}^{\circ}_{\mathcal{C}}\rightarrow{\mathscr{G}}^{\circ}_{\mathcal{F}}$ (3.2). The image of $\overline{\mathscr{G}}^{\circ}_{\mathcal{C}}$ in $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$, under the induced homomorphism of special fibers, is a minimal pseudo-parabolic $\kappa$-subgroup of $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$, and conversely, any minimal pseudo-parabolic $\kappa$-subgroup of the latter determines a chamber of $\mathscr{B}$ with $\mathcal{F}$ as a face (Corollary 3.13). Now as $\kappa$ is infinite, $\overline{\mathscr{G}}^{\circ}_{\mathcal{F}}$ contains infinitely many minimal pseudo-parabolic $\kappa$-subgroups. We conclude that $\mathcal{F}$ is a face of infinitely many chambers.

The second assertion follows at once from the following well-known result in algebraic topology: In any simplicial complex whose geometric realization is a topological manifold without boundary (such as an apartment $\mathscr{A}$ of $\mathscr{B}$), any simplex of codimension 1 is a face of exactly two chambers (i.e., maximal dimensional simplices). $\Box$

Proposition 3.16. Let $\mathscr{A}$ be an apartment of $\mathscr{B}$ and $S$ be the maximal $K$-split torus of $G$ corresponding to this apartment. $($Then $\mathscr{A}={\mathcal{B}}(Z_{H}(S)/K)^{\Theta}.)$ The group $N_{G}(S)(K)$ acts transitively on the set of chambers of $\mathscr{A}$.

Proof. According to Proposition 3.14, given any two chambers in $\mathscr{A}$, there exists a minimal gallery in $\mathscr{A}$ joining these two chambers. So to prove the proposition by induction on the length of a minimal gallery joining two chambers, it suffices to prove that given two different chambers $\mathcal{C}$ and $\mathcal{C}^{\prime}$ in $\mathscr{A}$ which share a panel $\mathcal{F}$, there is an element $n\in N_{G}(S)(K)$ such that $n\cdot\mathcal{C}=\mathcal{C}^{\prime}$. Let ${\mathscr{G}}:={\mathscr{G}}^{\circ}_{\mathcal{F}}$ be the Bruhat-Tits smooth affine $\mathcal{O}$-group scheme associated to the panel $\mathcal{F}$ and ${\mathscr{S}}\subset{\mathscr{G}}$ be the closed $\mathcal{O}$-torus with generic fiber $S$. Let $\overline{\mathscr{G}}$ be the special fiber of ${\mathscr{G}}$ and ${\overline{\mathscr{S}}}$ the special fiber of ${\mathscr{S}}$. Then ${\overline{\mathscr{S}}}$ is a maximal torus of $\overline{\mathscr{G}}$. The chambers $\mathcal{C}$ and $\mathcal{C}^{\prime}$ correspond to minimal pseudo-parabolic subgroups $\overline{\mathscr{P}}$ and $\overline{\mathscr{P}}^{\prime}$ of $\overline{\mathscr{G}}$ (Corollary 3.13). Both of these minimal pseudo-parabolic $\kappa$-subgroups contain ${\overline{\mathscr{S}}}$ since the chambers $\mathcal{C}$ and $\mathcal{C}^{\prime}$ lie in $\mathscr{A}$. But then by Theorems C.2.5 and C.2.3 of [CGP], there is an element $\overline{n}\in\overline{\mathscr{G}}(\kappa)$ that normalizes ${\overline{\mathscr{S}}}$ and conjugates $\overline{\mathscr{P}}$ onto $\overline{\mathscr{P}}^{\prime}$. Now from Proposition 2.1(iii) of [P2] we conclude that there is an element $n\in N_{{\mathscr{G}}}({\mathscr{S}})(\mathcal{O})$ lying over $\overline{n}$. It is clear that $n$ normalizes $S$ and hence it lies in $N_{G}(S)(K)$; it fixes $\mathcal{F}$ pointwise and $n\cdot\mathcal{C}=\mathcal{C}^{\prime}$. $\Box$

Now in view of Propositions 2.17, 3.4, 3.14 and 3.15, Theorem 3.11 of [Ro] (cf. also [P2, 1.8]) implies that $\mathscr{B}$ is an affine building if for any maximal $K$-split torus $S$ of $G$, ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ is taken to be the corresponding apartment, and $\mathscr{B}$ is given the polysimplicial structure described in 3.5. Thus we obtain the following:

