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\title[Analytic filtrations]{Bruhat--Tits theory from Berkovich's point of view. Analytic filtrations}
\alttitle{Point de vue de Berkovich sur l'immeuble et filtrations}

\subjclass{20G25, 20E42, 14G22}
\keywords{Berkovich spaces, Bruhat--Tits buildings, Moy--Prasad filtrations}


\author[\initial{A.} \lastname{Mayeux}]{\firstname{Arnaud} \lastname{Mayeux}}
\address{Université Clermont-Auvergne,\\
CNRS, LMBP, F-63000\\
Clermont-Ferrand (France)}
\email{arnaud.mayeux@uca.fr}
\thanks{This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 101002592). The author was previously funded by a Boya postdoctoral fellowship of BICMR and Peking University.}


\begin{abstract}
We define filtrations by affinoid groups, in the Berkovich analytification of a connected reductive group, related to Moy--Prasad filtrations. They are parametrized by a cone, whose basis is the Bruhat--Tits building and whose vertex is the neutral element, via the notions of Shilov boundary and holomorphically convex envelope.
\end{abstract}

\begin{altabstract}
Nous définissons des groupes affino{\"\i}des, dans l'analytifié d'un groupe réductif connexe, étroitement liés aux filtrations de Moy--Prasad. Ils sont paramétrés par un cône, dont la base est l'immeuble de Bruhat--Tits et dont le sommet est l'élément neutre, grâce aux notions de bord de Shilov et d'enveloppe convexe holomorphe.
\end{altabstract}

\datereceived{2020-01-08}
\daterevised{2020-11-16}
\dateaccepted{2021-08-07}

\editors{X. Caruso and J. Ayoub}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\dateposted{2022-11-14}
\begin{document}

\maketitle

\section{Introduction}
Let $G$ be a connected reductive group over a discretely valued complete non-Archimedean field with a perfect residue field $k$. Berkovich~\cite[Chapter~5]{berk} (split case) and Rémy--Thuillier--Werner~\cite{RTW} (general case) showed that there is a canonical embedding of the Bruhat--Tits building of $G$ into the Berkovich analytification $G^{an}$. This embedding and related ideas form what is called ``\emph{Berkovich's point of view on Bruhat--Tits buildings}''. In this article, we observe that Berkovich's point of view allows to define and parametrize natural $k$-analytic filtrations related to Moy--Prasad filtrations~\cite{Mopr}~\cite{Mopr2}. Let $x$ be a point in the reduced Bruhat--Tits building of $G$ over $k$. The group $G(k)$ acts on the reduced and enlarged buildings. This action is compatible with the canonical projection from the enlarged building to the reduced one. Let us consider the stabilizer of a preimage of $x$ in the enlarged building, this is a compact open subgroup of $G(k)$ independent of the preimage. One idea of Berkovich's point of view is to construct a $k$-affinoid group $G_x$ that realizes this stabilizer. The space $G_x$ is equipped with a partial order and has a maximal point: its Shilov Boundary denoted $\theta (x)$. The space $G_x$ can be recovered from $\theta (x)$ taking the holomorphically convex envelope. The preceding constructions and results for general connected reductive groups are done in~\cite{RTW}. Now our filtrations are some $k$-affinoid subgroups $\{G_{x,r}\} _{r\,\in\,\R _{\geq\,0}}$ of $G_x$ satisfying also that the Shilov boundary of $G_{x,r}$ is a singleton $\theta (x,r)$, and that the holomorphically convex envelope of $\theta (x,r)$ is $G_{x,r}$.

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The heuristic picture represents $G_x$ and its filtrations. Pictorially, the stabilizer of $x$ is the set of lower extremal points, $G_x$ is the whole picture and $G_{x,r}$ is the orange part. The $k$-points of $G_{x,r}$ are the orange lower extremal points. The neutral element is the central lower point. For $r >r ' $ we have
\[
G_{x,r} \underset{{\ne}}{\subset} G_{x,r'}.
\]
When $r$ goes to $ + \infty$, $\theta (x,r) $ goes to the neutral element. The set $\{ \theta (x,r) \mid r \geq 0 \}$ is a line segment joining $\theta (x) $ to the neutral element.

Our construction of $G_{x,r}$ use dilatations and Néron blowups~\cite{NBU}, generic fibers of formal completions of schemes over ring of integers and affinoid descent~\cite{RTW}. We also define similar filtrations for the Lie algebra.

We show that in tame situations $G_{x,r} (k)$ is the corresponding Moy--Prasad group for $r>0$. We prove that the map
\[
\theta : \BT(G,k) \times \R _{\geq\,0} \to G^{an}
\]
is continuous and injective. This gives birth to a topological cone in $G^{an}. $ We also compute some examples.

Let us now describe the structure of the document and of our construction. A posteriori, the formal definition of our filtrations is as follows. Given a point $x$ in the Bruhat--Tits $\BT (G,k)$ of $G$ and a positive real number $r$, we choose a $k$-affinoid extension $K/k$ such that $G$ is split, the image $\iota _{K/k} (x)$ of $x$ in $\BT (G,k)$ is special and $r $ is in $\ord (K)$. We consider the canonical Demazure group scheme $\GG$ over $K^{\circ}$ attached to $\iota _{K/k} (x)$. It is a split reductive group whose generic fiber is $G \times _k K $. We then consider the congruence subgroup $\GG _r$ of $\GG$, whose definition is given in Section~\ref{sectscheme}. Next, we consider the $K$-affinoid analytification (i.e the generic fiber of the formal completion along the special fiber) $\widehat{~\GG _r~}_{\eta}$ of $\GG_r $, this is a $K$-affinoid group. Then $G_{x,r}$ is defined as $\pr _{K/k} \widehat{~\GG _r~}_{\eta}$ where $\pr _{K/k}$ is the canonical projection from $(G \times _k K) ^{an} $ to $G^{an}$. We prove that it is a $k$-affinoid group, independent of the choice of $K$. In order to prove that we apply Rémy--Thuillier--Werner's descent theorem, see Section~\ref{sectanalytic} for the statement of this theorem. Applying this descent theorem requires to prove a certain identity:
\[
\pr _{K/k}^{-1} \pr _{K/k} \left(\widehat{~\GG _r~}_{\eta}\right) = \widehat{~\GG _r~}_{\eta}~~(*).
\]
Proving this identity is a guideline for many statements of this paper and is not a formal consequence of the definition given above. That is why our construction is done step by step. The first step (Section~\ref{spli}) deals with split groups. In this step, $(*)$ is proved using explicit computations together with the notion of peaked point. For non split groups, we reduce in Section~\ref{secgebuu} to the split case choosing a finite Galois extension $L/k$ that splits $G$. During this step, we need to show that objects defined at the first step are stable under $\Gal (L/k)$. In this occasion, we before define and study in Section~\ref{secraaa} filtrations for $k$-affinoid groups $H$ that are analytification of Demazure models after a finite Galois base change, here $(*)$ is obtained using Galois stability. Filtrations of Lie algebras are considered in Section~\ref{seclie}. In Section~\ref{seccompa} we compare our filtrations with Moy--Prasad ones and in Section~\ref{seccone} we define the cone. Sections~\ref{exgm} and~\ref{exwild} contain examples.

\subsection*{Acknowledgements}
I thank Anne-Marie Aubert for many helps and comments. I thank Antoine Ducros, Tasho Kaletha, Marco Maculan and Bertrand Rémy for answering some questions about Berkovich spaces, references, and~\cite{RTW}. I thank Jean-François Dat for some discussions. I thank the referees for their suggestion to generalize the construction to all points in the building instead of just the rational points as in~\cite{thmay}. I thank Amaury Thuillier for his precious help in order to generalize the construction to all points, in particular for suggesting me to use the notion of peaked points.



\section*{Notation and prerequisites}
%\begin{sloppypar}
Let $G$ be a connected reductive group scheme over a complete non-Archimedean field $k$. We assume here that $k$ is discretely valued with a perfect residue field. This implies that functoriality of buildings holds~\cite[\S~1.3.4]{RTW}. So for any non-Archimedean extension $K/k$ we have a canonical map between (reduced) Bruhat--Tits buildings $\iota _{K/k} : \BT (G,k) \to \BT (G,K)$. We fix a uniformizer $\pi _k$ of $k$. We denote by $|. | _K$ the norm on a non-Archimedean extension $K$ of $k$ and $\ord $ the extension to $K$ of the additive valuation on $k$ such that $\ord (\pi _k)=1$. We assume $|.|_K= e^{- \ord (.)}$. We sometimes use the notation $|~|$ instead of $|~|_K$. Moreover, we put $K^{\circ}= \{x \in K \mid~|x | _K \leq 1 \}$, $K^{\circ \circ } = \{ x \in K |~|x| _K <1 \}$ and $\tilde{K}= K^{\circ} / K^{\circ \circ}$. We assume that the reader is familiar with reductive group schemes, Bruhat--Tits theory and Berkovich spaces. One can read the necessary material in~\cite[\S~1]{RTW}. If $X$ is a Berkovich $k$-analytic space and $K$ is a non-Archimedean extension, we denote by $X \hat{\otimes }K $ the analytic base change to $K$. If $X$ is an affine scheme, then $\clO(X)$ denotes its coordinate ring. If $G$ is an affine group scheme over a ring, the ring $\clO(X) $ is canonically an Hopf algebra denoted $\Hopf (X)$.
%\end{sloppypar}

\section{Schematic congruence subgroups} \label{sectscheme}

In this section we recall some results about congruence subgroups for group schemes. Congruence subgroups are built using dilatations and Néron blowups~\cite{NBU}. Given 
\[
Z \underset{closed}{\subset} D \underset{closed}{\subset} X
\]
schemes such that $D$ is locally principal, the dilatation of $Z$ in $X$ along $D$ is a scheme $\Bl_Z ^D X$ defined in~\cite{NBU}. Now assume that $X=\GG$ is a smooth group scheme over $K^{\circ}$ where $K$ is a non-Archimedean field, $D= \GG \times _{K^{\circ}} K^{\circ} / \pii $ and $Z=e$ is the neutral section of the group scheme $D$, here $\pii $ is a principal ideal in $K^{\circ}$. In this case $\Bl_Z ^D \GG$ is a group scheme over $K^{\circ}$ called the $\pii$-congruence subgroup of $\GG$ and is denoted $\GG _{\pii}$. Let $\pi$ be a generator of $\pii$.

\pagebreak
\begin{prop}[{\cite{NBU}}]\label{NBU}\leavevmode
\begin{enumerate}
\item\label{prop2.1.1} Assume $\GG = \spec (\fA)$ and let $J \subset \fA$ be the augmentation ideal of the Hopf algebra $\fA$. Then
\[
\GG _{\pii} = \spec \left(\fA \left[\pii ^{-1} J\right]\right)
\]
where $ \fA [\pii ^{-1} J] $ is the ring generated by $\fA$ and $\pii ^{-1} J = \{ \pi^{-1} j | j \in J\} $ inside $\fA \otimes _{K^{\circ}} K$.
\item\label{prop2.1.2} Assume $\pii \ne K^{\circ}$, then the scheme $\GG _{\pii} \times _{K^{\circ}} \tilde{K} $ is a vector group over $\tilde{K} $, in particular it is irreducible.

\item\label{prop2.1.3} We have $\GG _{\pii} (K^{\circ}) = \ker (\GG (K^{\circ}) \to \GG (K^{\circ} / \pii)) $.
\end{enumerate}
\end{prop}

\begin{proof}
\begin{itemize}
\item[\eqref{prop2.1.1}] By~\cite[Remark~2.2]{NBU}, the ring of $\GG _{\pii}$ is $\fA [\frac{I}{\pi}]$ where $I$ is the ideal in $\fA$ that defines the closed subscheme $e$ and $\fA[\frac{I}{\pi}] $ is the $\fA$-subalgebra of $\fA[\pi ^{-1}]$ generated by fraction $i/\pi$ with $i \in I$. It is clear that $\fA[\pi ^{-1}] = \fA \otimes _{K^{\circ} } K$ and that $I = J + \pi \fA$. We deduce that $\fA [\frac{I}{\pi}]= \fA [\pii ^{-1} J]$.

