unique Shilov boundary point of $G_{x}$ to that of $G_{gx}$, whence

$\vartheta(gx)=g\vartheta(x)g^{-1}.$

∎

######  Lemma 24.

Let $K/k$ be a finite Galois extension. We have the commutative diagram

${\mathcal{B}_{\mathbb{Q}}(G,K)}$${G_{K}^{\operatorname{an}}}$${\mathcal{B}_{\mathbb{Q}}(G,k)}$${G^{\operatorname{an}}}$$\scriptstyle{\vartheta}$$\scriptstyle{\operatorname{pr}_{K/k}}$$\scriptstyle{p_{K/k}}$$\scriptstyle{\vartheta}$

.

###### Proof.

Let $x\in\mathcal{B}_{\mathbb{Q}}(G,k)$ and $L/K$ be a finite Galois extension
such that $G_{L}$ is pseudo-split and $x_{L}$ is a special point. Then, we
have

$G_{x_{K}}=\operatorname{pr}_{L/K}((\mathcal{G}_{x_{L}}^{\circ})^{\operatorname{an}})$

as well as

$G_{x}=\operatorname{pr}_{L/k}((\mathcal{G}_{x_{L}}^{\circ})^{\operatorname{an}})=\operatorname{pr}_{K/k}(G_{x_{K}}).$

Because the projection $\operatorname{pr}_{K/k}$ maps Shilov boundaries to
Shilov boundaries, we have
$\vartheta(x)=\operatorname{pr}_{K/k}(\vartheta(x_{K}))$. ∎

###  2.3 Extending to the building

Our goal is now to extend the map $\vartheta$ to the whole of
$\mathcal{B}(G,k)$. Given that we have the commutative diagram established in
Lemma 24

${\mathcal{B}_{\mathbb{Q}}(G,K)}$${G_{K}^{\operatorname{an}}}$${\mathcal{B}_{\mathbb{Q}}(G,k)}$${G^{\operatorname{an}}}$$\scriptstyle{\vartheta}$$\scriptstyle{\operatorname{pr}_{K/k}}$$\scriptstyle{p_{K/k}}$$\scriptstyle{\vartheta}$

for each finite Galois extension $K/k$, it is sufficient to work in the case
where $G$ is pseudo-split, which we henceforth assume.

We begin by establishing a formula for the norm $\vartheta(x)$ associated to a
point of the building.

Let $T$ be a split maximal torus and let $\Phi=\Phi(G,T)$ be the root system
corresponding to $T$. Consider a minimal pseudo-parabolic subgroup $P$
containing $T$ and let $\Phi^{+}=\Phi(P,T)$ be the set of positive roots
corresponding to $P$. Denote by $\Phi_{\operatorname{nd}}$ the set of non
divisible roots in $\Phi$ and by
$\Phi^{+}_{\operatorname{nd}}=\Phi^{+}\cap\Phi_{\operatorname{nd}}$. Then the
multiplication map yields an open immersion

$\prod_{\alpha\in\Phi^{+}_{\operatorname{nd}}}U_{-\alpha}\times
Z_{G}(T)\times\prod_{\alpha\in\Phi^{+}_{\operatorname{nd}}}U_{\alpha}\hookrightarrow
G$

whose image is independent of the order on $\Phi^{+}_{\operatorname{nd}}$. We
denote this image by $\Omega(T,P)$ and call it the open Bruhat cell
corresponding to the pair $(T,P)$.

If $o$ is a special point in $A(G,T)$, then, $o$ determines integral models
$\mathcal{U}_{a,o}$ for each $U_{a}$ and an integral model $\mathcal{G}$ for
$G$ defined in [Lou21, 4.2]. Letting $\mathcal{Z}^{\circ}$ be the open
subgroup of the Néron model of $Z=Z_{G}(T)$ containing its generic fibre and
the connected component of its special fibre, it follows from [Lou21, Théorème
4.5] that the above open immersion descends to an open immersion of
$k^{\circ}$-schemes

$\prod_{\alpha\in\Phi^{+}_{\operatorname{nd}}}\mathcal{U}_{-\alpha,o}\times\mathcal{Z}^{\circ}\times\prod_{\alpha\in\Phi^{+}_{\operatorname{nd}}}\mathcal{U}_{\alpha,o}\hookrightarrow\mathcal{G}.$

We denote by $\Omega(\mathcal{T},\mathcal{P})$ the image of this immersion.

For each non-divisible root $\alpha\in\,\Phi_{\operatorname{nd}}$, we have an
$\mathcal{T}$-equivariant isomorphism of $k^{\circ}$-schemes:

$\mathcal{U}_{\alpha}\simeq\mathcal{U}_{2{\alpha}}\times\mathcal{U}_{\alpha}/\mathcal{U}_{2{\alpha}},$

where
$\mathcal{T}=\operatorname{Spec}(k^{\circ}[X^{*}(T)])\subset\mathcal{Z}^{\circ}$.
Moreover, each $\mathcal{U}_{2{\alpha}}$ (resp.
$\mathcal{U}_{\alpha}/\mathcal{U}_{2{\alpha}}$) is an affine space over
$k^{\circ}$ on which $\mathcal{T}$ acts linearly via the character $2{\alpha}$
(resp. ${\alpha}$). Fix $\mathcal{T}$-equivariant isomorphisms

$u_{\alpha}:\mathbb{A}^{N_{\alpha}}_{k^{\circ}}\overset{\sim}{\to}\mathcal{U}_{\alpha}/\mathcal{U}_{2{\alpha}}\text{
and
}u_{2{\alpha}}:\mathbb{A}^{N_{2{\alpha}}}_{k^{\circ}}\overset{\sim}{\to}\mathcal{U}_{2{\alpha}}$

for each non-divisible root ${\alpha}$. These yield a
$\mathcal{T}$-equivariant open immersion of $k^{\circ}$-schemes

$\prod_{{\alpha}\in\
\Phi^{+}}\mathbb{A}^{N_{-\alpha}}_{k^{\circ}}\times\mathcal{Z}^{\circ}\times\prod_{{\alpha}\in\
\Phi^{+}}\mathbb{A}^{N_{\alpha}}_{k^{\circ}}\overset{\sim}{\to}\Omega(\mathcal{T},\mathcal{P})\subset\mathcal{G}.$

Lastly, we denote by $\vartheta_{Z}(\ast)$ the only point in the Shilov
boundary of $\mathcal{Z}^{\operatorname{an}}$, a multiplicative norm on
$k[Z]$. The uniqueness of that point follows from Proposition 22 applied to
$Z$. Note that the Bruhat-Tits building of the quasi-reductive group $Z$
[CGP15, Remark C.2.12 (1)] reduces to a point, which makes this notation
consistent with the above.

######  Proposition 25.

Assume that $G$ is pseudo-split. We keep the above notation. Let

${\mathbf{\Phi}}=\\{(a,i)\in\,\Phi\times\mathbb{N},1\leq i\leq N_{a}\\}.$

  1. 1.

The point $\vartheta(o)$ lies in $\Omega(T,P)^{\operatorname{an}}$ and
satisfies for each
$f=\sum_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}f_{\nu}\xi^{\nu}\in
k[\Omega(T,P)]=k[Z][(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Phi}}}]$,

$|f|(\vartheta(o))=\max_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}|f_{\nu}|(\vartheta_{Z}(\ast)).$

  2. 2.

Given the identification between $A(G,T)$ and $V(G,T)$ given by $o$, we have
for each $x\in A_{\mathbb{Q}}(G,T)$ and $f\in k[\Omega(T,P)]$ :

$|f|(\vartheta(x))=\max_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}|f_{\nu}|(\vartheta_{Z}(\ast))\prod_{\begin{subarray}{c}\alpha\in\Phi\\\
1\leq i\leq N_{\alpha}\end{subarray}}e^{\nu(\alpha,i)\langle\alpha,x\rangle}.$
(2)

###### Proof.

  1. 1.

The proof is analogous to that of [RTW10, Proposition 2.6].

Recall that $\vartheta(o)$ is the only point of the Shilov boundary of the
affinoid subgroup $\mathcal{G}^{\operatorname{an}}$ of
$G^{\operatorname{an}}$, that is, by [RTW10, 1.2.4], the only point in
$\mathcal{G}^{\operatorname{an}}$ that is mapped to the generic point of
$\mathcal{G}_{\widetilde{k}}$ by the reduction map

$\rho:\mathcal{G}^{\operatorname{an}}\to\mathcal{G}_{\widetilde{k}}.$

Because $\Omega(\mathcal{T},\mathcal{P})$ is an open subset of $\mathcal{G}$
that meets the special fibre, its special fibre
$\Omega(\mathcal{T},\mathcal{P})_{\widetilde{k}}$ contains the generic point
of $\mathcal{G}$. As a consequence of the commutativity of the following
diagram

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the open subset $\Omega(\mathcal{T},\mathcal{P})^{\operatorname{an}}$ of
$\mathcal{G}^{\operatorname{an}}$ must contain $\vartheta(o)$. Moreover, the
condition that $\vartheta(o)$ reduces to the generic point of
$\Omega(\mathcal{T},\mathcal{P})_{\widetilde{k}}$ may be rewritten as the
claim that, for any function $f\in
k[\Omega(T,P)]=k[Z][(\xi_{a,i})_{a\in\,_{k}\Phi,1\leq i\leq N_{a}}]$, we have
the equivalences:

$|f|(\vartheta(o))\leq 1\Longleftrightarrow f\in
k^{\circ}[\mathcal{Z}][(\xi_{a,i})_{(\alpha,i)\in{\mathbf{\Phi}}}]$

and

$|f|(\vartheta(o))<1\Longleftrightarrow f\text{ maps to zero in
}(\widetilde{k}\otimes_{k^{\circ}}k^{\circ}[\mathcal{Z}])[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Phi}}}].$

It follows that the restriction of $\vartheta(o)$ to $k[Z]$ is the Shilov
boundary of the affinoid subgroup $\mathcal{Z}^{\operatorname{an}}\subset
Z^{\operatorname{an}}$.

  2. 2.

By Proposition 20 there exist a finite Galois extension $K$ and $t\in T(K)$
such that $x_{K}=t\cdot o_{K}$. By Proposition 23, we have
$\vartheta(x_{K})=t\vartheta(o_{K})t^{-1}$. In particular, because
$\operatorname{int}(t)$ preserves $\Omega(T,P)$, we must have
$\vartheta(x_{K})\in\Omega(T,P)^{\operatorname{an}}$. Moreover, we have

$|f|(\vartheta(x))=|f|(\vartheta(x_{K}))=|\tau^{*}(f)|(\vartheta(o_{K})),$

where $\tau$ is the only map that makes the following diagram commute

${\prod_{{\alpha}\in\
\Phi^{+}}\mathbb{A}^{N_{-\alpha}}_{K}\times\mathcal{Z}\times\prod_{{\alpha}\in\
\Phi^{+}}\mathbb{A}^{N_{\alpha}}_{K}}$${\prod_{{\alpha}\in\
\Phi^{+}}\mathbb{A}^{N_{-\alpha}}_{K}\times\mathcal{Z}\times\prod_{{\alpha}\in\
\Phi^{+}}\mathbb{A}^{N_{\alpha}}_{K}}$${\Omega(T,P)_{K}}$${\Omega(T,P)_{K}}$$\scriptstyle{\tau}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{int}(t_{K})_{|\Omega(T,P)_{K}}}$

.

Now, because $T_{K}$ normalises $U_{\alpha,K}$ for each
$\alpha\in\Phi_{\operatorname{nd}}$ and, further, preserves the weight space
decomposition $U_{\alpha,K}\simeq U_{\alpha,K}/U_{2\alpha,K}\times
U_{2\alpha,K}$, it follows that $\tau$ acts on each
$\mathbb{A}_{K}^{N_{\alpha}}$ by multiplication by the character $\alpha$. In
other words,

$\tau^{*}:K[Z][(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Phi}}}]\to
K[Z][(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Phi}}}]$

is the only morphism of $K[Z]$-algebras that maps $\xi_{\alpha,i}$ to
$\alpha(t)\xi_{\alpha,i}$ for each $\alpha\in\Phi$ and
$i\in[\\![1,N_{\alpha}]\\!]$. Consequently, if
$f=\sum_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}f_{\nu}\xi^{\nu}$, then

$\tau^{*}(f)=\sum_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}f_{\nu}\prod_{(\alpha,i)\in{\mathbf{\Phi}}}\alpha(t)^{\nu(a,i)}\xi^{\nu}$

and therefore

$|f|(\vartheta(x))=\max_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}|f_{\nu}|(\vartheta_{Z}(\ast))\prod_{(\alpha,i)\in{\mathbf{\Phi}}}|\alpha(t)|^{\nu(a,i)}.$

Now, given that, $o_{K}$ is taken as origin of $A(G_{K},T_{K})$ and that, by
definition of the apartment [Lou21, Définition 3.3], $t$ acts on
$A(G_{K},T_{K})$ by translation by the vector $\nu(t)\in V(G_{K},T_{K})$ such
that $\langle\alpha,\nu(t)\rangle=-\omega(\alpha(t))$, we have

$|f|(\vartheta(x))=\max_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}|f_{\nu}|(\vartheta_{Z}(\ast))\prod_{(\alpha,i)\in{\mathbf{\Phi}}}e^{\nu(\alpha,i)\langle\alpha,x\rangle}.$

∎

######  Theorem 26.

The map $\vartheta$ extends uniquely to a continuous $G(k)$-equivariant map

$\vartheta:\mathcal{B}(G,k)\to G^{\operatorname{an}}.$

###### Proof.

Note that, because $\vartheta$ is already $G(k)$-equivariant, the continuity
of the $G(k)$-action on $\mathcal{B}(G,k)$ and $G^{\operatorname{an}}$ will
immediately imply that the extension is equivariant. Consequently, all we need
to do is prove that, for any point $x\in\mathcal{B}(G,k)$ and any sequence
$(x_{n})\in\mathcal{B}_{\mathbb{Q}}(G,k)^{\mathbb{N}}$ converging to $x$, the
sequence $\vartheta(x_{n})$ converges in $G^{\operatorname{an}}$ and that the
limit depends only on $x$.

The proof comprises two steps, one mainly using Berkovich, and one using
Bruhat-Tits theory. First of all, we treat the case of a sequence of virtually
special points in an apartment converging to a point in that apartment, which
follows easily from Proposition 25. Next, we reduce the general case to the
first step using the geometry of the building. The main observation is the
fact that, from any sequence of points in the building converging to a point
$x$, we may extract a subsequence lying in a single apartment containing $x$,
a consequence of the local finiteness of the building.

Step 1: Assume that $x\in A(G,T)$. Then, it follows from Proposition 25 that,
for any sequence $(x_{n})\in A_{\mathbb{Q}}(G,T)^{\mathbb{N}}$ converging to
$x$, the sequence $\vartheta(x_{n})$ converges in
$\Omega(T,P)^{\operatorname{an}}\subset G^{\operatorname{an}}$ and that the
limit depends only on $x$. Namely, for any $f\in k[\Omega(T,P)]$, we have

$|f|(\vartheta(x_{n}))=\max_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}|f_{\nu}|(\vartheta_{Z}(\ast))\prod_{(\alpha,i)\in{\mathbf{\Phi}}}e^{\nu(\alpha,i)\langle\alpha,x_{n}\rangle}\underset{n\to\infty}{\longrightarrow}\max_{\nu\in\mathbb{N}^{\mathbf{\Phi}}}|f_{\nu}|(\vartheta_{Z}(\ast))\prod_{(\alpha,i)\in{\mathbf{\Phi}}}e^{\nu(\alpha,i)\langle\alpha,x\rangle}.$

Step 2: Now, consider an arbitrary sequence
$(y_{n})\in\mathcal{B}_{\mathbb{Q}}(G,k)^{\mathbb{N}}$ converging to $x$.
Consider $\mathbf{f}\subset\mathcal{B}(G,k)$ the facet of $\mathcal{B}(G,k)$
containing $x$, that is the only relatively open polysimplex of
$\mathcal{B}(G,k)$ containing $x$. We may associate to it the parahoric
$k^{\circ}$-group scheme $\mathcal{G}^{0}_{\mathbf{f}}$. By [Lou21, Théorème
4.7], the star of $\mathbf{f}$, that is the set of facets of
$\mathcal{B}(G,k)$ whose closure contains $\mathbf{f}$, is in one-to-one
correspondence with the set of parabolic subgroups of
$\mathcal{G}^{0}_{\mathbf{f},\widetilde{k}}$. In particular, because $k$ is
local, the residue field $\widetilde{k}$ is finite and the set of parabolic
$\widetilde{k}$-subgroups of $\mathcal{G}^{0}_{\mathbf{f},\widetilde{k}}$ is
finite and the star of $\mathbf{f}$ is therefore finite.

Because the union of the facets in the star of $\mathbf{f}$ is a neighbourhood
of $x$ [Bro89, Lemma VI.3A], it follows that it contains the $y_{n}$ for all
sufficiently large $n$. Because the star of $\mathbf{f}$ is a finite
collection, there must exist a facet $\mathbf{f}^{\prime}$ such that
$\mathbf{f}\subset\overline{\mathbf{f}^{\prime}}$ and a subsequence
$(y_{n_{j}})$ such that $y_{n_{j}}\in\mathbf{f}^{\prime}$ for all
$j\in\mathbb{N}$. Moreover, by [BT72, 7.4.9], the group
$\mathcal{G}^{0}_{\mathbf{f}}(k^{\circ})$ acts transitively on the apartments
containing $\mathbf{f}$. Therefore, there exists
$g\in\mathcal{G}^{0}_{\mathbf{f}}(k^{\circ})$ such that
$\mathbf{f}^{\prime}\subset g\cdot A(G,T)$. Consequently, we may write
$y_{n_{j}}=gx_{n_{j}},$ with $x_{n_{j}}\in A_{\mathbb{Q}}(G,T)$ for all
$j\in\mathbb{N}$. Note that the sequence $(x_{n_{j}})$ also converges to $x$.
It will suffice to prove that
$\lim_{j\to\infty}\vartheta(y_{n_{j}})=\lim_{j\to\infty}\vartheta(x_{n_{j}})$.

Because $\vartheta$ is $G(k)$-equivariant, we have
$\vartheta(y_{n_{j}})=g\vartheta(x_{n_{j}})g^{-1}$ for all $j\in\mathbb{N}$.
By continuity of the action, we have

$\lim_{j\to\infty}\vartheta(y_{n_{j}})=g\left(\lim_{j\to\infty}\vartheta(x_{n_{j}})\right)g^{-1}.$

By the first step, we know that the limit
$\lim_{j\to\infty}\vartheta(x_{n_{j}})$ is independent of the choice of the
sequence of points in $A_{\mathbb{Q}}(G,T)$ converging to $x$. Moreover, by
Proposition 19, we know that there exists a sequence $(u_{n})$ of virtually
special points in $\mathbf{f}\cap\mathcal{B}_{\mathbb{Q}}(G,k)$ that converges
to $x$. Therefore, we have

$\lim_{j\to\infty}\vartheta(y_{n_{j}})=g\left(\lim_{j\to\infty}\vartheta(u_{j})\right)g^{-1}=\lim_{j\to\infty}\vartheta(gu_{j})=\lim_{j\to\infty}\vartheta(u_{j}).$

The exact same argument applies to any subsequence of $(y_{n})$, which
establishes the fact that from any subsequence of $(\vartheta(y_{n}))$, we may
extract a subsequence that converges to the common limit of the sequences
$(\vartheta(x_{n}))$, for $(x_{n})$ an arbitrary sequence of points in
$A_{\mathbb{Q}}(G,T)$ that converges to $x$. Hence, $(\vartheta(y_{n}))$
converges to that same limit.