Theorem 3.17. $\mathscr{B}={\mathcal{B}}(H/K)^{\Theta}$* is an affine building. Its apartments are the affine spaces ${\mathcal{B}}(Z_{H}(S)/K)^{\Theta}$ under $V(S):={\mathbb{R}}\otimes_{{\mathbb{Z}}}{\rm{X}}_{\ast}(S)$ for maximal $K$-split tori $S$ of $G$. Its facets are as in $3.5$. The group $G(K)$ acts on $\mathscr{B}$ by polysimplicial isometries.*

From Propositions 2.18 and 3.16 we obtain the following.

Proposition 3.18. $G(K)$* acts transitively on the set of ordered pairs $(\mathscr{A},\mathcal{C})$ consisting of an apartment $\mathscr{A}$ of $\mathscr{B}$ and a chamber $\mathcal{C}$ of $\mathscr{A}$.*

Remark 3.19. (i) As in [P2, 3.16], using the preceding proposition we can obtain Tits systems in suitable subgroups of $G(K)$.

(ii) As in [P2, §5], we can obtain filtration of root groups and a valuation of root datum for $G/K$.

§4. Tamely-ramified descent

We begin by proving the following proposition in which $\kappa$ is any field of characteristic $p\geqslant 0$.

Proposition 4.1. Let $\mathcal{H}$ be a noncommutative pseudo-reductive $\kappa$-group, $\theta$ a $\kappa$-automorphism of $\mathcal{H}$ of finite order not divisible by $p$, and ${\mathcal{G}}:={(\mathcal{H}^{\langle\theta\rangle})}^{\circ}$. Then

(i) *No maximal torus of ${\mathcal{G}}$ is central in $\mathcal{H}$. *

(ii) The centralizer in $\mathcal{H}$ of any maximal torus of ${\mathcal{G}}$ is commutative.

(iii) Given a maximal $\kappa$-torus $\mathcal{S}$ of ${\mathcal{G}}$, there is a $\theta$-stable maximal $\kappa$-torus of $\mathcal{H}$ containing $\mathcal{S}$.

(iv) If $\kappa$ is separably closed, then $\mathcal{H}$ contains a $\theta$-stable proper pseudo-parabolic $\kappa$-subgroup.

Proof. We fix an algebraic closure $\overline{\kappa}$ of $\kappa$. Let $\mathcal{H}^{\prime}$ be the maximal reductive quotient of $\mathcal{H}_{\overline{\kappa}}$. As $\mathcal{H}$ is noncommutative, $\mathcal{H}^{\prime}$ is also noncommutative (see [CGP, Prop. 1.2.3]). The automorphism $\theta$ induces a $\overline{\kappa}$-automorphism of $\mathcal{H}^{\prime}$ which we will denote again by $\theta$. According to a theorem of Steinberg [St, Thm. 7.5], $\mathcal{H}_{\overline{\kappa}}$ contains a $\theta$-stable Borel subgroup $\mathscr{B}$, and this Borel subgroup contains a $\theta$-stable maximal torus $\mathcal{T}$. The natural quotient map $\pi:\mathcal{H}_{\overline{\kappa}}\rightarrow\mathcal{H}^{\prime}$ carries $\mathcal{T}$ isomorphically onto a maximal torus of $\mathcal{H}^{\prime}$. We endow the root system of $\mathcal{H}^{\prime}$ with respect to the maximal torus $\mathfrak{T}^{\prime}:=\pi(\mathcal{T})\cap\mathscr{D}(\mathcal{H}^{\prime})$ of the derived subgroup $\mathscr{D}(\mathcal{H}^{\prime})$ of $\mathcal{H}^{\prime}$ with the ordering determined by the Borel subgroup $\pi({\mathcal{B}})$. Let $a$ be the sum of all positive roots. Then as $\pi({\mathcal{B}})$ is $\theta$-stable, $a$ is fixed under $\theta$ acting on the character group ${\rm X}(\mathfrak{T}^{\prime})$ of $\mathfrak{T}^{\prime}$. Therefore, ${\rm X}(\mathfrak{T}^{\prime})$ admits a nontrivial torsion-free quotient on which $\theta$ acts trivially. This implies that $\mathcal{T}$ contains a nontrivial subtorus ${\mathscr{T}}$ that is fixed pointwise under $\theta$ and is mapped by $\pi$ into $\mathfrak{T}^{\prime}\,(\subset\mathscr{D}(\mathcal{H}^{\prime}))$. The subtorus ${\mathscr{T}}$ is therefore contained in ${\mathcal{G}}_{\overline{\kappa}}$. Since the center of the semi-simple group $\mathscr{D}(\mathcal{H}^{\prime})$ does not contain a nontrivial smooth connected subgroup, we infer that ${\mathscr{T}}$ is not central in $\mathcal{H}_{\overline{\kappa}}$. Thus the subgroup ${\mathcal{G}}_{\overline{\kappa}}$ contains a noncentral torus of $\mathcal{H}_{\overline{\kappa}}$. Now by conjugacy of maximal tori in ${\mathcal{G}}_{\overline{\kappa}}$, we see that no maximal torus of this group can be central in $\mathcal{H}_{\overline{\kappa}}$. This proves (i).