\item[\eqref{prop2.1.2}] We have $\GG _{\pii} \times _{K^{\circ}} K^{\circ} / \pii = \GG _{\pii} \times _D Z $, by~\cite[Proposition~2.9]{NBU} this is a vector group over $K^{\circ} / \pii$. We have 
\[
{\GG _{\pii} \times _{K^{\circ}} \tilde{K}= \left(\GG _{\pii} \times _{K^{\circ}} K^{\circ} / \pii\right) \times _{ K^{\circ} / \pii} \tilde{K}},
\]
which is a vector group over $\tilde{K}$.

\item[\eqref{prop2.1.3}] This is~\cite[Lemma~4.1]{NBU}.\qedhere
\end{itemize}
\end{proof}

Let $G=\spec(A)$ be an affine $k$-group scheme of finite type. Let $K/k$ be a Galois extension and $\fA$ be a flat sub-Hopf-$\Ko$-algebra of finite type of the Hopf $K$-algebra $A_K= A \otimes _k K$ such that $\fA\otimes _{\Ko} K = A_K$. In this situation, we say that $\GG= \spec (\fA)$ is $\Gal(K/k)$-stable if $\fA $ is $\Gal(K/k)$-stable in $ A \otimes _k K$. The following Proposition shows that Galois stability is preserved under the operation of taking congruence subgroups.


\begin{lemm}\label{gammastab}
Assume that $\GG $ is $\Gal(K/k)$-stable.
Then for any principal ideal ${\pii \subset K^{\circ}}$, the congruence subgroup $\GG _{\pii}$ is $\Gal(K/k)$-stable.
\end{lemm}

\begin{proof}
Let $\varepsilon _{\fA} : \fA \to \Ko$ be the augmentation, $J=\ker(\varepsilon _{\fA})$. Let us remark that $\varepsilon _ {\fA} $ is the restriction to $\fA$ of the augmentation $\varepsilon_A \otimes \Id : A \otimes _k K \to K$ of $A_K$. So $J=\ker(\varepsilon _A \otimes \Id) \cap \fA$. The set $\ker(\varepsilon _A \otimes \Id)$ is $\Gal(K/k)$-stable, and $\fA$ is stable by hypothesis, so $J$ is $\Gal(K/k)$-stable as the intersection of two $\Gal(K/k)$-stable subsets of $A \otimes _k K$. By Proposition~\ref{NBU}, the ring of $\GG _{\pii}$ is $\fA [\pii ^{-1} J] \subset A \otimes _k K $ and so it is $\Gal(K/k)$-stable.
\end{proof}



\section{Berkovich \texorpdfstring{$k$}{k}-analytic spaces} \label{sectanalytic}
In this section $k$ is a non-Archimedean field. References for Berkovich analytic spaces are~\cite{berk} and~\cite{berkoetale}. To each scheme $X$ of finite type over $k$, Berkovich \cite[\S~3.4]{berk} associated a $k$-analytic space $X^{an}$ such that for any non-Archimedean field $K/k$, there is a bijection~$X^{an}(K) \simeq X(K)$.

\begin{prop}[{\cite[\S~3.4.2]{berk}\label{analyk}}]
If $X=\spec(A)$, where $A$ is a finitely generated ring over $k$, then the underlying topological space $X^{an}$ coincides with the set of all multiplicative seminorms on $A$ whose restriction to $k$ is the norm on $k$.\linebreak
A point $x$ in $X^{an}$ is also denoted $\lvert~\rvert _x $.
\end{prop}

If $x \in X^{an}$, we define
\[
\Hol (x) := \left\{ y \in X^{an} \middle|\; |f|_y \leq |f|_x \forall f \in A \right\}.
\]
The following definition/proposition is extracted from Rémy--Thuillier--Werner's work~\cite[\S~1.2.4]{RTW}~\cite[\S~2.1.1]{thuithe} (see also~\cite[\S~5.3.2]{berk}).

\begin{defi}[{Analytification of $k^{\circ}$-schemes\label{compfo}}]
Let $\fA$ be a flat topologically finitely presented $\ko$-algebra whose spectrum $\M (\clA)$ we denote $\fX$. Let $X=\spec(\fA \otimes _{\ko} k)$ be the generic fiber of $\fX$. The map
\[ 
|~|_{\fA}: \fA \otimes _{\ko} k \to \R _{\geq\,0}, a\mapsto \inf \left\{ \lvert \lambda \rvert\, \middle|\lambda \in k ^{\times} \;\text{ and }\; a \in \lambda \left(\fA\otimes 1\right)\right\}
\]
is a norm on $\fA\otimes _{\ko} k$. The Banach algebra $\clA=\overline{\fA\otimes _{\ko} k}^{\lvert~\rvert _{\fA}}$ obtained by completion is a strictly $k$-affinoid algebra whose spectrum is denoted by $\widehat{\fX}_{\eta}$ and is called the generic fiber of the formal completion of $\fX$ along its special fiber. This affinoid space is naturally an affinoid domain in $X^{an}$ (whose points are multiplicative seminorms on $\fA\otimes _{\ko} k $ which are bounded with respect to the seminorm $\lvert. \rvert _{\fA}$).
Moreover, there is a reduction map $\tau : \hat{\fX}_{\eta} \to \fX \times _{\ko} \tilde{k}$ defined as follows: a point $x$ in $\hat{\fX}_{\eta}$ gives a sequence of ring homomorphisms:
\[
\fA\to \clH(x)^{\circ} \to \widetilde{\clH(x)}
\]
whose kernel $\tau (x)$ defines a prime ideal of $\fA \otimes _{\ko} \tilde{k}$, i.e a point in $\fX \times_{\ko} \tilde{k}$. If the scheme $\fX $ is integrally closed in its generic fiber — in particular if $\fX $ is smooth — then $\tau$ is the reduction map of Berkovich (see~\cite[\S~2.4]{berk}). And so the Shilov Boundary of $\hat{\fX}_{\eta}$ is in bijection with the irreducible components of the special fiber $\fX \times _{\ko} \tilde{k}$. Moreover, the spectral norm $\rho$ on $\clA$ is equal to $\lvert~\rvert _{\fA}$ if and only if the algebra $\fA \otimes _{\ko} k $ is reduced~\cite[Proposition~2.1.1]{thuithe}.
\end{defi}

\begin{coro}\label{analyko}
Let $\fX= \spec (\fA)$ be a smooth $\ko$-scheme with irreducible special fiber. Let $X$ be the generic fiber of $\fX$. Then
\begin{enumerate}
\item\label{coro3.3.1} $\widehat{\fX}_{\eta}$ is a strictly $k$-affinoid domain of $X^{an}$,
\item\label{coro3.3.2} the Shilov boundary of $\widehat{\fX}_{\eta}$ is a singleton equal to $\lvert~\rvert _{\fA}$,
\item\label{coro3.3.3} $\widehat{\fX}_{\eta}$ is the holomorphically convex envelope of $\lvert~\rvert _{\fA} \in X^{an}$.
\end{enumerate}
\end{coro}

\begin{proof}
\begin{itemize}
\item[\eqref{coro3.3.1}] This is contained in Proposition~\ref{compfo}.
\item[\eqref{coro3.3.2}] Since the special fiber of $\fX$ is irreducible, the Shilov boundary of $\widehat{\fX}_{\eta}$ is a singleton by Proposition~\ref{compfo}. The algebra $ \fA \otimes _{\ko} \tilde{k}$ is reduced since $\fX$ is smooth, thus by Proposition~\ref{compfo}, $\lvert~\rvert _{\fA}$ is the spectral norm. This implies that $\Shi (\widehat{\fX}_{\eta})=\lvert~\rvert _{\fA}$.
\item[\eqref{coro3.3.3}] Put $A= \fA \otimes _{k ^{\circ} } k $. Recall that the holomorphically convex envelope of $\lvert~\rvert _{\fA}$ is
\[
\Hol\left(\lvert~\rvert _{\fA}\right) = \left\{ x \in G^{an} \middle|\lvert f \rvert _x \leq \lvert f \rvert _{\fA}\quad\forall f \in A \right\}.
\]


By Proposition~\ref{compfo} the $k$-affinoid algebra $\clA$ of $ \widehat{\fX}_{\eta}$ is the completion of $A$ relatively to the norm $\lvert~\rvert _{\fA}.$ Let $i$ denote the natural corresponding injective $k$-algebras morphism $A \to \clA$. The inclusion $ \widehat{\fX}_{\eta}\subset G^{an}$ is given by
\begin{align*}
\iota: \M(\clA)&\to G^{an}\\
~~\lvert~\rvert _x~~&\mapsto \lvert~\rvert _x \circ i~.
\end{align*}
Since $\M(\clA)$ is the set of all multiplicative seminorms on $\clA$ bounded by $\lvert~\rvert _{\fA}$,~$\iota (\M(\clA)) $ is contained in the holomorphically convex envelope of $\lvert~\rvert _{\fA}$. Reciprocally, let $x \in \Hol (\lvert~\rvert _{\fA})$, $x= \lvert~\rvert _x$ is a multiplicative seminorm $A \to \R_{\geq\,0}$ such that $\rvert f \lvert _x \leq \lvert f \rvert _{\fA} \forall f \in A$. Since $\clA $ is the completion of $A$ relatively to $\lvert~\rvert _{\fA}$, $\lvert~\rvert _x $ induces a multiplicative seminorm on $\clA $ bounded by $\lvert~\rvert _{\fA}$. This ends the proof of Corollary~\ref{analyko}.\qedhere
\end{itemize}
\end{proof}

If $K/k$ is an affinoid extension, and $X$ is a $k$-analytic space, we denote by $\pr_{K/k}$ the canonical surjective map $X \hat{\otimes} K \to X$. If $K/k$ is a finite Galois extension and $X$ is a $k$-analytic space, the group $\Gal(K/k)$ acts on $X \hat{\otimes} K$ and $\pr_{K/k}$ induces an isomorphism $(X \hat{\otimes} K) / \Gal(K/k) \simeq X$. This implies that if $D_K$ is a subset of $X \hat{\otimes} K$ then $D_K$ is $\Gal(K/k)$-stable if and only if $\pr_{K/k}^{-1} (\pr _{K/k} (D_K))=D_K$. We now state a descent theorem, due to Rémy--Thuillier--Werner.

\begin{theo}[{\cite[Appendix~A]{RTW}\label{descent}}]
Let $X$ be a $k$-affinoid space. Let $K$ be a $k$-affinoid extension. Let $D $ be a subset of $X$, then $D$ is a $k$-affinoid domain of $X$ if and only if the subset $\pr _{K/k}^{-1} (D)$ is a $K$-affinoid domain in $X \hat{\otimes} K$.
\end{theo}

\begin{coro}\label{galdescentdirect}
Let $X$ be a $k$-affinoid space. Let $K/k$ be a finite Galois extension. Let $D_K $ be a $\Gal (K/k)$-stable $K$-affinoid domain of $X \hat{\otimes} K$, put $D=\pr_{K/k}(D_K)$. Then $D$ is a $k$-affinoid domain of $X$.
\end{coro}

\begin{proof}
Since $D_K $ is $\Gal (K/k)$-stable, $\pr_{K/k}^{-1} (\pr _{K/k} (D_K))=D_K$, so $\pr_{K/k}^{-1} (D)$ is $K$-affinoid, so by Theorem~\ref{descent}, $D$ is $k$-affinoid.
\end{proof}

We now show that Galois stability is preserved by taking the generic fiber of the formal completion along the special fiber.