This proves that, for any sequence of points $(x_{n})$ in
$\mathcal{B}_{\mathbb{Q}}(G,k)$ converging to a point $x$ in $A(G,T)$, the
limit $\lim_{n\to\infty}\vartheta(x_{n})$ exists and depends only on $x$. The
general case easily follows. Indeed, if $y\in\mathcal{B}(G,k)$, then there
exists $g\in G(k)$ such that $gy\in A(G,T)$ and, for any sequence of virtually
special points $(y_{n})$ converging to $y$, the sequence $(\vartheta(y_{n}))$
converges to a limit depending only on $y$ if and only if the same holds for
the sequence $(\vartheta(gy_{n}))=(g\vartheta(y_{n})g^{-1})$. ∎

Note that the density of virtually special points and the continuity of
$\vartheta$ yield the following as a direct corollary of Lemma 24.

######  Proposition 27.

Let $K/k$ be a finite Galois extension. We have the commutative diagram

${\mathcal{B}(G,K)}$${G_{K}^{\operatorname{an}}}$${\mathcal{B}(G,k)}$${G^{\operatorname{an}}}$$\scriptstyle{\vartheta}$$\scriptstyle{\operatorname{pr}_{K/k}^{\operatorname{an}}}$$\scriptstyle{p_{K/k}}$$\scriptstyle{\vartheta}$

.

##  3 Embedding in pseudo-flag varieties

In this section, our goal is to prove that we can recover the family of
polyhedral compactifications of $\mathcal{B}(G,k)$ constructed in [Cha10,
Chap. 1] – the abstract building-theoretic analog of Satake’s
compactifications of symmetric spaces – by mapping $\mathcal{B}(G,k)$ in the
$(G/P_{G}(\mu))^{\operatorname{an}}$ for various $\mu\in X_{*}(G)$.

###  3.1 Dependence on the pseudo-parabolic subgroup

Let $P$ be a pseudo-parabolic subgroup. We check that the composition

$\vartheta_{P}:\mathcal{B}(G,k)\overset{\vartheta}{\longrightarrow}G^{\operatorname{an}}\overset{\lambda_{P}}{\longrightarrow}(G/P)^{\operatorname{an}}$

depends only on the conjugacy class of $P$, in the following sense.

######  Proposition 28.

Let $Q$ be a pseudo-parabolic subgroup of $G$ conjugate to $P$ and $h\in G(k)$
such that $Q=hPh^{-1}$. Then, setting the isomorphism

$\iota_{Q,P}:\begin{array}[]{ccc}G/P&\to&G/Q\\\
gP&\mapsto&gh^{-1}Q\end{array},$

induced by the orbit map $g\mapsto gh^{-1}Q$, we have
$\vartheta_{Q}=\iota_{Q,P}^{\operatorname{an}}\circ\vartheta_{P}$.

###### Proof.

Because both sides of the equation are continuous maps on $\mathcal{B}(G,k)$,
it is sufficient to prove that they agree on the dense subset of virtually
special points. Moreover, Proposition 27 allows us to reduce to proving the
equality for special vertices in the case when $G$ is pseudo-split over $k$.

Now, assume that $G$ is pseudo-split and consider a special vertex
$x\in\mathcal{B}(G,k)$. Denote by $\mathcal{G}$ the parahoric
$k^{\circ}$-group scheme $\mathcal{G}_{x}^{\circ}$ associated to $x$. Let
$\mu$ be a cocharacter of $G$ such that $P=P_{G}(\mu)$. Then, there exists a
maximal split torus $T\subset G$ such that $\mu$ factors through $T$. Now, by
construction, $\mathcal{G}$ contains the split $k^{\circ}$-torus
$\mathcal{T}=\operatorname{Spec}(k^{\circ}[X^{*}(T)])$ and $\mu$ descends to a
cocharacter $\mu^{\circ}$ of $\mathcal{T}$. Let $\mathcal{P}$ be the subgroup
of $\mathcal{G}$ defined by $\mathcal{P}=P_{\mathcal{G}}(\mu^{\circ})$. Then,
the image $\vartheta_{P}(x)\in(G/P)^{\operatorname{an}}$ is the unique
preimage of the generic point of $(\mathcal{G}/\mathcal{P})_{\widetilde{k}}$
under the reduction map

$\rho_{\mathcal{P}}:(\mathcal{G}/\mathcal{P})^{\operatorname{an}}\to(\mathcal{G}/\mathcal{P})_{\widetilde{k}}.$

Indeed, the commutative diagram

${G^{\operatorname{an}}}$${(G/P)^{\operatorname{an}}}$${\mathcal{G}^{\operatorname{an}}}$${(\mathcal{G}/\mathcal{P})^{\operatorname{an}}}$${\mathcal{G}_{\widetilde{k}}}$${(\mathcal{G}/\mathcal{P})_{\widetilde{k}}}$$\scriptstyle{\lambda_{P}^{\operatorname{an}}}$$\scriptstyle{\subset}$$\scriptstyle{\rho}$$\scriptstyle{\lambda_{\mathcal{P}}^{\operatorname{an}}}$$\scriptstyle{\rho_{\mathcal{P}}}$$\scriptstyle{\subset}$$\scriptstyle{(\lambda_{\widetilde{P}})_{\widetilde{k}}}$

and the surjectivity of the map
$(\lambda_{\mathcal{P}})_{\widetilde{k}}:\mathcal{G}_{\widetilde{k}}\to(\mathcal{G}/\mathcal{P})_{\widetilde{k}}$
imply that $\vartheta(x)$, which by construction is the unique preimage of the
generic point of $\mathcal{G}_{\widetilde{k}}$ under the reduction map $\rho$,
maps to a preimage of the generic point of
$(\mathcal{G}/\mathcal{P})_{\widetilde{k}}$ in
$(\mathcal{G}/\mathcal{P})^{\operatorname{an}}$. To establish the uniqueness
of the preimage, recall that, for any non-empty affine open subset
$\mathcal{U}$ of $\mathcal{G}/\mathcal{P}$, the space
$\mathcal{U}^{\operatorname{an}}$ contains a unique preimage of the generic
point of $(\mathcal{G}/\mathcal{P})_{\widetilde{k}}$ by [Ber90, Proposition
2.4.4]. The fact that, by construction,
$(\mathcal{G}/\mathcal{P})^{\operatorname{an}}$ is covered by subsets of the
form $\mathcal{U}^{\operatorname{an}}$ and that any two of these meet along a
subset of the same type allows one to conclude.

Next, observe that due to the Iwasawa decomposition [Sol18, Theorem 3.12], we
may write $h=kp$, with $k\in\mathcal{G}(k^{\circ})$ and $p\in P(k)$. Hence, we
have $Q=kPk^{-1}=P_{G}(k\cdot\mu)$. Set
$\mathcal{Q}=P_{\mathcal{G}}(k\cdot\mu^{\circ})$. Then, we have
$\mathcal{Q}=k\mathcal{P}k^{-1}$ and the map $\iota_{Q,P}$ descends to a map

$\iota_{\mathcal{Q},\mathcal{P}}:\begin{array}[]{ccc}\mathcal{G}/\mathcal{P}&\to&\mathcal{G}/\mathcal{Q}\\\
g\mathcal{P}&\mapsto&gk^{-1}\mathcal{Q}\end{array}.$

Therefore, the diagram

${(G/P)^{\operatorname{an}}}$${(G/Q)^{\operatorname{an}}}$${(\mathcal{G}/\mathcal{P})^{\operatorname{an}}}$${(\mathcal{G}/\mathcal{Q})^{\operatorname{an}}}$${(\mathcal{G}/\mathcal{P})_{\widetilde{k}}}$${(\mathcal{G}/\mathcal{Q})_{\widetilde{k}}\par}$$\scriptstyle{\iota_{Q,P}^{\operatorname{an}}}$$\scriptstyle{\subset}$$\scriptstyle{\iota_{\mathcal{Q},\mathcal{P}}^{\operatorname{an}}}$$\scriptstyle{\rho_{\mathcal{P}}}$$\scriptstyle{\rho_{\mathcal{Q}}}$$\scriptstyle{\subset}$$\scriptstyle{(\iota_{\mathcal{Q},\mathcal{P}})_{\widetilde{k}}}$

commutes. Because $(\iota_{\mathcal{Q},\mathcal{P}})_{\widetilde{k}}$ maps
generic points to generic points, the map $\iota_{Q,P}^{\operatorname{an}}$
maps the unique preimage in $(\mathcal{G}/\mathcal{P})^{\operatorname{an}}$ of
the generic point of $(\mathcal{G}/\mathcal{P})_{\widetilde{k}}$ to the unique
preimage in $(\mathcal{G}/\mathcal{Q})^{\operatorname{an}}$ of the generic
point of $(\mathcal{G}/\mathcal{Q})_{\widetilde{k}}$. In other words,

$\vartheta_{Q}(x)=\iota_{Q,P}^{\operatorname{an}}(\vartheta_{P}(x)).$

∎

######  Remark 29.

The map $\iota_{Q,P}$ does not depend on the choice of $h$. Indeed, if
$h^{\prime}\in G(k)$ also satisfies $Q=h^{\prime}Ph^{\prime-1}$, then, by
[CGP15, Proposition 3.5.7], we have $h^{\prime}h^{-1}\in N_{G}(Q)(k)=Q(k)$,
and thus $h^{-1}Q=h^{\prime-1}Q$.

######  Corollary 30.

The map $\vartheta_{P}$ is $G(k)$-equivariant.

###### Proof.

Let $g\in G(k)$. We have the following commutative diagram

${G}$${G}$${G/P}$${G}$${G/g^{-1}Pg}$$\scriptstyle{\cdot
g^{-1}}$$\scriptstyle{\lambda_{P}}$$\scriptstyle{\lambda_{g^{-1}Pg}}$$\scriptstyle{\iota_{P,g^{-1}Pg}}$

,

in which all maps are $G(k)$-equivariant for the action by left-translation.
Now, recall from Proposition 23 that, for each $x\in\mathcal{B}(G,k)$, we have

$\vartheta(gx)=g\vartheta(x)g^{-1}.$

Consequently, we have

$\vartheta_{P}(gx)=\lambda^{\operatorname{an}}_{P}(g\vartheta(x)g^{-1})=g\lambda^{\operatorname{an}}_{P}(\vartheta(x)g^{-1}).$

By the above diagram, we have

$\lambda_{P}^{\operatorname{an}}(\vartheta(x)g^{-1})=\iota_{P,gPg^{-1}}^{\operatorname{an}}(\vartheta_{g^{-1}Pg}(x))=\vartheta_{P}(x).$

Finally, we find

$\vartheta_{P}(gx)=g\vartheta_{P}(x),$

which completes the proof. ∎

###  3.2 Restricting to an apartment: An explicit formula

The following sections will establish that, if $P$ is a suitable pseudo-
parabolic subgroup of $G$, then $\vartheta_{P}$ is an embedding with
relatively compact image. The argument hinges on an explicit formula for the
restriction of $\vartheta_{P}$ to an apartment $A(G,T)$ of the building such
that $T\subset P$ in the case where $G$ is pseudo-split. The argument is
essentially identical – yet not reducible – to that of Proposition 25.

Assume that $G$ is pseudo-split over $k$ and let $A$ be an apartment in
$\mathcal{B}(G,k)$ associated to a maximal split torus $T$. We denote by
$\Phi=\Phi(G,T)$ the root system of $G$ corresponding to $T$. Let $o$ be a
special vertex in $A$ and denote by $\mathcal{G}$ the parahoric
$k^{\circ}$-model of $G$ associated to $o$. For each
$\alpha\in\Phi_{\operatorname{nd}}$, we denote by
$\mathcal{U}_{\alpha}\subset\mathcal{G}$ the $k^{\circ}$-model of the root
group $U_{\alpha}$ associated to $o$.

Let $P=P_{G}(\mu)$ be a pseudo-parabolic subgroup containing $T$ and
$U^{-}=U_{G}(-\mu)$. Arguing as in the proof of Proposition 28, we may find a
$k^{\circ}$-cocharacter $\mu^{\circ}:\mathbb{G}_{m,k^{\circ}}\to\mathcal{G}$
such that $\mu^{\circ}\otimes_{k^{\circ}}k=\mu$. Then, let
$\mathcal{P}=P_{\mathcal{G}}(\mu^{\circ})$ and
$\mathcal{U}^{-}=U_{\mathcal{G}}(-\mu^{\circ})$. Let

$\Psi=\Phi(U^{-},T)=\\{\alpha\in\Phi,\langle\alpha,\mu\rangle<0\\}.$

Then, we claim that the multiplication map

$m:\prod_{\alpha\in\Psi_{\operatorname{nd}}}\mathcal{U}_{\alpha}\to\mathcal{U}^{-}$

is a $\mathcal{T}$-equivariant isomorphism of $k^{\circ}$-schemes.

For each non-divisible root $\alpha\in\,\Psi_{\operatorname{nd}}$, we have a
$\mathcal{T}$-equivariant isomorphism of $k^{\circ}$-schemes:

$\mathcal{U}_{\alpha}\simeq\mathcal{U}_{2{\alpha}}\times\mathcal{U}_{\alpha}/\mathcal{U}_{2{\alpha}}.$

Moreover, each $\mathcal{U}_{2{\alpha}}$ (resp.
$\mathcal{U}_{\alpha}/\mathcal{U}_{2{\alpha}}$) is an affine space over
$k^{\circ}$ on which $\mathcal{T}$ acts linearly via the character $2{\alpha}$
(resp. ${\alpha}$). For each $\alpha\in\Phi$, set
$N_{\alpha}=\dim_{k}(U_{\alpha}/U_{2\alpha})$ and fix
$\mathcal{T}$-equivariant isomorphisms

$u_{\alpha}:\mathbb{A}^{N_{\alpha}}_{k^{\circ}}\overset{\sim}{\to}\mathcal{U}_{\alpha}/\mathcal{U}_{2{\alpha}}\text{
and
}u_{2{\alpha}}:\mathbb{A}^{N_{2{\alpha}}}_{k^{\circ}}\overset{\sim}{\to}\mathcal{U}_{2{\alpha}}$

for each non-divisible root ${\alpha}$. These yield a
$\mathcal{T}$-equivariant isomorphism of $k^{\circ}$-schemes

$\prod_{\alpha\in\Psi}\mathbb{A}^{N_{\alpha}}_{k^{\circ}}\overset{\sim}{\to}\mathcal{U}^{-}.$

In the following, we identify $k[U^{-}]$ to the polynomial algebra
$k[(\xi_{\alpha,i})_{(\alpha,i)\in\,\Psi\times\mathbb{N},1\leq i\leq
N_{\alpha}}]$ using this isomorphism.

######  Remark 31.

We call a system of coordinates
$(\xi_{\alpha,i})_{(\alpha,i)\in\mathbf{\Psi}}$ on $U^{-}$ so constructed
adapted to the point $o$.

Denote by $\Omega(\mathcal{T},\mathcal{P})$ the image of the open immersion
defined by the multiplication map [CGP15, Proposition 2.1.8 (3)]

$m:\mathcal{U}^{-}\times\mathcal{P}\to\mathcal{G}.$

and let $\Omega(T,P)=\Omega(\mathcal{T},\mathcal{P})\otimes_{k^{\circ}}k$.

######  Proposition 32.

We keep all the above notation. Let $x$ be a point in $A$. We let
$\Psi=\Phi(U_{G}(-\mu),T)$ and

$\mathbf{\Psi}=\\{(\alpha,i)\in\,\Psi\times\mathbb{N},1\leq i\leq
N_{\alpha}\\}.$

  1. 1.

The image $\vartheta_{P}(x)$ lies in
$(U^{-})^{\operatorname{an}}\simeq(\Omega(T,P)/P)^{\operatorname{an}}\subset(G/P)^{\operatorname{an}}$.

  2. 2.

For each $f=\sum_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}f_{\nu}\xi^{\nu}\in
k[U_{G}(-\mu)]=k\left[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Psi}}}\right]$,
we have

$|f(\vartheta_{P}(x))|=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}|f_{\nu}|\prod_{(\alpha,i)\in{\mathbf{\Psi}}}e^{\nu(\alpha,i)\langle\alpha,x\rangle}.$

###### Proof.

The arguments are essentially the same as those of Proposition 25.

The statement will follow from the fact that $\vartheta(x)$ lies in
$\Omega(T,P)^{\operatorname{an}}$ for all $x\in A$. To establish this, we
proceed in three steps, first proving it for the origin $o$, then extending to
any virtually special point, and finally concluding by density.

  1. 1.

First of all, because $\Omega(\mathcal{T},\mathcal{P})$ is an open subset of
$\mathcal{G}$ that meets the special fibre of $\mathcal{G}$, the special fibre
$\Omega(\mathcal{T},\mathcal{P})_{\widetilde{k}}$ contains the generic point
of $\mathcal{G}_{\widetilde{k}}$. Therefore, the analytification
$\Omega(\mathcal{T},\mathcal{P})^{\operatorname{an}}$ contains $\vartheta(o)$.
Now, because the diagram

${\Omega(\mathcal{T},\mathcal{P})^{\operatorname{an}}}$${(\Omega(\mathcal{T},\mathcal{P})/\mathcal{P})^{\operatorname{an}}}$${\Omega(\mathcal{T},\mathcal{P})_{\widetilde{k}}}$${(\Omega(\mathcal{T},\mathcal{P})/\mathcal{P})_{\widetilde{k}}}$$\scriptstyle{\rho}$$\scriptstyle{\lambda_{\mathcal{P}}^{\operatorname{an}}}$$\scriptstyle{\rho}$$\scriptstyle{\lambda_{\mathcal{P}_{\widetilde{k}}}}$

commutes and the bottom arrow is surjective (and therefore dominant), the
image $\vartheta_{P}(o)$ is the unique Shilov boundary of
$(\mathcal{G}/\mathcal{P})^{\operatorname{an}}$. Identifying
$(\Omega(\mathcal{T},\mathcal{P})/\mathcal{P})^{\operatorname{an}}$ to
$(\mathcal{U}^{-})^{\operatorname{an}}$ via
$\lambda_{\mathcal{P}}^{\operatorname{an}}$, the point $\vartheta_{P}(o)$ may
be understood as the only seminorm such that, for each $f\in
k[U^{-}]=k[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Psi}}}]$, we have

$|f(\vartheta_{P}(o))|\leq 1\iff f\in
k^{\circ}[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Psi}}}]$

and

$|f(\vartheta_{P}(o))|<1\iff f\text{ maps to zero in
}\widetilde{k}[(\xi_{\alpha,i})_{(\alpha,i)\in\mathbf{\Psi}}].$

In other words, if
$f=\sum_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}f_{\nu}\xi^{\nu}\in
k[(\xi_{\alpha,i})]$, we have

$|f(\vartheta_{P}(o))|=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}|f_{\nu}|.$

  2. 2.