To prove (ii), it would suffice to show that the centralizer $Z_{\mathcal{H}}(\mathcal{S})$ in $\mathcal{H}$ of a maximal $\kappa$-torus $\mathcal{S}$ of ${\mathcal{G}}$ is commutative. This centralizer is a $\theta$-stable pseudo-reductive subgroup of $\mathcal{H}$, and ${(Z_{\mathcal{H}}(\mathcal{S})^{\langle\theta\rangle})}^{\circ}=Z_{{\mathcal{G}}}(\mathcal{S})$. As $\mathcal{S}$ is a maximal torus of $Z_{{\mathcal{G}}}(\mathcal{S})$ that is central in $Z_{\mathcal{H}}(\mathcal{S})$, if $Z_{\mathcal{H}}(\mathcal{S})$ were noncommutative, we could apply (i) to this subgroup in place of $\mathcal{H}$ to get a contradiction.

To prove (iii), we consider the centralizer $Z_{\mathcal{H}}(\mathcal{S})$ of $\mathcal{S}$ in $\mathcal{H}$. This centralizer is $\theta$-stable and commutative according to (ii). The unique maximal $\kappa$-torus of it contains $\mathcal{S}$ and is a $\theta$-stable maximal torus of $\mathcal{H}$.

To prove (iv), we assume now that $\kappa$ is separably closed and let $\mathcal{S}$ be a maximal torus of ${\mathcal{G}}$. Then $\mathcal{S}$ is $\kappa$-split, and in view of (i), there is a 1-parameter subgroup $\lambda:{\rm{GL}}_{1}\rightarrow\mathcal{S}$ whose image is not central in $\mathcal{H}$. Then $P_{\mathcal{H}}({\lambda})$ is a $\theta$-stable proper pseudo-parabolic $\kappa$-subgroup of $\mathcal{H}$. $\Box$

In the following proposition we will use the notation introduced in §§1, 2. As in 2.13, we will assume that $H$ is semi-simple and the central torus of $G$ is $K$-anisotropic. We will further assume that $H$ is $K$-isotropic, $\Theta$ is a finite cyclic group of automorphisms of $H$, and $p$ does not divide the order of $\Theta$.

Proposition 4.2. The Bruhat-Tits building ${\mathcal{B}}(H/K)$ of $H(K)$ contains a $\Theta$-stable chamber.