\begin{prop}\label{galcompl}
Let $K/k$ be a finite Galois extension. Let $X=\spec(A)$ be an affine $k$-scheme of finite type and let $\fX=\spec(\fA)$ be a smooth $\Ko$-scheme of finite type such that $\fX\times _{\Ko} K = X \times _k K$ and such that $\fX\times _{K^{\circ}} \tilde{K}$ is irreducible. Assume that $\fA$ is a $\Gal(K/k)$-stable subalgebra of $A \otimes _k K$. Then the generic fiber of the formal completion of $\fX$ along its special fiber is a $\Gal(K/k)$-stable $K$-affinoid domain $\hat{\fX}_{\eta}$ of $X^{an} \hat{\otimes} K$.
\end{prop}

\begin{proof}
Let $\lvert~\rvert _x \in \hat{\fX}_{\eta} \subset X^{an} \hat{\otimes} K$, it is a seminorm on $A \otimes _k K$ bounded by $\lvert~\rvert _{\fA}$. Let $\gamma \in \Gal (K/k)$, we need to show that $\lvert~\rvert _x. \gamma $ stay in $\hat{\fX}_{\eta}$. Let $f \in A \otimes _k K$, then $(\lvert~\rvert _x. \gamma) (f) = \lvert \gamma.f \rvert _x$. By definition of $\lvert~\rvert _x$, we have $\lvert \gamma.f \rvert _x \leq \lvert \gamma.f \rvert _{\fA} $.
Since $\fA$ is $\Gal (K/k)$ stable in $A \otimes _k K$, we have $\gamma. \fA= \fA$ for all $\gamma \in \Gal (K/k)$ and we deduce the following.

\begin{align*}
\lvert \gamma.f \rvert _{\fA} &= \inf_{\lambda\,\in\, K^{\times}} \left\{ \lvert \lambda \rvert \middle| \gamma. f \in \fA \subset \fA \otimes _{K^{\circ}} K \right\} \\
&= \inf_{\lambda\,\in\,K^{\times}} \left\{ \lvert \lambda \rvert \middle| f \in \gamma ^{-1} \fA \subset \fA \otimes _{K^{\circ}} K \right\} \\
&= \inf_{\lambda\,\in\,K^{\times}} \left\{ \lvert \lambda \rvert \middle| f \in \fA \subset \fA \otimes _{K^{\circ}} K \right\} \\
&=\lvert f \rvert _{\fA} \\
\end{align*}
Consequently, we have $(\lvert~\rvert _x. \gamma) (f) = \lvert \gamma.f \rvert _x \leq \lvert \gamma.f \rvert _{\fA} = \lvert f \rvert _{\fA}$. Thus $\lvert~\rvert _x. \gamma \leq \lvert~\rvert _{\fA} $, and so $(\lvert~\rvert _x. \gamma) \in \hat{\fX}_{\eta}$ by Corollary~\ref{analyko}. The proof ends here.
\end{proof}


\section{The split case} \label{spli}

Let $G= \spec (\Hopf (G))$ be a split connected reductive group scheme over a non-Archimedean discretely valued field $k$ with a perfect residue field. Let $\BT (G,k)$ be the reduced Bruhat--Tits building of $G$. Let $x$ be a special point in $\BT(G,k)$ and $r \in \R _{\geq\,0}$. Since $G$ is split and $x $ is special, the canonical scheme $\GG $ attached to $x $ by Bruhat--Tits is a Demazure (i.e. split, reductive and connected) $k^{\circ}$-group scheme, as remarked in~\cite[Page~19]{RTW}. The scheme $\GG$ is smooth. We fix a $k$-affinoid extension $K$ such that the real number $r $ is contained in $\ord (K)$. Let $\pii _r \subset K^{\circ}$ be the unique ideal of $K^{\circ}$ generated by elements $\pi _r$ with $\ord (\pi _r) = r $. We now consider the $\pii_r$-congruence subgroup $\GG _{\pii _r}$ of $\GG \times _{k ^{\circ}} K^{\circ}$, we also denote it as $\GG _{r}$. We have identifications
\begin{align*}
\GG_{r} \times _{K^{\circ}} K &= G \times _k K\\
\intertext{and}
\Hopf (\GG_{r}) \otimes _{K^{\circ}} K &=\Hopf(G)\otimes _k K.
\end{align*}
We now consider $\widehat{~\GG _r~} _{\eta}$, the generic fiber of the formal completion of $\GG _r $ along its special fiber. Since $\GG_{r}$ is smooth by~\cite[Theorem~3.2]{NBU}, the Shilov boundary of the $K$-affinoid group $\widehat{~\GG_{r}~}_{\eta}$ is in bijection with the set of generic points of irreducible components of the special fiber of $\GG_{r}$ (cf Proposition~\ref{compfo} and Corollary~\ref{analyko}). So by Proposition~\ref{NBU}, it is a singleton, let us denote it by $x_r$. Moreover by Corollary~\ref{analyko}, the holomorphically convex envelope $\Hol (x_r)$ of $x_r$ is $\widehat{~\GG_{r}~}_{\eta}$. We want an explicit formula for $x_r$. Let $\fT$ be a maximal $\ko$-split torus of $\GG$ and $\Phi$ be the corresponding set of roots. Let $\fB$ be a Borel subgroup such that $\fT$ is a Levi subgroup of $\fB$. Let $\Phi ^-, \Phi ^+$ be the corresponding sets of negative and positive roots. For each $\alpha \in \Phi$, we have a canonical $\ko$-root subgroup $\fU_{\alpha} \subset \GG$. Choose an ordering on $\Phi ^-, \Phi ^+$, then the multiplication morphism of $\ko$-schemes
\begin{equation}\label{cellull}
\prod _{\alpha\,\in\,\Phi ^-} \fU_{\alpha} \times_{\ko} \fT \times_{\ko} \prod_{\alpha\,\in\,\Phi ^+} \fU_{\alpha}\to \GG
\end{equation}
is an open immersion. Its image, which does not depend on the choice of the ordering, is denoted $\underline{\Omega}$ and is called the open cell of $\GG$. Taking generic fibers, we obtain similar objects for $G$. The objects
\begin{align*}
T :=& \; \fT \times_{\ko} k\\
U_{\alpha} :=& \; \fU_{\alpha} \times_{\ko} k \\
B :=& \; \fB \times_{\ko} k
\end{align*}
are respectively a maximal split torus, a root subgroup, and a Borel subgroup of $G = \GG \times _{ \ko} k$. We can identify canonically $\Phi$ with the set of roots associated to $G,T$. Moreover~\eqref{cellull} induces an open immersion
\[
\prod _{\alpha\,\in\,\Phi ^-} {U}_{\alpha} \times _k {T} \times _k \prod _{\alpha\,\in\,\Phi ^+} {U}_{\alpha}\to G
\]
whose image, independent of the ordering, is denoted $\Omega$ and is called the open cell of $G$. We can identify $\Omega $ and $ \underline{\Omega} \times _{\ko } k$. The open cell $\Omega$ is affine and the open immersion $\Omega \to G$ corresponds to an injective morphism of Hopf algebras from $\clO(G)$ to $\clO(\Omega)$ (see~\cite[line~24 page~103]{berk}). The torus $\fT$ is split, so is isomorphic to $(\bbG_{m} / \ko)^{s}$ for some integer $s$. Fix an isomorphism
\[
\fT \simeq \spec \left(\ko \left[X_1,\,\ldots,\,X_s, Y_1, \,\ldots,\,Y_s\right] / \left(X_i Y_i =1~\text{for}~1 \leq i \leq s\right)\right).
\]
Fix an integral Chevalley basis of $\Lie (\GG, \ko)$, it induces, for each root $\alpha \in \Phi$, a $\ko$-isomorphism $\fU_{\alpha} \simeq\bbG_{a}$, where $\bbG_{a}$ is the additive group over $\ko$. Thus we have fixed an isomorphism $\fU_{\alpha} \simeq \spec (\ko [Z_{\alpha}])$, i.e. we have fixed an isomorphism $ \clO (\fU_{\alpha}) \simeq \ko [Z_{\alpha}]$, for any root $\alpha$. Since
\begin{align*}
\Omega &=  \prod _{\alpha\,\in\,\Phi^-} {U}_{\alpha} \times_k {T} \times _k  \prod_{\alpha\,\in\,\Phi^+} {U}_{\alpha},
\\
\intertext{we obtain}
\clO(\Omega) &=\bigotimes_{\alpha\,\in\,\Phi ^-} \clO({U}_{\alpha})\otimes_k \clO({T}) \otimes_k \bigotimes_{\alpha\,\in\,\Phi^+} \clO({U}_{\alpha}).
\end{align*}
The torus $T$ is equal to $\fT \times_{\ko} k$. The previously fixed isomorphism
\[
\fT \simeq \spec \left(\ko \left[X_1,\,\ldots,\,X_s, Y_1, \ldots,\,Y_s\right] / \left(X_i Y_i =1\;\text{ for }\;1 \leq i \leq s\right)\right)
\]
induces a similar isomorphism over $k$ for $T$.
The set
\[
\left\{X^{\rmk} Y^l\,\middle|\,\rmk,l \in \bbN ; \rmk \not = 0 \Rightarrow l=0 \right\}
\]
is a basis of the $k$-vector space $k[X,Y]/XY-1$. We need an other basis of $\clO (\bbG_m)$, ``centered at unity''. The set
\[
\left\{ (X-1)^{\rmk} (Y-1)^l \,\middle|\, \rmk,l \in \bbN ; \rmk \not = 0 \Rightarrow l=0 \right\}
\]
is a basis of the $k$-vector space $k[X,Y]/XY-1$.
The previously fixed isomorphisms $ \{ \clO (\fU_{\alpha}) \simeq \ko [Z_{\alpha}] \}_{\alpha\,\in\,\Phi}$ induce isomorphisms $\{\clO (U_{\alpha}) \simeq k[Z_{\alpha}]\}$. We identify the corresponding objects. The set $\{ {Z_{\alpha}}^{m_{\alpha}} \mid m_{\alpha}\,\in\,\bbN \}$ is a basis of the $k$-vector space $\clO (U_{\alpha}) $. These considerations allow us to fix an isomorphism
\begin{align*}
\clO({\Omega}) &\simeq \left(\bigotimes _{\alpha\,\in\,\Phi ^-} k\left[Z_{\alpha}\right] \right) \otimes _k \left(\bigotimes_{i=1}^s k\left[X_i, Y_i\right] / X_i Y_i -1 \right) \otimes_k \left( \bigotimes_{\alpha\,\in\,\Phi ^+} k\left[Z_{\alpha}\right]) \right)\\
&\simeq k\left[X_1,\,\ldots,\,X_s, Y_1,\,\ldots,\,Y_s, \{Z_{\alpha} \}_{\alpha\,\in\,\Phi}\right]/ \left(X_i Y_i-1, 1 \leq i \leq s\right).
\end{align*}
Moreover the set
\[
\left\{ \prod_{i=1}^s \left(X_i -1\right)^{k_i} \left(Y_i -1\right)^{l_i} \prod _{\alpha\,\in\,\Phi } {Z _{\alpha}}^{m _{\alpha}}\, \middle|\, k_i, l_i, m_{\alpha} \in \bbN;\forall 1 \leq i \leq s, k_i \not = 0 \Rightarrow l_i =0 \right\}
\]
is a $k$-basis of the $k$-vector space $\clO(\Omega)$. So given $f \in \clO (\Omega)$, $f$ can be written uniquely as
\[
f = \sum _{k_1,\,\ldots,\,k_s, l_1,\,\ldots,\,l_s, m_{\alpha} \alpha\,\in\,\Phi } a_{k_1\,\ldots\,k_s l_1\,\ldots\,l_s, m_{\alpha} \alpha\,\in\,\Phi } \prod_{i=1}^s \left(X_i -1\right)^{k_i} \left(Y_i -1\right)^{l_i}\prod _{\alpha\,\in\,\Phi } Z_{\alpha}^{m _{\alpha}}.
\]
In order to simplify formulas, let us introduce some notation. We denote a parameter $(k_1,\, \dots,\, k_s, l_1,\,\ldots,\, l_s,\{ m_{\alpha}\}_{\alpha\,\in\,\Phi})$ with $ k_i, l_i, m_{\alpha} \in \bbN $ and ${ k_i \not = 0 \Rightarrow l_i =0} $, as appearing above, by the symbol $u$. Let $U$ the set of all such parameters $u$. Moreover, the element $ \prod_{i=1}^s (X_i -1)^{k_i} (Y_i -1)^{l_i} \prod _{\alpha\,\in\,\Phi } Z_{\alpha}^{m_{\alpha}} $ is denoted by the symbol $((X-1)(Y-1) Z)^{u}$. With these notations, an element $f \in \clO (\Omega)$ is written uniquely as
\[
f = \sum _{u\,\in\,U} a_u \big((X-1)(Y-1) Z\big)^{u}.
\]