Next, consider a virtually special point $x\in A_{\mathbb{Q}}(T,k)$. By
Corollary 21, we have at our disposal a finite Galois extension $K/k$ and a
$t\in T(K)$ such that $x_{K}=to_{K}$. Then, we have
$\vartheta(x_{K})=t\vartheta(o_{K})t^{-1}$ and, because $\Omega(T,P)$ is
normalised by $T$, we have
$\vartheta(x_{K})\in\Omega(T,P)_{K}^{\operatorname{an}}$ and, consequently,
$\vartheta(x)=\operatorname{pr}_{K/k}^{\operatorname{an}}(\vartheta(x_{K}))\in\Omega(T,P)^{\operatorname{an}}$,
and finally $\vartheta_{P}(x)\in(\Omega(T,P)/P)^{\operatorname{an}}$.
Moreover, for $f\in k[U^{-}]$, we have

$|f|(\vartheta_{P}(x))=|f|(\vartheta_{P}(x_{K}))=|\tau^{*}(f)|(\vartheta_{P}(o_{K})),$

where $\tau$ is the only map that makes the following diagram commute

${\prod_{{\alpha}\in\Psi}\mathbb{A}^{N_{\alpha}}_{K}}$${\prod_{{\alpha}\in\Psi}\mathbb{A}^{N_{\alpha}}_{K}}$${(U^{-})_{K}}$${(U^{-})_{K}}$$\scriptstyle{\tau}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{int}(t_{K})}$

.

In other words, $\tau^{*}$ is the only $k$-automorphism of
$k[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Psi}}}]$ that maps
$\xi_{\alpha,i}$ to $\alpha(t)\xi_{\alpha,i}$ for each $\alpha\in\Psi$ and
$i\in[\\![1,N_{\alpha}]\\!]$. We deduce that if
$f=\sum_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}f_{\nu}\xi^{\nu}\in
k[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Psi}}}]$, we have

$\tau^{*}(f)=\sum_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}\left(f_{\nu}\prod_{(\alpha,i)\in{\mathbf{\Psi}}}\alpha(t)^{\nu(\alpha,i)}\right)\xi^{\nu}$

and thus

$|f|(\vartheta_{P}(x_{K}))=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}|f_{\nu}|\prod_{(\alpha,i)\in{\mathbf{\Psi}}}|\alpha(t)|^{\nu(\alpha,i)}=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}|f_{\nu}|\prod_{(\alpha,i)\in{\mathbf{\Psi}}}e^{\nu(\alpha,i)\langle\alpha,x\rangle}$

using point 1.

  3. 3.

If $x$ is an arbitrary point in $A$, then there exists a sequence $(x_{n})$ of
points in $A_{\mathbb{Q}}(T,k)$ that converges to $x$. Then, it is clear from
the continuity in $x$ of the formula in point 2. that the sequence
$(\vartheta_{P}(x_{n}))$ converges to a seminorm on $k[U^{-}]$, that is that
$\vartheta_{P}(x)\in\Omega(T,P)^{\operatorname{an}}$ and that the formula for
$\vartheta_{P}(x)$ is as announced.

∎

###  3.3 Injectivity of the map

This section establishes the injectivity of the map $\vartheta_{P}$ under the
condition that $P$ is of non-degenerate type, a property we define below.

######  Definition 33.

Let $G$ be a quasi-reductive group over a field $k$ and $P$ be a pseudo-
parabolic subgroup of $G$. We say that $P$ is of non-degenerate type if it
does not contain any minimal nontrivial $k$-isotropic perfect smooth connected
normal subgroup of $G$.

The property of being of non-degenerate type admits the following Lie-
theoretic characterisations.

######  Proposition 34.

Let $G$ be a quasi-reductive group over a field $k$ and $P$ be a pseudo-
parabolic subgroup of $G$. Fix a $k$-split maximal torus $S\subset P$ and
$P_{0}$ a minimal pseudo-parabolic subgroup such that $S\subset P_{0}\subset
P$. Denote by $P=L_{P}\ltimes U$ the Levi decomposition of $P$ corresponding
to $S$. Let $\Phi=\Phi(G,S)$ be the root system associated to $S$ and
$\Delta\subset\Phi$ be the system of simple roots corresponding to $P_{0}$.
The following are equivalent.

  1. 1.

$P$ is of non-degenerate type.

  2. 2.

The type $I=\Phi(L_{P},S)\cap\Delta$ of $P$ contains no connected components
of the Dynkin diagram $\Delta$.

  3. 3.

The subset $-\Psi=\Phi(U,S)$ spans
$V^{*}(G,S)=\operatorname{Span}_{\mathbb{R}}(\Phi)$ as a vector space.

###### Proof.

It follows from [CGP15, Proposition C.2.32] that the minimal nontrivial
$k$-isotropic perfect smooth connected normal subgroup of $G$ are the
$N_{\Phi_{i}}=\langle U_{\alpha},\alpha\in\Phi_{i}\rangle$ for $\Phi_{i}$
ranging in the set of irreducible components of the root system $\Phi$.
Because $P$ contains the root group $U_{\alpha}$ if and only if
$\alpha\in\Phi(P,S)$, we have the equivalences

$N_{\Phi_{i}}\subset
P\iff\Phi_{i}\subset\Phi(P,S)\iff\Phi_{i}\subset\Phi(L_{P},S)=\Phi(P,S)\cap-\Phi(P,S).$

The equivalence $1\iff 2$ immediately follows as the connected components of
the Dynkin diagram $\Delta$ are precisely the $\Delta\cap\Phi_{i}$ for
$\Phi_{i}$ ranging among the irreducible components of $\Phi$.

We now prove the equivalence $2\iff 3$. Assume that $I$ contains a connected
component $\Delta_{i}=\Delta\cap\Phi_{i}$ of the Dynkin diagram $\Delta$.
Then, we have $\Phi_{i}\subset\Phi(L_{P},S)$ and therefore
$\Phi(U,S)=\Phi(P,S)\setminus\Phi(L_{P},S)\subset\bigsqcup_{j\neq i}\Phi_{j}$
and therefore $\operatorname{Span}_{\mathbb{R}}(\Phi(U,S))$ is contained in
$\operatorname{Span}_{\mathbb{R}}\left(\bigsqcup_{j\neq i}\Phi_{j}\right)$,
which is a proper subspace of $\operatorname{Span}_{\mathbb{R}}(\Phi)$ because
the $\Phi_{i}$ are pairwise orthogonal. Conversely, if $I$ contains no
connected component of $\Delta$, let $\Phi_{i}$ be an irreducible component of
$\Phi$. By assumption, there exists an $\alpha\in\Phi_{i}$ such that
$\alpha\in\Delta\setminus I$. Because $P_{0}\subset P$, that is
$\Delta\subset\Phi(P,S)$, we must have $\alpha\in\Phi(U,S)$. Then, for each
$\beta\in I\cap\Phi_{i}$, the root
$s_{\alpha}(\beta)=\beta-\underbrace{\langle\alpha,\beta\rangle}_{<0}\alpha$
is a sum of elements of $\Delta$ and is not contained in the span of $I$,
hence it lies in $\Phi(U,S)$. Therefore, the subspace
$\operatorname{Span}_{\mathbb{R}}(\Phi(U,S))$ contains $I\cap\Phi_{i}$.
Because it contains $(\Delta\setminus I)\cap\Phi_{i}$, it actually contains
$\Delta\cap\Phi_{i}$ and therefore contains $\Phi_{i}$.

∎

######  Proposition 35.

The map $\vartheta_{P}$ is injective if and only if $P$ is of non-degenerate
type

###### Proof.

Observe that, because $\vartheta_{P}$ is $G(k)$-equivariant and because any
two points lie in an apartment, the map $\vartheta_{P}$ is injective if and
only if there exists an apartment $A$ such that $\vartheta_{P|A}$ is
injective.

Let $S$ be a maximal $k$-split torus contained in $P$. Let $o$ be a special
point of $A(G,k)$ and $K$ be a finite Galois extension such that $G_{K}$ is
pseudo-split and $o_{K}$ is special. Let $T$ be a maximal $K$-split torus that
contains $S_{K}$ such that $p_{K/k}(A(G,S))\subset A(G_{K},T)$. By
Propositions 27 and 32, we have the commutative diagram

${A(G_{K},T)}$${(U^{-})_{K}^{\operatorname{an}}}$${A(G,S)}$${(U^{-})^{\operatorname{an}}}$$\scriptstyle{\vartheta_{P_{K}}}$$\scriptstyle{\operatorname{pr}_{K/k}^{\operatorname{an}}}$$\scriptstyle{p_{K/k}}$$\scriptstyle{\vartheta_{P}}$

,

where $U^{-}$ is the $k$-split unipotent radical of the opposite pseudo-
parabolic subgroup to $P$ with respect to $S$. With the notations of Section
3.2 associated to the point $o_{K}$, we have, for each
$f=\sum_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}f_{\nu}\xi^{\nu}\in
K[U^{-}]=K[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Psi}}}]$ and $x\in
A(G_{K},T)$:

$|f(\vartheta_{P_{K}}(x))|=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}|f_{\nu}|\prod_{(\alpha,i)\in{\mathbf{\Psi}}}e^{\nu(\alpha,i)\langle\alpha,x-o_{K}\rangle}.$
(3)

Note also that, for each character $\chi\in X^{*}(T)$ and each cocharacter
$\lambda\in X_{*}(S)$, we have the following compatibility property between
duality pairings

$\langle\chi,\lambda_{K}\rangle=\langle\chi_{|S},\lambda\rangle.$

Now, let $a\in\Phi(U^{-},S)$ be a relative weight of the Lie algebra of
$U^{-}$, and $\phi\in\operatorname{Lie}(U^{-})^{*}\subset k[U^{-}]$ be a
nonzero weight vector of weight $-a$. Then, we may write

$\phi=\sum_{\begin{subarray}{c}(\alpha,i)\in{\mathbf{\Psi}}\\\
\alpha_{|S}=a\end{subarray}}\phi_{\alpha,i}\xi_{\alpha,i},$

with $\phi_{\alpha,i}\in K$ and not all $\phi_{\alpha,i}$ equal to zero.
Hence, we have, for all $x\in A(G,S)$:

$|\phi(\vartheta_{P}(x))|=|\phi_{K}(\vartheta_{P_{K}}(x_{K}))|=\left(\max_{\begin{subarray}{c}(\alpha,i)\in{\mathbf{\Psi}}\\\
\alpha_{|S}=a\end{subarray}}|\phi_{\alpha,i}|\right)e^{\langle a,x-o\rangle}.$

Therefore, if $x,y\in A(G,S)$ satisfy $\vartheta_{P}(x)=\vartheta_{P}(y)$,
then $\langle a,x-o\rangle=\langle a,y-o\rangle$ for each $a\in\Phi(U^{-},S)$.
Conversely, if $x$ and $y$ in $A(G,S)$ satisfy $\langle a,x-y\rangle=0$ for
each $a\in\Phi(U^{-},S)$, then the observation following Equation (3) that
$\vartheta_{P_{K}}(x)=\vartheta_{P_{K}}(y)$, and therefore
$\vartheta_{P}(x)=\vartheta_{P}(y)$. In other words, for $x$ and $y$ in
$A(G,S)$, we have $\vartheta_{P}(x)=\vartheta_{P}(y)$ if and only if $x-y$
lies in the annihilator

$\Phi(U^{-},S)^{0}=\\{v\in V(G,S),\forall a\in\Phi(U^{-},S),\langle
a,v\rangle=0\\}.$

Hence, the map $\vartheta_{P}$ is injective if and only if the subset
$\Phi(U^{-},S)$ spans the dual space $V^{*}(G,S)$, that is if and only if $P$
is of non-degenerate type by Proposition 34. ∎

###  3.4 Relative compactness

In this section, we establish that, if $P$ is a pseudo-parabolic subgroup of
$G$, then the image of the map $\vartheta_{P}$ is relatively compact. As
usual, it will suffice to consider the case when $G$ is pseudo-split. The main
insight is that the Cartan decomposition [Sol18, Theorem 3.13 (d)] allows one
to restrict to proving that sequences taking values in a suitably chosen cone
admit a convergent subsequence, which will follow from the previous section.

######  Proposition 36.

The map $\vartheta_{P}$ has relatively compact image. In particular, if $P$ is
of non-degenerate type, then $\vartheta_{P}$ factors through a
$G(k)$-equivariant compactification of $\mathcal{B}(G,k)$.

###### Proof.

First of all, thanks to Proposition 27, we have the following commutative
diagram of continuous maps

${\mathcal{B}(G,K)}$${(G/P)_{K}^{\operatorname{an}}}$${\mathcal{B}(G,k)}$${(G/P)^{\operatorname{an}}}$$\scriptstyle{\vartheta_{P_{K}}}$$\scriptstyle{\operatorname{pr}_{K/k}^{\operatorname{an}}}$$\scriptstyle{p_{K/k}}$$\scriptstyle{\vartheta_{P}}$

for any finite Galois extension $K/k$. It follows that, if $\vartheta_{P_{K}}$
has relatively compact image, then so does $\vartheta_{P}$.

It is therefore sufficient to prove our claim in the case where $G$ is pseudo-
split over $k$, which we will assume in what follows. Let $T$ be a maximal
split torus contained in $P$ and $\mu\in X_{*}(T)$ be a cocharacter such that
$P=P_{G}(\mu)$. Set $U^{-}=U_{G}(-\mu)$. Fix a special vertex $o$ in $A(G,T)$,
set $\Psi=\Phi(U^{-},T)$, and consider the cone

$C(P)=\\{x\in A(G,T)\ |\ \forall\alpha\in\Psi,\langle\alpha,x-o\rangle\leq
0\\}.$

Observe that we have

$A(G,T)=W(G,T)(k)\cdot C(P),$

where the action of $W(G,T)(k)=\dfrac{N_{G}(T)(k)}{Z_{G}(T)(k)}$ on $A(G,T)$
is given by the identification of $A(G,T)$ with $V(G,T)$ setting $o$ as
origin. Moreover, because $o$ is special, the projection

$N_{G}(T)(k)_{o}\to W(G,T)(k)$

is onto, hence we have

$A(G,T)=N_{G}(T)(k)_{o}\cdot C(P).$

Moreover, by [Sol18, Theorem 3.11], we have

$\mathcal{B}(G,k)=G(k)_{o}\cdot A(G,T),$

and therefore

$\mathcal{B}(G,k)=G(k)_{o}\cdot C(P).$

Because $\vartheta_{P}$ is $G(k)$-equivariant and continuous and because
$G(k)_{o}$ acts through its compact quotient $G(k)_{o}/Z(G)(k)$, proving that
$\vartheta_{P}$ has relatively compact image reduces to proving that its
restriction to $C(P)$ has relatively compact image. Now, let $(x_{n})$ be a
sequence in $C(P)$. Identify $A(G,T)$ to $V(G,T)$ using $o$ as origin and
assume the notations of Section 3.2. Then, by Proposition 32, we have
$\vartheta_{P}(x_{n})\in(U^{-})^{\operatorname{an}}$ for each $n\in\mathbb{N}$
and, for each $f=\sum_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}f_{\nu}\xi^{\nu}\in
k[U^{-}]=k[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Psi}}}]$, we have

$|f|(\vartheta_{P}(x_{n}))=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}|f_{\nu}|\prod_{(\alpha,i)\in{\mathbf{\Psi}}}e^{\nu(\alpha,i)\langle\alpha,x_{n}\rangle}.$

For each $\alpha\in\Psi$, the sequence $(e^{\langle\alpha,x_{n}\rangle})$ is
bounded. Consequently, there exists a subsequence $(x_{n_{j}})$ such that
$(e^{\langle\alpha,x_{n_{j}}\rangle})$ converges for each $\alpha\in\Psi$.
Then, for each $f\in k[U^{-}]$, the sequence $(|f|(\vartheta_{P}(x_{n_{j}})))$
converges and thus the subsequence $(\vartheta_{P}(x_{n_{j}}))$ converges in
$(U^{-})^{\operatorname{an}}$. ∎

The proof actually yields the following more precise statement.

######  Proposition 37.

Let $S$ be a maximal split torus in $P$ and $o$ a special vertex in $A(G,S)$.
Consider the cone

$C(P)=\\{x\in
A(G,S),\forall\alpha\in\Phi(R_{u,k}(P),S),\langle\alpha,x-o\rangle\geq 0\\}.$

Then, the image $\vartheta_{P}(C(P))$ is relatively compact in the open
subspace
$(\Omega(S,P)/P)^{\operatorname{an}}\subset(G/P)^{\operatorname{an}}$.

###  3.5 Openness of the corestriction

In this section, we check that the corestriction

$\vartheta_{P}:\mathcal{B}(G,k)\to\overline{\vartheta_{P}(\mathcal{B}(G,k))}$

is an open map. In particular, if $P$ is of non-degenerate type, we prove that
$\vartheta_{P}$ is an embedding.

######  Proposition 38.

The corestricted map
$\vartheta_{P}:\mathcal{B}(G,k)\to\overline{\vartheta_{P}(\mathcal{B}(G,k))}$
is open. In particular, whenever $P$ is of non-degenerate type, the map
$\vartheta_{P}$ is an embedding.

The proof goes in three steps. First of all, we prove that the restriction of
$\vartheta_{P}$ to an apartment $A(G,S)$ (and thus any apartment) induces an
open map to the closure of its image. Secondly, we prove that the action map

$G(k)\times\overline{\vartheta_{P}(A(G,S))}\longrightarrow\overline{\vartheta_{P}(\mathcal{B}(G,k))}$

is a topological quotient map. Lastly, we deduce that the map
$\vartheta_{P}:\mathcal{B}(G,k)\to\overline{\vartheta_{P}(\mathcal{B}(G,k))}$
is open.

######  Lemma 39.

Let $S$ be a maximal split torus of $G$. The image $\vartheta_{P}(A(G,S))$ is
open in its closure $\overline{\vartheta_{P}(A(G,S))}$.

###### Proof.

Without loss of generality, we may assume $S$ to be contained in $P$.

Let $o$ be a special vertex in $A(G,S)$. Let $T$ be a maximal torus of $G$ and
$K/k$ a finite Galois extension such that

  1. 1.

The torus $T_{K}$ is split.

  2. 2.

The apartment $A(G_{K},T_{K})$ contains $p_{K/k}(A(G,S))$.

  3. 3.

The point $o_{K}\in A(G_{K},T_{K})$ is a special vertex.