Proof. Let $F$ be a $\Theta$-stable facet of ${\mathcal{B}}(H/K)$ that is maximal among the $\Theta$-stable facets. Let ${\mathscr{H}}:={\mathscr{H}}^{\circ}_{F}$ be the Bruhat-Tits smooth affine $\mathcal{O}$-group scheme with generic fiber $H$, and connected special fiber $\overline{\mathscr{H}}$, corresponding to $F$. Let $\mathcal{H}:=\overline{\mathscr{H}}/{\mathscr{R}}_{u,\kappa}(\overline{\mathscr{H}})$ be the maximal pseudo-reductive quotient of $\overline{\mathscr{H}}$. In case $\mathcal{H}$ is commutative, $\overline{\mathscr{H}}$ does not contain a proper pseudo-parabolic $\kappa$-subgroup and so $F$ is a chamber of ${\mathcal{B}}(H/K)$. We assume, if possible, that $\mathcal{H}$ is not commutative. As $F$ is stable under the action of $\Theta$, there is a natural action of this finite cyclic group on ${\mathscr{H}}$ by $\mathcal{O}$-group scheme automorphisms (2.5(i)). This action induces an action of $\Theta$ on $\overline{\mathscr{H}}$, and so also on its pseudo-reductive quotient $\mathcal{H}$. Now taking $\theta$ to be a generator of $\Theta$, and using Proposition 4.1(iv), we conclude that $\mathcal{H}$ contains a $\Theta$-stable proper pseudo-parabolic $\kappa$-subgroup. The inverse image $\overline{\mathscr{P}}$ in $\overline{\mathscr{H}}$ of any such pseudo-parabolic subgroup of $\mathcal{H}$ is a $\Theta$-stable proper pseudo-parabolic $\kappa$-subgroup of $\overline{\mathscr{H}}$. The facet $F^{\prime}$ corresponding to $\overline{\mathscr{P}}$ is $\Theta$-stable and $F\prec F^{\prime}$. This contradicts the maximality of $F$. Hence, $\mathcal{H}$ is commutative and $F$ is a chamber. $\Box$

To prove the next theorem (Theorem 4.4), we will use the following:

Proposition 4.3. Let $\mathfrak{K}$ be a field complete with respect to a discrete valuation and with separably closed residue field. Let $\mathfrak{G}$ be a connected absolutely simple $\mathfrak{K}$-group of inner type $A$ that splits over a finite tamely-ramified field extension $\mathfrak{L}$ of $\mathfrak{K}$. Then $\mathfrak{G}$ is $\mathfrak{K}$-split.

Proof. We may (and do) assume that $\mathfrak{G}$ is simply connected. Then $\mathfrak{G}$ is $\mathfrak{K}$-isomorphic to ${\rm{SL}}_{n,\mathfrak{D}}$, where $\mathfrak{D}$ is a finite dimensional division algebra with center $\mathfrak{K}$ that splits over the finite tamely-ramified field extension $\mathfrak{L}$ of $\mathfrak{K}$. By Propositions 4 and 12 of [S, Ch. II] the degree of $\mathfrak{D}$ is a power of $p$, where $p$ is the characteristic of the residue field of $\mathfrak{K}$. But a noncommutative division algebra of degree a power of $p$ cannot split over a field extension of degree prime to $p$. So, $\mathfrak{D}=\mathfrak{K}$, hence $\mathfrak{G}\simeq{\rm{SL}}_{n}$ is $\mathfrak{K}$-split. $\Box$

Theorem 4.4. A semi-simple $K$-group $G$ that is quasi-split over a finite tamely-ramified field extension of $K$ is already quasi-split over $K$.

This theorem has been proved by Philippe Gille in [Gi] by an entirely different method.

Proof. We assume that all field extensions appearing in this proof are contained in a fixed separable closure of $K$. To prove the theorem, we may (and do) replace $G$ by its simply-connected central cover and assume that $G$ is simply connected. Let $S$ be a maximal $K$-split torus of $G$. Then $G$ is quasi-split over a (separable) extension $L$ of $K$ if and only if the derived subgroup $Z_{G}(S)^{\prime}$ of the centralizer $Z_{G}(S)$ of $S$ is quasi-split over $L$. Moreover, $G$ is quasi-split over $K$ if and only if $Z_{G}(S)^{\prime}$ is trivial. Therefore, to prove the theorem we need to show that any semi-simple simply connected $K$-anisotropic $K$-group that is quasi-split over a finite tamely-ramified field extension of $K$ is necessarily trivial. Let $G$ be any such group. In view of Proposition 1.1, we may replace $K$ by its completion and assume that $K$ is complete.