\begin{lemm}\label{lemm4.1}
The point $\pr_{K/k}(x_r)$ belongs to $\Omega ^{an}$ and corresponds to the norm
\begin{align*}
\clO(\Omega) &\to \R _{\geq\,0} \\
\sum _{u\,\in\,U} a_u \big((X-1)(Y-1) Z\big)^u &\mapsto  \max _{u\,\in\,U} \lvert a _u \rvert e^{-r \lvert u \rvert}.
\end{align*}
\end{lemm}

\begin{proof}
Since $G$ is split and $\pr_{K/k} (x_r) $ is the restriction of $x_r$ to $\clO(G)$, we can assume $K=k$. By~\cite[\S~1.2.4]{RTW}, $x_r$ is the unique point in $\widehat{~\GG _r~} _{\eta}$ sent to the generic point of $\GG _r \times _{\ko} \tilde{k}$ by the reduction map. Let $\sigma$ denote the generic point of $\GG _r \times _{\ko} \tilde{k} $. Let $\underline{\Omega} _r$ be
\[
\prod _{\alpha\,\in\,\Phi^-} \fU_{\alpha,r} \times _{\ko} \fT_r \times _{\ko} \prod _{\alpha\,\in\,\Phi^+} \fU_{\alpha,r}.
\]
The multiplication map on $\GG _r$ induces an open immersion $\underline{\Omega} _r \to \GG _r$. The special fiber $\underline{ \Omega} _r\times _{\ko} \tilde{k}$ is open in $\GG_r \times _{\ko} \tilde{k}$ (and non empty), consequently $\sigma$ is contained in $\underline{ \Omega} _r \times _{\ko} \tilde{k} $. The commutative diagram
\[
\xymatrix{\widehat{~{\underline{\Omega }} _r~}_{\eta} \ar[d] \ar[r]^-\pi & \underline{ \Omega} _r \times _{ \ko} \tilde{k} \ni \sigma \ar[d] \\
\widehat{~{\GG _r}~}_{\eta} \ar[r]& \GG _r \times _{\ko} \tilde{k}~~~~~}
\]
whose vertical arrows are inclusions shows that $\Shi (\widehat{~\GG _r~}_{\eta})= \pi ^{-1} (\sigma).$ So $\Shi (\widehat{~\GG_r~} _{\eta}) = \Shi (\widehat{~\underline{\Omega }_r~}_{\eta})$. By~\cite[\S~1.2.4]{RTW} the only point in $\Shi (\widehat{~\underline{\Omega } _r~}_{\eta})$ is the norm $\lvert~\rvert _{\clO(\underline{\Omega} _r)}$ on $\clO (\Omega)$ given by
\[
\lvert f \rvert _{\clO\left(\underline{\Omega} _r\right)}= \inf \left\{ \lvert \lambda \rvert\, \middle| \lambda \in k \quad\text{and}\quad f \in \lambda \left(\clO\left(\underline{\Omega} _r\right) \otimes 1\right) \right\}.
\]
Let us describe $\clO(\underline{\Omega} _r)$ explicitly. Let us fix an element $\pi _r \in k^{\circ}$ such that $\ord (\pi _r) = r$. We have
\[
\clO\left(\underline{\Omega}_r\right) =  \bigotimes _{\alpha\,\in\,\Phi^-} \clO\left(\fU_{\alpha, r }\right)\otimes _{\ko}\clO\left(\fT_r\right) \otimes _{\ko} \bigotimes _{\alpha\,\in\,\Phi ^+} \clO\left(\fU_{\alpha, r}\right).
\]
By Proposition~\ref{NBU}, we have
\begin{align*}
\clO\left(\fU_{\alpha,r}\right) &= \ko \left[{\pi _r }^{-1}Z _{\alpha}\right] \subset k\left[Z _{\alpha}\right]
\\
\intertext{and}
\clO \left(\fT _r\right) &= \ko \left[{\pi _r} ^{-1} \left(X_1 -1\right),\,\ldots,\,{\pi _r} ^{-1} \left(X_s -1\right),{\pi _r}^{-1} \left(Y_1 -1\right),\,\ldots,\,{\pi _r}^{-1} \left(Y_s -1\right)\right].
\end{align*}
Finally, we get the formula
\[
\clO\left(\underline{\Omega} _r\right) = \ko \left[\left\{{\pi _r}^{-1} Z_{\alpha}\right\}_{\alpha\,\in\,\Phi}, \left\{ {\pi_r}^{-1} \left(X_i -1\right), {\pi_r}^{-1} \left(Y_i -1\right)\right\}_{1\,\leq\,i\,\leq\,s}\right] \subset \clO (\Omega).
\]
For $f \in \clO(\Omega)$, write $f = \sum_{u\,\in\,U} a_u ((X-1)(Y-1) Z)^{u}$. Now we obtain
\begin{align*}
\lvert f \rvert_{\clO\left(\underline{\Omega} _r\right)}&= \inf \left\{ \lvert \lambda \rvert\, \middle| \lambda \in k \quad\text{and}\quad f \in \lambda \left(\clO\left(\underline{\Omega}_r\right) \otimes 1\right) \right\}\\
&= \inf \left\{ \lvert \lambda \rvert\, \middle| \lambda \in k \quad\text{and}\quad a_u \in \lambda \left(\pi_r^{-1}\right)^{\lvert u \rvert} \ko~~\forall u \in U \right\}\\
&= \inf \left\{ \lvert \lambda \rvert\, \middle| \lambda \in k \quad\text{and}\quad \lvert a_u \rvert \leq \lvert \lambda \rvert \lvert \pi _r ^{-1} \rvert^{\lvert u \rvert}~~\forall u \in U \right\}\\
&= \inf \left\{ \lvert \lambda \rvert \middle| \lambda \in k \quad\text{and}\quad \lvert a_u \rvert \lvert \pi _r \rvert^{\lvert u \rvert} \leq \lvert \lambda \rvert~~\forall u \in U \right\}\\
&=  \max_{u\,\in\,U } \lvert a_u \rvert\,\lvert \pi _r \rvert ^{\lvert u \rvert}\\
&= \max_{u\,\in\,U } \lvert a_u \rvert e^{-r{\lvert u \rvert}}
\end{align*}
This ends the proof of Lemma~\ref{lemm4.1}.
\end{proof}

Now let us fix a point $y $ in $ A (G,T,k)$. We choose an affinoid extension $E/k$ such that firstly the point $\iota_{E/k} (y)$ is a special point in the building $\BT (G,E)$ and secondly the real number $r$ is contained in $\ord (E)$, it is easy to see that such an extension exists using~\cite[Proposition~1.6]{RTW}. Since $\iota_{E/k} (x)$ is also a special point, there exists $t \in T(E)$ such that $t.x=y$. Let $\GG _y$ be the canonical $K^{\circ}$-Demazure scheme attached to $\iota_{E/k} (y)$. Let $y_r \in (G \times_k K) ^{an}$ be the unique point in the Shilov boundary of $\widehat{~\GG_{y,r}~} _{\eta}$, and let $\theta(y,r)$ be the image of $y_r$ under the canonical projection $(G \times _k K)^{an} \to G^{an}$. Let us use the point $x$ to identify the apartment $A(G,T,k)$ with $V(T) = \Hom_{Ab} (X^*(T),\R)$.

\begin{prop}\label{peaked}
The point $\theta (y,r)$ belongs to $\Omega ^{an}$ and corresponds to the norm
\begin{align*}
\clO(\Omega) &\to \R _{\geq\,0} \\
\sum _{u\,\in\,U} a_u \big((X-1)(Y-1) Z\big)^u &\mapsto \max _{u\,\in\,U} \lvert a _u \rvert e^{-r \lvert u \rvert} \prod_{\alpha\,\in\,\Phi} e^{ m_{\alpha} \,<y, \alpha>}
\end{align*}
where $<.,.>$ is the map $V(T) \times X^*(T) \to \R$, $(y, \alpha) \mapsto <y, \alpha > = y(\alpha)$.
\end{prop}

\begin{proof}
Here again, we can assume $E=k$. Let $t \in T(k)$ such that $y=t.x$. The element $t $ normalizes the root group $U _{\alpha}$ and conjugation by $t$ induces an automorphism of $U_{\alpha}$ which is just the homothety of ratio $\alpha (t) \in k^{\times}$. If we read it through the isomorphisms $\spec(k[Z_{\alpha}]) \simeq U _{\alpha}$, we have a commutative diagram
\[
\xymatrix{ \spec\left(\clO(T)\left[\left\{Z_{\alpha}\right\}_{\alpha\,\in\,\Phi}\right]\right) \ar[r]\ar[d]^-{\tau} &\Omega \ar[d]^-{int(t)} \\
\spec\left(\clO(T)\left[\left\{Z_{\alpha}\right\}_{\alpha\,\in\,\Phi}\right]\right)\ar[r] & \Omega }
\]
where $\tau$ is induced by the $\clO(T)$-automorphism $\tau^*$ of $\clO(T)[\{Z_{\alpha}\}_{\alpha\,\in\,\Phi}$ mapping $Z_{\alpha}$ to $\alpha (t) Z_{\alpha}$ for any $\alpha \in \Phi$. It follows that $\theta (t.x, r) $ is the point of $G^{an}$ defined by the multiplicative norm on $\clO(\Omega)$ mapping $f = \sum_{u\,\in\,U} a_u((X-1) (Y-1) Z_{\alpha})^u $ to

\begin{align*}
\left\lvert \tau^* (f) \right\rvert_{\theta (x,r) }&= \left\lvert  \sum_{u\,\in\,U} \left(a_u \prod_{\alpha\,\in\,\Phi} \alpha (t)^{m_{\alpha}} \right) \big((X-1)(Y-1) Z_{\alpha}\big)^u \right\rvert_{\theta (x,r)} \\
&= \max_{u\,\in\,U} \lvert a_u \rvert e^{-r \lvert u \rvert} \prod_{\alpha\,\in\,\Phi} \lvert \alpha (t) \rvert^{m_{\alpha}}\\
&= \max_{u\,\in\,U} \lvert a_u \rvert e^{-r \lvert u \rvert} \prod_{\alpha\,\in\,\Phi } e^{ m_{\alpha} < y, \alpha > }.\qedhere
\end{align*}
\end{proof}

\begin{prop}\label{pe}
The point $\theta (y,r)\in \Omega ^{an} $ is peaked (in the sense of~\cite[\S~5]{berk}).
\end{prop}

\begin{rema}
The point $\theta (y,r)$ is peaked as a point in $\Omega ^{an}$ and also as a point in $G^{an}$ by~\cite[Lemma~2.2.3]{pp}.
\end{rema}

\begin{proof}
Let $K/k$ be a non-Archimedean extension. We have to show that the norm $||.||:= |.|_{\theta (y,r)} \otimes |.| _K $ on the algebra $ \clO (\Omega) \otimes _k K $ is multiplicative. Recall that $\|.\|$ is the norm defined as $ \|f\|= \inf \max_i |g_i|_{\theta (y,r)} |\lambda_i|_K $ where the infimum is taken over all representatives $ f = \sum _i g_i \otimes \lambda _i $. The set $\{((X-1) (Y-1) Z )^u \otimes 1 | u \in U \} $ is a $K$-basis of $\clO (\Omega) \otimes _k K $. Let $f \in \clO (\Omega) \otimes _k K $ and let $\{ a_u^K \}_{u \,\in\,U }$ be the coordinates of $f$ in the previous basis i.e. such that $f =\sum _{u\,\in\,U} ((X-1) (Y-1) Z)^u \otimes a_u^K$. By definition of $\|.\|$, we have
\[
\| f \| \leq \max_{u\,\in\,U} \left|\big((X-1)(Y-1)Z\big)^u\right|_{\theta (y,r)} \left| a_u^K \right| = \max _{u\,\in\,U} \left|a_u^K\right| e^{-r \lvert u \rvert} \prod_{\alpha\,\in\,\Phi } e^{ m_{\alpha} < y, \alpha > }.
\]
\pagebreak