Then, as in the proof of Proposition 34, we have the diagram

${A(G_{K},T)}$${(U^{-})_{K}^{\operatorname{an}}}$${A(G,S)}$${(U^{-})^{\operatorname{an}}}$$\scriptstyle{\vartheta_{P_{K}}}$$\scriptstyle{\operatorname{pr}_{K/k}^{\operatorname{an}}}$$\scriptstyle{p_{K/k}}$$\scriptstyle{\vartheta_{P}}$

,

where $U^{-}$ is the $k$-split unipotent radical of the pseudo-parabolic
subgroup of $G$ opposite to $P$ with respect to $S$. We may rewrite the
explicit formula in Proposition 32 as the statement that, for each
$x\in\vartheta_{P}(A(G,S))$ and $f\in
K[U^{-}]=K[(\xi_{\alpha,i})_{(\alpha,i)\in{\mathbf{\Psi}}}]$, we have

$|f(x)|=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}}}|f_{\nu}|\prod_{(\alpha,i)\in{\mathbf{\Psi}}}|\xi_{\alpha,i}(x)|^{\nu(\alpha,i)},$

where, for each $\alpha\in\Phi(U^{-},S)$, the family $(\xi_{\alpha,i})$ is a
choice of coordinates on $U_{\alpha}$ given by $o_{K}$ through the procedure
described in the introduction to Section 3.2. It follows from the proof of
Proposition 36 that this formula extends to all points
$x\in\overline{\vartheta_{P}(A(G,S))}\cap(U^{-})^{\operatorname{an}}$ and that
a point $x\in\overline{\vartheta_{P}(A(G,S))}\cap(U^{-})^{\operatorname{an}}$
lies in $\vartheta_{P}(A(G,S))$ if and only if $|\xi_{\alpha,i}(x)|>0$ for all
$(\alpha,i)\in{\mathbf{\Psi}}$.

It follows that the image $\vartheta_{P}(A(G,S))$ is open in
$\overline{\vartheta_{P}(A(G,S))}\cap(\Omega(S,P)/P)^{\operatorname{an}}$.
Note that the Bruhat decomposition [CGP15, Theorem C.2.8] implies that
$G/P=\bigcup_{w\in W}\dot{w}\Omega(S,P)/P$, where, for each $w\in W$, we
choose an arbitrary lift $\dot{w}$ of $w$ in $N_{G}(S)(k)$, see [Jan87,
II.1.9.7]. By $N_{G}(S)(k)$-equivariance of $\vartheta_{P}$, we deduce that
$\vartheta_{P}(A(G,S))$ is open in
$\overline{\vartheta_{P}(A(G,S))}\cap(\dot{w}\Omega(S,P)/P)^{\operatorname{an}}$
for each $w\in W$, and therefore open in $\overline{\vartheta_{P}(A(G,S))}$.

∎

We now conclude our first step.

######  Lemma 40.

Let $S$ be a maximal split torus of $G$. The map

$\vartheta_{P}:A(G,S)\to\overline{\vartheta_{P}(A(G,S))}$

is open.

###### Proof.

As in the previous lemma, we may assume that $S$ is contained in $P$. Let
$U^{-}$ be the $k$-split unipotent radical of the pseudo-parabolic subgroup
opposite to $P$ with respect to $S$. Let $o$ be a special vertex in $A(G,S)$.
Recall from Proposition 35 that $\vartheta_{P}$ factors through the quotient
map

$A(G,S)\to A(G,S)/\langle\Phi(U^{-},S)\rangle^{0},$

a surjective affine map, hence open. We may therefore assume that $P$ is of
non-degenerate type, ie. that we have the implication

$(\forall a\in\Phi(U^{-},S),\langle a,x\rangle=0)\implies x=0$

for each $x\in V(G,S)$. Under this assumption, the open half-spaces

$H_{a,r}^{+}=\\{x\in A(G,S),\langle a,x-o\rangle>r\\}\text{ and
}H_{a,r}^{-}=\\{x\in A(G,S),\langle a,x-o\rangle<r\\}$

for $a\in\Phi(U^{-},S)$ form a subbase of open sets for the topology of
$A(G,S)$. Because $\vartheta_{P}$ is injective, we need only check that these
half-spaces have an open image. Consider a system of coordinates
$(\xi_{\alpha,i})_{(\alpha,i)\in\mathbf{\Psi}}$ on $U^{-}_{K}$ as in Section
3.2. Let $\phi=\sum_{\begin{subarray}{c}(\alpha,i)\in{\mathbf{\Psi}}\\\
\alpha_{|S}=a\end{subarray}}\phi_{\alpha,i}\xi_{\alpha,i}\in k[U^{-}]$ be a
nonzero weight vector of weight $-a$ in $\operatorname{Lie}(U^{-})^{*}$. Then,
it follows from Proposition 32 that, for each $r\in\mathbb{R}$, we have

$\vartheta_{P}(H_{a,r}^{+})=\vartheta_{P}(A(G,S))\cap\\{x\in(U^{-})^{\operatorname{an}},|\phi(x)|>Ce^{r}\\}$

and

$\vartheta_{P}(H_{a,r}^{-})=\vartheta_{P}(A(G,S))\cap\\{x\in(U^{-})^{\operatorname{an}},|\phi(x)|<Ce^{r}\\},$

where $C=\max_{\begin{subarray}{c}(\alpha,i)\in{\mathbf{\Psi}}\\\
\alpha_{|S}=a\end{subarray}}|\phi_{\alpha,i}|$. Consequently, for each
$r\in\mathbb{R}$, the images $\vartheta_{P}(H_{a,r}^{+})$ and
$\vartheta_{P}(H_{a,r}^{-})$ are open in $\vartheta_{P}(A(G,S))$, and thus
open in $\overline{\vartheta_{P}(A(G,S))}$ by Lemma 39. ∎

Now, let $o$ be a special vertex in $A(G,S)$. By [Sol18, Theorem 3.12], we
have a coarse Cartan decomposition

$\mathcal{B}(G,k)=G(k)_{o}\cdot A(G,S),$

and therefore

$\overline{\vartheta_{P}(\mathcal{B}(G,k))}=G(k)_{o}\cdot\overline{\vartheta_{P}(A(G,S))}.$

Because $G(k)_{o}\times\overline{\vartheta_{P}(A(G,S))}$ and
$\overline{\vartheta_{P}(\mathcal{B}(G,k))}$ are compact Hausdorff by
Proposition 36, the surjective action map

$\pi_{P}:G(k)_{o}\times\overline{\vartheta_{P}(A(G,S))}\to\overline{\vartheta_{P}(\mathcal{B}(G,k))}$

is in fact a quotient map.

In order to conclude that $\vartheta_{P}$ is an open map, we need to make one
last observation.

######  Lemma 41.

Let $S$ be a maximal split torus of $G$ and
$x\in\overline{\vartheta_{P}(A(G,S))}$. If, for some $g\in G(k)$, we have
$gx\in\vartheta_{P}(\mathcal{B}(G,k))$, then $x\in\vartheta_{P}(A(G,S))$.

###### Proof.

The proof is completely analogous to that of [RTW10, Lemma 3.33]. The key
point is the fact that, using Proposition 32, we may characterise the points
of $\vartheta_{P}(A(G,S))$ as the points $x$ of
$\overline{\vartheta_{P}(A(G,S))}$ that induce norms on their local ring
$\mathcal{O}_{(G/P)^{\operatorname{an}},x}$, a property that is clearly
preserved by the action of $G(k)$.

First of all, reasoning as in the proof of Lemma 39, we observe that there
exists a $\dot{w}\in N_{G}(S)(k)$ such that
$\dot{w}x\in(\Omega(S,P)/P)^{\operatorname{an}}$. Moreover, the lemma holds
for $x$ if and only if it does for $\dot{w}x$. We may therefore assume that
$x$ lies in
$\overline{\vartheta_{P}(A(G,S))}\cap(\Omega(S,P)/P)^{\operatorname{an}}$. Let
$o$ be a special vertex in $A(G,S)$. Let $T$ be a maximal torus of $G$ and
$K/k$ a finite Galois extension such that

  1. 1.

The torus $T_{K}$ is split.

  2. 2.

The apartment $A(G_{K},T_{K})$ contains $p_{K/k}(A(G,S))$.

  3. 3.

The point $o_{K}\in A(G_{K},T_{K})$ is a special vertex.

Denote by $U^{-}$ the $k$-split unipotent radical of the opposite pseudo-
parabolic to $P$ with respect to $S$. Identify $A(G,S)$ with $V(G,S)$ (resp.
$A(G_{K},T_{K})$ with $V(G_{K},T_{K})$) setting $o$ (resp. $o_{K}$) as origin.
Under this identification, the map $p_{K/k}:A(G,S)\to A(G_{K},T_{K})$ is the
map induced by base change on cocharacters. Consider the sets
$\Psi=\Phi(U^{-},S)$ and $\Psi_{K}=\Phi(U-,T_{K})$. Let
$(\xi_{\alpha,i})_{(\alpha,i)\in\mathbf{\Psi}_{K}}$ be a system of coordinates
on $U^{-}_{K}$ that is adapted to $o_{K}$ (Remark 31). Let $(x_{n})$ be a
sequence in $A(G,S)$ such that $x=\lim_{n\to\infty}\vartheta_{P}(x_{n})$. The
sequence $(\vartheta_{P_{K}}((x_{n})_{K}))$ converges to a point $x_{K}$ in
$\overline{\vartheta_{P_{K}}(A(G_{K},T_{K}))}\cap(\Omega(T,P)/P)_{K}^{\operatorname{an}}$
by Proposition 32. Moreover, we have the equivalences

$\begin{array}[]{ccl}x_{K}\in\vartheta_{P_{K}}(A(G_{K},T_{K}))&\iff&\forall(\alpha,i)\in\mathbf{\Psi}_{K},|\xi_{\alpha,i}(x_{K})|>0\\\
&\iff&x_{K}\text{ defines a norm on
}\mathcal{O}_{(G/P)_{K}^{\operatorname{an}},x_{K}}\end{array}.$

To conclude, first observe that, by Proposition 32, for each weight
$\alpha\in\Psi_{K}$ and $a=\alpha_{|S}\in\Psi$ its restriction to $S$, we have

$|\xi_{\alpha,i}(x_{K})|=\lim_{n\to\infty}e^{\langle a,x_{n}\rangle}.$

Therefore, we have the equivalence

$x_{K}\in\vartheta_{P_{K}}(A(G_{K},T_{K}))\iff x\in\vartheta_{P}(A(G,S)),$

and in particular the implication

$x\in\vartheta_{P}(A(G,S))\implies x\text{ defines a norm on
}\mathcal{O}_{(G/P)^{\operatorname{an}},x},$

as the seminorm defined by $x$ on $\mathcal{O}_{(G/P)^{\operatorname{an}},x}$
is the restriction of the one defined by $x_{K}$ on
$\mathcal{O}_{(G/P)_{K}^{\operatorname{an}},x_{K}}$, hence a norm. On the
other hand, if $a\in\Psi$ and $\phi\in\operatorname{Lie}(U^{-})^{*}$ is a
nonzero weight vector of weight $-a$, then as observed in the proof of Lemma
40, there exists $C>0$ such that

$|\phi(x)|=|\phi_{K}(x_{K})|=\lim_{n\to\infty}Ce^{\langle a,x_{n}\rangle}.$

It follows that, if $x$ defines a norm on
$\mathcal{O}_{(G/P)^{\operatorname{an}},x}$, then by the above computation, we
have

$\forall(\alpha,i)\in\mathbf{\Psi}_{K},|\xi_{\alpha,i}(x_{K})|>0,$

and therefore $x\in\vartheta_{P}(A(G,S))$, by the above observations.

∎

We now carry out the proof of Proposition 38.

###### Proof of Proposition 38.

Let $U$ be an open subset of $\mathcal{B}(G,k)$. Consider the diagram

${G(k)_{o}\times
A(G,S)}$${G(k)_{o}\times\overline{\vartheta_{P}(A(G,S))}}$${\mathcal{B}(G,k)}$${\overline{\vartheta_{P}(\mathcal{B}(G,k))}}$$\scriptstyle{\operatorname{id}\times\vartheta_{P}}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{P}}$$\scriptstyle{\vartheta_{P}}$

.

Then, the subset $V=(\operatorname{id}\times\vartheta_{P})(\pi^{-1}(U))$ is
open in $G(k)_{o}\times\overline{\vartheta_{P}(A(G,S))}$ by Lemma 40. As we
established above, the rightmost map $\pi_{P}$ is a quotient map. Therefore,
in order to prove that $\vartheta_{P}(U)=\pi_{P}(V)$ is open in
$\overline{\vartheta_{P}(\mathcal{B}(G,k)}$, it is sufficient to check that
$V$ is $\pi_{P}$-saturated, which is a direct consequence of Lemma 41. ∎

##  4 Description of the boundary

In this section, we establish that the family of compactifications of
$\mathcal{B}(G,k)$ constructed above corresponds to the one constructed in
[Cha10] and specialises to the one constructed in [RTW10] in the reductive
case. More precisely, we will prove that, if $P$ is a pseudo-parabolic
subgroup of $G$ of non-degenerate type and $S$ a maximal split torus in $P$,
then the compactification
$\overline{\vartheta_{P}(\mathcal{B}(G,k))}\subset(G/P)^{\operatorname{an}}$
is $G(k)$-equivariantly homeomorphic to the compactification
$\overline{\mathcal{B}(G,k)}$ defined in [Cha10, Section 5] for a suitable
choice of fan. We deduce that the compactifications are stratified and
interpret the strata in terms of group-theoretic data.

###  4.1 Polyhedral compactifications of affine buildings

In this section, we introduce terminology and results from [Cha10] that will
be in constant use in what follows.

The article [Cha10] associates to each affine building $\mathcal{I}$, modeled
on an affine Coxeter complex $\Sigma$, and each fan $\mathcal{F}$ in the
direction $\vec{\Sigma}$ of $\Sigma$ subject to some compatibility relations
[Cha10, 2.3] a canonical compactification $\overline{I}^{\mathcal{F}}$ of
$\mathcal{I}$. We give an overview of the construction, first compactifying
each apartment and then glueing the compactifications of the various
apartments together to get a compactification of the building.

#### Fans and compactifications of affine spaces

We first describe the compactifications of apartments in [Cha10, Section 3].

The setting is that of a Euclidean space $V$ whose dual is endowed with a root
system $\Phi\subset V^{*}$. Denote by $W$ the Weyl group of $\Phi$. Recall
that a cone in $V$ is said to be polyhedral if it can be defined by a finite
number of linear inequalities. We call a cone relatively open if it is open in
its linear span. A polyhedral cone is relatively open if and only if it can be
defined by a finite number of linear equations and strict linear inequalities.
Then, a fan in the sense of [Cha10] is a family $\mathcal{F}$ of relatively
open polyhedral cones of $V$ satisfying the following conditions:

  1. 1.

The family $\mathcal{F}$ is a finite partition of $V$.

  2. 2.

$\\{0\\}\in\mathcal{F}$.

  3. 3.

For each $f\in\mathcal{F}$, the boundary $\partial f=\overline{f}\setminus f$
is a union of cones of $\mathcal{F}$, called the proper faces of $f$ (we also
consider $g$ to be a face of itself).

  4. 4.

For all cones $f$ and $g$ in $\mathcal{F}$, if $f$ is a face of $g$, then
$\overline{f}=\operatorname{Span}_{\mathbb{R}}(f)\cap\overline{g}$.

######  Example 42.

The root system $\Phi\subset V^{*}$ determines a hyperplane arrangement in
$V$. The facets, in the sense of [Bou81, Ch. V, §1.3], then form a fan
$\mathcal{F}$ in $V$ if the root system $\Phi$ is essential. In the following,
we call this fan the Weyl fan. Its facets will be called Weyl facets, and its
open facets called Weyl chambers.

######  Remark 43.

The above definition differs slightly from the one coming from toric geometry
adopted in e.g. [Ful93, 1.1] or [RTW10, Appendix B], which we shall call a
closed fan, namely a finite set $\mathcal{F}$ of closed strictly convex
polyhedral cones (ie. containing no lines) such that any two cones in
$\mathcal{F}$ meet along a face, and such that any face of a cone in
$\mathcal{F}$ belongs to $\mathcal{F}$. In that context, a face of a closed
polyhedral convex cone $f$ is defined as the intersection of $f$ with a
supporting hyperplane. The operations of taking closures and relative
interiors allow one to pass from fans on the one hand to closed fans covering
$V$ and vice versa. These operations also establish a correspondence between
both notions of faces, as we check below.

######  Lemma 44.

Let $\mathcal{F}$ be a fan in $V$. For each cone $g$ in $\mathcal{F}$, the
closure $\overline{g}$ is strictly convex.

###### Proof.

Let $g\in\mathcal{F}$ be a cone of minimal dimension such that there exists
$x\in V\setminus\\{0\\}$ such that $x\in\overline{g}$ and $-x\in\overline{g}$.
Note that $x$ is in $g$. Indeed, otherwise, there exists a proper face $f$ of
$g$ such that $x\in f$. Then, by Condition 4, we have $-x\in\overline{f}$,
which contradicts the minimality of $g$. Now, since $g$ is relatively open and
$x\in g$, it follows from [Roc72, Theorem 6.1] that $0=\frac{1}{2}(x+(-x))\in
g$, which contradicts Condition 2.

∎

######  Lemma 45.

Let $\mathcal{F}$ be a fan in $V$ and $g$ a cone in $\mathcal{F}$. Let $f$ be
a relatively open polyhedral cone in $V$. The following are equivalent:

  1. 1.

The cone $f$ is a face of $g$.

  2. 2.

There exists a linear form $\alpha\in V^{*}$ such that $\alpha(x)\geq 0$ for
each $x\in g$ and such that $\overline{f}=(\ker\alpha)\cap\overline{g}$.

###### Proof.

(Step 1: $(i)\implies(ii)$) Assume that $f$ is a face of $g$. Let us first
assume that $g$ is open. Then because $\overline{g}$ is strictly convex, the
dual cone

$g^{\vee}=\left\\{\alpha\in V^{*},\forall x\in g,\alpha(x)\geq 0\right\\}$

has non-empty interior by [Roc72, Corollary 14.6.1]. Because the annihilator

$f^{0}=\\{\alpha\in V^{*},f\subset\ker\alpha\\}$

meets the interior of $g^{\vee}$ (that is, because $f$ lies in a supporting
hyperplane of $g$, see [Roc72, Theorem 11.6]), the intersection $f^{0}\cap
g^{\vee}$ has non-empty relative interior. Let $\alpha\in f^{0}\cap g^{\vee}$
be a relative interior point. Let
$x\in\overline{g}\setminus\operatorname{Span}_{\mathbb{R}}(f)$. Because
$\alpha$ is in the relative interior of $f^{0}\cap g^{\vee}$, it lies in the
relative interior of the half-space

$f^{0}\cap\\{x\\}^{\vee}=\\{\alpha\in f^{0},\alpha(x)\geq 0\\}$

by [Roc72, Corollary 6.5.2], that is the (relatively) open half-space
$\\{\alpha\in f^{0},\alpha(x)>0\\},$ hence $\alpha(x)>0$. Finally, we have

$(\ker\alpha)\cap\overline{g}=\operatorname{Span}_{\mathbb{R}}(f)\cap\overline{g}=\overline{f}.$

In the general case, let $H_{0}$ be a supporting hyperplane of $g$ in
$\operatorname{Span}_{\mathbb{R}}(g)$ such that
$H_{0}\cap\overline{g}=\overline{f}$. Then, the subspace
$H=H_{0}\oplus\operatorname{Span}_{\mathbb{R}}(g)^{\perp}$ is a supporting
hyperplane of $g$ in $V$ such that $H\cap\overline{g}=\overline{f}$.