There exists a finite indexing set $I$, and for each $i\in I$, a finite separable field extension $K_{i}$ of $K$ and an absolutely almost simple simply connected $K_{i}$-anisotropic $K_{i}$-group $G_{i}$ such that $G=\prod_{i\in I}{\rm{R}}_{K_{i}/K}(G_{i})$. Now $G$ is quasi-split over a finite separable field extension $L$ of $K$ if and only if for each $i$, ${\rm{R}}_{K_{i}/K}(G_{i})$ is quasi-split over $L$. But ${\rm{R}}_{K_{i}/K}(G_{i})$ is quasi-split over $L$ if and only if $G_{i}$ is quasi-split over the compositum $L_{i}:=K_{i}L$. For $i\in I$, the finite extension $K_{i}$ of $K$ is complete and its residue field is separably closed, and if $L$ is a finite tamely-ramified field extension of $K$, then $L_{i}$ is a finite tamely-ramified field extension of $K_{i}$. So to prove the theorem, we may (and do) replace $K$ by $K_{i}$ and $G$ by $G_{i}$ to assume that $G$ is an absolutely almost simple simply connected $K$-anisotropic $K$-group that is quasi-split over a finite tamely-ramified field extension of $K$. We will show that such a group $G$ is trivial.

Since the residue field $\kappa$ of $K$ is separably closed, any finite tamely-ramified field extension of $K$ is a cyclic Galois extension. Let $L$ be a finite tamely-ramified field extension of $K$ of minimal degree over which $G$ is quasi-split. The Galois group $\Theta$ of $L/K$ is a finite cyclic group and its order is not divisible by $p$ (= char($\kappa$)).

As $G_{L}$ is quasi-split, Bruhat-Tits theory is available for $G$ over $L$ [BrT2, §4]. Let $H={\rm{R}}_{L/K}(G_{L})$. Then $H$ is quasi-split over $K$ and hence Bruhat-Tits theory is also available for $H$ over $K$. Let ${\mathcal{B}}(H/K)$ be the Bruhat-Tits building of $H(K)\,(=G(L))$. Elements of $\Theta$ act by $K$-automorphisms on $H$ and so on ${\mathcal{B}}(H/K)$ by polysimplicial isometries; moreover, $G=H^{\Theta}$. The Galois group $\Theta$ acts on $G(L)$ by continuous automorphisms and so it acts on the Bruhat-Tits building ${\mathcal{B}}(G/L)$ of $G(L)$ by polysimplicial isometries. There is a natural $\Theta$-equivariant identification of the building ${\mathcal{B}}(H/K)$ with the building ${\mathcal{B}}(G/L)$. (Note that $K$-rank $H$ = $L$-rank $G_{L}$, and there is a natural bijective correspondence between the set of maximal $K$-split tori of $H$ and the set of maximal $L$-split tori of $G_{L}$, see [CGP, Prop. A.5.15(2)]. This correspondence will be used below.) The results of §3 imply that Bruhat-Tits theory is available for $G$ over $K$ and $\mathscr{B}:={\mathcal{B}}(H/K)^{\Theta}\,(={\mathcal{B}}(G/L)^{\Theta})$ is the Bruhat-Tits building of $G(K)$.

Since $G$ is $K$-anisotropic, the building $\mathscr{B}$ of $G(K)$ consists of a single point (Proposition 2.17), hence $\Theta$ fixes a unique point of  ${\mathcal{B}}(G/L)$. Let $C$ be the facet of ${\mathcal{B}}(G/L)$ that contains this point. Then $C$ is stable under $\Theta$. According to Proposition 4.2, $C$ is a chamber. Let ${\mathscr{H}}:={\mathscr{H}}_{C}^{\circ}$ be the Bruhat-Tits smooth affine $\mathcal{O}$-group scheme associated to $C$ with generic fiber $H$ and connected special fiber $\overline{\mathscr{H}}$. As $C$ is a chamber, the maximal pseudo-reductive quotient $\overline{\mathscr{H}}^{\rm{pred}}$ of $\overline{\mathscr{H}}$ is commutative [P2, 1.10]. Now using Proposition 2.6 for $\Omega=C$ we obtain a $\Theta$-stable maximal $K$-split torus $T$ of $H$ such that $C$ lies in the apartment $A(T)$ corresponding to $T$ (and the special fiber of the schematic closure of $T$ in ${\mathscr{H}}$ maps onto the maximal torus of $\overline{\mathscr{H}}^{\rm{pred}}$). Let $T^{\prime}$ be the image of $T_{L}$ under the natural surjective homomorphism $q:\,H_{L}={\rm{R}}_{L/K}(G_{L})_{L}\rightarrow G_{L}$ (for the proof of surjectivity of $q$, see [CGP, Prop. A.5.11(1)]). Then $T^{\prime}$ is a $L$-torus of $G_{L}$ and according to [CGP, Prop. A.5.15(2)] it is the unique maximal $L$-split torus of $G_{L}$ such that ${\rm{R}}_{L/K}(T^{\prime})\,(\subset{\rm{R}}_{L/K}(G_{L})=H)$ contains the maximal $K$-split torus $T$ of $H$.