Now let $ f = \sum _{i=1}^N (\sum _{u\,\in\,U} a^i_u ((X-1)(Y-1)Z)^u ) \otimes \lambda _i $ be an other representative of $f$. We have $ f = \sum _{u\,\in\,U} ((X-1) (Y-1) Z) ^u \otimes (\sum_{i=1}^N a_u^i \lambda _i)$ and so for all $u \in U $, $ \sum_{i=1}^N a_u ^i \lambda_i = a_u^K$ and $\max _{i=1}^N |a_u^i \lambda_i | \geq |a_u^K| $. Let $u \in U $, we have
\begin{align*}
\max _{i=1}^N& \left| \sum_{u\,\in\,U} a^i_u \big((X-1)(Y-1)Z\big)^u \right|_{\theta (y,r)} |\lambda_i |_K\\
&= \max _{i=1}^N \left(\max_{u\,\in\,U} \left|a_u^i \right| e^{-r |u|} \prod_{\alpha\,\in \,\Phi} e^{m_{\alpha}\,< y, \alpha > } \right) \left|\lambda_i \right| _K\\
&\geq  \max _{i=1}^N \left(\left|a_u^i \right| e^{-r |u|} \prod_{\alpha\,\in\,\Phi } e^{ m_{\alpha}\,< y, \alpha > }\right) \left|\lambda_i \right| _K\\
&\geq  \left|a_u^K \right|_K e^{-r |u|} \prod_{\alpha\,\in\,\Phi} e^{ m_{\alpha} < y, \alpha > }.
\end{align*}

%\begin{sloppypar}
We deduce that
\[
 \| f\| \geq \max _{u\,\in\,U} \left| a_u^K \right|_K e^{-r |u|} \prod _{\alpha\,\in\,\Phi} e^{ m_{\alpha} < y, \alpha >}. 
\] 
Consequently 
\[
\| f\| = \max _{u\,\in\,U} \left| a_u^K \right|_K e^{-r |u|} \prod_{\alpha\,\in\,\Phi} e^{ m_{\alpha}\,< y, \alpha >}.
\] 
So $\|.\|$ is the norm on $\clO (\Omega \times _k K) $ given by the ``same formula'' as the norm $\theta (y,r)$ on $\clO (\Omega)$, so $\|.\|$ is in particular multiplicative.
%\end{sloppypar}
\end{proof}

Let us put $G_{y,r}= \pr _{E/k}(\widehat{~\GG _{y,r}~}_{\eta} )$.

\begin{prop}\leavevmode
\begin{enumerate}
\item\label{prop4.5.1} $G_{y,r}$ is a $k$-affinoid domain of $G^{an}$.
\item\label{prop4.5.2} $G_{y,r}$ is a $k$-affinoid group.
\item\label{prop4.5.3} The Shilov boundary of $G_{y,r}$ is a singleton equal to the point $\theta (y,r)$ considered above, moreover $\Hol (\theta (y,r)) = G_{y,r}$.
\end{enumerate}
\end{prop}

\begin{proof}
\begin{itemize}
\item[\eqref{prop4.5.1}] By Theorem~\ref{descent}, it is enough to prove that $\pr_{E/k}^{-1} (G_{y,r})$ is $E$-affinoid.
By Proposition~\ref{pe}, the point $\theta (y,r)$ is peaked. So by~\cite[Corollary~5.2.4]{berk} taking holomorphically convex envelope commutes with base change and we have $\pr _{E/k}^{-1} (\pr_{E/k} (\Hol (y_r)))= \Hol (y_r)$. So $\pr _{E/k}^{-1} (G_{y,r})$ is $E$-affinoid since $\Hol (y_r) = \widehat{~\GG _{y,r}~} _{\eta}$. This ends the proof of the first assertion.
\item[\eqref{prop4.5.2}] This is a consequence of the following lemma.


\begin{lemm}\label{groudesc}
Let $G$ be a $k$-analytic group, let $K/k$ be an affinoid extension, let $H_K$ be a $K$-affinoid subgroup of $G_K= G\hat{\otimes} K$. Let $H=\pr_{K/k} (H_K)$, if it is a $k$-affinoid domain of $G$ then it is a $k$-affinoid subgroup of $G$.
\end{lemm}

\begin{proof}
Let $m : G\times G \to G $ be the multiplication map and $ inv : G \to G $ be the inversion map coming from the analytic group structure on $G$.
We have to show that the restriction and inversion maps factor through $H$. Consider the following diagram whose four squares are commutative.
\[
\centerline{ \xymatrix { H_K \times H_K \ar[rr]^m \ar@{->>}[dd]^p \ar[dr]^i && H_K \ar@{->>}[dd]^p \ar[drr]^i \\
& G_K \times G_K \ar[rrr]^{m~~~~~~~~~~} \ar@{->>}[dd]^p &&& G_K \ar@{->>}[dd]^p \\
H \times H \ar[dr]^i && H \ar[drr]^i \\
& G \times G \ar[rrr]^m &&& G \\
}}
\]

Let $x$ be in $H\times H$, it is enough to show that there is $y$ in $H$ such that $m \circ i (x) = i (y)$. Let $z$ in $H _K \times H_ K $ such that $p(z)=x$, then
\[
m \circ i (x) = m \circ p \circ i (z) = p \circ m \circ i (z) = p \circ i \circ m (z) = i \circ p \circ m (z).
\]
So $y = p \circ m (z)$ works. The same argument works for $inv$.
\end{proof}

\item[\eqref{prop4.5.3}] By~\cite[Proof of Proposition~2.4.4]{berk}, the Shilov boundary of $ \widehat{~\GG _{y,r}~} _{\eta}$ surjects to the Shilov boundary of $G_{y,r}$, so $\Shi (G_{y,r})= \{ \theta (y,r) \}$. By Corollary~\ref{analyko}, $\Hol(y_r)$ equals $\widehat{~\GG _{y,r}~} _{\eta}$. Now since $\theta (y,r)$ is peaked, \cite[Corollary~5.2.4]{berk} implies that $G_{y,r}=\Hol (\theta (y,r)) $.\qedhere
\end{itemize}
\end{proof}

\section{The rational potentially Demazure case} \label{secraaa}

In this section $G= \spec (\Hopf (G))$ is a connected reductive group scheme over $k$. Let $H \subset G^{an}$ be a $k$-affinoid group such that there exists a finite Galois extension $K/k$ such that $ G \times _k K $ is split and $H \hat{\otimes} K$ is the generic fiber of the formal completion $\widehat{\GG}_{\eta}$ of a Demazure group scheme $\GG$ over $K^{\circ}$ satisfying $\GG \times _k K = G $. Let $K$ be such an extension. We call such a $H$ a rational potentially Demazure $k$-affinoid group in $G^{an}$. Let $\Gamma \subset \Q _{\geq\,0}$ be $\ord(({k^{sep}})^{\circ})$ where ${k}^{sep}$ is a separable closure of $k$. The set $\Gamma$ is dense in $\R _{\geq\,0}$. Fix $r \in \Gamma$. We can assume that $r \in \ord (K)$ and we now fix such a $K$.


\begin{lemm}\label{lemde}
There exists a unique $K^{\circ}$-Demazure group scheme $\GG$ such that $H \hat{\otimes} K =\widehat{\GG}_{\eta}$, moreover it is Galois stable.
\end{lemm}


\begin{proof}
Assume $\GG= \spec (\fA)$ and $\GG ' = \spec (\fA ')$ are two $K^{\circ}$-Demazure group schemes satisfying $H \hat{\otimes} K =\widehat{\GG}_{\eta}=\widehat{\GG '}_{\eta}$. By Proposition~\ref{compfo}, we have $\Shi (\widehat{\GG '} _{\eta})= \Shi (\widehat{\GG } _{\eta}) = \lvert~\rvert _{\fA} = \lvert~\rvert _{\fA '}$.
By definition $\lvert~\rvert _{\fA} $ is a norm on $\clO(G\times _k K)$ given by the formula
\[
\lvert f \rvert _{\fA} = \underset{\lambda\,\in\, K^{\times}}{\inf} \left\{ \lvert \lambda \rvert \middle| f \in \lambda \left(\fA \otimes 1\right) \right\}.
\]
The valuation of $K$ is discrete, so we have
\begin{align*}
f \in \fA &\Leftrightarrow 1 \in \left\{ \lambda \in K^{\times} \,\middle|\, f \in \lambda \left(\fA \otimes 1\right) \right\}\\
&\Leftrightarrow \underset{\lambda \in K^{\times}}{\inf} \left\{ \lvert \lambda \rvert \,\middle| f \in \lambda \left(\fA \otimes 1\right) \right\} \leq 1\\
&\Leftrightarrow \lvert f \rvert _{\fA} \leq 1.
\end{align*}

Similarly we have $f \in \fA ' \Leftrightarrow \lvert f \rvert _{\fA '} \leq 1$. So finally $f \in \fA \Leftrightarrow f \in \fA '$, as required.


Now let us prove that $\fA$ is Galois stable. Let $\sigma \in \Gal (K/k)$. On one hand, we have 
\[
\sigma \left(H \hat{\otimes} K\right) = H \hat{\otimes} K,\quad\text{so}\quad\sigma \left(\widehat{\spec\left(\fA\right)}_{\eta}\right) =\widehat{\spec \left(\fA\right)}_{\eta}.
\]
On the other hand, we have 
\[
\widehat{\spec \left(\sigma (\fA)\right)} _{\eta} = \sigma \left(\widehat{\spec (\fA)} _{\eta}\right).
\]
So we have 
\[
\widehat{\spec \left(\sigma (\fA)\right)}_{\eta} =\widehat{\spec \left(\fA\right)}_{\eta}.
\]
Thus by the previous assertion, we have $\sigma (\fA)= \fA$.
\end{proof}

Let $\pii_r$ be an ideal in $K^{\circ}$ generated by elements $\pi _r $ such that $\ord (\pi_r) =r$. Let $\widehat{~\GG_{\pii_r}~}_{\eta}$ be the generic fiber of the formal completion of $\GG_{\pii_r}$ along its special fiber.

\begin{lemm}\label{rastabshi}
The Shilov boundary of $\widehat{~\GG _{\pii_r}~}_{\eta}$ is a singleton in $(G \times_k K)^{an}$ that is stable under the Galois group $\Gal (K/k)$.
\end{lemm}

\begin{proof}
The Shilov boundary of $\widehat{~\GG _{\pii_r}~}_{\eta}$ is a singleton by Corollary~\ref{analyko}. By Lemma~\ref{lemde} $\GG$ is Galois stable. So by Lemma~\ref{gammastab} $\GG _{\pii_ r } $ is Galois stable. Consequently by Proposition~\ref{galcompl} $\widehat{~\GG _{\pii_r}~}_{\eta}$ is Galois stable and so is its Shilov boundary.
\end{proof}

Let $H_r $ be $\pr_{K/k} \widehat{~\GG _{\pii_r}~}_{\eta} $.

\begin{prop}\label{jkl}
The set $H_r$ is a $k$-affinoid group independent of the choice of the extension $K/k$ used in order to define it, moreover its Shilov boundary is a singleton $\sigma_{r}$ and $\Hol (\sigma_{r }) = H_r$.
\end{prop}

\begin{proof}
This is proved in the same way as Proposition~\ref{holoxr} (we do not use Proposition~\ref{jkl} in order to prove Proposition~\ref{holoxr}).
\end{proof}


\begin{rema}\label{rema}
A point $x \in \BT (G,k)$ is called rational if there exists a finite Galois extension $K/k$ such that $\iota _{K/k} (x)$ is a special point in $\BT (G,K)$ and $G$ is split over $K$. The set of rational points is denoted $\underline{\BT (G,k)}$. Using~\cite[Theorem~2.1 and its proof]{RTW}, we see that each rational point $x$ gives birth canonically to a rational potentially Demazure $k$-affinoid group $G_x$ ($G_x = \pr_{K/k} \widehat{~\GG~}_{\eta}$ where $\GG$ is the canonical Demazure $K^{\circ}$-group-scheme attached to $\iota_{K/k} (x)$).
\end{rema}
The following lemma will be useful in the next section.

\begin{lemm}\label{dense}
\leavevmode
\begin{enumerate}
\item The set $\underline{\BT(G,k)}$ is dense in $\BT (G,k)$.