(Step 2: $(ii)\implies(i)$) Let $\alpha\in V^{*}$ be a linear form such that
$\alpha(x)\geq 0$ for each $x\in g$ and such that
$\overline{f}=(\ker\alpha)\cap\overline{g}$. Because each face of $g$ is
relatively open, the form $\alpha$ is either identically zero or only assumes
positive values on it. Consequently, the intersection
$(\ker\alpha)\cap\overline{g}$ is a union of faces of $g$, say

$(\ker\alpha)\cap\overline{g}=f_{1}\cup\dots\cup f_{r}.$

Because the intersection is closed, we in fact have

$(\ker\alpha)\cap\overline{g}=\overline{f}_{1}\cup\dots\cup\overline{f}_{r}.$

One of the $\overline{f}_{i}$ must then have non-empty interior in
$\operatorname{Span}_{\mathbb{R}}((\ker\alpha)\cap\overline{g})$, from which
we deduce that

$\overline{f}_{i}=\operatorname{Span}_{\mathbb{R}}(f_{i})\cap\overline{g}=\operatorname{Span}_{\mathbb{R}}((\ker\alpha)\cap\overline{g})\cap\overline{g}=(\ker\alpha)\cap\overline{g}=\overline{f}.$

Because both $f_{i}$ and $f$ are convex and relatively open, we conclude that
$f=f_{i}$, hence $f$ is indeed a face of $g$. ∎

We now assume that we are given an affine space $A$ directed by $V$. Two
affine convex cones $f=x+\vec{f}$ and $g=y+\vec{g}$ in $A$, directed by cones
$\vec{f}$ and $\vec{g}$ in $V$, are said to be parallel ($f/\\!/g$) if they
have the same direction, that is if $\vec{f}=\vec{g}$. The cones $f$ and $g$
are said to be equivalent ($f\sim g$) if they are parallel and have non-empty
intersection or, equivalently, if they share a parallel subcone, ie. if there
exists a cone $h\subset f\cap g$ that is parallel to $f$ and $g$. Call
$f\subset A$ an affine $\mathcal{F}$-cone if its direction $\vec{f}$ belongs
to $\mathcal{F}$. Denote by $\mathcal{F}_{A}$ the set of affine
$\mathcal{F}$-cones in $A$.

Then, the compactification of $A$ associated to a fan $\mathcal{F}$ in $V$ is
the quotient of the set of affine $\mathcal{F}$-cones in $A$ under the
equivalence relation defined above:

$\overline{A}^{\mathcal{F}}=\underbrace{\\{x+\vec{f},x\in
A,\vec{f}\in\mathcal{F}\\}}_{\mathcal{F}_{A}}/\sim.$

The equivalence classes can be grouped by direction, thereby partitioning the
compactification:

$\overline{A}^{\mathcal{F}}=\bigsqcup_{\vec{f}\in\mathcal{F}}A_{\vec{f}}.$

For each $\vec{f}\in\mathcal{F}$, the subspace
$A_{\vec{f}}=\\{[x+\vec{f}],x\in A\\}$ is called the façade of
$\overline{A}^{\mathcal{F}}$ of type $\vec{f}$. This façade is naturally an
affine space under $V/\operatorname{Span}_{\mathbb{R}}(\vec{f})$.
Geometrically, setting an origin at some point $o\in A$, we compactify each
open cone $o+\vec{f}=\\{x\in A,\forall i\in I,\alpha_{i}(x-o)>0\\}$ by
allowing the $\alpha_{i}(x-o)$ for $i\in I$, which we think of as the distance
from $x$ to the walls of the cone $o+\vec{f}$, to assume infinite values.

Precisely, we have the following statement

######  Proposition 46 (Corollary of [Cha10, Proposition 3.2.4]).

Let $(x_{n})$ be a sequence of points in $A$, and
$l=[a+\vec{f}]\in\overline{A}^{\mathcal{F}}$. Then, the sequence $(x_{n})$
converges to $l$ if and only if, for each linear form $\alpha\in V^{*}$ such
that $\vec{f}\subset\ker\alpha$, we have

$\alpha(x_{n}-a)\underset{n\to+\infty}{\longrightarrow}0$

and, for each $\alpha\in V^{*}$ with $\alpha(x)>0$ for each $x$ in $\vec{f}$,
we have

$\alpha(x_{n}-a)\underset{n\to+\infty}{\longrightarrow}+\infty.$

These façades form a stratification of $\overline{A}^{\mathcal{F}}$,
specifically we have

$A_{\vec{g}}\subset\overline{A_{\vec{f}}}\iff\vec{f}\text{ is a face of
}\vec{g}.$

#### Fans associated to types of pseudo-parabolic subgroups

We now define the family of fans that will concern us in the following.

Weyl chambers are in one-to-one correspondence with bases of the root system:
Given a basis $\Delta$ of the root system, the cone

$C(\Delta)=\\{x\in V,\forall\alpha\in\Delta,\alpha(x)>0\\}$

is a Weyl chamber. Conversely, given a Weyl chamber $C$, the intersection
$C^{\vee}\cap\Phi$ of the dual cone of $C$ with $\Phi$ is a positive system of
roots that gives rise to a unique basis. Moreover, given a Weyl chamber $C$
corresponding to a basis $\Delta$, the faces of $C$ are in one-to-one
correspondence with the subsets of $\Delta$: If $J\subset\Delta$, then the set

$\\{x\in V,\forall\alpha\in J,\alpha(x)=0\text{ and
}\forall\alpha\in\Delta\setminus J,\alpha(x)>0\\}$

is a face of $C$, said to be of type $J$.

######  Example 47.

Fix a Weyl chamber $C$ corresponding to a basis $\Delta$ of $\Phi$, and a
subset $J\subset\Delta$ containing no connected component of $\Delta$,
considered as a Dynkin diagram. Say a face of $C$ is admissible if its type
contains no connected component of $\Delta$. Then, for each admissible face
$f$ of $C$ of type $I$, we let $\overline{f}^{J}$ denote the union of $f$ and
its faces of type included in $I\cup(J\cap I^{\perp})$. Then, the set

$J\cdot f=\bigcup_{w\in\langle s_{\alpha},\alpha\in J\cap
I^{\perp}\rangle}w\overline{f}^{J}$

is a relatively open polyhedral cone, and the set

$\mathcal{F}^{J}=\bigcup_{w\in W}w\cdot\\{J\cdot f,f\text{ admissible face of
}C\\}$

is a fan in $V$ by [Cha10, Proposition 2.5.3].

When the root system is given by a maximal split torus $S$ in a quasi-
reductive group $G$ over a field $k$, we have a natural group-theoretic
interpretation for this fan.

Let $P$ is a pseudo-parabolic subgroup of non-degenerate type in a quasi-
reductive group $G$ over a field $k$. Let $P_{0}$ be a minimal pseudo-
parabolic subgroup of $G$ and $S$ a maximal split torus such that $S\subset
P_{0}\subset P$. Set $V=V(G,S)$, and $\Phi=\Phi(G,S)$. Let $W$ be the Weyl
group of $\Phi$. Let $\Delta$ be the basis of the root system $\Phi$
corresponding to $P_{0}$ and $J=\Delta\cap-\Phi(P,S)$ be the type of $P$. Set

$C(P)=\\{x\in
V(G,S),\forall\alpha\in\Phi(\mathcal{R}_{u,k}(P),S),\langle\alpha,x\rangle\geq
0\\},$

and let

$C=\\{x\in V(G,S),\forall\alpha\in\Delta,\langle\alpha,x\rangle>0\\}$

be the Weyl chamber associated to $P_{0}$. These cones are related as follows.

######  Proposition 48.

With the above notation, we have

$C(P)=\overline{J\cdot C}.$

Consequently, the open cones of $\mathcal{F}^{J}$ are the
$\overset{\circ}{\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{23.54584pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle C(Q)$}}}{\vbox{
\hbox{\leavevmode\resizebox{23.54584pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle C(Q)$}}}{\vbox{
\hbox{\leavevmode\resizebox{16.48207pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle C(Q)$}}}{\vbox{
\hbox{\leavevmode\resizebox{11.7729pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle C(Q)$}}}}$, where $Q$
ranges among the pseudo-parabolic subgroups containing $S$ conjugate to $P$.
Moreover, the correspondence $P\mapsto C(P)$ is $W$-equivariant.

###### Proof.

Observe that

$\overline{J\cdot C}=\bigcup_{w\in\langle s_{\alpha},\alpha\in
J\rangle}w(\overline{C}),$

which is the union of the closed Weyl chambers associated to the bases
$w(\Delta)$ for $w$ in $W_{J}=\langle s_{\alpha},\alpha\in J\rangle$. Let
$L_{P}$ be the Levi subgroup of $P$ corresponding to $S$. Then, the reflection
group $W_{J}$ is the Weyl group $W(L_{P},S)$. In particular, because the
conjugacy action of $L_{P}$ on $P$ preserves $\mathcal{R}_{u,k}(P)$, the group
$W_{J}$ stabilises the cone $C(P)$. Hence, because $\overline{C}\subset C(P)$,
we must in turn have $\overline{J\cdot C}\subset C(P)$.

On the other hand, the cone $C(P)$ is a union of Weyl facets with non-empty
interior and is therefore the closure of the union of the Weyl chambers that
it contains. Further, a Weyl chamber is contained in $C(P)$ if and only if it
corresponds to a minimal pseudo-parabolic subgroup of $G$ containing $S$ and
contained in $P$ by [CGP15, Proposition 3.5.14]. Now, by [CGP15, Proposition
2.2.10], the map

$Q\mapsto Q\cap L_{P}$

establishes an $L_{P}(k)$-equivariant and order-preserving bijection between
pseudo-parabolic subgroups of $P$ containing $S$ and pseudo-parabolic
subgroups of $L_{P}$ containing $S$. By [CGP15, Theorem C.2.3] and [CGP15,
Theorem C.2.5], the normaliser $N_{L_{P}}(S)(k)$ acts transitively on the
minimal pseudo-parabolic subgroups of $L_{P}$ containing $S$, hence it acts
transitively on the minimal pseudo-parabolic subgroups of $P$ containing $S$.
Hence, the Weyl chambers contained in $C(P)$ are precisely the $w(\Delta)$ for
$w\in W(L_{P},S)(k)=W_{J}$. We conclude that $C(P)$ is in fact equal to
$\bigcup_{w\in W_{J}}w(\overline{C})=\overline{J\cdot C}$, which completes the
proof. ∎

#### Compactifications of affine buildings

To conclude these preliminary considerations, we extend the above construction
to yield compactifications of affine buildings. In [Lan96], where the affine
buildings are Bruhat-Tits buildings of reductive groups, this is obtained by
way of a glueing procedure, much like the building itself: stabilisers are
defined for points of the compactification of a given apartment with respect
to the Weyl fan, then the compactification of $\mathcal{B}(G,k)$ is defined to
be the quotient of $G(k)\times\overline{A}^{\mathcal{F}}$ by an equivalence
relation similar to that defining the building $\mathcal{B}(G,k)$.

A salient feature of [Cha10] is that it performs the same construction while
solely relying on the internal geometry of the building. For this purpose, the
author introduces the notion of a cone in the building, which in some sense is
a glueing of affine cones in various apartments containing a common subcone of
a Weyl facet, called its core.

Consider an apartment $A$ in an affine building $\mathcal{B}$. We denote by
$V$ the direction of $A$ and by $\Phi\subset V^{*}$ its vectorial root system.
We also assume that we are given a fan $\mathcal{F}$ in $V$ that is invariant
under the action of the Weyl group $W=W(\Phi)$. Under this assumption, the
notion of an affine $\mathcal{F}$-cone can be defined in the whole building:
An affine $\mathcal{F}$-cone is a subset $f\subset\mathcal{B}$ such that there
exist an apartment $B$ containing $f$ and an isomorphism of affine Coxeter
complexes $\phi_{B}:A\to B$ such that $\phi_{B}^{-1}(f)$ is an affine
$\mathcal{F}$-cone in $A$. For each apartment $B$, denote by $\mathcal{F}_{B}$
the set of affine $\mathcal{F}$-cones contained in $B$.

Then, for each cone $f\in\mathcal{F}$, the core of $f$ is the convex cone

$\delta(f)=f\cap\operatorname{Fix}(\operatorname{Stab}_{W}(f)).$

######  Example 49.

In the case when $\mathcal{F}=\mathcal{F}^{J}$ with the notations of the
previous example, then, for each face $f$ of the Weyl cone $C$ of admissible
type $I$, the core of $J\cdot f$ is the face of $C$ of type $I\cup(J\cap
I^{\perp})$.

Likewise, given any affine $\mathcal{F}$-cone $f=x+\vec{f}$ in $A$, the core
of $f$ is defined as $\delta(f)=x+\delta(\vec{f})$.

We then define the cone of $\mathcal{B}$ associated to an affine cone $f$ in
$A$ to be

$\widetilde{f}=\bigcup_{\delta(f)\subset B}\phi_{B}(f),$

where $B$ ranges among all apartments containing the core $\delta(f)$, and
$\phi_{B}:A\to B$ is an isomorphism of Coxeter complexes fixing the
intersection $A\cap B$. Denote by $\mathcal{F}_{\mathcal{B}}$ the set

$\mathcal{F}_{\mathcal{B}}=\\{\widetilde{f},f\in\mathcal{F}_{A},A\text{
apartment of }\mathcal{B}\\}.$

It can then be checked that the $\phi_{B}(f)$ all have the same core, which we
also call the core of the cone $\widetilde{f}$ and denote by
$\delta(\widetilde{f})$.

Finally, the compactification of $\mathcal{B}$ is constructed in the same way
as that of $A$, ie. as equivalence classes of cones. The only problem is to
define the notion of parallelism for cones in $\mathcal{B}$, which is where
cores play a crucial role. Indeed, it follows from [Cha10, Corollaire 4.5.9]
that, given any two cones $F$ and $G$ in $\mathcal{B}$, there exists an
apartment $A$ that contains parallel subcones of $\delta(F)$ and $\delta(G)$.
We then say that $F$ and $G$ are parallel ($F/\\!/G$) if there exist parallel
subcones $f\subset\delta(F)$ and $g\subset\delta(G)$ and an apartment $A$
containing $f$ and $g$ such that $f/\\!/g$ in $A$ (this parallelism does not
depend on the specific $A$). Likewise, we say that $F$ and $G$ are equivalent
($F\sim G$) if $F$ and $G$ are parallel and have a non-empty intersection.
Then, we set

$\overline{\mathcal{B}}^{\mathcal{F}}=\mathcal{F}_{\mathcal{B}}/\sim.$

We call $\overline{\mathcal{B}}^{\mathcal{F}}$ the polyhedral compactification
of $\mathcal{B}$ with respect to the fan $\mathcal{F}$.

As in the previous case, the equivalence classes can be grouped in parallelism
classes:

$\overline{\mathcal{B}}^{\mathcal{F}}=\bigcup_{F\in\mathcal{F}_{\mathcal{B}}}\mathcal{B}_{F}.$

For each $F\in\mathcal{F}_{\mathcal{B}}$, the subspace
$\mathcal{B}_{F}=\\{G\in\mathcal{F}_{\mathcal{B}},G/\\!/F\\}$ is called the
(outer) façade of $\mathcal{B}$ of type $F$. The façade $\mathcal{B}_{F}$ is
naturally an affine building whose apartments are the façades $A_{f}$, for
apartments $A$ containing affine $\mathcal{F}$-cones $f$ such that
$\widetilde{f}/\\!/F$. These façades are either equal or disjoint, and form
the strata of a topological stratification of
$\overline{\mathcal{B}}^{\mathcal{F}}$ [Cha10, §8.2.2].

###  4.2 A comparison

Because the building $\mathcal{B}(G,k)$ arises from valuated root data in
$G(k)$ [Sol18, Section 3.2], Proposition 7.4 of [Cha10] applies. It is thus
sufficient, in order to identify the aforementioned compactifications, to
prove that the procedure of taking the closure of a given apartment in one
space and the other yields homeomorphic spaces.

######  Proposition 50.

Let $P$ be a pseudo-parabolic subgroup of non-degenerate type, let
$P_{0}\subset P$ be a minimal pseudo-parabolic subgroup, and let $S\subset
P_{0}$ a maximal split torus. Denote by $\Delta$ the basis of $\Phi(G,S)$
associated to $P_{0}$ and let $J\subset\Delta$ be the type of $P$.

The map $\vartheta_{P}$ extends to a continuous map

$\overline{A(G,S)}^{\mathcal{F}^{J}}\to(G/P)^{\operatorname{an}}$

that is a homeomorphism onto its image.

###### Proof.

By [GJT98, Lemma 3.28], it suffices to prove that, for any two sequences
$(x_{n})$ and $(x^{\prime}_{n})$ of points in $A(G,S)$, the following are
equivalent:

  1. (i)

The sequences $(x_{n})$ and $(x^{\prime}_{n})$ converge and have the same
limit in $\overline{A(G,S)}^{\mathcal{F}^{J}}$.

  2. (ii)

The sequences $(\vartheta_{P}(x_{n}))$ and $(\vartheta_{P}(x^{\prime}_{n}))$
converge and have the same limit in $(G/P)^{\operatorname{an}}$.

We use Propositions 32 and 46 to characterise convergence in one space and the
other. Both characterisations can be translated into convergence conditions on
the sequences $(\langle a,x_{n}\rangle)$ and $(\langle
a,x^{\prime}_{n}\rangle)$ for certain roots $a$.

Fix a special vertex $o\in A(G,S)$. Identify $A(G,S)$ with its direction
$V(G,S)$ setting $o$ as origin. Let

$\begin{array}[]{ccl}C(P)&=&\\{x\in A(G,S),\forall
a\in\Phi(\mathcal{R}_{u,k}(P),S),\langle a,x\rangle\geq 0\\}\end{array}.$

(Step 1: (i) $\implies$ (ii)) Let $(x_{n})$ and $(x^{\prime}_{n})$ be two
sequences of points in $A(G,S)$. Assume that both $(x_{n})$ and
$(x^{\prime}_{n})$ converge in $\overline{A(G,S)}^{\mathcal{F}^{J}}$ and have
the same limit $l$ lying in the façade $A(G,S)_{\vec{f}}$, where
$\vec{f}\in\mathcal{F}^{J}$.

Without loss of generality, we may assume that $\vec{f}$ is a face of
$\overset{\circ}{\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{23.44933pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{23.44933pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{16.41452pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{11.72464pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle C(P)$}}}}$. Indeed, by
construction, $\vec{f}$ is a face of an open cone of $\mathcal{F}^{J}$ and, by
Proposition 48, the Weyl group $W(G,S)(k)$ acts transitively on these open
cones. If $w\in W(G,S)(k)$ maps $\vec{f}$ to a face of
$\overset{\circ}{\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{23.44933pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{23.44933pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{16.41452pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{11.72464pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle C(P)$}}}}$, and if
$\dot{w}\in N_{G}(S)(k)$ is a lift of $w$, then $(\dot{w}x_{n})$ and
$(\dot{w}x^{\prime}_{n})$ both converge and have the same limit $\dot{w}l$
lying in $A(G,S)_{w\vec{f}}$, and $w\vec{f}$ is a face of
$\overset{\circ}{\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{23.44933pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{23.44933pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{16.41452pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{11.72464pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle C(P)$}}}}$. Furthermore,
$(\vartheta_{P}(x_{n}))$ and $(\vartheta_{P}(x^{\prime}_{n}))$ converge and
have the same limit in $(G/P)^{\operatorname{an}}$ if and only if the same
holds for $(\vartheta_{P}(\dot{w}x_{n}))$ and
$(\vartheta_{P}(\dot{w}x^{\prime}_{n}))$ by $G(k)$-equivariance of
$\vartheta_{P}$ and continuity of the action of $G(k)$ on
$(G/P)^{\operatorname{an}}$.

