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\title{Compactly supported cohomology of a tower of graphs and generic representations of $\mathrm{PGL}_{n}$ over a local field}

\author{\firstname{Anis} \lastname{Rajhi}}
\address{Department of quantitative methods, Higher business school of Tunis, Manouba University, Manouba 2010, Tunisia}
\address{Department of mathematics, Faculty of Sciences and Humanities in Dawadmi, Shaqra University, 11911 Shaqra, Saudi Arabia}
\email{anis.rajhi.maths@gmail.com}

\subjclass{22E50, 20E42}

\begin{abstract}
Let $\mathrm{F}$ be a non-archimedean locally compact field and let $\mathrm{G}_{n}$ be the group $\mathrm{PGL}_{n}(\mathrm{F})$. In this paper we construct a tower $(\tilde{\mathrm{X}}_{k})_{k\geqslant0}$ of graphs fibred over the one-skeleton of the Bruhat--Tits building of $\mathrm{G}_{n}$. We prove that a non-spherical and irreducible generic complex representation of $\mathrm{G}_{n}$ can be realized as a quotient of the compactly supported cohomology of the graph $\tilde{\mathrm{X}}_{k}$ for $k$ large enough. Moreover, when the representation is cuspidal then it has a unique realization in a such model.
\end{abstract}

\begin{document}
\maketitle

\section{Introduction}\label{sec1}

Let $\F$ be a non-archimedean locally compact field and let $\G_{n}$ be the locally profinite group $\PGL_{n}(\F)$. In~\cite{Br04}, P. Broussous has constructed a projective tower of simplicial complexes fibred over the Bruhat--Tits building of $\G_{n}$. The idea (due to P. Schneider) consists of constructing simplicial complexes whose structure is very related to that of the Bruhat--Tits building. The goal of a such construction is to try to find geometric interpretation of certain classes of irreducible smooth representations of $\G_{n}$. Such a geometric interpretation exists for example for the Steinberg representation of $\G_{n}$ which can be realized (see~\cite[Thm.~3]{BS71}) as the cohomology with compact support in top dimension of the Bruhat--Tits building. In a second work (see~\cite{Br09}), P. Broussous has constructed in the case $n=2$ a slightly modified version of his previous construction. More precisely, he construct a tower of directed graphs $(\widetilde{\X}_{k})_{k\geqslant0}$ fibred over the Bruhat--Tits tree of $\G_{2}$. Based on the existence of new vectors for irreducible generic representations of $\G_{2}$, he proves that an irreducible generic representation $\pi$ of $\G_{2}$ can be realized as a quotient of the compactly supported cohomology space $H_{c}^{1}(\widetilde{\X}_{c(\pi)},\mathbb{C})$, where $c(\pi)$ is an integer related to the conductor of the representation $\pi$. He proves moreover that if $\pi$ is cuspidal then it can be realized as a subrepresentation of the last cohomology space and that a such realization is unique. In a parallel direction, the author has constructed a simplicial complex fibred over the Bruhat--Tits building of $\G_{n}$ whose top compactly supported cohomology realize as subquotient all the irreducible cuspidal level zero representations of $\G_{n}$, see~\cite{Ra18}.

In this paper our aim is to generalize the construction of Broussous given in~\cite{Br09} to the case $n\geqslant3$. More precisely we construct a projective tower $(\widetilde{\X}_{k})_{k\geqslant0}$ of directed graphs fibred over the 1-skeleton of the Bruhat--Tits building of $\G_{n}$. In our construction, the graphs considered will be defined in terms of combinatorial geodesic paths of the Bruhat--Tits building of $\G_{n}$.

Let $\pi$ be an irreducible smooth generic and non-spherical representation of $\G_{n}$. We prove that there exists an injective intertwining operator
\[
\Psi^{\vee}_{\pi}:V^{\vee}\longrightarrow \mathcal{H}_{\infty}(\widetilde{\X}_{c(\pi)},\mathbb{C}),
\]
where $V^{\vee}$ is the contragredient representation of $\pi$ and $\mathcal{H}_{\infty}(\widetilde{\X}_{c(\pi)},\mathbb{C})$ is the space of smooth harmonic forms on the graph $\widetilde{\X}_{c(\pi)}$. By applying contragredients to this intertwining operator and then by restriction to $H_{c}^{1}(\widetilde{\X}_{c(\pi)},\mathbb{C})$ we obtain a nonzero intertwining operator
\[
\Psi_{\pi}:H_{c}^{1}(\widetilde{\X}_{c(\pi)},\mathbb{C})\longrightarrow V.
\]
That is the representation $\pi$ is isomorphic to a quotient of $H_{c}^{1}(\widetilde{\X}_{c(\pi)},\mathbb{C})$. In the case when $\pi$ is cuspidal, the $\G_{n}$-equivariant map $\Psi_{\pi}$ splits so that $\pi$ injects in $H_{c}^{1}(\widetilde{\X}_{c(\pi)},\mathbb{C})$. We prove that such an injection is unique, that is :
\[
\dim_{\mathbb{C}}\Hom_{\G_{n}}(\pi,H_{c}^{1}(\widetilde{\X}_{c(\pi)},\mathbb{C}))=1.
\]


\section{Notations and preliminaries}\label{sec2}

In this article, $\F$ will be a non-archimedean locally compact field. We write $\of_{\F}$ for the ring of integers of $\F$, $\pf_{\F}$ for the maximal ideal of $\of_{\F}$, $k_{\F}:=\of_{\F}/\pf_{\F}$ for the residue class field of $\F$ and $q_{\F}$ for the cardinal of $k_{\F}$. We fix a normalized uniformizer $\varpi_\F$ of $\of_{\F}$ and we denote by $\upsilon_\F$ the normalized valuation of $\F$.

\subsection{The projective general linear group \texorpdfstring{$\PGL_{n}(\F)$}{PGLn(F)}}

For every integer $n\geqslant2$, the projective general linear group $\PGL_{n}(\F)$ will be denoted by $\G_{n}$. If $k\geqslant1$ is an integer, we write $\tilde{\Gamma}_{0}(\pf_{\F}^{k})$ for the following subgroup of $\GL_{n}(\F)$
\begin{equation}\label{11}
\tilde{\Gamma}_{0}(\pf_{\F}^{k})=\left\{
\matrice{a}{b}{c}{d}\in \GL_{n}(\of_{\F}) \,\middle|\, a\in \GL_{n-1}(\of_{\F}),\ d\in \of_{\F}^\times,\ c\equiv0\bmod\pf_{\F}^{k}
\right\}
\end{equation}
and we write $\Gamma_{0}(\pf_{\F}^{k})$ for its image in $\G_{n}$. We denote also the image in $\G_{n}$ of the standard maximal compact subgroup of $\GL_{n}(\F)$ by $\Gamma_{0}(\pf_{\F}^{0})$.

\subsection{The Bruhat--Tits building of \texorpdfstring{$\G_{n}$}{Gn}}

In this section we fix some notations and recall some well-known facts. For more details the reader may refer to~\cite{AB08}, \cite{Ga97} or~\cite{Ro09}. Recall that a lattice of the vector space $\F^{n}$ is an open compact subgroup of the additive group of $\F^{n}$. A such lattice is an $\of_{\F}$-lattice if moreover it is an $\of_{\F}$-submodule of $\F^{n}$. Equivalently, an $\of_{\F}$-lattice of $\F^{n}$ is a free $\of_{\F}$-submodule $\Lrm$ of $\F^{n}$ of rank $n$. If $\Lrm$ is an $\of_{\F}$-lattice of $\F^{n}$ then $\Lrm=\of_{\F}f_{1}+\dots+\of_{\F}f_{n}$ for some $\F$-basis of $\F^{n}$. More generally if $\Lrm$ and $\M$ are two $\of_{\F}$-lattices of $\F^{n}$ then there exist an $\F$-basis $(f_{1},\dots,f_{n})$ of $\F^{n}$ and $(\alpha_{1},\dots,\alpha_{n})\in\mathbb{Z}^{n}$, with $\alpha_{1}\leqslant\dots\leqslant\alpha_{n}$, such that
\[
\Lrm=\of_{\F}f_{1}+\dots+\of_{\F}f_{n}\quad\text{and}\quad\M=\pf^{\alpha_{1}}_{\F}f_{1}+\dots+\pf^{\alpha_{n}}_{\F}f_{n}.
\]
For two $\of_{\F}$-lattices $\Lrm$ and $\M$ of $\F^{n}$, we say that $\Lrm$ and $\M$ are equivalent if $\Lrm=\lambda\M$ for some $\lambda\in\F^{\times}$, and we denote the class of $\Lrm$ by $[\Lrm]$. The Bruhat--Tits building of $\G_{n}$, denoted by $\BT$, can be defined as the simplicial complex whose vertices are the equivalence classes of $\of_{\F}$-lattices in $\F^{n}$ and in which a collection $\Lambda_{0},\Lambda_{1},\dots,\Lambda_{q}$ of pairwise distinct vertices form a $q$-simplex if we can choose representatives $\Lrm_{i}\in\Lambda_{i}$, for $i\in\{0,\dots,q\}$, such that
\[
\varpi_{\F}\Lrm_{0}<\Lrm_{q}<\Lrm_{q-1}<\dots<\Lrm_{0}
\]
A $q$-simplex as above define the following flag of the $k_{\F}$-vector space $\Lrm_{0}/\varpi_{\F}\Lrm_{0}$
\[
\{0\}<\Lrm_{q}/\varpi_{\F}\Lrm_{0}<\Lrm_{q-1}/\varpi_{\F}\Lrm_{0}<\dots<\Lrm_{0}/\varpi_{\F}\Lrm_{0}
\]
The type of a such $q$-simplex is defined to be the type of the corresponding flag of the $k_{\F}$-vector space $\Lrm_{0}/\varpi_{\F}\Lrm_{0}\simeq k^{n}_{\F}$. Note that the maximal dimension of the flag corresponding to a simplex $\sigma$ of $\BT$ is equal to $n-2$. Thus $\BT$ is a simplicial complex of dimension $n-1$. The group $\GL_{n}(\F)$ acts naturally on $\BT$ by simplicial automorphisms and its center $\Z(\GL_{n}(\F))\simeq\F^{\times}$ acts trivially. So the group $\G_{n}$ acts simplicially on $\BT$ and the action is transitive on vertices (resp. chambers, $q$-simplices of a fixed type). Let's recall that a labelling of $\BT$ is a map from the set $\BTs{0}$ of vertices of $\BT$ to the set $\{0,\dots,n-1\}$ whose restriction to every chamber is injective. We can construct a labelling $\lambda:\BTs{0}\longrightarrow\{0,\dots,n-1\}$ of $\BT$ as follows (see~\cite[19.3]{Ga97}). Let $\Lrm_{0}$ be a fixed $\of_{\F}$-lattices of $\F^{n}$. If $v$ is a vertex of $\BT$, we can choose a representative $\Lrm$ such that $\Lrm_{0}\subset \Lrm$. Since $\of_{\F}$ is a principal ideal domain, the finitely generated torsion $\of_{\F}$-module $\Lrm/\Lrm_0$ is isomorphic to
\[
\of_{\F}/\pf_{\F}^{k_1} \oplus \dots \oplus \of_{\F}/\pf_{\F}^{k_n}
\]
for some $n$-tupe of integers $0\leqslant k_1 \leqslant k_2 \leqslant
\dots \leqslant k_n$. Then
\[
\lambda (v)= \sum_{i=0}^{n} k_i\bmod n\ .
\]