We identify $H(K)$ with $G(L)$. Then for $x\in H(K)(\subset H(L))$ and $\theta\in\Theta$, we have $q(\theta(x))=\theta(x)$. Since $T(K)$ is $\Theta$-stable, for $t\in T(K)$ and $\theta\in\Theta$, $\theta(t)$ lies in $T^{\prime}(L)$. Now as $T(K)$ is Zariski-dense in $T$, its image in $T^{\prime}(L)$ is Zariski-dense in $T^{\prime}$. Since this image is stable under the action of $\Theta={\rm{Gal}}(L/K)$ on $G(L)$, from the Galois criterion [Bo, Ch. AG, Thm. 14.4(3)] we infer that $T^{\prime}$ descends to a $K$-torus of $G$, i.e., there is a $K$-torus $\mathcal{T}$ of $G$ such that $T^{\prime}=\mathcal{T}_{L}$. In the natural identification of ${\mathcal{B}}(H/K)$ with ${\mathcal{B}}(G/L)$, the apartment $A(T)$ of the former is $\Theta$-equivariantly identified with the apartment $A(T^{\prime})$ of the latter. We will view the chamber $C$ as a $\Theta$-stable chamber in $A(T^{\prime})$.

Let $\Delta$ be the basis of the affine root system of the absolutely almost simple, simply connected quasi-split $L$-group $G_{L}$ with respect to $T^{\prime}\,(=\mathcal{T}_{L})$, determined by the $\Theta$-stable chamber $C$ [BrT2, §4]. Then $\Delta$ is stable under the action of $\Theta$ on the affine root system of $G_{L}$ with respect to $T^{\prime}$. There is a natural $\Theta$-equivariant bijective correspondence between the set of vertices of $C$ and $\Delta$. Since $\mathscr{B}$, and hence $C^{\Theta}$, consists of a single point, $\Theta$ acts transitively on the set of vertices of $C$ so it acts transitively on $\Delta$. Now from the classification of irreducible affine root systems [BrT1, §1.4.6], we see that $G_{L}$ is a split group of type $A_{n}$ for some $n$. Proposition 4.3 implies that $G$ cannot be of inner type $A_{n}$ over $K$. On the other hand, if $G$ is of outer type $A_{n}$, then over a quadratic Galois extension $K^{\prime}\,(\subset L)$ of $K$ it is of inner type. Now, according to Proposition 4.3, $G$ splits over $K^{\prime}$. We conclude that $L=K^{\prime}$ and hence $\#\Theta=2$. As $\Theta$ acts transitively on $\Delta$ and $\#\Delta=n+1$, we infer that $n+1=2$, i.e., $n=1$, and then $G$ is of inner type, a contradiction. $\Box$

4.5. Now let $k$ be a field endowed with a nonarchimedean discrete valuation. We assume that the valuation ring of $k$ is Henselian. Let $K$ be the maximal unramified extension of $k$, and $L$ be a finite tamely-ramified field extension of $K$ with Galois group $\Theta:={\rm{Gal}}(L/K)$. Let $G$ be a connected reductive $k$-group that is quasi-split over $K$ and $H={\rm{R}}_{L/K}(G_{L})$. Then $G=H^{\Theta}$, and by Theorem 3.17, the Bruhat-Tits building ${\mathcal{B}}(G/K)$ of $G(K)$ can be identified with the subspace of points in the Bruhat-Tits building of $G(L)\,(=H(K))$ that are fixed under $\Theta$ (with polysimplicial structure on ${\mathcal{B}}(G/K)$ as in 3.5). Now by “unramified descent” [P2], Bruhat-Tits theory is available for $G$ over $k$ and the Bruhat-Tits building of $G(k)$ is ${\mathcal{B}}(G/K)^{{\rm{Gal}}(K/k)}$.

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University of Michigan

Ann Arbor, MI 48109.

e-mail: gprasad@umich.edu