Let us therefore assume that $\vec{f}$ is a face of
$\overset{\circ}{\mathchoice{\vbox{
\hbox{\leavevmode\resizebox{23.44933pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\displaystyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{23.44933pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\textstyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{16.41452pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptstyle C(P)$}}}{\vbox{
\hbox{\leavevmode\resizebox{11.72464pt}{0.0pt}{{\char 62\relax}}}
\nointerlineskip\hbox{\set@color$\scriptscriptstyle C(P)$}}}}$. Each
$a\in\Phi(\mathcal{R}_{u,k}(P),S)$, assumes only nonnegative values on
$\vec{f}$. In fact, because $\vec{f}$ is relatively open, each $a$ is either
identically zero or positive on $\vec{f}$. Therefore, by Proposition 46, the
limits

$\lim_{n\to\infty}\langle a,x_{n}\rangle\text{ and }\lim_{n\to\infty}\langle
a,x^{\prime}_{n}\rangle$

both exist in $\mathbb{R}\cup\\{+\infty\\}$ and coincide for each
$a\in\Phi(\mathcal{R}_{u,k}(P),S)$.

We now deduce that $(\vartheta_{P}(x_{n}))$ and
$(\vartheta_{P}(x^{\prime}_{n}))$ converge in $(G/P)^{\operatorname{an}}$ and
have the same limit. We use Proposition 32. Let $T$ be a maximal torus of $G$
containing $S$ and $K/k$ be a finite Galois extension such that:

  1. 1.

The torus $T_{K}$ is split.

  2. 2.

The apartment $A(G_{K},T_{K})$ contains $p_{K/k}(A(G,S))$.

  3. 3.

The point $o_{K}\in A(G_{K},T_{K})$ is a special vertex.

Identify $A(G_{K},T_{K})$ with its direction $V(G_{K},T_{K})$ by setting the
origin at $o_{K}$. Let $\mu:\mathbb{G}_{m}\to S$ such that $P=P_{G}(\mu)$ and
set $U^{-}=U_{G}(-\mu)$. By Proposition 32, both $(\vartheta_{P}(x_{n}))$ and
$(\vartheta_{P}(x^{\prime}_{n}))$ take their values in
$(\Omega(T,P)/P)^{\operatorname{an}}$. Moreover, identifying $\Omega(T,P)/P$
with $U^{-}$ and choosing a system of coordinates
$(\xi_{\alpha,i})_{(\alpha,i)\in\mathbf{\Psi}_{K}}$ on $(U^{-})_{K}$ adapted
to $o_{K}$, we have, for each
$f=\sum_{\nu\in\mathbb{N}^{J}}f_{\nu}\xi^{\nu}\in K[U^{-}]$:

$|f(\vartheta_{P_{K}}((x_{n})_{K}))|=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}_{K}}}|f_{\nu}|\prod_{(\alpha,i)\in\mathbf{\Psi}_{K}}e^{\nu(\alpha,i)\langle\alpha,(x_{n})_{K}\rangle},$

which we may rewrite

$|f(\vartheta_{P_{K}}((x_{n})_{K}))|=\max_{\nu\in\mathbb{N}^{\mathbf{\Psi}_{K}}}|f_{\nu}|\prod_{a\in\Psi}\prod_{\begin{subarray}{c}(\alpha,i)\in\mathbf{\Psi}_{K}\\\
\alpha_{|S}=a\end{subarray}}e^{\nu(\alpha,i)\langle a,x_{n}\rangle}.$

Evidently, for each $f\in K[U^{-}]$, the quantities
$|f(\vartheta_{P_{K}}((x_{n})_{K}))|$ and
$|f(\vartheta_{P_{K}}((x^{\prime}_{n})_{K}))|$ converge in $\mathbb{R}$ with
the same limit, hence the sequences $(\vartheta_{P_{K}}((x_{n})_{K}))$ and
$(\vartheta_{P_{K}}((x^{\prime}_{n})_{K}))$ converge in
$(U^{-})^{\operatorname{an}}_{K}$ and have the same limit. By continuity of
$\operatorname{pr}_{K/k}^{\operatorname{an}}$, so do the sequences
$(\vartheta_{P}(x_{n}))$ and $(\vartheta_{P}(x^{\prime}_{n}))$, by
commutativity of the diagram

${A(G_{K},T)}$${(U^{-})_{K}^{\operatorname{an}}}$${A(G,S)}$${(U^{-})^{\operatorname{an}}}$$\scriptstyle{\vartheta_{P_{K}}}$$\scriptstyle{\operatorname{pr}_{K/k}^{\operatorname{an}}}$$\scriptstyle{p_{K/k}}$$\scriptstyle{\vartheta_{P}}$

.

(Step 2: (ii) $\implies$ (i)) Let $(x_{n})$ and $(x^{\prime}_{n})$ be two
sequences of points of $A(G,S)$. Assume that $(\vartheta_{P}(x_{n}))$ and
$(\vartheta_{P}(x^{\prime}_{n}))$ both converge in
$(G/P)^{\operatorname{an}}$. We first prove that the common limit of the two
sequences may be assumed to lie in $(\Omega(S,P)/P)^{\operatorname{an}}$.

Recall from Proposition 48 that

$A(G,S)=\bigcup_{\begin{subarray}{c}Q\text{ conjugate to }P\\\ S\subset
Q\end{subarray}}C(Q).$

Because only finitely many subgroups $Q$ conjugate to $P$ contain $S$, there
exists such a $Q$ and a subsequence $(x_{n_{j}})_{j\in\mathbb{N}}$ such that
$\vartheta_{P}(x_{n_{j}})\in C(Q)$ for each $j\in\mathbb{N}$. In particular,
we have
$\lim_{n\to\infty}\vartheta_{P}(x_{n})\in\overline{\vartheta_{P}(C(Q))}$. By
Proposition 37, for each pseudo-parabolic subgroup $Q$ containing $S$
conjugate to $P$, the image $\vartheta_{Q}(C(Q))$ is relatively compact in
$(\Omega(S,Q)/Q)^{\operatorname{an}}$, ie.
$\overline{\vartheta_{Q}(C(Q))}\subset(\Omega(S,Q)/Q)^{\operatorname{an}}$.
Let $w\in W(G,S)(k)$ be such that $wC(Q)=C(P)$ and $\dot{w}$ be a lift of $w$
in $N_{G}(Q)(k)$. Then, we have $P=\dot{w}Q\dot{w}^{-1}$, and therefore, by
Proposition 28:

$\overline{\vartheta_{P}(C(Q))}=\iota_{P,Q}^{\operatorname{an}}(\vartheta_{Q}(C(Q)))\subset\iota_{P,Q}^{\operatorname{an}}((\Omega(S,Q)/Q)^{\operatorname{an}})=\dot{w}(\Omega(S,P)/P)^{\operatorname{an}}.$

Consequently, the sequences $(\vartheta_{P}(\dot{w}^{-1}x_{n}))$ and
$(\vartheta_{P}(\dot{w}^{-1}x^{\prime}_{n}))$ converge to a point in
$(\Omega(S,P)/P)^{\operatorname{an}}$. Moreover, the sequences $(x_{n})$ and
$(x^{\prime}_{n})$ converge to the same limit if and only if the same holds
for $(\dot{w}^{-1}x_{n})$ and $(\dot{w}^{-1}x^{\prime}_{n})$.

Let us therefore reduce to the case when $(\vartheta_{P}(x_{n}))$ and
$(\vartheta_{P}(x^{\prime}_{n}))$ both converge in
$(\Omega(S,P)/P)^{\operatorname{an}}\simeq(U^{-})^{\operatorname{an}}$. Choose
a special vertex $o$ in $A(G,S)$, a maximal torus $T$ of $G$ containing $S$
and a finite Galois extension $K/k$ as in Step 1 and make the same choice of
notations. Let $a\in-\Phi(\mathcal{R}_{u,k}(P),S)$ and let
$\phi=\sum_{\begin{subarray}{c}(\alpha,i)\in{\mathbf{\Psi}_{K}}\\\
\alpha_{|S}=a\end{subarray}}\phi_{\alpha,i}\xi_{\alpha,i}\in k[U^{-}]\subset
K[U^{-}]$ be a $k$-linear form on the root group $U_{a}$. Then, we have

$|\phi(\vartheta_{P}(x_{n}))|=|\phi_{K}(\vartheta_{P_{K}}((x_{n})_{K}))|=\left(\max_{\begin{subarray}{c}(\alpha,i)\in{\mathbf{\Psi}_{K}}\\\
\alpha_{|S}=a\end{subarray}}|\phi_{\alpha,i}|e^{\langle\alpha,(x_{n})_{K}\rangle}\right),$

which rewrites as

$|\phi(\vartheta_{P}(x_{n}))|=\left(\max_{\begin{subarray}{c}(\alpha,i)\in{\mathbf{\Psi}_{K}}\\\
\alpha_{|S}=a\end{subarray}}|\phi_{\alpha,i}|\right)e^{\langle
a,x_{n}\rangle}.$

Because the sequences $(|\phi(\vartheta_{P}(x_{n}))|)$ and
$(|\phi(\vartheta_{P}(x^{\prime}_{n}))|)$ converge with the same limit, the
sequences $(\langle a,x_{n}\rangle)$ and $(\langle a,x^{\prime}_{n}\rangle)$
must converge in $\mathbb{R}\cup\\{-\infty\\}$ with the same limit. We deduce
that, for each $a\in\Phi(\mathcal{R}_{u,k}(P),S)$, we have

$\lim_{n\to\infty}\langle a,x_{n}\rangle=\lim_{n\to\infty}\langle
a,x^{\prime}_{n}\rangle\in\mathbb{R}\cup\\{+\infty\\}.$

Let

$I_{c}=\left\\{a\in\Phi(\mathcal{R}_{u,k}(P),S),\lim_{n\to\infty}\langle
a,x_{n}\rangle\in\mathbb{R}\right\\}$

and

$I_{d}=\left\\{a\in\Phi(\mathcal{R}_{u,k}(P),S),\lim_{n\to\infty}\langle
a,x_{n}\rangle=+\infty\right\\}.$

Then, denoting by $\vec{f}$ the set

$\vec{f}=\\{x\in V(G,S),\forall a\in I_{c},\langle a,x\rangle=0\text{ and
}\forall a\in I_{d},\langle a,x\rangle>0\\},$

the cone $\vec{f}$ is a face of $C(P)$ by Lemma 45 and it follows from
Proposition 46 that $(x_{n})$ and $(x^{\prime}_{n})$ converge to a common
limit lying in the façade $A(G,S)_{\vec{f}}$, which completes the proof of the
equivalence. ∎

By the criterion [Cha10, Proposition 7.4], the compactification
$\overline{\vartheta_{P}(\mathcal{B}(G,k))}$ is thus $G(k)$-equivariantly
homeomorphic to the polyhedral compactification of $\mathcal{B}(G,k)$
associated to the fan $\mathcal{F}^{J}$.

###  4.3 Identifying the strata

It now follows from [Cha10, Corollaire 8.2.11] that the compactification
$\overline{\vartheta_{P}(\mathcal{B}(G,k))}$ is stratified by affine
buildings. Precisely, we have a $G(k)$-equivariant homeomorphism

$\overline{\vartheta_{P}(\mathcal{B}(G,k))}=\bigcup_{F\in(\mathcal{F}^{J})_{\mathcal{B}(G,k)}}\mathcal{B}(G,k)_{F},$

where the right-hand side is indexed by cones in $\mathcal{B}(G,k)$ in the
sense of [Cha10, Sections 5.2]. The stratum $\mathcal{B}(G,k)_{F}$
corresponding to a cone $F$ is in turn defined as the set of equivalence
classes of cones parallel to $F$ for the relation [Cha10, Définition 5.4.1].

We begin by proving that these strata – which are affine buildings – can be
interpreted as Bruhat-Tits buildings of certain Levi subgroups of pseudo-
parabolic subgroups of $G$.

######  Theorem 51.

Let $F$ be a cone in $(\mathcal{F}^{J})_{\mathcal{B}(G,k)}$. There exists a
unique pseudo-parabolic subgroup $Q$ of $G$ such that $\mathcal{B}(G,k)_{F}$
is stable under the action of $Q(k)$. Moreover, this stratum identifies
$Q(k)$-equivariantly to the Bruhat-Tits building of the quasi-reductive group
$Q/\mathcal{R}_{us,k}(Q)$.

The proof of this theorem will take up most of this section and will therefore
be broken down into several steps, namely Propositions 54, 56, and 57. The
outline goes as follows: We begin by associating to an apartment $A(G,S)$
containing the core of $F$ a pseudo-parabolic subgroup $Q$ containing $S$. We
prove that this pseudo-parabolic subgroup $Q$ depends only on the parallelism
class of $F$ (thus neither on the specific representative $F$ nor the specific
apartment containing $\delta(F)$ that we choose). We deduce from these
observations that $Q(k)$ stabilises the façade $\mathcal{B}(G,k)_{F}$ and that
the $k$-split unipotent radical $\mathcal{R}_{us,k}(Q)(k)$ acts trivially on
that same façade. Lastly, we check that, given a Levi factor $L_{Q}\subset Q$,
the projections $A(G,S)\to A(G,S)_{F}$ with $S\subset L_{Q}$ glue together to
a $L_{Q}(k)$-equivariant map $L_{Q}(k)\cdot
A(G,S)\twoheadrightarrow\mathcal{B}(G,k)_{F}$ that factors through an
equivariant, simplicial homeomorphism
$\mathcal{B}(L_{Q},k)\to\mathcal{B}(G,k)_{F}$, which will complete the proof.

#### Associating a pseudo-parabolic subgroup to a façade

Let us now carry out the first step, namely identifying the pseudo-parabolic
subgroup $Q$ and establishing its sole dependency on the parallelism class of
$F$. The argument hinges on the following preliminary observation.

######  Proposition 52.

Let $X$ be a smooth irreducible variety over a field $k$ complete with respect
to a non-trivial absolute value. Then, any non-empty open subset of $X(k)$ for
its analytic topology is Zariski-dense in $X$.

###### Proof.

A proof is given in [Poo17, Proposition 3.5.75 and Remark 3.5.76] in the case
of local fields, but a careful inspection shows this proof only to depend on
the completeness of $k$ (specifically in the use of the implicit function
theorem over $k$ in the proof of [Poo17, Proposition 3.5.73 (i)]). ∎

We take note of the following special case.

######  Corollary 53.

Let $k$ be a complete discretely valued field. Let $\mathfrak{X}$ be a smooth
irreducible variety over the valuation ring $k^{\circ}=\\{x\in k,\omega(x)\geq
0\\}$ and let $X=\mathfrak{X}_{k}$. Then, the set of integral points
$\mathfrak{X}(k^{\circ})$ is either empty or Zariski-dense in $X$.

From this analytic observation, we draw the following consequence.

######  Proposition 54.

Let $Q$ be the Zariski closure in $G$ of the pointwise stabiliser
$P_{\delta(F)}$ of $\delta(F)$ in $G(k)$. Then $Q$ is a pseudo-parabolic
subgroup in $G$ and, for any cone $F^{\prime}$ in $\mathcal{B}(G,k)$ parallel
to $F$, the pointwise stabiliser $\hat{P}_{\delta(F^{\prime})}$ is Zariski-
dense in $Q$.

###### Proof.

We give an indirect proof, namely we first construct a suitable pseudo-
parabolic subgroup $Q$ then prove that, for any cone $F^{\prime}$ parallel to
$F$, the pointwise stabiliser

$\hat{P}_{\delta(F^{\prime})}=\\{g\in G(k),\forall
x\in\delta(F^{\prime}),gx=x\\}$

is Zariski-dense in $Q$.

Let $S$ be a maximal split torus of $G$ such that the apartment $A(G,S)$
contains the core $\delta(F)$ of $F$ [Cha10, Définition 3.3.2]. Because the
direction $\vec{\delta}(F)$ is a Weyl facet in $V(G,S)$ (Example 49) , it
corresponds to a unique parabolic subset $\Psi$ of $\Phi(G,S)$, namely the
intersection of the root system with the dual cone of $\vec{\delta}(F)$

$\Psi=\Phi(G,S)\cap\vec{\delta}(F)^{\vee}=\\{\alpha\in\Phi(G,S)\ |\ \forall
x\in\vec{\delta}(F),\langle\alpha,x\rangle\geq 0\\}.$

This subset is indeed parabolic as it is clear that, whenever
$\alpha,\beta\in\Psi$ and $\alpha+\beta\in\Phi(G,S)$, we have
$\alpha+\beta\in\Psi$ and, if $\alpha\in\Phi(G,S)$, then because the
hyperplane $\\{x\in V(G,S),\langle\alpha,x\rangle=0\\}$ is part of the
hyperplane arrangement that defines $\Sigma(G,S)$, the root $\alpha$ assumes a
constant sign on each facet of $\Sigma(G,S)$, hence either $\alpha\in\Psi$ or
$-\alpha\in\Psi$. By [CGP15, Theorem C.2.15], the subset $\Psi$ then
determines a unique pseudo-parabolic subgroup $Q$ of $G$ containing $S$ such
that $\Psi$ is the set of $S$-weights in the Lie algebra of $Q$, namely
$Q=P_{G}(\mu)$ for any cocharacter $\mu\in\vec{\delta}(F)$.