The simplicial complex $\BT$ is the union of a family of subcomplexes, called apartments, defined as follows. A frame is a set $\mathcal{F}=\{d_{1},\dots,d_{n}\}$ of one-dimensional $\F$-vector subspaces of $\F^{n}$ so that $\F^{n}=d_{1}+\dots+d_{n}$. The apartment corresponding to the frame $\mathcal{F}$ is formed by all simplices $\sigma$ with vertices $\Lambda$ which are equivalence classes of lattices with representatives $\Lrm\in\Lambda$ such that
\[
\Lrm=\Lrm_{1}+\dots+\Lrm_{n},
\]
where $\Lrm_{i}$ is a lattice of the $\F$-vector space $d_{i}$. If we fix an $\F$-basis $(f_{1},\dots,f_{n})$ of $\F^{n}$ adapted to the decomposition $\F^{n}=d_{1}+\dots+d_{n}$, then a vertex $[\Lrm]$ is in the apartment corresponding to the frame $\mathcal{F}$ if and only if
\[
\Lrm=\pf_{\F}^{\alpha_{1}}f_{1}+\dots+\pf_{\F}^{\alpha_{n}}f_{n},
\]
where $(\alpha_{1},\dots,\alpha_{n})\in\mathbb{Z}^{n}$. Note that the set of frames of $\F^{n}$ can be identified with the set of maximal $\F$-split torus of $\G_{n}$. To a frame $\mathcal{F}=\{d_{1},\dots,d_{n}\}$ we can associate the maximal $\F$-split torus $\Srm\subset\G_{n}$ acting diagonally with respect the decomposition of $\F^{n}$ as direct sum of vectorial lines. Under this identification, for every maximal $\F$-split torus $\Srm$ of $\G_{n}$, we denote by $\mA_{\Srm}$ the corresponding apartment of $\BT$. The apartment corresponding to the diagonal torus $\T$ will be called \emph{the standard apartment} of $\BT$ and denoted by $\mA_{0}$.

The geometric realization $\vert\BT\vert$ of the building $\BT$ is equipped by a metric defined, up to a multiplicative scalar, as follows. The geometric realization of each apartment $\vert\mA\vert$ can be identified to the euclidian space
\[
\mathbb{R}_{0}^{n}:= \bigl\{(x_1,\dots,x_n)\in\mathbb{R}^{n}
\,\big|\, x_1+\dots +x_n =0\bigr\}
\]
via the map defined by the following way. We fix an $\F$-basis $(f_{1},\dots,f_{n})$ of $\F^{n}$ corresponding to the apartment $\mA$. The set $\mA^{0}$ of vertices of $\mA$ is then embedded in $\mathbb{R}_{0}^{n}$ via the map $\varphi:\mA^{0}\longrightarrow \mathbb{R}_{0}^{n}$ defined by
\[
\varphi([\pf_{\F}^{x_{1}}f_{1}+\dots+\pf_{\F}^{x_{n}}f_{n}])=x-\frac{1}{n}\sigma(x)e,
\]
where for $x=(x_{1},\dots,x_{n})\in\mathbb{Z}^{n}$, $\sigma(x)=x_{1}+\dots+x_{n}$ and where $e=(1,\dots,1)$. This map extends to a bijection $\varphi:\vert\mA\vert\longrightarrow \mathbb{R}_{0}^{n}$. Via this identification we can then equip $\vert\mA\vert$ by an euclidian metric. More explicitly, if $[\Lrm]$ and $[\M]$ are two vertices of $\mA$ with
\[
\Lrm=\pf_{\F}^{x_{1}}f_{1}+\dots+\pf_{\F}^{x_{n}}f_{n}\quad\text{and}\quad\M=\pf_{\F}^{y_{1}}f_{1}+\dots+\pf_{\F}^{y_{n}}f_{n},
\]
then
\[
d_{\mA}([\Lrm],[\M])=\frac{1}{\sqrt{1-\frac{1}{n}}}d_{0}\left(x-\frac{1}{n}\sigma(x)e,y-\frac{1}{n}\sigma(y)e\right)
\]
where $d_{0}$ is the euclidian metric of $\mathbb{R}_{0}^{n}$. We note that in the above formula the term $1/\sqrt{1-1/n}$ is just used to normalize the metric of the building. The metric $d$ of $\vert\BT\vert$ is then defined as follows. If $x,y\in\BT$ then $d(x,y)=d_{\mA}(x,y)$ for any apartment $\mA$ containing $x$ and $y$ and this is independent of the choice of apartment containing them. Finally we recall that the action of the group $\G_{n}$ on $\vert\BT\vert$ is by isometries.

\subsection{Smooth representations of a locally profinite group}

Let $G$ be a locally profinite group. By a representation of $G$ we mean a pair $(\pi,V)$ formed by a $\mathbb{C}$-vector space $V$ and by a group homomorphism $\pi:G\longrightarrow\GL\/_{\mathbb{C}}(V)$. A such representation is called smooth if for every $v\in V$ the stabilizer
\[
\Stab_{G}(v):=\bigl\{g\in\mathcal{G}\,\big|\,\pi(g).v=v\bigr\}
\]
is an open subgroup of $G$. \emph{In this paper all the representations will be assumed to be smooth and complex}. A representation $(\pi,V)$ of $G$ is called admissible if for every compact open subgroup $K$ of $G$ the space $V^{K}=\{v\in V\,|\forall k\in K, \pi(k)v=v\}$ of $K$-fixed vectors is finite dimensional. If $(\pi,V)$ is a representation of $G$, its contragredient $\pi^{\vee}$ is the representation of $G$ in the subspace $V^{\vee}$ of the algebraic dual $V^{*}$ formed by the linear forms whose stabilizers in $G$ is open.

Let $H$ be a closed subgroup of $G$ and $(\rho,W)$ a representation of $H$. We recall that the induced representation from $H$ to $G$ of $(\rho,W)$, denoted by $\Ind_{H}^{G}\rho$, is the representation of $G$ on the space $\Ind_{H}^{G}W$ formed by the locally constant functions $f:G\longrightarrow W$ such that $f(hg)=\rho(h).f(h)$ for every $g\in G$ and $h\in H$, where the action of $G$ on $\Ind_{H}^{G}\rho$ is by left translation. The compactly induced representation $\cind_{H}^{G}\rho$ is defined as the subrepresentation of $\Ind_{H}^{G}\rho$ formed by the functions $f\in\Ind_{H}^{G}W$ whose support is compact modulo $H$.


\subsection{Locally profinite group acting on directed graphs}

Throughout this paper, we call graph every one dimensional simplicial complex. If $\Y$ is a graph, the set of vertices (resp. edges) of $\Y$ will be denoted by $\Y^{0}$ (resp. $\Y^{1}$). A locally finite graph is a graph $\Y$ for which every vertex belongs to a finite number of edges. All graphs in this paper will be assumed to be locally finite. A directed graph is a graph $\Y$ with a map $\Y^{1}\longrightarrow\Y^{0}\times\Y^{0}$, $a\longmapsto(a^{-},a^{+})$, such that for every edge $a$ one has $a=\{a^{-},a^{+}\}$, where for any edge $a$ we denote by $a^+$ and $a^-$ its head and tail respectively. A path in a graph $Y$ is a sequence $(s_{0},\dots,s_{m})$ of vertices such that two consecutive vertices are linked by an edge. The graph $Y$ is called connected if every pair of vertices are linked by a path. A cover of a graph $Y$ is a family $(Y_{\alpha})_{\alpha\in\Delta}$ of subgraphs such that
\[
Y=\bigcup_{\alpha\in\Delta}Y_{\alpha}.
\]
The nerve of a such cover, denoted $\mathcal{N}(Y,(Y_{\alpha})_{\alpha\in\Delta})$ or just $\mathcal{N}(Y)$ if there is no risk of confusion, is the simplicial complex whose vertex set is $\Delta$ and in which a finite number of vertices $\alpha_{0},\dots,\alpha_{r}$ form a simplex if
\[
\bigcap_{i=0}^{r}\Y_{\alpha_{i}}\neq\emptyset.
\]

In the remainder of this section the notations and definitions are taken from~\cite{Br09}. If $Y$ is a graph, we denote by $C_{0}(Y,\mathbb{C})$ (resp. $C_{1} (Y,\mathbb{C})$) the $\mathbb{C}$-vector space with basis $Y^0$ (resp. $Y^1$). Let $C_{c}^{i}(Y,\mathbb{C})$, $i=1,2$, be the $\mathbb{C}$-vector space of $1$-cochains with finite support~: $C_{c}^{i}(Y,\mathbb{C})$ is the subspace of the algebraic dual of $C_{i}(Y,\mathbb{C})$ formed of those linear forms whose restrictions to the basis $Y^i$ have finite support. The coboundary map
\[
d\ : \ C_{c}^{0}(Y,\mathbb{C})\longrightarrow C_{c}^{1} (Y,\mathbb{C})
\]
is defined by $d(f)(a)=f(a^+)-f(a^-)$. Then the compactly supported cohomology space $H_{c}^{1}(Y,\mathbb{C})$ of the graph $Y$ is defined by
\[
H_{c}^{1}(Y,\mathbb{C})=C_{c}^{1}(Y,\mathbb{C})/dC_{c}^{0}(Y,\mathbb{C}).
\]

Let $G$ be a locally profinite group and $Y$ be a directed graph. We assume that $G$ acts on $Y$ by automorphisms of directed graphs. For all $s\in Y^0$, $a\in Y^1$, the incidence numbers are defined by $[a:a^+]=+1$, $[a:a^-]=-1$, and $[a:s]=0$ if $s\not\in \{a^+,a^-\}$. These incidence numbers are equivariant in the sense that $[g.a:g.s]=[a:s]$, for all $g\in G$. The group $G$ acts on $C_{i}(\Y,\mathbb{C})$ and $C_{c}^i (\Y,\mathbb{C})$. If the action of $G$ on $Y$ is proper, that is for every $s\in Y^{0}$, the stabilizer $\Stab_{G}(s):=\{g\in G \,|\, g.s=s\}$ is open and compact, then the spaces $C_{i}(Y,\mathbb{C})$ and $C_{c}^i (Y,\mathbb{C})$ are smooth $G$-modules. The coboundary map is $G$-equivariant so that $H_{c}^{1}(Y,\mathbb{C})$ have a structure of a smooth $G$-module.