\item The set $\underline{\BT(G,k) }\times \Gamma $ is dense in $\BT (G,k) \times \R _{\geq 0}$ for product topologies.
\end{enumerate}
\end{lemm}

\begin{proof}
Remark first that if $G$ is split over $k$, it is obvious that $\underline{\BT(G,k)}$ is dense in $\BT (G,k)$, because for any maximal split torus $S$ over $k$ and any finite extension $K/k$, the apartment $A(G,S,K)$ is obtained from $A(G,S,k)$ adding regularly $e(K:k)$ times more walls.
Let us now prove the proposition. It is enough to show that for any maximal split torus $S$ of $G$ over $k$, $\underline{A(G,S,k)}$ is dense in $A(G,S,k)$. Let $L$ be a finite Galois extension such that $G$ is split over $L$. By~\cite[\S~4.1.1, \S~4.1.2, \S~5.1.12]{brti2}, there exists a torus $T \supset S$ defined over $k$ such that $T \times _k L$ is a maximal split torus of $G
\times _k L$. There exists a facet $F$ in $A(G,T,L)$ which is $\Gal (L/k)$-stable. The barycentre $x$ of $F$ is $\Gal (L/k)$-stable and so $x \in A(G,S,k)$ (since $A(G,T,L) ^{\Gal(L/k)}= A(G,S,k)$). By~\cite[\S~6.3.4, lines~8-9]{Cornut}, the point $x$ becomes special over a finite extension $K/L$. So we have proved that there exists one rational point $x$ in $A(G,S,k)$. Now the set of points $\{ g.x \mid g \in S (k^{sep}) \}$ consists in a dense subset of $A(G,S,k)$ constituted of rational points. Indeed, let us first show that this set consists in rational points. So let $g \in S (k^{sep})$, there exists a finite extension $K/L$ such that $ g \in S(K)$. The point $x$ is special in the building $\BT(G,K)$ (since $G$ is split over $L$ and $x$ is special in the building $\BT(G,L)$), so $g.x$ is special in $\BT(G,K)$. By definition $ T(k^{sep})$ acts on $A(G,T,L)$ by translation (the translation vector $v$ associated to $t \in T(k^{sep})$ is given by the formula ``$<v, \alpha> = -\ord(\alpha(t))~\forall \alpha$'', see~\cite[\S~4.2.3$\MK$(I)]{brti2}) and for any $g \in S(k^{sep}) \subset T(k^{sep})$, we have $g.x \in A(G,S,k)$, so $g.x$ is a rational point in $\BT(G,k)$. Since $\ord(k^{sep})$ is dense in $\R$, the first assertion follows. The second assertion is a direct consequence of the first one since $\Gamma $ is dense in $\R _{\geq 0}$ for the archimedean topology.
\end{proof}


\section{The general case for points in the Bruhat--Tits buildings} \label{secgebuu}


Let $G$ be a connected reductive group over $k$. There exists a finite Galois extension $L/k$ such that $G \times _k L$ is split. Let $(x,r) \in \BT(G,k) \times \R _{\geq\,0} $. Consider the point $(\iota_{L/k} (x), r) \in \BT (G,L) \times \R_{\geq\,0}$. Let $\theta _L (\iota_{L/k} (x), r) $ in $ (G \times _k L)^{an} $ be the Shilov boundary of the $L$-affinoid group attached to $(\iota_{L/k} (x), r)$ using the construction of Section~\ref{spli}.

\begin{lemm}\label{STABLE}
The point $\theta_L (\iota_{L/k} (x), r) $ is $\Gal (L/k)$-stable.
\end{lemm}

\begin{proof}
Using Lemma~\ref{rastabshi} and Remark~\ref{rema}, we see that $\theta_L (\iota_{L/k} (\underline{\BT (G,k)}) \times \Gamma)$ is fixed by $\Gal (L/k)$. Lemma~\ref{dense} implies that $\theta_L (\iota_{L/k} ({\BT (G,k)}, \R _{\geq\,0})$ is fixed by $\Gal (L/k)$, since $\theta_L$ is continuous by the explicit formula given in Proposition~\ref{peaked}.
\end{proof}

\begin{rema}
It is possible to prove Lemma~\ref{STABLE} without using Section~\ref{secraaa}. Indeed let $ S \subset T$ be torus over $k$ such that $S$ is a maximal split torus over $k$, $x \in A(G,S,k)$, and $T$ is a maximal split torus over $K$. The point $ \iota_{L/k} (x)$ is in $A(G,T,L)$. Moreover, we have an explicit formula for $\theta _L (\iota_{L/k} (x), r)$ (Proposition~\ref{peaked}) and we can check on the formula that it is $\Gal (L/k)$-stable using the fact that the action of $\Gal (L/k)$ on $X^* (T,L)$ stabilizes $\Phi(G,T, L)$ and associated objects.
\end{rema}

So $G_{\iota_{L/k} (x), r } $ is $\Gal (L/k)$-stable and so $\pr_{L/k}^{-1}(\pr_{L/k} (G_{\iota_{L/k}(x), r }))= G_{\iota_{L/k}(x), r }$. So using Theorem~\ref{descent} and Lemma~\ref{groudesc}, we obtain that $\pr_{L/k} (G_{\iota_{L/k}(x), r }) $ is a $k$-affinoid group that we denote by $G_{x,r}$. Moreover, it is easy to see that the Shilov boundary of $G_{x,r}$ is equal to $\pr_{K/l} (\theta_L (\iota_{L/k} (x), r))$, we denote it as $\theta (x,r)$.

\begin{rema}\label{remadef}
For any $(x,r) \in \BT (G,k) \times \R_{\geq\,0}$, we defined a $k$-affinoid group $G_{x,r}$ as $\pr _{L/k} \pr _{K/L} (\widehat{~\GG _r~}_{\eta})$ where $L/k$ is a finite Galois extension splitting $G$ and $K/L$ is an affinoid extension. Using the compatibility of dilatations under base change~\cite[Theorem~3.2$\MK$(6)]{NBU}, it is a formal computation to check that $G_{x,r}$ is well-defined, i.e it does not depend on the choice of extensions $L$ and $K$ used in order to define it. So we see that $G_{x,r} =\pr_{K/k} (\widehat{~\GG_r~}_{\eta}) $ where $K/k$ is any $k$-affinoid extension such that $G$ is split over $K$, $\iota_{K/k} (x) $ is special and $r \in \ord (K)$. Moreover the identity $G_{x,0} =\pr _{K/k} (\widehat{~\GG~}_{\eta}) $ shows that $G_{x,0}= G_x$, where $G_x$ is the $k$-affinoid group defined in~\cite[Theorem~2.1 and its proof]{RTW}.
\end{rema}



\begin{prop}\label{holoxr}\ \\*[-2em]
\begin{enumerate}
\item\label{prop6.4.1} The holomorphically convex envelope of $\theta (x,r) $ is $G_{x,r}$.
\item\label{prop6.4.2} The $k$-affinoid algebra of $G_{x,r}$ is the completion of $\clO(G)$ relatively to the norm $\theta (x,r)$.
\end{enumerate}
\end{prop}

\begin{proof}\ \\*[-2em]
\begin{itemize}
%\begin{sloppypar}
\item[\eqref{prop6.4.1}] Since $\theta _L (\iota_{L/k} (x), r) $ is $\Gal (L/k)$-stable and $\pr_{L/k} (\theta _L (\iota_{L/k} (x), r)) = \theta (x,r)$, we have $\pr _{L/k}^{-1} (\theta (x,r)) = \theta _L(\iota_{L/k} (x), r)$. So by~\cite[Proposition~4.4]{PM}, we obtain $\Hol (\theta _L (\iota_{L/k} (x), r)) = \pr _{L/k}^{-1} (\Hol (\theta (x,r)))$, this implies $G_{x,r}= \Hol (\theta (x,r))$ as required.

\item[\eqref{prop6.4.2}] Let $\clA$ be the $k$-affinoid algebra of $G_{x,r}$. This is reduced and we can assume that its norm equals its spectral norm~\cite[Proposition~2.1.4]{berk} and so equals its unique Shilov boundary point $\theta (x,r)$. Let $\clA (x,r) $ be the completion of $\clO (G) $ relatively to the norm $ \theta (x,r)$. The immersion $G_{x,r} \to G^{an}$ corresponds to an injective morphism of $k$-algebras $\clO (G) \to \clA$. This morphism extends to an isometric embedding $i: \clA(x,r) \to \clA$. Let $K/k$ be an affinoid extension such that we can write $G_{x,r}= \pr_{K/k} (\widehat{~\GG _{r}~} _{\eta})$ (see Remark~\ref{remadef}). Let $\clA_K$ be the $K$-affinoid algebra of $ \widehat{~\GG _{r}~}_{\eta}$. The algebra $\clA_K$ is equal to the completion of $\clO(G) \otimes _k K$ relatively to the norm $|.| _{\GG_r}$, in particular $\clO(G) \otimes_k K $ is dense in $\clA_K$. Moreover by definition, $\clA \hat{\otimes}_k K = \clA _K $. So $\clO (G) \otimes_k K $ is dense in both $\clA \hat{\otimes }_k K $ and $ \clA(x,r) \hat{\otimes}_k K $. So $i \hat{\otimes} \Id_K : \clA(x,r) \hat{\otimes}_k K \to \clA \hat{\otimes}_k K $ is an isomorphism of Banach algebras, and so $\clA(x,r) = \clA$ by~\cite[Appendix~A, Lemma~A.5]{RTW}.
%\end{sloppypar}
\end{itemize}
\end{proof}


\begin{prop}\label{action}
Let $g \in G(k)$, then $G_{g.x,r} = g G_{x,r} g^{-1}$ and $\theta (g.x, r)\linebreak = g \theta (x,r) g^{-1}$.
\end{prop}

\begin{proof}
The assertions are equivalent by Proposition~\ref{holoxr}. Let us prove the first one. Choose a $k$-affinoid extension $K/k$ such that we can write $G_{x,r}=\pr _{K/k} (\widehat{~\GG _{x,r}~}_{\eta})$. The sequence of equalities

\begin{align*}
g G_{x,r} g^{-1} &=g~{\pr _{K/k} \left(\widehat{~\GG _{x,r}~}_{\eta} \right)} g^{-1} \\
&= \pr _{K/k} \left(g \widehat{~\GG _{x,r}~}_{\eta}~g^{-1} \right)\\
&= \pr_{K/k} \left(\widehat{~g{\GG_{x,r}} g^{-1}~} _{\eta} \right)\\
&= \pr_{K/k} \left(\widehat{~\GG_{g.x, r}~}_{\eta} \right)\\
&=G_{g.x, r}
\end{align*}
ends the proof of Proposition~\ref{action}.
\end{proof}

We end this section with the following result.

\begin{prop}
Assume $G$ splits over a tamely ramified extension. Then for all $(x,r) \in \BT (G, k) \times \R _{\geq\,0}$, the point $\theta (x,r) \in G ^{an}$ is peaked.
\end{prop}

\begin{proof}
We use notation of the beginning of this section. The point $\theta_L (\iota_{L/k} (x), r)$ is peaked by Proposition~\ref{pe}. Now~\cite[Lemma~A.10]{RTW} and~\cite[Part~II, Chapter on Buildings, Section~6 about erratum]{RTWduc} end the proof.
\end{proof}

\begin{rema}
If $G$ does not split over a tamely ramified extension, then in general $\theta (x,r)$ is not peaked (see Proposition~\ref{counter} for a counter-example.)
\end{rema}


\section{Filtration of Lie algebras} \label{seclie}
Let $\GG$ be the Lie algebra of $G$ and $\GG ^{an}$ its analytification. Let $(x,r) \in \BT(G,k) \times \R _{\geq\,0}$. Let $K/k$ be a $k$-affinoid extension such that we can write $G_{x,r} = \pr _{K/k}(\widehat{~\GG_r~}_{\eta})$ (see Remark~\ref{remadef}). We define 
\[
\GG_{x,r}= \pr_{K/k} \left(\widehat{\Lie (\GG _r)}_{\eta} \right).
\]
Using similar arguments as for $G_{x,r}$, we see that $\GG_{x,r}$ is a $k$-affinoid subgroup of $\GG^{an}$ equal to the holomorphically convex envelope of its unique Shilov boundary point. We can also define similar filtrations in the context of rational potentially Demazure $k$-affinoid groups.