We now prove that the pointwise stabiliser $\hat{P}_{\delta(F)}$ is Zariski-
dense in $Q$. Fix a special vertex $o$ in $A(G,S)$ and the associated
valuation

$(\varphi_{\alpha}:U_{\alpha}(k)\to\mathbb{R})_{\alpha\in\Phi(G,S)}$

of the generating root datum
$(Z_{G}(S)(k),(U_{\alpha}(k))_{\alpha\in\Phi(G,S)})$. Denote by $x$ the vertex
of the cone $\delta(F)$. By [BT72, Définition 7.1.8 and Proposition 7.4.4],
the pointwise stabiliser $\hat{P}_{\delta(F)}$ is spanned by the subgroup
$\hat{N}_{\delta(F)}=N_{G}(S)(k)\cap\hat{P}_{\delta(F)}$ and the root
subgroups $U_{\alpha,f_{\delta(F)}(\alpha)}$ for $\alpha\in\Phi$, where

$f_{\delta(F)}(\alpha)=\inf\\{k\in\mathbb{R}\,|\,\forall
z\in\delta(F),\langle\alpha,z-o\rangle+k\geq 0\\}.$

First of all, observe that if $n\in N_{\delta(F)}$, then $n$ fixes the vertex
$x$ of $\delta(F)$ as well as the point $x+\mu$. Hence, the linear part of $n$
fixes $\mu$, that is $n$ commutes with $\mu$. Hence, we have the inclusion
$\hat{N}_{\delta(F)}\subset Z_{G}(\mu)(k)\subset Q(k)$. Next, recall from
[BT72, Proposition 7.4.5] that, if $\alpha\in\Phi(G,S)$ and $u\in
U_{\alpha}(k)\setminus\\{1\\}$, then the set of points of $A(G,S)$ that are
fixed under $u$ is the affine root

$\\{z\in A(G,S),\langle\alpha,z-o\rangle+\varphi_{\alpha}(u)\geq 0\\}.$

It follows that, if $\langle\alpha,\mu\rangle<0$ and $u\in
U_{\alpha}(k)\setminus\\{1\\}$, there exists $t>0$ such that $x+t\mu$ is not
fixed under $u$ and therefore $u$ does not fix $\delta(F)$. Hence, if
$\langle\alpha,\mu\rangle<0$, we have $U_{\alpha,\delta(F)}=\\{1\\}$.
Therefore $\hat{P}_{\delta(F)}$ is spanned by a subgroup of $Z_{G}(\mu)(k)$
and subgroups of $U_{\alpha}(k)$ for $\langle\alpha,\mu\rangle\geq 0$, hence
$\hat{P}_{\delta(F)}\subset Q(k)$. Consequently, the Zariski closure of
$\hat{P}_{\delta(F)}$ lies in the Zariski closure of $Q(k)$, ie. $Q$ by
Proposition 52. On the other hand, if $\langle\alpha,\mu\rangle\geq 0$, we
have
$\langle\alpha,x-o\rangle=\inf\\{\langle\alpha,z-o\rangle,z\in\delta(F)\\}$
and therefore
$f_{\delta(F)}(\alpha)=\inf(\Gamma^{\prime}_{\alpha}\cap[-\langle\alpha,x-o\rangle,+\infty[)<\infty$.
Hence, for each nondivisible $\alpha\in\Phi(Q,S)$, the stabiliser
$\hat{P}_{\delta(F)}$ contains the subgroup
$U_{\alpha,f_{\delta(F)}(\alpha)}U_{2\alpha,f_{\delta(F)}(2\alpha)}$, which is
the set of $k^{\circ}$-points of a smooth connected $k^{\circ}$-model of
$U_{\alpha}$ by [Lou21, Lemme 4.4], and therefore Zariski-dense in
$U_{\alpha}$ by Corollary 53. Moreover, because the maximal compact subgroup
$Z_{G}(S)(k)_{b}$ of $Z_{G}(S)(k)$ acts trivially on the apartment $A(G,S)$,
it fixes $\delta(F)$ and is thus contained in $\hat{N}_{\delta(F)}$. Because
$Z_{G}(S)(k)_{b}$ is the set of integral points of a smooth connected
$k^{\circ}$-model of $Z_{G}(S)$ by [Lou21, Proposition 4.3], it is Zariski-
dense in $Z_{G}(S)$. Consequently, the Zariski closure of
$\hat{P}_{\delta(F)}$ contains $U_{\alpha}$ for each nondivisible
$\alpha\in\Phi(Q,S)$ as well as $Z_{G}(S)$, hence it contains $Q$, which
completes the proof.

It is clear that the above argument is independent of the specific vertex
$x\in A(G,S)$ of the affine cone $\delta(F)$. Hence, for any affine cone
$F^{\prime}\subset A(G,S)$ parallel to $F$, the pointwise stabiliser
$\hat{P}_{\delta(F^{\prime})}$ is Zariski-dense in $Q$.

Lastly, if $\mathfrak{F}\subset\mathcal{B}(G,k)$ is an affine cone that is
parallel to $F$, then there exist parallel subcones $F^{\prime}\subset F$ and
$\mathfrak{F}^{\prime}\subset\mathfrak{F}$ and an apartment $A(G,S^{\prime})$
that contains both $\delta(F^{\prime})$ and $\delta(\mathfrak{F}^{\prime})$.
Then, several applications of the above result yield that the Zariski closures
of the pointwise stabilisers of
$\delta(F),\delta(F^{\prime}),\delta(\mathfrak{F})$, and
$\delta(\mathfrak{F}^{\prime})$ are all equal to $Q$.

∎

######  Corollary 55.

For any maximal split torus $S^{\prime}$ in $G$ such that $A(G,S^{\prime})$
contains $\delta(F^{\prime})$ for some cone $F^{\prime}$ in $\mathcal{B}(G,k)$
parallel to $F$, we have the inclusion $S^{\prime}\subset Q$.

###### Proof.

The pointwise stabiliser $\hat{P}_{\delta(F^{\prime})}$ contains the maximal
compact subgroup $Z_{G}(S^{\prime})(k)_{b}$ of $Z_{G}(S^{\prime})$, which is
the set of $k^{\circ}$-points of a smooth connected $k^{\circ}$-model of
$Z_{G}(S^{\prime})$ by [Lou21, Lemme 4.4] and therefore Zariski-dense in
$Z_{G}(S^{\prime})$ by Corollary 53. It follows that the Zariski closure $Q$
of $\hat{P}_{\delta(F^{\prime})}$ contains $Z_{G}(S^{\prime})$, and therefore
$S^{\prime}$. ∎

#### Action of the pseudo-parabolic subgroup on the façade

We now deduce that $Q(k)$ acts on the façade $\mathcal{B}(G,k)_{F}$ through
its quotient $(Q/\mathcal{R}_{us,k}(Q))(k)=Q(k)/\mathcal{R}_{us,k}(Q)(k)$, the
first step towards identifying the façade with the Bruhat-Tits building of
$Q/\mathcal{R}_{us,k}(Q)$.

######  Proposition 56.

The group $Q(k)$ is the stabiliser of the façade $\mathcal{B}(G,k)_{F}$ in
$G(k)$. Moreover, the subgroup $\mathcal{R}_{us,k}(Q)(k)$ fixes
$\mathcal{B}(G,k)_{F}$ pointwise.

###### Proof.

It follows from the previous proposition that, if $F^{\prime}$ is an affine
cone in an apartment $A(G,S^{\prime})$ such that
$\delta(F^{\prime})/\\!/\delta(F)$, then we have $S^{\prime}\subset Q$. We
check that $Q(k)$ maps the façade $A(G,S^{\prime})_{F^{\prime}}$ into
$\mathcal{B}(G,k)_{F}$. To this end, it is sufficient to check that the
centraliser $Z_{G}(S^{\prime})(k)$ and the root groups $U_{\alpha}(k)$ for
$\alpha\in\Phi(Q,S^{\prime})$ all map $A(G,S^{\prime})_{F^{\prime}}$ into
$\mathcal{B}(G,k)_{F}$. Let $\mu\in X_{*}(S^{\prime})$ be such that
$Q=P_{G}(\mu)$, so that

$\Phi(Q,S^{\prime})=\\{\alpha\in\Phi(G,S^{\prime})\,|\,\langle\alpha,\mu\rangle\geq
0\\}.$

First of all, the subgroup $Z_{G}(S^{\prime})(k)$ acts on $A(G,S^{\prime})$ by
translations, and therefore permutes affine cones in $A(G,S^{\prime})$
parallel to $\delta(F^{\prime})$, hence $Z_{G}(S^{\prime})(k)$ maps
$A(G,S^{\prime})_{F^{\prime}}$ to itself. Next, we deal with root groups. Let
$\alpha\in\Phi(G,S^{\prime})$ such that $\langle\alpha,\mu\rangle\geq 0$ and
$u\in U_{\alpha}(k)$. By [BT72, Proposition 7.4.5], there exists a half-space
$H$ in $A(G,S^{\prime})$ with direction

$\vec{H}=\\{x\in A(G,S^{\prime})\,|\,\langle\alpha,x\rangle\geq 0\\}$

that $u$ fixes pointwise. Denote by $x^{\prime}$ the vertex of $F^{\prime}$.
There exists $v\in X_{*}(S^{\prime})$ such that $x+v\in H$ and therefore
$x+v+\vec{\delta}(F^{\prime})\subset H$. Then, because $u$ induces an affine
map from $A(G,S^{\prime})$ to $A(uS^{\prime}u^{-1},k)$ that fixes $H$, the
cone $u(\delta(F^{\prime}))=\delta(u(F^{\prime}))$ is parallel to
$u(x+v+\vec{\delta}(F^{\prime}))=x+v+\vec{\delta}(F^{\prime})$, and therefore
to $\delta(F^{\prime})$. Therefore, $u(F^{\prime})$ is parallel to
$F^{\prime}$ and we have $u([F^{\prime}])\in\mathcal{B}(G,k)_{F}.$ Moreover,
if $\langle\alpha,\mu\rangle>0$, then we may choose the cocharacter $v$ of the
form $n\mu$ for some sufficiently large $n\in\mathbb{N}$. Then, the cone
$x+n\mu+\vec{\delta}(F^{\prime})$ is a parallel subcone of $F^{\prime}$ that
is fixed by $u$, hence, in that case, $u$ fixes the equivalence class
$[F^{\prime}]$.

We have thus proved that the group $Q(k)$ leaves $\mathcal{B}(G,k)_{F}$
globally invariant and that the subgroup $\mathcal{R}_{us,k}(Q)(k)$, which is
spanned by the $U_{\alpha}(k)$ for $\langle\alpha,\mu\rangle>0$, fixes
$\mathcal{B}(G,k)_{F}$ pointwise.

Assume that the stabiliser $\operatorname{Stab}_{G(k)}(\mathcal{B}(G,k)_{F})$
properly contains $Q(k)$. Then, by [CGP15, Theorem C.2.23], there exists a
pseudo-parabolic subgroup $Q^{\prime}\supsetneq Q$ such that

$Q^{\prime}(k)=\operatorname{Stab}_{G(k)}(\mathcal{B}(G,k)_{F}).$

Then, there exists a root $\alpha\in\Phi(Q^{\prime},S)$ such that
$\alpha\not\in\Phi(Q,S)$, ie. $\langle\alpha,\mu\rangle<0$. Then, we have
$-\alpha\in\Phi(Q,S)\subset\Phi(Q^{\prime},S)$, hence $Q^{\prime}(k)$ contains
the root groups $U_{\alpha}(k)$ and $U_{-\alpha}(k)$. Consequently, by [CGP15,
Proposition C.2.24], the subgroup $Q^{\prime}(k)$ contains an element $m\in
N_{G}(S)(k)$ that lifts the reflection $s_{\alpha}\in W(G,S)(k)$ with respect
to the hyperplane $H_{\alpha}=\\{x\in V(G,S)\,|\,\langle\alpha,x\rangle=0\\}$.
But the cone $\vec{\delta}(F)$ is contained in the open half-space $\\{x\in
V(G,S)\,|\,\langle\alpha,x\rangle<0\\}$ and is thus not invariant under
$s_{\alpha}$, hence $m$ maps $\delta(F)$ to a cone with direction
$s_{\alpha}(\vec{\delta}(F))\neq\vec{\delta}(F)$, hence $m$ maps the
equivalence class $[F]\in\mathcal{B}(G,k)_{F}$ outside of the façade
$\mathcal{B}(G,k)_{F}$, a contradiction. ∎

#### Recovering the façade

Lastly, we identify the façade $\mathcal{B}(G,k)_{F}$ with the Bruhat-Tits
building of $Q/\mathcal{R}_{us,k}(Q)$ (under the guise of a Levi subgroup
$L_{Q}$ of $Q$).

######  Proposition 57.

Let $S$ be a maximal split torus such that $\delta(F)\subset A(G,S)$ and
$L_{Q}$ be the Levi subgroup of $Q$ that contains $S$. Let $f=F\cap A(G,S)$.

  1. 1.

There exists a unique continuous $L_{Q}(k)$-equivariant surjection

$p:L_{Q}(k)\cdot A(G,S)\twoheadrightarrow\mathcal{B}(G,k)_{F}$

that extends the projection $A(G,S)\twoheadrightarrow A(G,S)_{f}$.

  2. 2.

The map $p$ factors through a $L_{Q}(k)$-equivariant simplicial homeomorphism

$\bar{p}:\mathcal{B}(L_{Q},k)\overset{\sim}{\to}\mathcal{B}(G,k)_{F}.$

###### Proof.

  1. 1.

Uniqueness follows directly from the $L_{Q}(k)$-equivariance of $p$. Our goal
is now to prove that the $L_{Q}(k)$-equivariant map

$\begin{array}[]{ccc}p:L_{Q}(k)\cdot
A(G,S)&\longrightarrow&\mathcal{B}(G,k)_{F}\\\
x&\longmapsto&\lim\limits_{t\to+\infty}\left(x+t\mu\right)\end{array}$

is well-defined, where the operator $+$ denotes the action of the vector space

$V=\\{x\in
V(G,S)\,|\,\forall\alpha\in\Phi(G,S),\langle\alpha,\mu\rangle=0\implies\langle\alpha,x\rangle=0\\}$

on $L_{Q}(k)\cdot A(G,S)$ defined in [BT72, Proposition 7.6.4 (iii)] and
described in the Appendix.

Let $S^{\prime}$ be a maximal split torus in $L_{Q}$ and $x\in
A(G,S^{\prime})$. For each root $\alpha\in\Phi(G,S^{\prime})$, we have

$\begin{array}[]{ccc}\langle\alpha,x+t\mu\rangle\underset{t\to+\infty}{\longrightarrow}+\infty&\text{
if }&\langle\alpha,\mu\rangle>0\end{array}$

and

$\begin{array}[]{ccc}\langle\alpha,x+t\mu\rangle\underset{t\to+\infty}{\longrightarrow}\langle\alpha,x\rangle&\text{
if }&\langle\alpha,\mu\rangle=0\end{array}$

hence, by [Cha10, Proposition 3.2.4] , the limit
$p(x)=\lim_{t\to+\infty}(x+t\mu)$ exists and lies in the façade
$A(G,S^{\prime})_{\vec{f}^{\prime}}\subset\mathcal{B}(G,k)_{F}$, where
$\vec{f}^{\prime}\subset X_{*}(S^{\prime})_{\mathbb{R}}$ is the cone in the
fan $\mathcal{F}^{P}$ that contains $\vec{\delta}(F)$. More precisely, for
each $S^{\prime}\subset L_{Q}$, the restriction of $p$ to the apartment
$A(G,S^{\prime})$ is the projection $x\mapsto[x+\vec{f}^{\prime}]$ of
$A(G,S^{\prime})$ onto its façade of type $\vec{f}^{\prime}$ [Cha10,
Définition 3.1.3]. In particular, the image of the map $p$ is the union
$\bigcup_{S^{\prime}\subset L_{Q}}A(G,S^{\prime})_{\vec{f}^{\prime}}$ over
maximal split tori $S^{\prime}\subset L_{Q}$. To prove that this union is the
entire façade $\mathcal{B}(G,k)_{F}$, recall from [Cha10, Proposition 8.1.5]
that the façade $\mathcal{B}(G,k)_{F}$ is the union of the façades
$A(G,S^{\prime})_{f^{\prime}}$ for apartments $A(G,S^{\prime})$ containing a
cone $f^{\prime}$ such that $\widetilde{f}^{\prime}/\\!/F$. Let
$A(G,S^{\prime})$ be an apartment containing a cone $f^{\prime}$ such that
$\widetilde{f}^{\prime}/\\!/F$. By Corollary 55, there exists $q\in Q(k)$ such
that $A(G,S^{\prime})=qA(G,S)$. Then, because $Q(k)$ maps
$\mathcal{B}(G,k)_{F}$ to itself, we must have
$A(G,S^{\prime})_{f^{\prime}}=qA(G,S)_{f}$. Write $q=mu$ with $m\in L_{Q}(k)$
and $u\in\mathcal{R}_{u,k}(Q)(k)$. Because $\mathcal{R}_{u,k}(Q)(k)$ acts
trivially on the $\mathcal{B}(G,k)_{F}$ by Proposition 56, we then have
$A(G,S^{\prime})_{f^{\prime}}=mA(G,S)_{f}$. Hence, we have

$\mathcal{B}(G,k)_{F}=L_{Q}(k)\cdot A(G,S)_{f}$

and the map $p$ is therefore surjective. Finally, the map $p$ is
$L_{Q}(k)$-equivariant because the $V$-action on $L_{Q}(k)\cdot A(G,S)$ is
continuous and commutes with the action of $L_{Q}(k)$.

  2. 2.

Recall from [BT72, Proposition 7.6.4 (i)] that we have a canonical
$L_{Q}(k)$-equivariant map $\widetilde{\pi}:L_{Q}(k)\cdot
A(G,S)\twoheadrightarrow\mathcal{B}(L_{Q},k)$ that extends the projection
$A(G,S)\twoheadrightarrow A(L_{Q},S)$. Moreover, the map $\widetilde{\pi}$ is
invariant under the action of $V$ on $L_{Q}(k)\cdot A(G,S)$ and it follows
from [BT72, Proposition 7.6.4 (iii)] that any $V$-invariant map with domain
$L_{Q}(k)\cdot A(G,S)$ factors through $\widetilde{\pi}$. Now, observe that,
if $S^{\prime}\subset L_{Q}$ is a maximal split torus, $x\in A(G,S^{\prime})$,
$\mu\in X_{*}(S^{\prime})$ such that $Q=P_{G}(\mu)$, and $v\in V$, then for
each $\alpha\in\Phi(G,S^{\prime})$, we have

$\begin{array}[]{ccc}\langle\alpha,x+v+t\mu\rangle\underset{t\to+\infty}{\longrightarrow}+\infty&\text{
if }&\langle\alpha,\mu\rangle>0\end{array}$

and

$\begin{array}[]{ccc}\langle\alpha,x+v+t\mu\rangle\underset{t\to+\infty}{\longrightarrow}\langle\alpha,x\rangle&\text{
if }&\langle\alpha,\mu\rangle=0\end{array},$

and therefore $p(x+v)=p(x)$. The map $p$ is thus invariant under $V$ and, by
the above discussion, there exists a map

$\overline{p}:\mathcal{B}(L_{Q},k)\to\mathcal{B}(G,k)_{F}$

such that $p$ admits the factorisation

${L_{Q}(k)\cdot
A(G,S)}$${\mathcal{B}(G,k)_{F}}$${\mathcal{B}(L_{Q},k)}$$\scriptstyle{\widetilde{\pi}}$$\scriptstyle{p}$$\scriptstyle{\overline{p}}$

.