The space of harmonic forms of the graph $Y$ is defined as the subspace of $C^{1}(Y,\mathbb{C})$ formed by the elements $f\in C^{1}(Y,\mathbb{C})$ verifying the following harmonicity condition (see~\cite[\S(1.3)]{Br09}):
\[
\sum_{a\in Y^{1}}[a:s]f(a)=0\quad\text{for all }s\in Y^{0}.
\]
This space will be denoted by $\mathcal{H}(Y,\mathbb{C})$. It is naturally provided by a linear action of $G$. The smooth part of $\mathcal{H}(Y,\mathbb{C})$ under the action of $G$, i.e. the space of \emph{smooth harmonic forms} is denoted by $\mathcal{H}_{\infty}(Y,\mathbb{C})$.

\begin{lemm}[{\cite[(1.3.2)]{Br09}}]\label{lem1}
The algebraic dual of $H_{c}^{1}(Y,\mathbb{C})$ naturally identifies with $\mathcal{H}(Y,\mathbb{C})$. Under this isomorphism, the contragredient representation of $H_{c}^{1}(Y,\mathbb{C})$ corresponds to $\mathcal{H}_{\infty}(Y,\mathbb{C})$.
\end{lemm}



\section{Combinatorial geodesic paths in \texorpdfstring{$\mathcal{BT}_{n}$}{BTn}}\label{sec3}


The aim of this section is to define a class of combinatorial paths in $\BT$ and to study the action of the group $\G_{n}$ on this class of paths. The pointwise stabilisers of such paths will be related to the new-vectors subgroups of $\GL_{n}(\F)$ (the subgroups defined in \eqref{11}), see~\cite{JSS81}.

\subsection{Geodesic paths of \texorpdfstring{$\BT$}{BT} and their prolongations}

\begin{defi}\label{def2}
Let $k\geqslant0$ be an integer. A \emph{geodesic path} of length $k$ in $\BT$ (or more simply \emph{geodesic $k$-path}) is a path $\alpha=(\alpha_{0},\alpha_{1},\dots,\alpha_{k})$ of $\BT$ such that for every $i,j\in\{0,\dots,k\}$, $d(\alpha_{i},\alpha_{j})=\vert i-j\vert$. We denote the set of geodesic $k$-paths of $\BT$ by $\mathcal{C}_{k}(\BT)$.
\end{defi}

\begin{remark}\label{rem3}
We notice that when $n\geqslant4$ the edges of $\BT$ are not all of length one, but in the particular cases $n=2$ and $n=3$ all the edges of $\BT$ are of length one. We also note that every geodesic $k$-path of $\BT$ lies in a same apartment. In fact if $\alpha\in\mathcal{C}_{k}(\BT)$ is a geodesic $k$-path as previously, then the geometric realization of any apartment containing the vertices $\alpha_{0}$ and $\alpha_{k}$ contain the segment $[\alpha_{0},\alpha_{k}]$ and then all the vertices of $\alpha$ are contained in the apartment $\mA$.
\end{remark}

In the following, if $s$ is a vertex of $\BT$ we write $\mathcal{V}(s)$ for its combinatorial neighborhood. That is $\mathcal{V}(s)$ is the set of vertices of $\BT$ which are linked to $s$ by an edge.

\begin{defi}\label{def4}
Let $\alpha=(\alpha_{0},\dots,\alpha_{k})\in\mathcal{C}_{k}(\BT)$. A vertex $s$ of $\BT$ is called a right (resp. left) prolongation of $\alpha$ if $s\in\mathcal{V}(\alpha_{k})$ (resp. $s\in\mathcal{V}(\alpha_{0})$) and the sequence $(\alpha_{0},\dots,\alpha_{k},s)$ (resp. $(s,\alpha_{0},\dots,\alpha_{k})$) is a geodesic $(k+1)$-path. We denote the set of right and left prolongation of a geodesic $k$-path $\alpha$ respectively by $\mathcal{P}^{+}(\alpha)$ and $\mathcal{P}^{-}(\alpha)$.
\end{defi}

\begin{prop}\label{prop5}
Let $k\geqslant1$ be an integer and let $\alpha=(\alpha_{0},\dots,\alpha_{k})$ be a geodesic $k$-path of $\BT$. Then for every apartment $\mathcal{A}$ containing $\alpha$, there exists a unique right (resp. left) prolongation of $\alpha$ in the apartment $\mathcal{A}$.
\end{prop}

\begin{proof}
Let $\mA$ be an apartment containing the path $\alpha$. Assume that $\alpha$ have two right prolongations $x$ and $y$ in $\mA$, that is $x,y\in\mathcal{V}(\alpha_{k})$ and the two sequences $(\alpha_{0},\dots,\alpha_{k},x)$ and $(\alpha_{0},\dots,\alpha_{k},y)$ are geodesic $(k+1)$-paths of $\mA$. So in the geometric realization $\vert\mA\vert$ of the apartment $\mA$ we have $\alpha_{k}\in[\alpha_{0},x]\cap[\alpha_{0},y]$. Therefore we have $\alpha_{k}=tx+(1-t)\alpha_{0}$ and $\alpha_{k}=sy+(1-s)\alpha_{0}$ for same $t$ and $s$ in $]0,1[$. Moreover the two vertices $x$ and $y$ are of the same distance from $\alpha_{k}$, that is $d(x,\alpha_{k})=d(y,\alpha_{k})$. So we have $\|x-\alpha_{k}\|=\|y-\alpha_{k}\|$ (here $\|{\,\cdot\,}\|$ is the euclidian norm of $\vert\mA\vert\simeq\mathbb{R}_{0}^{n}$). From this we obtain $(1-t)\|x-\alpha_{0}\|=(1-s)\|y-\alpha_{0}\|$. But $\|x-\alpha_{0}\|=\|y-\alpha_{0}\|$ so we get $t=s$ and then $x=y$.
\end{proof}

Let $\alpha=(\alpha_{0},\dots,\alpha_{k})$ be a geodesic path of $\BT$. The inverse of $\alpha$, denoted by $\alpha^{-1}$, is defined by $\alpha^{-1}:=(\alpha_{k},\dots,\alpha_{0})$. It is clear that $\alpha^{-1}$ is a geodesic path of $\BT$. If $k\geqslant1$, the tail and the head of $\alpha$ are the two geodesic paths defined respectively by
\[
\alpha^{-}:=(\alpha_{0},\dots,\alpha_{k-1})\quad\text{and}\quad\alpha^{+}:=(\alpha_{1},\dots,\alpha_{k}).
\]
We define also the initial and terminal directed edge of $\alpha$ respectively by $e^{-}(\alpha):=(\alpha_{0},\alpha_{1})$ and $e^{+}(\alpha):=(\alpha_{k-1},\alpha_{k})$.

\begin{prop}\label{prop6}
Let $k\geqslant1$ be an integer and let $\alpha,\beta\in\mathcal{C}_{k}(\BT)$. If $\alpha$ and $\beta$ are contained in a same apartment and if $e^{-}(\alpha)=e^{-}(\beta)$ (resp. $e^{+}(\alpha)=e^{+}(\beta)$), then $\alpha=\beta$.
\end{prop}

\begin{proof}
By induction on $k$, let $\alpha=(\alpha_{0},\dots,\alpha_{k+1})$ and $\beta=(\beta_{0},\dots,\beta_{k+1})$ two geodesic $(k+1)$-paths such that $e^{-}(\alpha)=e^{-}(\beta)$. Assume that $\alpha$ and $\beta$ are contained in a same apartment $\mA$. Since the two geodesic $k$-paths $\alpha^{-}$ and $\beta^{-}$ are contained in the same apartment $\mA$ and as they have the same initial directed edges then by induction hypothesis we have $\alpha^{-}=\beta^{-}$, that is $\alpha_{i}=\beta_{i}$ for each $i\in\{0,\dots,k\}$. So the two vertices $\alpha_{k+1}$ and $\beta_{k+1}$ are two right prolongation of the geodesic $k$-paths $\alpha^{-}$ which are contained in the same apartment $\mA$. Then by the previous proposition we obtain $\alpha_{k+1}=\beta_{k+1}$ and then $\alpha=\beta$ as required.
\end{proof}

\subsection{Action of\/ \texorpdfstring{$\G_{n}$}{Gn} on the sets \texorpdfstring{$\mathcal{C}_{k}(\BT)$}{Ck(BT)}}

The group $G_{n}$ acts on its building $\BT$ by isometries, so $\G_{n}$ acts naturally on the sets $\mathcal{C}_{k}(\BT)$ for each integer $k\geqslant0$. The action is given by
\[
g.(\alpha_{0},\dots,\alpha_{k})=(g.\alpha_{0},\dots,g.\alpha_{k})
\]
for every $g\in G_{n}$ and for every $(\alpha_{0},\dots,\alpha_{k})\in\mathcal{C}_{k}(\BT)$. Note that since the set $\mathcal{C}_{0}(\BT)$ may be identified with the set of vertices of $\BT$, then the action of $\G_{n}$ on $\mathcal{C}_{0}(\BT)$ is transitive. In the particular case $n=2$, the action of $\G_{2}$ on the sets $\mathcal{C}_{k}(\mathcal{BT}_{2})$ is transitive for every integer $k\geqslant0$, see~\cite{Br09}. The situation is slightly different when $n\geqslant3$. We are going to prove that in this last case, the sets $\mathcal{C}_{k}(\BT)$ (for $k\geqslant1$) have exactly two $\G_{n}$-orbits. We first define the type of a directed edge of $\BT$ and we will prove in the lemma bellow that two geodesic $1$-paths are in the same $\G_{n}$-orbit if and only if they have the same type. Let $e=([L_{0}],[L_{1}])$ be a directed edge of $\BT$, where $L_{0}$ and $L_{1}$ are two $\of_{\F}$-lattices such that
\[
\varpi_{\F} L_{0}<L_{1}<L_{0}.
\]
The type of the directed edge $e$, denoted $\xi(e)$, is defined by
\[
\xi(e)=\dim_{k_{\F}}\bigl(L_{1}/\varpi_{\F} L_{0}\big).
\]
This definition is clearly independent of the choice of representatives. For every directed edge $e$ of $\BT$, we write $e^{-1}$ for the inverse of $e$ which is obtained from $e$ by interchanging its vertices.