\section{Comparison with Moy--Prasad filtrations} \label{seccompa}
Let $G$ be a connected reductive $k$-group scheme that splits over a tamely ramified extension. Let $G(k)_{x,r}^{MP} $ denote the normalized Moy--Prasad filtration as used in~\cite{YU}. We will use the following facts

\begin{itemize}
\item \cite[Corollary~8.8]{smoothYu} if $G$ is split and $x$ is special, then Moy--Prasad filtrations are obtained by taking set-theoretic congruence subgroups of the integral points of the attached integral Demazure group $\GG _x$;

\item \cite[Line~15 Page~278]{Kim} or~\cite[Section Tamely ramified descent]{KP} $~~~$ Moy--Prasad filtrations are compatible relatively to field extensions in the tame case; in order to prove the following proposition.
\end{itemize}

\begin{prop}\label{tamesituation}
Assume that we can choose a finite and tamely ramified extension $K/k$ in order to define (as in Remark~\ref{remadef}) the $k$-affinoid group $G_{x,r}$ and assume $r>0$, then $G_{x,r }(k) = G(k)^{MP}_{x,r}$.
\end{prop}

\begin{proof}
Let $K/k$ be a finite tamely ramified extension such that we can write $G_{x,r} = \pr_{K/k} (\widehat{~\GG _r~}_{\eta})$. The following equalities
\begin{align*}
G_{x,r}(k) &= G_{x,r} (K) \cap G(k) \\
& = \widehat{~\GG _r~}_{\eta} (K) \cap G (k) \\
&= \GG _r (K^{\circ}) \cap G(k)\\
&= G(K)_{x,r}^{MP} \cap G(k)\\
&= G(k)^{MP}_{x,r}
\end{align*}
ends the proof.
\end{proof}



\section{A cone in \texorpdfstring{$G^{an}$}{Gan}} \label{seccone}


In the previous section, we constructed for each pair $(x,r) \subset \BT (G,k) \times \R _{\geq\,0}$ a $k$-affinoid group $G_{x,r}$ whose Shilov boundary is a singleton $\theta (x,r)$. If $r' >r$, it is easy to see that $G_{x,r'} \underset{\ne }{\subset} G_{x,r}$. We now introduce a map $\theta$.


\begin{defi}
Let $\theta$ be the map
\begin{align*}
\theta : \BT (G,k) \times \R _{\geq\,0} &\to G^{an} \\
(x,r) &\mapsto \theta (x,r).
\end{align*}
\end{defi}


\begin{theo}\leavevmode
\begin{enumerate}
\item\label{theo9.2.1} The map $\theta$ is $G(k)$-equivariant relatively to the actions $g.(x,r)=(g.x,r)$ and $g.p=gpg^{-1}$, for $g \in G(k)$, $x \in \BT (G,k)$, $r \in \R_{\geq \,0}$, and $p \in G^{an}.$

\item\label{theo9.2.2} For any finite extension $k'/k$, the diagram
\[
\xymatrix{ \BT\left(G,k'\right) \times \R _{\geq\,0} \ar[r]^{\theta '} & \left(G\times_k k'\right)^{an} \ar[d]^{\pr_{K/k}} \\ \BT(G,k) \times \R _{\geq\,0} \ar[r]^{\theta} \ar[u]^{\iota_{k'/k} \times \Id} & G^{an}}
\]
is commutative. Here $\theta '$ is defined as $\theta$. In other words, it is the map sending $(x,r) \in \BT (G,k') \times \R _{\geq\,0}$ to $\Shi (\pr_{K/k'} (\widehat{~\GG _r~}_{\eta}))$ where $K/k'$ is an affinoid extension such that $G$ is split over $K$, $\iota_{K/k'} (x) $ is special and $r \in \ord (K)$, moreover $\GG_r$ is the $\pii_r$-congruence subgroup of the canonical Demazure $K^{\circ}$-group scheme attached to $\iota_{K/k'} (x)$.
\item\label{theo9.2.3} The map $\theta$ is continuous and injective.
\end{enumerate}
\end{theo}

\begin{proof}
\begin{itemize}
\item[\eqref{theo9.2.1}] This is a reformulation of Proposition~\ref{action}.
\item[\eqref{theo9.2.2}] This is a direct consequence of definitions.
\item[\eqref{theo9.2.3}] Let us first assume that $G$ is split over $k$. It is enough to prove that the map $A(G,T,k) \times \R_{\geq\,0} \to G^{an}$ is continuous and injective for all apartments $A(G,T,k)$. This is a direct consequence of the formula of Proposition~\ref{peaked}. In general, we choose a finite Galois extension such that $G$ is split over $L$ and conclude directly using the commutative diagram of the second assertion of this theorem.\qedhere
\end{itemize}
\end{proof}

The set $\theta (\BT (G,k), \R_{\geq\,0}) \cup e_{G^{an}}$ is a topological cone in $G^{an}$ whose basis is the Bruhat--Tits building and whose vertex is the neutral element $e_{G^{an}}$.



\section{Picture for the split torus of rank one} \label{exgm}

Let $G$ be $\spec(k[X,Y]/XY-1)$, it is a split torus of rank one over $k$. The reduced Bruhat--Tits building of $G$ is a singleton $\{x\}$. The point $x$ is special and $G$ is split over $k$. The grosse cellule of $G$ is $G$. Let $r \geq 0$ and choose a $k$-affinoid extension $K/k$ such that $r \in \ord (K)$. Let $\GG $ be the $K^{\circ}$-Demazure group scheme attached to $\iota_{K/k}(x)$. It is equal to $\spec(K^{\circ} [X,Y] / XY-1).$ By definition $G_{x,r}$ is equal to $\pr_{K/k} (\widehat{~\GG _r~} _{\eta})$. The ring $\Hopf (\GG _r)$ is equal to $K^{\circ} [{\pi _r }^{-1} (X-1),{\pi _r}^{-1} (Y-1)] \subset K[X,Y] /XY-1$. Writing $f \in K[X,Y]/XY-1$ as $\sum _{(k_1, k_2)\, \in \,U \,} a_{k_1k_2} (X-1)^{k_1}(Y-1)^{k_2}$ ($U$ is the set of parameters for the basis of $K[X,Y]/XY-1$ centered at unity), the norm $\vert~\rvert _{\Hopf (\GG _r)} $ is explicitly given by the map
\begin{align*}
K[X,Y]/ XY-1 &\to \R _{\geq\,0} \\
f &\mapsto\max _{(k_1, k_2)\,\in\,U} \left\lvert a_{k_1k_2} \right\rvert e^{-r \left(k_1+k_2\right)}~_.
\end{align*}

The Shilov boundary of $\widehat{~\GG _r~} _{\eta}$ is $\vert~\rvert _{\Hopf (\GG _r)} $. The Shilov boundary $\theta (x,r)$ of $\pr_{K/k}\linebreak(\widehat{\GG _r} _{\eta})$ is $ \lvert~\rvert _{\Hopf (\GG _r)} $ restricted to the $k$-algebra $\Hopf(G)$. The point $\theta (x,r) \in G^{an}$ is thus equal to the norm on $k[X,Y]/XY-1$ which map
\[
\sum_{\left(k_1, k_2\right)\,\in\,U} a_{k_1k_2} (X-1)^{k_1} (Y-1)^{k_2} \quad \text{to}\quad\max _{\left(k_1,k_2\right)\,\in\,U} \left\lvert a_{k_1k_2} \right\rvert e^{-r \left(k_1+k_2\right)}.
\]
It corresponds via the embedding $G^{an} \to (\bbA^1_k)^{an}\setminus 0 $ to the norm usually denoted $\lvert~\rvert _{1, e^{-r}}$ inside $(\bbA^1_k)^{an}$.
We have the picture


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{flushleft}

\begin{tikzpicture}

\draw[red, thick] (0,-12)-- (0,0);

\draw[gray, thick] (6,-12) -- (0,-6);
\draw[gray, thick] (3,-9) -- (4.5,-12);
\draw[gray, thick] (3,-12) -- (0,-9);


\draw[gray, thick] (1.5,-10.5) -- (1.69,-12);
\draw[gray, thick] (0,0) -- (0,8.1);
\draw[gray, thick] (11,-12) -- (0,7.8);
\draw[gray, thick] (0,0) -- (7.25,-12);

\draw[gray, thick] (0,-10.5) -- (0.9,-12);

\draw[gray, thick] (0,-11.25) -- (0.45,-12);

\filldraw[black] (0,0) circle (2pt) node[anchor=west] {{\footnotesize Reduced Bruhat-Tits Building}};
\filldraw[black] (0,0) circle (2pt) node[anchor=east] {$\theta(x,0)=\lvert~\rvert_{1,1}$};
\filldraw[black] (0,-10.5) circle (2pt) node[anchor=east] {$\theta(x,3)=\lvert~\rvert_{1,e^{-3}}$};
\filldraw[black] (0,-12) circle (2pt) node[anchor=east] {$\theta(x,+\infty)=\lvert~\rvert_{1,0}$};
\filldraw[black] (0,-6) circle (2pt) node[anchor=east] {$\theta(x,1)=\lvert~\rvert_{1,e^{-1}}$};
\filldraw[black] (0,-9) circle (2pt) node[anchor=east] {$\theta(x,2)=\lvert~\rvert_{1,e^{-2}}$};
\filldraw[black] (0,-11.25) circle (2pt) node[anchor=east] {$\theta(x,4)=\lvert~\rvert_{1,e^{-4}}$};
\filldraw[black] (0,-12) circle (1pt) node[anchor=north] {{\tiny $1$}};

\filldraw[black] (1.69,-12) circle (1pt) node[anchor=north] {{\tiny $1+\pi^2 $}};

\filldraw[black] (0.45,-12) circle (1pt) node[anchor=north] {{$\dots$}};
\filldraw[black] (0.9,-12) circle (1pt) node[anchor=north] {{\tiny $1+\pi^3 $}};
\filldraw[black] (3,-12) circle (1pt) node[anchor=north] {{\tiny $1+\pi^2 + \pi ^3$}};
\filldraw[black] (4.5,-12) circle (1pt) node[anchor=north] {{\tiny $1+\pi +\pi^2$}};
\filldraw[black] (6,-12) circle (1pt) node[anchor=north] {{\tiny $1+\pi$ }};
\filldraw[black] (7.25,-12) circle (1pt) node[anchor=north] {{\tiny $\delta$ }};
\filldraw[black] (11,-12) circle (1pt) node[anchor=north] {{\tiny $1+\pi ^{-1}$ }};
\end{tikzpicture}
\end{flushleft}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\pagebreak
\noindent giving some points (of course it is not exhaustive) of $G^{an}$ inside $(\bbA^1_k)^{an}$. Here $\delta$ is an element in $(\ko) ^{\times} \setminus 1 + k^{\circ \circ}$. The point $\theta (x,0)$ is mapped to the so-called Gauss point, and corresponds to the reduced Bruhat--Tits building. When $r \geq 0$ is increasing the point $\theta (x,r)$ is getting closer to $1$, the neutral element of $G^{an}$. The holomorphically convex envelope $G_{x,r}$ of $\theta (x,r)$ should be thought as all the points under (attainable by going only down) $\theta (x,r)$ and the $k$-rational points of $G_{x,r}$ as certain lower extremities. In this situation the cone is the red line.