The map $\overline{p}$ is surjective because $p$ is, and
$L_{Q}(k)$-equivariant because $p$ and $\widetilde{\pi}$ are
$L_{Q}(k)$-equivariant and surjective. The restriction of $\overline{p}$ to
the apartment $A(L_{Q},S)$ is a surjective affine map between affine spaces of
the same dimension, and therefore one-to-one. The $L_{Q}(k)$-equivariance of
$\overline{p}$ then implies that its restriction to each apartment is one-to-
one, and therefore that $\overline{p}$ is globally one-to-one – as any two
points in $\mathcal{B}(L_{Q},k)$ are contained in an apartment – and finally
that $\overline{p}$ is bijective. To prove that $\overline{p}$ is a
homeomorphism, we first observe that the restriction of $\overline{p}$ to
$A(L_{Q},S)$, a bijective affine map, is a homeomorphism onto its image. Next,
we recall that if $x\in A(L_{Q},S)$, then, by [Sol18, Theorem 3.11 (d) and
(e)], we have

$\mathcal{B}(L_{Q},k)=L_{Q}(k)_{x}\cdot A(L_{Q},S).$

Further, because $Z(L_{Q})(k)$ acts trivially on the building
$\mathcal{B}(L_{Q},k)$ and is contained and cocompact in the stabiliser
$L_{Q}(k)_{x}$ by [Sol18, Proposition 3.9 (a) and (b)], we may write

$\mathcal{B}(L_{Q},k)=\underbrace{\frac{L_{Q}(k)_{x}}{Z(L_{Q})(k)}}_{\text{compact}}\cdot
A(L_{Q},S).$

Now, because $\bar{p}$ is $L_{Q}(k)$-equivariant, the action of $Z(Q)(k)$ on
$\mathcal{A}(G,S)_{f}$ is also trivial and we also have

$\mathcal{B}(G,k)_{F}=\frac{L_{Q}(k)_{x}}{Z(L_{Q})(k)}\cdot A(G,S)_{f}.$

Let $(u_{n})_{n\in\mathbb{N}}$ be a sequence of points in
$\mathcal{B}(L_{Q},k)$ that converges to a point $u\in\mathcal{B}(L_{Q},k)$
and, for each $n\in\mathbb{N}$, write $u_{n}=k_{n}x_{n}$ with $k_{n}\in
L_{Q}(k)_{x}/Z(L_{Q})(k)$ and $x_{n}\in A(G,S)$. Let
$\varphi:\mathbb{N}\to\mathbb{N}$ be an increasing function. Then, because
$L_{Q}(k)_{x}/Z(L_{Q})(k)$ is compact, we may extract from $k_{\varphi(n)}$ a
subsequence $k_{\varphi\circ\psi(n)}$ that converges to a point $k\in
L_{Q}(k)_{x}/Z(L_{Q})(k)$. Then, the sequence $x_{\varphi\circ\psi(n)}$
converges to $x=k^{-1}u$ and, by continuity of the $L_{Q}(k)$-action and of
the restriction of $\overline{p}$ to $A(G,S)$, we may write

$\overline{p}(u_{\varphi\circ\psi(n)})=k_{\varphi\circ\psi(n)}\overline{p}(x_{\varphi\circ\psi(n)})\underset{n\to\infty}{\longrightarrow}k\overline{p}(x)=\overline{p}(u).$

We have thus proved that, from any subsequence of $(\overline{p}(u_{n}))$ we
may extract a subsubsequence that converges to $\overline{p}(u)$, hence
$(\overline{p}(u_{n}))$ converges to $\overline{p}(u)$. Hence, the map
$\overline{p}$ is continuous. Because the above argument only relied on
$L_{Q}(k)$-equivariance and the continuity of the restriction to an apartment,
it also applies to $(\overline{p})^{-1}$, hence $\overline{p}$ is a
homeomorphism.

Lastly, we check that $\overline{p}$ is simplicial, which amounts to proving
that it maps walls in $\mathcal{B}(L_{Q},k)$ to walls in
$\mathcal{B}(G,k)_{F}$. By $L_{Q}(k)$-equivariance, it is sufficient to check
this property in our distinguished apartment $A(L_{Q},S)$. By [Cha10, Section
3.3.4], the walls in $A(G,S)_{f}$ are the images under $p$ of the walls in
$A(G,S)$ whose direction contains the cone $\vec{f}$ or, equivalently the
subspace $V=\operatorname{Span}_{\mathbb{R}}(\vec{f})$. Now, if $H\subset
A(G,S)$ is a wall, then we have $\overline{p}(H)=p(\widetilde{\pi}^{-1}(H))$.
By [BT72, Proposition 7.6.4 (ii)], the affine subspace
$\widetilde{\pi}^{-1}(H)$ is a wall of $A(G,S)$ and its direction contains
$V$, as the linear part of $\widetilde{\pi}$ is the quotient map
$V(G,S)\twoheadrightarrow V(L_{Q},S)=V(G,S)/V$. Hence, the affine subspace
$\overline{p}(H)$ is a wall in $A(G,S)_{f}$, which completes the proof.

∎

#### Relevant pseudo-parabolic subgroups

We have thus proved that the stratification of
$\overline{\vartheta_{P}(\mathcal{B}(G,k))}$ is indexed by pseudo-parabolic
subgroups of $G$. In this last section, we characterise those pseudo-parabolic
subgroups that give rise to a stratum in terms of $P$, thereby completing our
description.

Fix a maximal split torus $S$ of $G$ and a minimal pseudo-parabolic subgroup
$P_{0}$ such that $S\subset P_{0}\subset P$. Let $\Phi=\Phi(G,S)$ be the root
system associated to $S$ and $\Delta\subset\Phi$ be the system of simple roots
corresponding to $P_{0}$. We endow $\Delta$ with its Dynkin diagram structure.
All subsets of $\Delta$ thus inherit an induced graph structure. Denote by $J$
the type of $P$, that is the set of simple roots $J=\Delta\cap-\Phi(P,S)$. We
have proved in Section 4.2 that $\overline{\vartheta_{P}(\mathcal{B}(G,k))}$
is the polyhedral compactification in the sense of [Cha10] associated with the
fan $\mathcal{F}^{J}$ of [Cha10, Définition 2.5.2].

In keeping with the terminology of [RTW10], we give the following definition:

######  Definition 58.

Let $Q$ be a pseudo-parabolic subgroup in $G$. By [CP17, Theorem 5.1.3 and
Proposition 5.1.4 (iii)], there exists a unique pseudo-parabolic subgroup $Q$
in the conjugacy class of $Q$ such that $P_{0}\subset Q$. Then, we say that
$Q$ is $J$-relevant if there exists $I\subset\Delta$, none of whose connected
components is contained in $J$, such that the type of $Q^{\prime}$ is
$I\sqcup(J\cap I^{\perp})$.

Then, we have

######  Proposition 59.

Let $Q$ be a pseudo-parabolic subgroup in $G$. The following are equivalent:

  1. 1.

The subgroup $Q$ is $J$-relevant.

  2. 2.

There exists a cone $F$ in $(\mathcal{F}^{J})_{\mathcal{B}(G,k)}$ such that
$Q(k)$ is the stabiliser in $G(k)$ of the stratum $\mathcal{B}(G,k)_{F}$.

###### Proof.

Because both statements are invariant under $G(k)$-conjugacy, we may assume
that $Q$ contains $P_{0}$. Then, the equivalence essentially follows from the
second example in [Cha10, Section 3.3.2]. Namely, the proof of Proposition 54
and Proposition 56 imply that $Q$ is the stabiliser of a façade
$\mathcal{B}(G,k)_{F}$ for some $F\in(\mathcal{F}^{J})_{\mathcal{B}(G,k)}$ if
and only if, for any apartment $A(G,S^{\prime})$ with $S^{\prime}\subset Q$,
the Weyl facet

$\vec{\delta}=\\{x\in
V(G,S^{\prime})\,|\,\forall\alpha\in\Phi(Q,S^{\prime}),\langle\alpha,x\rangle\geq
0\\}$

is the direction of the core of an affine cone in
$(\mathcal{F}^{J})_{A(G,S^{\prime})}$.

Now, by [Cha10, Section 3.3.2], the directions of the cores of the cones in
$(\mathcal{F}^{J})_{A(G,S)}$ are precisely the Weyl facets whose type is of
the form $I\cup(J\cap I^{\perp})$ for some $I\subset\Delta$ none of whose
connected component is contained in $J$, from which we deduce both
implications. ∎

We have thus proved the following theorem.

######  Theorem 60.

The compactification $\overline{\vartheta_{P}(\mathcal{B}(G,k))}$ is
$G(k)$-equivariantly homeomorphic to the disjoint union

$\overline{\vartheta_{P}(\mathcal{B}(G,k))}=\bigsqcup_{\begin{subarray}{c}Q\subset
G\\\ Q\,J\text{-relevant pseudo-
parabolic}\end{subarray}}\mathcal{B}(Q/\mathcal{R}_{us,k}(Q),k).$

For each $J$-relevant pseudo-parabolic subgroup $Q$ of $G$, the group $Q(k)$
is the stabiliser of the stratum $\mathcal{B}(Q/\mathcal{R}_{us,k}(Q),k)$.

For the sake of completeness, we now include the comparison between our
definition of $J$-relevant and the characterisation given in [RTW10,
Proposition 3.24 (ii)], thereby proving that the terminology is consistent in
the reductive case.

######  Proposition 61.

Let $Q$ be a pseudo-parabolic subgroup of $G$ containing $P_{0}$. Denote by
$T\subset\Delta$ its type. Endow $T$ with the graph structure induced by the
Dynkin diagram structure on $\Delta$. Let $\widetilde{T}$ be the union of the
connected components of $T$ that meet $\Delta\setminus J$. The following are
equivalent:

  1. 1.

$Q$ is $J$-relevant

  2. 2.

For any simple root $\alpha\in\Delta$, if $\alpha\in J$ and
$\alpha\perp\widetilde{T}$, then $\alpha\in T$.

###### Proof.

Assume that $Q$ is $J$-relevant. Then, there exists $I$ such that
$T=I\sqcup(J\cap I^{\perp})$. Observe that $I$ and $J\cap I^{\perp}$ are
disconnected for the graph structure on $T$, so that the connected components
of $T$ are precisely those of $I$ and those of $J\cap I^{\perp}$. Then,
because no connected component of $I$ is contained in $J$, the connected
components of $T$ that meet $\Delta\setminus J$ are precisely the connected
components of $I$. Therefore, a simple root $\alpha$ that is in $J$ and
orthogonal to $\widetilde{T}=I$ is in $J\cap I^{\perp}$ and thus in $T$.

Conversely, if $T$ satisfies condition 2, then setting $I=\widetilde{T}$, no
connected component of $I$ is contained in $J$ by construction, and we have
$T=I\sqcup(J\cap I^{\perp})$. Indeed, condition 2 immediately implies that
$J\cap I^{\perp}\subset T$. On the other hand, if $\alpha\in T\setminus I$
then, because $I$ is a union of connected components, we have $\alpha\in
I^{\perp}$. Moreover, by construction, $T\setminus I$ is the union of the
connected components of $T$ that are contained in $J$, hence $\alpha\in J$,
hence $\alpha\in J\cap I^{\perp}$, hence $T\subset I\sqcup(J\cap I^{\perp})$,
which completes the proof. ∎

##  5 Appendix: Levi subgroups and their inner façades

In the following, we fix a quasi-reductive group $G$ over a discretely valued
field $k$ with a perfect residue field. Let $S$ be a $k$-split torus and
$G_{1}=Z_{G}(S)$ be its centraliser. In this section, we relate the Bruhat-
Tits building of $G_{1}$ to that of the ambient group $G$, which essentially
amounts to rephrasing the results of [BT72, §7.6] in our particular context.

### The inner façade

Observe that the apartments in the Bruhat-Tits building $\mathcal{B}(G_{1},k)$
are in one-to-one correspondence with the maximal $k$-split tori of $G_{1}$,
that is the maximal $k$-split tori of $G$ that contain $S$. Therefore, a
maximal $k$-split torus $T$ of $G$ containing $S$ gives rise to two
apartments, namely $A(G,T)$ and $A(G_{1},T)$, and two root systems
$\Phi=\Phi(G,T)$, and $\Phi_{1}=\Phi(G_{1},T)$. Moreover, these apartments are
related by the map

$\pi:A(G,T)\twoheadrightarrow A(G_{1},T)$

that associates to each valuation $\varphi=(\varphi_{\alpha})_{\alpha\in\Phi}$
in $A(G,T)$ the family $\varphi_{1}=(\varphi_{\alpha})_{\alpha\in\Phi_{1}}$,
which is established in [BT72, Proposition 7.6.3] to be a valuation of the
generating root data $(Z_{G}(T)(k),(U_{\alpha}(k))_{\alpha\in\Phi_{1}})$ in
$G_{1}(k)$. The map $\pi$ is well-defined as it preserves equipollence,
affine, and its linear part is the quotient map

$V(G,T)=\frac{X_{*}(T)_{\mathbb{R}}}{\operatorname{Span}_{\mathbb{R}}(\Phi)^{\perp}}\twoheadrightarrow
V(G_{1},T)=\frac{X_{*}(T)_{\mathbb{R}}}{\operatorname{Span}_{\mathbb{R}}(\Phi_{1})^{\perp}}.$

Further, if $T$ is a maximal $k$-split torus in $G_{1}$, then the subspace
$\mathcal{B}(G_{1},G,k)$ defined as

$\mathcal{B}(G_{1},G,k)=G_{1}(k)\cdot
A(G,T)=\bigcup_{\begin{subarray}{c}T^{\prime}\text{ max. }k\text{-split torus
of }G\\\ S\subset T^{\prime}\end{subarray}}A(G,T^{\prime})$

may be thought of as a Euclidean extension of the Bruhat-Tits building of
$G_{1}$, in the following sense. By [BT72, Proposition 7.6.4], there exists a
unique $G_{1}(k)$-equivariant map

$\widetilde{\pi}:\mathcal{B}(G_{1},G,k)\to\mathcal{B}(G_{1},k)$

that extends $\pi$. Moreover, it enjoys the following properties:

  1. 1.

We have $\widetilde{\pi}^{-1}(A(G_{1},T))=A(G,T)$, and the inverse image of
each apartment of $\mathcal{B}(G_{1},k)$ is an apartment in
$\mathcal{B}(G,k)$.

  2. 2.

There exists a unique action of the vector space

$V=\ker\big{(}V(G,T)\to
V(G_{1},T)\big{)}=\frac{\operatorname{Span}_{\mathbb{R}}(\Phi_{1})^{\perp}}{\operatorname{Span}_{\mathbb{R}}(\Phi)^{\perp}}$

on $\mathcal{B}(G_{1},G,k)$ that extends the action of $V$ on $A(G,T)$ by
translation and commutes with the action of $G_{1}(k)$ in the sense that, for
each $x\in\mathcal{B}(G_{1},G,k)$, $g\in G_{1}(k)$, and $v\in V$, we have

$g\cdot(x+v)=g\cdot x+v.$

Then, the map $\widetilde{\pi}$ is constant on $V$-orbits and factors through
a bijection
$\mathcal{B}(G_{1},G,k)/V\overset{\sim}{\longrightarrow}\mathcal{B}(G_{1},k)$.

To understand the action of $V$ more concretely, note that we have

$V=\frac{X_{*}(Z(G_{1}))\otimes_{\mathbb{Z}}\mathbb{R}}{X_{*}(Z(G))\otimes_{\mathbb{Z}}\mathbb{R}}.$

Hence, $V$ naturally sits inside $V(G,T^{\prime})$ for each maximal $k$-split
torus $T^{\prime}$ containing $S$. In particular, the vector space $V$ acts on
each $A(G,T^{\prime})$ by translation. Then, the action of $V$ on
$\mathcal{B}(G_{1},G,k)$ restricts to the translation action on
$A(G,T^{\prime})$ for each maximal $k$-split torus $T^{\prime}$ in $G_{1}$.

######  Remark 62.

With the terminology of [Rou20], the subspace $\mathcal{B}(G_{1},G,k)$ endowed
with the polysimplicial structure given by the affine roots with direction in
$\Phi_{1}$ is an inessential affine building [Rou20, 2.3.4]. It is the inner
façade associated to the sector face support $V$ [Rou20, 3.3.14]. Point 2
translates in that language to the statement that the map $\widetilde{\pi}$
realises the essentialisation map of $\mathcal{B}(G_{1},G,k)$ [Rou20, 2.1.12].

### Inner façades and field extensions

For each discretely valued extension $K/k$ with perfect residue field, we set

$\mathcal{B}(G_{1},G,K):=\mathcal{B}((G_{1})_{K},G_{K},K)=G_{1}(K)\cdot
A(G_{K},T)$

for $T$ any maximal $K$-split torus of $G_{K}$ containing $S_{K}$.

In the case where $K/k$ is Galois, we establish that the action of
$\operatorname{Gal}(K/k)$ on $\mathcal{B}(G,K)$ stabilises
$\mathcal{B}(G_{1},G,K)$, and that both the essentialisation map and the
$V$-action are compatible with the Galois action.

######  Proposition 63.

Let $K/k$ be a Galois extension of discretely valued fields with perfect
residue fields. Let $T$ be a maximal $K$-split torus containing $S_{K}$. Let

$V^{K}=\ker(V(G_{K},T)\to V((Z_{G}(S))_{K},T)).$

Then

  1. 1.

The action of $\operatorname{Gal}(K/k)$ on $\mathcal{B}(G,K)$ stabilises
$\mathcal{B}(G_{1},G,K)$.

  2. 2.

The map $\widetilde{\pi}_{K}:\mathcal{B}(G_{1},G,K)\to\mathcal{B}(G_{1},K)$ is
$\operatorname{Gal}(K/k)$-equivariant.

  3. 3.

The action of $V^{K}$ on $\mathcal{B}(G_{1},G,K)$ commutes with the action of
$\operatorname{Gal}(K/k)$ in the following sense: For each $v\in V^{K}$, each
$x\in\mathcal{B}(G_{1},G,K)$, and each $\sigma\in\operatorname{Gal}(K/k)$, we
have

$\sigma(x+v)=\sigma(x)+\sigma(v).$

###### Proof.

  1. 1.

This is clear as the action of $\operatorname{Gal}(K/k)$ stabilises the set of
maximal $K$-split tori containing $S_{K}$.

  2. 2.

Let $\sigma\in\operatorname{Gal}(K/k)$ and consider the automorphisms

$p_{\sigma}:\mathcal{B}(G,K)\to\mathcal{B}(G,K)$

and

$p^{G_{1}}_{\sigma}:\mathcal{B}(G_{1},K)\to\mathcal{B}(G_{1},K).$

Then, the map $p_{\sigma}^{G_{1}}\circ\widetilde{\pi}^{K}\circ
p_{\sigma}^{-1}$ is $G(K)$-equivariant and restricts to an affine map from
$A(G_{K},T)$ to $A((G_{1})_{K},T)$. Moreover, its restriction to $A(G_{K},T)$
is the map

$\pi^{K}:A(G_{K},T)\to A((G_{1})_{K},T)$

described in the previous section. Indeed, if we fix a valuation
$(\varphi_{\alpha})_{\alpha\in\Phi(G_{K},T)}$ corresponding to a point in
$A(G_{K},T)$, then it follows from Proposition 2 that
$p_{\sigma}\circ\widetilde{\pi}^{K}\circ(p^{G_{1}}_{\sigma})^{-1}$ maps
$(\varphi_{\alpha})$ to $(\varphi_{\alpha})_{\alpha\in\Phi((G_{1})_{K},T)}$.
Moreover, it follows from Point 2 of Theorem 1 that the linear part of
$p_{\sigma}\circ\widetilde{\pi}^{K}\circ(p^{G_{1}}_{\sigma})^{-1}$ is the same
as that of $\pi^{K}$, which completes the proof. We thus find that

$p_{\sigma}^{G_{1}}\circ\widetilde{\pi}^{K}=\widetilde{\pi}^{K}\circ
p_{\sigma}.$

  3. 3.

This follows directly from the fact that the action of
$\operatorname{Gal}(K/k)$ on each apartment is affine and that $V^{K}$ acts on
each apartment by translation.

∎

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