\begin{lemm}\label{lem7}\ 
\begin{enumerate}\romanenumi
\item \label{7.1} For every directed edge $e$ of $\BT$, $\xi(e^{-1})=n-\xi(e)$,
\item \label{7.2} For every $e\in\mathcal{C}_{1}(\BT)$, $\xi(e)\in\{1,n-1\}$,
\item \label{7.3} Two elements $e,e^{'}\in\mathcal{C}_{1}(\BT)$ are in the same $\G_{n}$-orbit if and only if they have the same~type.
\end{enumerate}
\end{lemm}

\begin{proof}
In the proof of the three statements we use the following notations. For each integer $n\geqslant1$, we write $\Delta_{n}$ for the set of integers $\{1,\dots,n\}$. If $e=([L_{0}],[L_{1}])$ is a directed edge of $\BT$ with $\varpi_{\F} L_{0}<L_{1}<L_{0}$ and if $(f_{1},\dots,f_{n})$ is a basis of $\F^{n}$ for which
\[
L_{0}=\of_{\F}f_{1}+\dots+\of_{\F}f_{n}\quad \text{and}\quad L_{1}=\pf^{k_{1}}_{\F}f_{1}+\dots+\pf^{k_{n}}_{\F}f_{n},
\]
where $(k_{1},\dots,k_{n})\in\mathbb{Z}^{n}$ and $k_{1}\leqslant\dots\leqslant k_{n}$, we put $A_{0}=\{i\in\Delta_{n}\,|\,k_{i}=0\}$ and $A_{1}=\{i\in\Delta_{n}\,|\,k_{i}=1\}$ and we write $p$ and $q$ respectively for their cardinality. The condition $\varpi_{\F} L_{0}< L_{1}< L_{0}$ implies that $k_{i}\in\{0,1\}$ for each $i\in\Delta_{n}$ and that $p,q\in\{1,\dots,n-1\}$ and $p+q=n$.

\begin{proof}[\meqref{7.1}]
Let $e=([L_{0}],[L_{1}])$ be a directed edge with $\varpi_{\F} L_{0}<L_{1}<L_{0}.$ The inverse of $e$ is then given by $e^{-1}=([\varpi_{\F}^{-1}L_{1}],[L_{0}])$ with $L_{1}< L_{0}< \varpi_{\F}^{-1}L_{1}.$ Let $(f_{1},\dots,f_{n})$ be a basis of $\F^{n}$ for which
\[
L_{0}=\of_{\F}f_{1}+\dots+\of_{\F}f_{n}\quad \text{and}\quad L_{1}=\pf^{k_{1}}_{\F}f_{1}+\dots+\pf^{k_{n}}_{\F}f_{n},
\]
where $(k_{1},\dots,k_{n})\in\mathbb{Z}^{n}$ with $k_{1}\leqslant\dots\leqslant k_{n}$. With the previous notations we have the identifications of $k_{\F}$-vector spaces
\begin{equation}\label{321}
L_{1}/\varpi_{\F} L_{0}\simeq\bigoplus_{i=1}^{n}\pf^{k_{i}}_{\F}/\pf_{\F}\simeq\bigoplus_{i\in A_{0}}\of_{\F}/\pf_{\F}\oplus\bigoplus_{i\in A_{1}}\pf_{\F}/\pf_{\F}\simeq k_{\F}^{p}
\end{equation}
and similarly
\begin{equation}\label{322}
L_{0}/L_{1}\simeq\bigoplus_{i=1}^{n}\of_{\F}/\pf^{k_{i}}_{\F}\simeq\bigoplus_{i\in A_{0}}\of_{\F}/\of_{\F}\oplus\bigoplus_{i\in A_{1}}\of_{\F}/\pf_{\F}\simeq k_{\F}^{q}.
\end{equation}
So we obtain $\dim_{k_{\F}}\big(L_{0}/L_{1}\big)=n-\dim_{k_{\F}}\big(L_{1}/\varpi_{\F} L_{0}\big),$ and then $\xi(e^{-1})=n-\xi(e)$.
\let\qed\relax
\end{proof}

\begin{proof}[\meqref{7.2}]
Let $e=([L_{0}],[L_{1}])$ be a directed edge of $\BT$ with $\varpi_{\F} L_{0}< L_{1}< L_{0}$ and let $\mA$ be an apartment containing $e$. To simplify, we can assume that in a some $\F$-basis $(f_{1},\dots,f_{n})$ of $\F^{n}$ we have $L_{0}=\of_{\F}f_{1}+\dots+\of_{\F}f_{n}$ and $L_{1}=\pf^{x_{1}}_{\F}f_{1}+\dots+\pf^{x_{n}}_{\F}f_{n}$, where $x=(x_{1},\dots,x_{n})$ is in $\mathbb{Z}^{n}$. As previously, the $x_{i}^{'}$s are in $\{0,1\}$.

Now if we assume that $e\in\mathcal{C}_{1}(\BT)$ then $d([L_{0}],[L_{1}])=1$. We have then
\[
d_{0}\left(0,x-\frac{1}{n}\sigma(x)e\right)=\frac{\sqrt{n-1}}{\sqrt{n}}
\]
that is
\[
\sum_{i=1}^{n}\left(x_{i}-\frac{1}{n}\sigma(x)\right)^{2}=\frac{n-1}{n}
\]
and then
\[
\left(\sum_{i=1}^{n}x_{i}^{2}\right)-\frac{1}{n}\sigma(x)^{2}=\frac{n-1}{n}.
\]
But since $x_{i}\in\{0,1\}$ then $\sigma(x)-\sigma(x)^{2}/n=(n-1)/n$ which implies that the values of $\sigma(x)$ are 1 or $n-1$. Moreover, from the isomorphisms~\eqref{321} and \eqref{322} we deduce that $\sigma(x)=n-\xi(e)$, so as desired we have $\xi(e)\in\{1,n-1\}$.
\let\qed\relax
\end{proof}

\begin{proof}[\meqref{7.3}]
Let $e\in\mathcal{C}_{1}(\BT)$ with $e=([L_{0}],[L_{1}])$ and $\varpi_{\F} L_{0}< L_{1}< L_{0}$. Let's prove firstly that if $\xi(e)=1$ then there exist an $\F$-basis $(f_{1},\dots,f_{n})$ of $\F^{n}$ such that $L_{0}=\pf^{-1}_{\F}f_{1}+\dots+\pf^{-1}_{\F}f_{n-1}+\of_{\F}f_{n}$ and $L_{1}=\of_{\F}f_{1}+\dots+\of_{\F}f_{n}$ and if $\xi(e)=n-1$ then there exist an $\F$-basis $(h_{1},\dots,h_{n})$ of $\F^{n}$ such that $L_{0}=\of_{\F}h_{1}+\dots+\of_{\F}h_{n}$ and $L_{1}=\of_{\F}h_{1}+\dots+\of_{\F}h_{n-1}+\pf_{\F}h_{n}$. Assume that $\xi(e)=n-1$ (the proof of the case $\xi(e)=1$ is similar). For a some $\F$-basis $(h_{1},\dots,h_{n})$ of $\F^{n}$ we have $L_{0}=\of_{\F}h_{1}+\dots+\of_{\F}h_{n}$ and $L_{1}=\pf^{k_{1}}_{\F}h_{1}+\dots+\pf^{k_{n}}_{\F}h_{n}$ where $(k_{1},\dots,k_{n})\in\mathbb{Z}^{n}$ with $k_{1}\leqslant\dots\leqslant k_{n}$.

As mentioned previously, for each $i\in\Delta_{n}$ the integer $k_{i}$ is in $\{0,1\}$. The fact that $k_{1}\leqslant\dots\leqslant k_{n}$ implies that $(k_{1},\dots,k_{n})=(0,\dots,0,1,\dots,1)$, where $0$ appear $p$-times and $1$ appear $q$-times.

So we have
\[
L_{1}/\varpi_{\F} L_{0}\simeq\bigoplus_{i=1}^{p}\of_{\F}/\pf_{\F}\oplus\bigoplus_{i=p+1}^{q}\pf_{\F}/\pf_{\F}\simeq k_{\F}^{p}.
\]
But since $\xi(e)=n-1$, that is $\dim_{k_{\F}}(L_{1}/\varpi_{\F} L_{0})=n-1$, then we have $L_{1}=\of_{\F}h_{1}+\dots+\of_{\F}h_{n-1}+\pf_{\F}h_{n}$. So as desired we have an $\F$-basis $(h_{1},\dots,h_{n})$ of $\F^{n}$ for which $L_{0}=\of_{\F}h_{1}+\dots+\of_{\F}h_{n}$ and $L_{1}=\of_{\F}h_{1}+\dots+\of_{\F}h_{n-1}+\pf_{\F}h_{n}$. Let's prove now that two elements $e,e'\in\mathcal{C}_{1}(\BT)$ are in the same $\G_{n}$-orbit if and only if they have the same type. Assume that $e=([L_{0}],[L_{1}])$ (resp. $e'=([L'_{0}],[L'_{1}])$) where $L_{0}$ and $L_{1}$ (resp. $L'_{0}$ and $L'_{1}$) are two $\of_{\F}$-lattices such that $\varpi_{\F} L_{0}< L_{1}< L_{0}$ (resp. $\varpi_{\F} L'_{0}< L'_{1}< L'_{0}$). If $e$ and $e'$ have the same type, say for example $\xi(e)=\xi(e')=1$, then by the previous point we can find two $\F$-basis $(f_{1},\dots,f_{n})$ and $(f'_{1},\dots,f'_{n})$ for which $L_{0}=\pf^{-1}_{\F}f_{1}+\dots+\pf^{-1}_{\F}f_{n-1}+\of_{\F}f_{n}$ and $L_{1}=\of_{\F}f_{1}+\dots+\of_{\F}f_{n}$ and likewise $L'_{0}=\pf^{-1}_{\F}f'_{1}+\dots+\pf^{-1}_{\F}f'_{n-1}+\of_{\F}f'_{n}$ and $L'_{1}=\of_{\F}f'_{1}+\dots+\of_{\F}f'_{n}$. So if $g\in \G_{n}$ is the unique element sending the $\F$-basis $(f_{1},\dots,f_{n})$ on $(f'_{1},\dots,f'_{n})$ we have $gL_{0}=L'_{0}$ and $gL_{1}=L'_{1}$, thus $g.e=e'$ and then $e$ and $e'$ are in the same $\G_{n}$-orbit. The converse is obvious.
\end{proof}
\let\qed\relax
\end{proof}

\begin{prop}\label{prop8}
Let $n\geqslant3$ be an integer. For every $k\geqslant1$, the set $\mathcal{C}_{k}(\BT)$ have two $\G_{n}$-orbits.
\end{prop}

\begin{proof}
Let us prove firstly that two elements $\alpha$ and $\beta$ of $\mathcal{C}_{k}(\BT)$ are in the same $\G_{n}$-orbit if and only if their initial directed edges $e^{-}(\alpha)$ and $e^{-}(\beta)$ are likewise. If $\alpha$ and $\beta$ are in the same $\G_{n}$-orbit then clearly $e^{-}(\alpha)$ and $e^{-}(\beta)$ are also in the same $\G_{n}$-orbit. Conversely, assume that $e^{-}(\alpha)$ and $e^{-}(\beta)$ are in the same $\G_{n}$-orbit, that is for same $g\in\G_{n}$ one has $e^{-}(\alpha)=g.e^{-}(\beta)$. So we have $e^{-}(\alpha)=e^{-}(g.\beta)$.