\section{Computation in a wild torus} \label{exwild}


In this section $k = \Q _2$. The polynomial $X^2-2$ does not have any solution in $k$. Let $\rd \in \overline{k}$ be a root of this polynomial and let $K$ be the field $k(\rd) \subset \overline{k}$. The extension $K/k$ is a wildly ramified Galois extension. We have $[K:k]=e(K:k)=2$. The element $\rd$ is a uniformizer of $K$. The $k$-vector space $K$ is $2$-dimensional and $\{1, \rd \}$ is a $k$-basis. So each element in $K$ can be written as $x+\rd y$ with $x,y \in k$. The norm of $x+\rd y$ is equal to $(x+ \rd y)(x- \rd y)= x^2 -2 y^2$. The set of norm $1$ elements is an algebraic group. Let us write the Hopf algebra of the corresponding affine $k$-group scheme $G$. The Hopf $k$-algebra of $G$ is $k[X,Y]/X^2-2Y^2-1$, moreover the comultiplication $\Delta$, the antipode $\tau$ and the augmentation $\varepsilon$ are
\begin{align*}
\Delta :\Hopf (G) &\to \Hopf (G) \otimes \Hopf (G)\\
X &\mapsto X \otimes X + 2 Y \otimes Y \\
Y &\mapsto X \otimes Y + Y \otimes X
\\[0.10em]
\tau : \Hopf (G) &\to \Hopf (G) \\
X& \mapsto X\\
Y &\mapsto -Y
\\[0.10em]
\varepsilon: \Hopf (G)&\to k \\
X &\mapsto 1\\
Y &\mapsto 0~.
\end{align*}

The $k$-group $G$ is a torus, indeed the equation
\begin{align*}
k[X,Y]/X^2-2Y^2-1 \otimes _k K& \simeq K[X,Y]/X^2-2Y^2-1 \\
&\simeq K[X,Y] /\left(X+ \rd Y\right)\left(X- \rd Y\right) -1 \\
& \simeq K[U,V] / UV-1
\end{align*}
shows that $G \times _{k} K \simeq \bbG_m /K$. The reduced Bruhat--Tits building $\BT(G,k)$ is a singleton $\{x\}$. The point $x$ is a special point of $\BT(G,k)$ and $\iota_{K/k} (x) \in \BT(G,K)$ is special for any finite extension $K/k$. The group $G$ is not split over $k$, it is split over $K$. Let us make explicit the group $G_{x,0}$. We need to find an extension such that $G$ is split over it and the image of $x$ over $K$ is special. The field $K$ works. By definition the $k$-analytic group $G_{x,0}$ is equal to $\pr_{K/k} (\widehat{\GG}_{\eta})$, where $\GG$ is the $K^{\circ}$-Demazure group scheme attached to $\iota_{K/k}(x)$. In the coordinates $U,V$, $\GG= \spec (K^{\circ} [U,V]/UV-1)$. Thus in the coordinates $X,Y$, $\Hopf (\GG)$ is equal to the $K^{\circ}$-subalgebra $K^{\circ}[X + \rd Y, X- \rd Y]$ of $K[X,Y]/X^2-2Y^2-1$ generated by $ X+\rd Y $ and $ X- \rd Y $. The $k$-affinoid algebra of $G_{x,0}$ is the completion of $\Hopf(G)$ relatively to the norm $\lvert~\rvert _{\Hopf (\GG)} \mid _{\Hopf (G)}$. So let us make as explicit as possible the norm $\lvert~\rvert _{\Hopf (\GG)} \mid _{\Hopf (G)}$. By definition, we have
\begin{multline*}
\lvert~\rvert_{\Hopf (\GG)}: \Hopf \left(G \times_k K\right) \to \R _{\geq\,0} \\
f \mapsto \inf_{\lambda\,\in\,K^{\times} } \left\{ \lvert \lambda \rvert \middle| f \in \lambda. K^{\circ}\left[X + \rd Y, X- \rd Y\right] \right\}.
\end{multline*}
And so, by restriction
\begin{multline*}
\lvert~\rvert _{\Hopf (\GG)} |_{\Hopf (G)}: k[X,Y]/X^2-2Y^2-1 \to \R _{\geq\,0} \\
f \mapsto \inf _{\lambda\,\in\,K^{\times} } \left\{ \lvert \lambda \rvert \middle| f \in \lambda. K^{\circ}\left[X + \rd Y, X- \rd Y\right] \right\}.
\end{multline*}
We have to complete $k[X,Y]/X^2-2Y^2-1$ relatively to this norm, in order to simplify notation let us put $ \lvert \lvert~\rvert \rvert =\lvert~\rvert _{\Hopf (\GG)} \mid _{\Hopf (G)} $. Let us compute the value $\lvert \lvert X \rvert \rvert$. Since $ \rd X \not \in K^{\circ}[X + \rd Y, X- \rd Y] $ and $2X =\rd ^2 X \in K^{\circ}[X + \rd Y, X- \rd Y]$, we deduce that
\[
\| X \| = e^{}.
\]
Let us now compute the value $\lvert \lvert Y \rvert \rvert$. Since $2Y =\rd^2 Y \not \in K^{\circ}[X- \rd Y, X+ \rd Y]$ and $ 2\rd Y =\rd^3 Y \in K^{\circ}[X- \rd Y, X+ \rd Y]$ we deduce that
\[
\| Y \| = e^{\frac{3}{2}}.
\]
Consider the algebra
\[
k\left\{e^{-1} X, \left(e^{\frac{3}{2}
}\right)^{-1}Y\right\} / X^2 - 2Y^2-1\;,\;\|~\|
\]
where $k\{ e^{-1} X, (e^{\frac{3}{2}})^{-1}Y\} $ is the $k$-algebra
\[
\left\{ \sum _{k_1, k_2} a_{k_1 k_2} X^{k_1} Y^{k_2} \middle|\,\lvert a _{k_1k_2} \rvert e^{k_1} \left(e^{\frac{3}{2}}\right)^{k_2} \to 0 \text{~as~} k_1+k_2 \to \infty \right\} \subset k[[X,Y]].
\]
We claim that it is the $k$-affinoid algebra of $G_{x,0}$. We need to check that
\[
\left(k\left\{ e^{-1} X, \left(e^{\frac{3}{2} }\right)^{-1}Y\right\} / X^2 - 2Y^2-1\right)\hat{\otimes} _{k} K
\]
is isomorphic to the $K$-affinoid algebra of $\widehat{\GG}_{\eta}$. In the coordinates $U,V$, the $K$-affinoid algebra of $\widehat{\GG}_{\eta}$ is $K\{U,V\}/ UV-1$. The $K$-algebra $(k\{ e^{-1} X, (e^{\frac{3}{2}})^{-1}Y\} / X^2 - 2Y^2-1)\hat{\otimes} _{k} K $ is isomorphic to $K\{ e^{-1} X, (e^{\frac{3}{2} })^{-1}Y\} / X^2 - 2Y^2-1) $. The isomorphism previously considered $K[X,Y]/X^2-2Y^2-1 \simeq K[U,V]/UV-1$ induces maps

\begin{align*}
K\left\{ e^{-1} X, \left(e^{\frac{3}{2}}\right)^{-1}Y\right\} /X^2-2Y^2-1 & \leftrightarrow K \{U,V\}/ UV-1 \\
X+ \rd Y &\mapsfrom U \\
X- \rd Y & \mapsfrom V \\
X &\mapsto \frac{U+V}{2} \\
Y & \mapsto \frac{U-V}{2 \rd}.
\end{align*}
These maps are mutual inverse $K$-Banach algebras isometries.


Now we are interested in the question: Is the Shilov boundary of $G_{x,0 }$ a peaked norm? In other words, is $\|~\|$ a peaked norm?


Recall that by definition $\|~\| \otimes |~| _K $ is the norm on $k[X,Y]/X^2-2Y^2-1 \otimes _k K $ defined by $\|~\| \otimes |~| _K (f) = \inf \max _{i} \| x_i \|. | \lambda _i | _K $ where $\inf $ is taken over all representatives $f =\sum _i x_i \otimes \lambda _i $. Let us start with a Lemma.

\begin{lemm}\label{lemm11.1}\leavevmode
\begin{enumerate}
\item Each $f \in k[X,Y]/X^2-2Y^2-1 \otimes _k K $ can be written uniquely as 
\[
f = x \otimes 1 + y \otimes \rd .
\]
\item Let $f \in k[X,Y]/X^2-2Y^2-1 \otimes _k K $ and write $f= x \otimes 1 + y \otimes \rd$ as in the previous assertion. Then $\|~\| \otimes |~|_K (f) = \max \{\|x\|, \|y\|. |\rd| \}$.
\end{enumerate}
\end{lemm}

\begin{proof}
The first assertion is a direct consequence of the fact that $\{1, \rd \}$ is a $k$-basis of $K$. Now let us prove the second assertion.
Let $Z \in k[X,Y]/X^2-2Y^2-1 \otimes _k K $. For a representative $R: Z = \sum_i x_i \otimes {\alpha_i } $, we use the notation $|Z|_{R}$ for $\max_i \|x_i\| | \alpha_i |_K $. So that we have $ \|~\| \otimes |~|_{K} (Z) = \inf_{R} |Z|_R$. Let $R: f = \sum_{i=1}^S x_i \otimes \alpha_i $ be a representative of $f$. Each $\alpha_i $ can be written as $\alpha _i = a_i + b_i \rd $. Since $a_i $ and $b_i$ are in $k$ we have $|a_i| \ne |b_i \rd|$. So $|\alpha _i | = \max \{ |a_i|,\;|b_i \rd | \}.$ We have
\begin{align*}
f &= \sum_{i=1} ^S x_i \otimes\left(a_i +b_i \rd\right)\\
&= \left(\sum_{i=1}^S x_i a_i\right)\otimes 1 +\left(\sum_{i=1}^S x_i b_i\right)\otimes \rd.
\end{align*}
Now let us denote by $R'$ this last representative, i.e
\[
R': f =\left(\sum _{i=1}^S x_i a_i\right)\otimes 1 +\left(\sum _{i=1}^S x_i b_i\right)\otimes \rd.
\]
We claim that $|f|_R \geq |f |_{R'}$. Indeed
\begin{align*}
|f|_R &= \max _{i=1} ^S \|x_i\|.|\alpha _i | \\
&= \max _{i=1}^S \left\{\|x_i\|. \max \left\{|a_i|~,~~|b_i \rd| \right\} \right\}\\
&= \max _{i=1}^S \left\{ \max \left\{\|x_i\|.|a_i|~,~\|x_i\|.|b_i \rd | \right\} \right\}\\
&= \max _{i=1}^S \left\{ \max \left\{\|x_i.a_i\|~,~\|x_i.b_i \|. | \rd | \right\} \right\}\\
&\geq \max \left\{ \left\|\left(\sum _{i=1}^S x_i a_i\right)\right\|. |1|,\quad \left\|\left(\sum _{i=1}^S x_i b_i\right) \right\|. \left|\rd\right|\right\}\\
&= |f|_{R'}.
\end{align*}
This ends the proof of Lemma~\ref{lemm11.1}, since $x= \sum_{i=1} ^S x_i a_i$ and $y = \sum_{i=1} ^S x_i b_i $.
\end{proof}


\begin{prop}\label{counter}
Let us see $U $ inside $k[X,Y]/ X^2 - 2Y^2 -1 \otimes _k K $ via the isomorphism above, i.e $U = {X} \otimes 1 + {Y} \otimes \rd $.
Then $ \|~\|\otimes _k |~|_K (U) = e $ and $|~| _{\Hopf (\GG)} (U) = 1.$ Moreover the norm $\|~\|$ is not universal (or peaked).
\end{prop}

\begin{proof}
First by definition of $ |~|_{\Hopf (\GG)}$, we have $ |~|_{\Hopf (\GG)} (U) = |1| = e^0 = 1.$
Now let us compute $\|~\| \otimes _k |~| _K (U)$ using the previous Lemma~\ref{lemm11.1}. We have $U = {X} \otimes 1 + {Y} \otimes \rd $. So we have
\begin{align*}
\|~\| \otimes _k |~| _K (U) &= \max \left\{ \|{X}\|~,~\|{Y}\| | \rd | \right\} \\
&= \max \left\{ e, e^{\frac{3}{2}}.e^{-\frac{1}{2}} \right\} \\
&= e\,. \\
\end{align*}
Now if $\|~\|$ was peaked, then we would have deduced $\|~\| \otimes _k |~|_K = |~| _{\Hopf (\GG)}$ by~\cite[Corollary~5.2.4]{berk}; so $\|~\|$ is not peaked.
\end{proof}

\vspace{50pt}

\bibliography{mayeux}
\end{document}