Let $\mA$ and $\mB$ two apartments containing $\alpha$ and $g.\beta$ respectively. Since the pointwise stabiliser $H_{0}$ of the edge $e^{-}(\alpha)$ acts transitively on the set of apartments containing $e^{-}(\alpha)$ (see~\cite[Cor.~(7.4.9)]{BT72}), then there exist $h\in H_{0}$ such that $h.\mB=\mA$. So the two geodesic $k$-paths $\alpha$ and $hg.\beta$ are contained in the same apartment $\mA$ and have the same initial directed edge (that is $e^{-}(\alpha)=e^{-}(hg.\beta)$). Thus the Proposition~\ref{prop6} implies that $\alpha=hg.\beta$ and then $\alpha$ and $\beta$ are in the same $\G_{n}$-orbit. Consequently, two elements $\alpha$ and $\beta$ of $\mathcal{C}_{k}(\BT)$ are in the same $\G_{n}$-orbit if and only if $e^{-}(\alpha)$ and $e^{-}(\beta)$ are likewise. The result follows then from Lemma~\ref{lem7}.
\end{proof}

One can prove that if $\alpha\in\mathcal{C}_{k}(\BT)$ then all the directed edges of $\alpha$ have the same type. So we can define the type of a geodesic $k$-path $\alpha$, denoted by $\xi(\alpha)$, as the type of any of its directed edges. The $\G_{n}$-orbit of $\mathcal{C}_{k}(\BT)$ corresponding to the type $n-1$ (resp. type 1) will be denoted by $\mathcal{C}^{+}_{k}(\BT)$ (resp. $\mathcal{C}^{-}_{k}(\BT)$). The Lemma~\ref{lem7} implies that if $\alpha\in\mathcal{C}_{k}^{+}(\BT)$ then its inverse $\alpha^{-1}$ is in $\mathcal{C}^{-}_{k}(\BT)$. So for every $\alpha\in\mathcal{C}_{k}(\BT)$ the pair $\{\alpha,\alpha^{-1}\}$ constitute a system of representatives of $\mathcal{C}_{k}(\BT)$ for the action of the group $\G_{n}$. The path $\gamma=([L_{0}],[L_{1}],\dots,[L_{k}])$, where for $i\in\{0,\dots,k\}$
\begin{equation}\label{323}
L_{i}=\of_{\F}e_{1}+\dots+\of_{\F}e_{n-1}+\pf_{\F}^{i}e_{n}
\end{equation}
is an element of $\mathcal{C}^{+}_{k}(\BT)$ contained in the standard apartment of $\BT$, this $k$-path will be called \emph{the standard geodesic $k$-path}.

\begin{lemm}\label{lem9}
For every $\alpha\in\mathcal{C}_{k}(\BT)$ the stabilizer $\Stab_{\G_{n}}(\alpha)$ acts transitively on $\mathcal{P}^{+}(\alpha)$ and $\mathcal{P}^{-}(\alpha)$.
\end{lemm}

\begin{proof}
Let $\alpha=(\alpha_{0},\dots,\alpha_{k})\in\mathcal{C}_{k}(\BT)$. We will prove that the action of $\Stab_{\G_{n}}(\alpha)$ is transitive on $\mathcal{P}^{+}(\alpha)$. By a similar way we get the same thing for $\mathcal{P}^{-}(\alpha)$. Let $s,t\in\mathcal{P}^{+}(\alpha)$, that is $\beta=(\alpha_{0},\dots,\alpha_{k},s)$ and $\gamma=(\alpha_{0},\dots,\alpha_{k},t)$ are two geodesic $(k+1)$-paths. Since every geodesic path of $\BT$ is contained in a some apartment, then there are two apartments $\mA$ and $\mB$ containing $\beta$ and $\gamma$ respectively. The stabilizer $\Stab_{\G_{n}}(\alpha)$ is also the pointwise stabilizer in $\G_{n}$ of the segment $[\alpha_{0},\alpha_{k}]$. So $\Stab_{\G_{n}}(\alpha)$ acts transitively on the set of apartments containing $\alpha$ (see~\cite[Cor.~(7.4.9)]{BT72}). Then there exist $g\in\Stab_{\G_{n}}(\alpha)$ such that $g.\mA=\mB$. So $g.s$ is a right prolongation of the geodesic path $\alpha$ contained in the apartment $\mB$. Hence, the two vertices $t$ and $g.s$ are two right prolongations of $\alpha$ contained in the apartment $\mB$. Then by the Proposition~\ref{prop5}, we obtain $g.s=t$ and then as desired the action of $\Stab_{\G_{n}}(\alpha)$ on $\mathcal{P}^{+}(\alpha)$ is transitive.
\end{proof}

\begin{coro}\label{cor10}
For every $\alpha\in\mathcal{C}_{k}(\BT)$ we have :
\[
\mathcal{P}^{+}(\alpha)=\mathcal{P}^{+}(e^{+}(\alpha))\quad\text{and}\quad\mathcal{P}^{-}(\alpha)=\mathcal{P}^{-}(e^{-}(\alpha)),
\]
that is the right (resp. left) prolongation of the geodesic path $\alpha$ are exactly the right (resp. left) prolongation of the directed edge $e^{+}(\alpha)$ (resp. $e^{-}(\alpha)$).
\end{coro}

\begin{proof}
Let's prove the first equality, the proof of the second is similar. It is clear that $\mathcal{P}^{+}(\alpha)\subset\mathcal{P}^{+}(e^{+}(\alpha))$. Since the two sets $\mathcal{P}^{+}(\alpha)$ and $\mathcal{P}^{+}(e^{+}(\alpha))$ are finite it suffice to prove that they have the same cardinality. If $\Gamma_{\alpha}$ denoted the subgroup $\Stab_{\G_{n}}(\alpha)$, then by the previous lemma $\Gamma_{\alpha}$ acts transitively on $\mathcal{P}^{+}(\alpha)$. So for any $s\in\mathcal{P}^{+}(\alpha)$ we can identify the set $\mathcal{P}^{+}(\alpha)$ with the quotient set $\Gamma_{\alpha}/\Stab_{\Gamma_{\alpha}}(s)$. Similarly, the set $\mathcal{P}^{+}(e^{+}(\alpha))$ identifies with the quotient set $\Gamma_{e^{+}(\alpha)}/\Stab_{\Gamma_{e^{+}(\alpha)}}(t)$ for any $t\in\mathcal{P}^{+}(e^{+}(\alpha))$. Now since the action of $\G_{n}$ on $\mathcal{C}_{k}(\BT)$ have two orbits and since an element $\beta\in\mathcal{C}_{k}(\BT)$ and its inverse $\beta^{-1}$ have the same stabilizers in $\G_{n}$ then we can assume that $\alpha$ is the standard geodesic $k$-path defined as previously by $([L_{0}],[L_{1}],\dots,[L_{k}])$, where $L_{i}=\of_{\F}e_{1}+\dots+\of_{\F}e_{n-1}+\pf_{\F}^{i}e_{n}$ for $i\in\{0,\dots,k\}$. If $s$ is the vertex $[L_{k+1}]$, it is clearly that $s\in\mathcal{P}^{+}(\alpha)$. By an easy computation we obtain that $\Gamma_{\alpha}=\Gamma_{0}(\pf_{\F}^{k})$ and $\Stab_{\Gamma_{\alpha}}(s)=\Gamma_{0}(\pf_{\F}^{k+1})$. Moreover, we can check that $\Gamma_{0}(\pf_{\F}^{k})/\Gamma_{0}(\pf_{\F}^{k+1})$ have cardinality $q_{\F}^{n-1}$. Similarly, we can check easily that the vertex $s$ whose equivalence class of $\of_{\F}$-lattice is represented by $L_{k+1}$ is in $\mathcal{P}^{+}(e^{+}(\alpha))$ and that $\Gamma_{e^{+}(\alpha)}=\Gamma_{0}(\pf_{\F})$ and $\Stab_{\Gamma_{e^{+}(\alpha)}}(s)=\Gamma_{0}(\pf_{\F}^{2})$. Furthermore, we can check that $\Gamma_{0}(\pf_{\F})/\Gamma_{0}(\pf_{\F}^{2})$ have also cardinality $q_{\F}^{n-1}$. So as desired we have the equality between the two sets $\mathcal{P}^{+}(\alpha)$ and $\mathcal{P}^{+}(e^{+}(\alpha))$.
\end{proof}

\begin{coro}
For every $\alpha,\beta\in\mathcal{C}_{k+1}(\BT)$, if $\alpha^{+}=\beta^{+}$ (resp. $\alpha^{-}=\beta^{-}$) then $\mathcal{P}^{+}(\alpha)=\mathcal{P}^{+}(\beta)$ (resp. $\mathcal{P}^{-}(\alpha)=\mathcal{P}^{-}(\beta)$).
\end{coro}

\begin{proof}
If $\alpha^{+}=\beta^{+}$ (resp. $\alpha^{-}=\beta^{-}$) then $e^{+}(\alpha)=e^{+}(\beta)$ (resp. $e^{+}(\alpha)=e^{+}(\beta)$) and then the equality $\mathcal{P}^{+}(\alpha)=\mathcal{P}^{+}(\beta)$ (resp. $\mathcal{P}^{-}(\alpha)=\mathcal{P}^{-}(\beta)$) follows from the previous corollary.
\end{proof}

\begin{lemm}
Let $s_{0}$ be a vertex of $\BT$. If $L_{0}\in s_{0}$ then for every vertex $x\in\mathcal{V}(s_{0})$ there is a unique representative $L\in x$ such that
\[
\varpi_{\F}L_{0}< L< L_{0}.
\]
\end{lemm}

\begin{proof}
Let us fix a representative $L_{0}\in s_{0}$. Let $L$ and $L'$ two representatives of $x$ such that $\varpi_{\F}L_{0}< L< L_{0}$ and $\varpi_{\F}L_{0}< L'< L_{0}$. Since $L$ and $L'$ are equivalent then $L'=\lambda L$ for some $\lambda\in\F^{\times}$. Put $\lambda=\varpi_{\F}^{m}u$ for some $m\in\mathbb{Z}$ and $u\in\of_{\F}^{\times}$. We have $\varpi_{\F}L_{0}< L< L_{0}$ which implies $\varpi_{\F}^{m+1}L_{0}< \lambda L< \varpi_{\F}^{m}L_{0}$, that is $\varpi_{\F}^{m+1}L_{0}< L'< \varpi_{\F}^{m}L_{0}$. The two inclusions $\varpi_{\F}L_{0}< L'< L_{0}$ and $\varpi_{\F}^{m+1}L_{0}< L'< \varpi_{\F}^{m}L_{0}$ implies then that $m=0$. Indeed, if we assume to the contrary that $m\neq0$, say for example $m>0$, then we have $\varpi_{\F}^{m}L_{0}\leqslant \varpi_{\F}L_{0}$. So from the two inclusions $\varpi_{\F}L_{0}< L'< L_{0}$ and $\varpi_{\F}^{m+1}L_{0}< L'< \varpi_{\F}^{m}L_{0}$ we obtain $L'< \varpi_{\F}^{m}L_{0}\leqslant \varpi_{\F}L_{0}<L'$ which is a contradiction. We deduce then that $L'=u L=L$.
\end{proof}

Let $s_{0}$ be a vertex of $\BT$ and $L_{0}\in s_{0}$ be a fixed representative. By the previous lemma to any vertex $x\in \mathcal{V}(s_{0})$ we can associate a non-trivial subspace of the $k_{\F}$-vector space $\tilde{V}_{s_{0}}:=L_{0}/\varpi_{\F}L_{0}$. Indeed, if $x\in\mathcal{V}(s_{0})$ and $L_{x}\in x$ is the unique representative such that $\varpi_{\F}L_{0}< L_{x}< L_{0}$, then $V_{x}$ is defined as $L_{x}/\varpi_{\F}L_{0}$. For every subspaces $X$ and $Y$ of $\tilde{V}_{s_{0}}$, we put
\[
\delta(X,Y)=\dim_{k_{\F}}(X+Y)-\dim_{k_{\F}}(X\cap Y).
\]
In the following proposition, we give two formulas for the metric of $\BT$ on the set of vertices in the neighborhood a fixed vertex $s_{0}$ of $\BT$ in terms of the corresponding $k_{\F}$-vector spaces.

\begin{prop}\label{prop13}
For every vertex $s_{0}$ of $\BT$ we have :
\begin{enumerate}\romanenumi
\item \label{13i} If $x\in\mathcal{V}(s_{0})$, then
\[
d(s_{0},x)=\frac{1}{\sqrt{n-1}}\Big(n\dim V_{x}-(\dim V_{x})^{2}\Big)^{\frac{1}{2}}.
\]
\item \label{13ii} If $x,y\in\mathcal{V}(s_{0})$, then
\[
d(x,y)=\frac{1}{\sqrt{n-1}}\Big(n\delta(V_{x},V_{y})-(\dim V_{x}-\dim V_{y})^{2}\Big)^{\frac{1}{2}}.
\]
\end{enumerate}
\end{prop}

\begin{proof}
\begin{proof}[\meqref{13i}]
Let us fix an $\of_{\F}$-lattice $L_{0}$ representing the vertex $s_{0}$. Let $x\in\mathcal{V}(s_{0})$. We can choose an apartment $\mA$ containing $s_{0}$ and $x$. Without loss of generality we can assume that $\mA$ is the standard apartment and that $L_{0}=\of_{\F}e_{1}+\dots+\of_{\F}e_{n}$, where $(e_{1},\dots,e_{n})$ is the standard basis of $\F^{n}$. Let $L_{x}$ be the unique representative of the vertex $x$ such that $\varpi_{\F}L_{0}<L_{x}<L_{0}$. Since the vertex $x$ lies in $\mA$ then for some $a=(a_{1},\dots,a_{n})\in \mathbb{Z}^{n}$ we can write $L_{x}=\pf_{\F}^{a_{1}}e_{1}+\dots+\pf_{\F}^{a_{n}}e_{n}$. As in the proof of Lemma~\ref{lem7}, the coordinates $a_{i}\in\{0,1\}$ and not all the $a_{i}'$s are zero or one. Moreover, if $A_{0}=\{i\in\Delta_{n}\,|\,a_{i}=0\}$ and $A_{1}=\{i\in\Delta_{n}\,|\,a_{i}=1\}$, then clearly $A_{0}\sqcup A_{1}=\Delta_{n}$. So we have
\[
L_{x}=\bigoplus_{i\in A_{0}}\of_{\F}\oplus\bigoplus_{i\in A_{1}}\pf_{\F}
\]
and then
\[
V_{x}=L_{x}/\varpi_{\F}L_{0}\simeq\bigoplus_{i\in A_{0}}\of_{\F}/\pf_{\F}\oplus\bigoplus_{i\in A_{1}}\pf_{\F}/\pf_{\F}\simeq k_{\F}^{\vert A_{0}\vert}.
\]
Consequently $\dim(V_{x})=\vert A_{0}\vert$. We have
\begin{align*}
d(s_{0},x)&=\sqrt{\frac{n}{n-1}}d_{0}(0,a-\frac{1}{n}\sigma(a)e)=\sqrt{\frac{n}{n-1}}\left\|a-\frac{\sigma(a)}{n}e\right\|
\\
&=\sqrt{\frac{n}{n-1}}\left(\sum_{i=1}^{n}\left(a_{i}-\frac{\sigma(a)}{n}\right)^{2}\right)^{\frac{1}{2}}=
\sqrt{\frac{n}{n-1}}\left(\sum_{i=1}^{n}a_{i}^{2}-\frac{2\sigma(a)}{n}a_{i}+\frac{\sigma(a)^{2}}{n^{2}}\right)^{\frac{1}{2}}
\\
&=\sqrt{\frac{n}{n-1}}\left(\sum_{i=1}^{n}a_{i}^{2}-\frac{2\sigma(a)^{2}}{n}+\frac{\sigma(a)^{2}}{n}\right)^{\frac{1}{2}}
\end{align*}
But as $a_{i}\in\{0,1\}$ for every $i\in \Delta_{n}$, then
\[
d(s_{0},x)=\sqrt{\frac{n}{n-1}}\left(\sigma(a)-\frac{\sigma(a)^{2}}{n}\right)^{\frac{1}{2}}.
\]
On the other hand
\[
\sigma(a)=\sum_{i=1}^{n}a_{i}=\sum_{i\in A_{1}}1=\vert A_{1}\vert=n-\dim V_{x}.
\]
So we get
\[
d(s_{0},x)=\sqrt{\frac{n}{n-1}}\left(n-\dim V_{x}-\frac{(n-\dim V_{x})^{2}}{n}\right)^{\frac{1}{2}},
\]
and then
\[
d(s_{0},x)=\frac{1}{\sqrt{n-1}}\Big(n\dim V_{x}-(\dim V_{x})^{2}\Big)^{\frac{1}{2}}.
\]
\let\qed\relax
\end{proof}

\begin{proof}[\meqref{13ii}]
The proof of the second formula is obtained by a similar way.
\end{proof}
\let\qed\relax
\end{proof}

If $x$ and $y$ are two vertices of $\BT$ we write $[x,y]^{0}$ for the combinatorial segment between $x$ and $y$. That is $[x,y]^{0}$ is the set of vertices $z$ of $\BT$ such that $d(x,z)+d(z,y)=d(x,y)$.

\begin{coro}\label{cor14}
If $x,y\in\mathcal{V}(s_{0})$, then $s_{0}\in[x,y]^{0}$ if and only if\/ $V_{x}\oplus V_{y}=\tilde{V}_{s_{0}}$.
\end{coro}

\begin{proof}
Follows from the previous proposition by an easy computation.
\end{proof}

If $\alpha=(\alpha_{0},\dots,\alpha_{k})$ is a $k$-path of $\BT$ (where $k\geqslant1$), the initial (resp. terminal) vertex of $\alpha$, that is $\alpha_{0}$ (resp. $\alpha_{k}$), will be denoted by $s^{-}(\alpha)$ (resp. $s^{+}(\alpha))$. If $\alpha$ and $\beta$ are respectively a $k$-path and an $\ell$-path with $s^{+}(\alpha)=s^{-}(\beta)$, then their concatenation $\alpha\beta$ is the $(k+\ell)$-path of $\BT$ defined by
\[
\alpha\beta:=(\alpha_{0},\dots,\alpha_{k},\beta_{1},\dots,\beta_{\ell}).
\]
It is not true in general that the concatenation of two geodesic paths of $\BT$ is a geodesic path. But we have the following result :

\begin{lemm}\label{lem15}
Let $\alpha=(\alpha_{0},\dots,\alpha_{k})$ and $\beta=(\beta_{0},\dots,\beta_{\ell})$ two geodesic paths of $\BT$ of length $k$ and $\ell$ respectively and with $s^{+}(\alpha)=s^{-}(\beta)$. Then $\alpha\beta$ is a geodesic $(k+\ell)$-path if and only if $\beta_{1}\in\mathcal{P}^{+}(e^{+}(\alpha))$ (resp. $\alpha_{k-1}\in\mathcal{P}^{-}(e^{-}(\beta))$).
\end{lemm}

\begin{proof}
If $\alpha\beta$ is geodesic then it is clear that $\beta_{1}\in\mathcal{P}^{+}(e^{+}(\alpha))$ (resp. $\alpha_{k-1}\in\mathcal{P}^{-}(e^{-}(\beta))$). For the converse, we will prove by induction on $\ell\geqslant1$ that for every geodesic path $\beta=(\beta_{0},\dots,\beta_{\ell})$ of length $\ell$ such that $s^{+}(\alpha)=s^{-}(\beta)$, if $\beta_{1}\in\mathcal{P}^{+}(e^{+}(\alpha))$ (resp. $\alpha_{k-1}\in\mathcal{P}^{-}(e^{-}(\beta))$) then the $(k+\ell)$-path $\alpha\beta$ is geodesic. For $\ell=1$ the property follows from Corollary~\ref{cor10}. Assume that the property is true for the order $\ell$. Let $\beta=(\beta_{0},\dots,\beta_{\ell+1})$ be a geodesic $(\ell+1)$-path of $\BT$ such that $s^{+}(\alpha)=s^{-}(\beta)$ and with $\beta_{1}\in\mathcal{P}^{+}(e^{+}(\alpha))$ (in the case when $\alpha_{k-1}\in\mathcal{P}^{-}(e^{-}(\beta))$ the proof is similar). From the induction hypothesis, the $(k+\ell)$-path $\alpha\beta^{-}$, that is the path $(\alpha_{0},\dots,\alpha_{k},\beta_{1},\dots,\beta_{\ell})$, is geodesic. Since moreover the vertex $\beta_{\ell+1}$ is a right prolongation of the directed edge $e^{+}(\alpha\beta^{-})$ then by Corollary~\ref{cor10} the path
\[
\alpha\beta=(\alpha_{0},\dots,\alpha_{k},\beta_{1},\dots,\beta_{\ell},\beta_{\ell+1})
\]
is also geodesic.
\end{proof}

\begin{coro}\label{cor16}
Let $\alpha\in\mathcal{C}_{k}(\BT)$ and $\beta\in\mathcal{C}_{\ell}(\BT)$, where $k,\ell\geqslant1$. If $\alpha$ is joined to $\beta$ by a nontrivial geodesic path, that is there exists an integer $0<m\leqslant\min(k,\ell)$ such that
\[
\alpha_{i}=\beta_{i-k+m},~\text{for~every}~i\in\{k-m,\dots,k\},
\]
then the sequence $\alpha\cup\beta:=(\alpha_{0},\dots,\alpha_{k},\beta_{m+1},\dots,\beta_{\ell})$ is a geodesic path. In particular if $\alpha,\beta\in\mathcal{C}_{k+1}(\BT)$ such that $\alpha^{+}=\beta^{-}$ (resp. $\alpha^{-}=\beta^{+}$) then $\alpha\cup\beta$ is a geodesic $(k+2)$-path.
\end{coro}

\begin{proof}
The case when $m=\min(k,\ell)$ is obvious since in this case $\alpha$ is a subpath of $\beta$ or $\beta$ is a subpath of $\alpha$. Assume then that $m<\min(k,\ell)$. Since $\widetilde{\alpha}=(\alpha_{0},\dots,\alpha_{k-m})$ is a subpath of $\alpha$ then $\widetilde{\alpha}$ is geodesic. Moreover it is clear that $s^{+}(\widetilde{\alpha})=s^{-}(\beta)$ (since from the hypothesis $\alpha_{k-m}=\beta_{0}$). So the concatenation $\widetilde{\alpha}\beta$ is a path of $\BT$. But $\widetilde{\alpha}\beta$ is nothing other than $\alpha\cup\beta$. The vertex $\beta_{1}$ is clearly a right prolongation of the directed edge $e^{+}(\widetilde{\alpha})$ as $\beta_{1}=\alpha_{k-m+1}$. So by the previous lemma $\alpha\cup\beta$ is geodesic.
\end{proof}


\section{The projective tower of graphs over \texorpdfstring{$\BTs{(1)}$}{BT(1)}}

In this section, our purpose is to give the construction of the tower of directed graphs lying equivariantly over the $1$-skeleton of the building $\BT$ and to give some basic properties of these tower of directed graphs. We note that our construction generalizes the construction of Broussous given in~\cite{Br09} for the case $n=2$. In the sequel, we will be interested then by the case $n\geqslant3$.

\subsection{The construction}

For every integer $k\geqslant0$, we define the graph $\tilde{\X}_{k}$ as the directed graph whose vertex (resp. edges) set is the set $\mathcal{C}^{+}_{k}(\BT)$ (resp. $\mathcal{C}^{+}_{k+1}(\BT)$). The structure of directed graph of $\tilde{\X}_{k}$ is given by :
\[
a^{-} =(\alpha_{0},\dots,\alpha_{k}), \ a^{+} =(\alpha_{1},\dots,\alpha_{k+1}), \ \text{ if }\ a=(\alpha_{0},\dots,\alpha_{k+1})\ .
\]
Let's notice firstly that the graph $\tilde{\X}_{0}$ is nothing other than the directed graph whose vertices are those of $\BT$ and for which the edges set is $\mathcal{C}^{+}_{1}(\BT)$. The action of $\G_{n}$ on the sets $\mathcal{C}^{+}_{k}(\BT)$ induce an action on the graph $\tilde{\X}_{k}$ by automorphisms of directed graphs. Moreover, since the stabilizers of the vertices of $\tilde{\X}_{k}$ are open and compact then the action is proper. From the previous section, the action of $\G_{n}$ on the graph $\tilde{\X}_{k}$ is transitive on vertices and edges. For every vertex $s$ (resp. edge $a$) of $\tilde{\X}_{k}$, we write $\Gamma_{s}$ (resp. $\Gamma_{a}$) for the stabilizer in $\G_{n}$ of $s$ (resp. $a$). The stabilizer in $\G_{n}$ of the standard vertex (resp. edge) of $\tilde{\X}_{k}$, that is the standard geodesic $k$-path (resp. $(k+1)$-path) given in \eqref{323}, is the subgroup $\Gamma_{0}(\pf_{\F}^{k})$ (resp. $\Gamma_{0}(\pf_{\F}^{k+1})$).

\begin{prop}\label{prop17}
For every vertex $s$ of $\tilde{\X}_{k}$ the stabilizer $\Gamma_{s}$ acts transitively on the two sets of neighborhoods :
\[
 \mathcal{V}^{-}(s)=\left\{a\in\tilde{\X}_{k}^{1}\,\middle|\,a^{-}=s\right\}\quad\text{and}\quad\mathcal{V}^{+}(s)=\left\{a\in\tilde{\X}_{k}^{1}\,\middle|\,a^{+}=s\right\}
\]
\end{prop}

\begin{proof}
Follows immediately from Lemma~\ref{lem9}.
\end{proof}

Recall that the $1$-skeleton of the building $\BT$, denoted by $\BTs{(1)}$, is the subcomplex of $\BT$ formed by the faces of dimension at most one. When $k=2m$ is even, there is a natural simplicial projection $p_{k}:\tilde{\X}_{k}\longrightarrow\BTs{(1)}$ defined on vertices by
\[
p_{k}(s_{-m},\dots,s_{0},\dots,s_{m})=s_{0}.
\]
Similarly, When $k=2m+1$ is odd, there is a natural simplicial projection $p_{k}:\tilde{\X}_{k}^{sd}\longrightarrow\mathcal{\widetilde{BT}}_{n}^{(1)}$, where $\tilde{\X}_{k}^{sd}$ and $\mathcal{\widetilde{BT}}_{n}^{(1)}$ are respectively the barycentric subdivision of the graphs $\tilde{\X}_{k}$ and $\BTs{(1)}$. The family of graphs $(\tilde{\X}_{k})_{k\geqslant0}$ constitute a tower of graphs over the graph $\BTs{(1)}$ in the sense that we have the following diagram of simplicial maps
\[
\cdots\longrightarrow\tilde{\X}_{k+1}\overset{\varphi^{\varepsilon}_{k}}{\longrightarrow}\tilde{\X}_{k}\longrightarrow\cdots\longrightarrow\tilde{\X}_{0}\overset{ p_{0}}{\longrightarrow}\BTs{(1)}
\]
where for $\varepsilon=\pm$ and for $k\geqslant0$, the map $\varphi^{\varepsilon}_{k}:\tilde{\X}_{k+1}\longrightarrow\tilde{\X}_{k}$ is the simplicial map defined on vertices by $\varphi^{\varepsilon}_{k}(s)=s^{\varepsilon}$.

\subsection{Connectivity of the graphs}

The aim of this section is the study of the connectivity of the graphs $\tilde{\X}_{k}$. We begin by defining a cover of $\tilde{\X}_{k+1}$ by finite subgraphs whose nerve is a graph isomorphic to $\tilde{\X}_{k}$. Assume that $k\geqslant0$ is an integer. For every vertex $s$ of $\tilde{\X}_{k}$ we define the subgraph $\tilde{\X}_{k+1}(s)$ of the graph $\tilde{\X}_{k+1}$ as the subgraph whose edges are the geodesic $(k+2)$-paths $\alpha\in\mathcal{C}^{+}_{k+2}(\BT)$ of the form $\alpha=(x,s_{0},\dots,s_{k},y),$ where $x$ (resp. $y$) is a left (resp. right) prolongation of the path $s$. The vertices of $\tilde{\X}_{k+1}(s)$ are exactly those $v\in\tilde{\X}^{0}_{k+1}$ such that $v^{-}=s$ or $v^{+}=s$. Obviously the subgraphs $\tilde{\X}_{k+1}(s)$, when $s$ range over the set of vertices of $\tilde{\X}_{k}$, form a cover the graph $\tilde{\X}_{k+1}$. That is
\begin{equation}\label{421}
\tilde{\X}_{k+1}=\bigcup_{s\in\tilde{\X}^{0}_{k}}\tilde{\X}_{k+1}(s).
\end{equation}
For every vertex $s_{0}$ of $\tilde{\X}_{0}$ (considered as a vertex of $\BT$) the subgraph $\tilde{\X}_{1}(s_{0})$ of $\tilde{\X}_{1}$ has two types of vertices : the directed edges $(x,s_{0})\in\mathcal{C}^{+}_{1}(\BT)$ and the directed edges $(s_{0},y)\in\mathcal{C}^{+}_{1}(\BT)$. Let us denote the $k_{\F}$-vector space $k_{\F}^{n}$ by $\bar{V}$. The Lemma~\ref{lem7} implies that the vertex set of $\tilde{\X}_{1}(s_{0})$ may be identified with the set $\mathbb{P}^{1}(\bar{V})\sqcup\mathbb{P}^{1}(\bar{V}^{\ast})$, where $\mathbb{P}^{1}(\bar{V})$ is the set of one dimensional subspaces and $\mathbb{P}^{1}(\bar{V}^{*})$ is the set of one codimensional subspaces of $\bar{V}$. By the Corollary~\ref{cor14} we deduce that the graph $\tilde{\X}_{1}(s_{0})$ is isomorphic to the graph $\Delta(\bar{V})$ whose vertex set is $\mathbb{P}^{1}(\bar{V})\sqcup\mathbb{P}^{1}(\bar{V}^{\ast})$ and in which a vertex $D\in\mathbb{P}^{1}(\bar{V})$ is linked to a vertex $H\in\mathbb{P}^{1}(\bar{V}^{\ast})$ if and only if $D\oplus H=\bar{V}$ and there is no edges between two distinct vertices of $\mathbb{P}^{1}(\bar{V})$ (resp. $\mathbb{P}^{1}(\bar{V}^{\ast})$). One can prove easily that $\Delta(\bar{V})$ is a connected bipartite graph so that $\tilde{\X}_{1}(s_{0})$ is connected and bipartite for every vertex $s_{0}$ of $\tilde{\X}_{0}$.

\begin{lemm}\label{lem18}
Let $k\geqslant1$ be an integer. Then we have :
\begin{enumerate}\romanenumi
\item \label{18i} For every $s\in\tilde{\X}^{0}_{k}$, the graph $\tilde{\X}_{k+1}(s)$ is a complete bipartite graph and hence connected,
\item \label{18ii} The nerve $\mathcal{N}(\tilde{\X}_{k+1})$ of the cover of $\tilde{\X}_{k+1}$ given in \eqref{421} is isomorphic to the graph $\tilde{\X}_{k}$.
\end{enumerate}
\end{lemm}

\begin{proof}
\begin{proof}[\meqref{18i}]
Let $s\in\tilde{\X}^{0}_{k}$. The set of vertices of $\tilde{\X}_{k+1}(s)$ is clearly partitioned into two subsets. The set $\mathcal{U}$ of vertices $v\in\tilde{\X}^{0}_{k+1}$ such that $v^{-}=s$ and the set $\mathcal{V}$ of vertices $v\in\tilde{\X}^{0}_{k+1}$ such that $v^{+}=s$. By Corollary~\ref{cor16} we deduce that every vertex in $\mathcal{U}$ is linked to every vertex in $\mathcal{V}$. So as desired the graph $\tilde{\X}_{k+1}(s)$ is a complete bipartite graph and then connected.
\let\qed\relax
\end{proof}

\begin{proof}[\meqref{18ii}]
Let $s$ and $t$ two distinct vertices of $\tilde{\X}_{k}$. If $s$ and $t$ are linked by an edge then by Corollary~\ref{cor16} the two subgraphs $\tilde{\X}_{k+1}(s)$ and $\tilde{\X}_{k+1}(t)$ have at least a common vertex, namely the vertex $s\cup t$. Conversely, if the two subgraphs $\tilde{\X}_{k+1}(s)$ and $\tilde{\X}_{k+1}(t)$ have at least a common vertex, say $v$, then we have $v^{-}=s$ or $v^{+}=s$ and $v^{-}=t$ or $v^{+}=t$. As $s$ and $t$ are distinct then we deduce that $v^{-}=s$ and $v^{+}=t$ or $v^{-}=t$ and $v^{+}=s$. The Corollary~\ref{cor16} implies then that $s$ and $t$ are linked by an edge. So the nerve of the cover of $\tilde{\X}_{k+1}$ by the subgraphs $\tilde{\X}_{k+1}(s)$, for $s\in\tilde{\X}^{0}_{k}$, is the graph $\tilde{\X}_{k}$.
\end{proof}
\let\qed\relax
\end{proof}

\begin{theo}\label{th19}
For every integer $k\geqslant0$, the geometric realization of $\tilde{\X}_{k}$ is connected and locally compact.
\end{theo}

\begin{proof}
The locally compactness of $\vert\tilde{\X}_{k}\vert$ follows from the fact that the graphs $\tilde{\X}_{k}$ are locally finite. For the connectedness, we will prove firstly that $\tilde{\X}_{0}$ is connected. Let $s=[L]$ and $t=[M]$ be two distinct vertices of $\tilde{\X}_{0}$, where $L$ and $M$ are two $\of_{\F}$-lattices. Let us choose an $\F$-basis $(v_{1},\dots,v_{n})$ of $\F^{n}$ for which $L=\of_{\F}v_{1}+\dots+\of_{\F}v_{n}$ and $M=\pf^{k_{1}}_{\F}v_{1}+\dots+\pf^{k_{n}}_{\F}v_{n}$, where $(k_{1},\dots,k_{n})\in\mathbb{Z}^{n}$ with $k_{1}\leqslant\dots\leqslant k_{n}$. By changing the representative $M\in[M]$ we can assume that $0<k_{1}$. Now let us consider the sequence $(L_{0},\dots,L_{m})$ of $\of_{\F}$-lattices, where $m=k_{1}+\dots+ k_{n}$, defined as follows. For every integer $0\leqslant i\leqslant m$, if $k_{1}+\dots+ k_{j-1}+1\leqslant i\leqslant k_{1}+\dots+ k_{j}$, where $1\leqslant j\leqslant n$, then
\[
L_{i}=\bigoplus_{\ell=1}^{j-1}\pf^{k_{\ell}}_{\F}v_{\ell}\oplus\pf^{i-(k_{1}+\dots+ k_{j-1})}_{\F}v_{j}\oplus\bigoplus_{\ell=j+1}^{n}\of_{\F}v_{\ell}.
\]
By a straightforward computation, we can check easily that the sequence $([L_{0}],\dots,[L_{m}])$ is a path of the graph $\tilde{\X}_{0}$ linking the vertex $s$ to the vertex $t$. So as desired $\tilde{\X}_{0}$ is connected. Now we will prove by induction that the graphs $\tilde{\X}_{k}$ are connected for every non-negative integer $k$. Let $k\geqslant0$ be an integer. Assume that the graph $\tilde{\X}_{k}$ is connected and let's prove that $\tilde{\X}_{k+1}$ is also connected. Let $u$ and $v$ be two distinct vertices of $\tilde{\X}_{k+1}$. Since $\tilde{\X}_{k+1}$ is covered by the subgraph $\tilde{\X}_{k+1}(s)$, when $s$ range over the set of vertices of $\tilde{\X}_{k}$, then there exist two vertices $s,t\in\tilde{\X}^{0}_{k}$ such that $u\in\tilde{\X}^{0}_{k+1}(s)$ and $v\in\tilde{\X}^{0}_{k+1}(t)$. As $\tilde{\X}_{k}$ is connected then there exist a path $p=(p_{0},\dots,p_{m})$ in $\tilde{\X}_{k}$ linking the two vertices $s$ and $t$ (say $p_{0}=s$ and $p_{m}=t$). For every integer $i\in\{1,\dots,m\}$, let $v_{i}$ be any vertex of the non-empty graph $\tilde{\X}_{k+1}(p_{i-1})\cap\tilde{\X}_{k+1}(p_{i})$. Let's also put $v_{0}=u$ and $v_{\ell+1}=v$. By the previous lemma the graphs $\tilde{\X}_{k+1}(p_{i})$ are connected. So for $i\in\{0,\dots,\ell\}$, since $p_{i}$ and $p_{i+1}$ are two vertices of the graph $\tilde{\X}_{k+1}(p_{i})$ then there exist a path in $\tilde{\X}_{k+1}$ from $p_{i}$ to $p_{i+1}$. Consequently there exist a path in $\tilde{\X}_{k+1}$ connecting the two vertices $u$ and $v$ and then the graph $\tilde{\X}_{k+1}$ is connected. We have then the connectedness of the graphs $\tilde{\X}_{k}$ for every integer $k\geqslant0$ which implies the connectedness of their geometric realization.
\end{proof}


\section{Realization of the generic representations of \texorpdfstring{$\G_{n}$}{Gn} in the cohomology of the tower of graphs}

\subsection{Generic representations of \texorpdfstring{$\G_{n}$}{Gn}}

Let us firstly recall some basic facts and introduce some notations. Let $\psi$ be a fixed additive smooth character of $\F$ trivial on $\pf_{\F}$ and nontrivial on $\of_{\F}$. We define a character $\theta_{\psi}$ of the group $\U_{n}$ of upper unipotent matrices as follows
\[
\theta_{\psi}\left(
\begin{pmatrix}
1&u_{1,2}&\dots&\dots&u_{1,n}{}\\
0&\ddots&\ddots& &\vdots\\
\vdots&\ddots&\ddots&\ddots&\vdots{}\\
\vdots & &\ddots &1&u_{n-1,n}\\
0& \dots & \dots &0&1\\
\end{pmatrix}\right)=\psi(u_{1,2}+\dots+u_{n-1,n}).
\]
Let $(\pi,\V)$ be an irreducible admissible representation of $\G_{n}$ considered as an irreducible admissible representation of $\GL_{n}(\F)$ with trivial central character. The representation $(\pi,\V)$ is called generic if
\[
\Hom_{\GL_{n}(\F)}(\pi,\Ind_{\U_{n}}^{\GL_{n}(\F)}\theta_{\psi})\neq0.
\]
By Frobenius reciprocity, this is equivalent to the existence of a nonzero linear form $\ell:\V\longrightarrow\mathbb{C}$ such that $\ell(\pi(u).v)=\theta_{\psi}(u)\ell(v)$ for every $v\in \V$ and $u\in \U_{n}$. Thus a generic representation $(\pi,V)$ of $\G_{n}$ can be realized on a same space of functions $f$ with the property $f(ug)=\theta_{\psi}(u)f(g)$ for every $u\in\U_{n}$ and $g\in \GL_{n}(\F)$ and for which the action of $\GL_{n}(\F)$ on the space of $\pi$ is by right translation. Such a realization is called the Whittaker model of $\pi$. The following theorem, due to Bernstein and Zelevinski, shows that generic representations have a unique Whittaker model.

\begin{theo}[{\cite[V.16]{BZ76}}]\label{th20}
Let $(\pi,V)$ be an irreducible admissible representation of $\G_{n}$. Then the dimension of the space $\Hom_{\GL_{n}(\F)}(\pi,\Ind_{\U_{n}}^{\GL_{n}(\F)}\theta_{\psi})$ is at most one, that is
\[
\dim_{\mathbb{C}}\Hom_{\GL_{n}(\F)}\left(\pi,\Ind_{\U_{n}}^{\GL_{n}(\F)}\theta_{\psi}\right)\leqslant1.
\]
In particular, if $\pi$ is generic then $\pi$ has a unique Whittaker model.
\end{theo}

We have the following result which is due to H.~Jacquet, J.~L.~Piatetski-Shapiro and J.~Shalika, see~\cite[Thm.~(5.1)]{JSS81}:

\begin{theo}\label{th21}
Let $(\pi,\V)$ be an irreducible generic representation of $\G_{n}$.
\begin{enumerate}\romanenumi
\item For $k$ large enough, the space of fixed vectors $V^{\Gamma_{0}(\pf_{\F}^{k})}$ is non-zero.
\item Let $c(\pi)$ the smallest integer such that $V^{\Gamma_{0}(\pf_{\F}^{c(\pi)+1})}\neq0$, then for every integer $k\geqslant c(\pi)$, we have :
\[
\dim_{\mathbb{C}}V^{\Gamma_{0}(\pf_{\F}^{k+1})}=k-c(\pi)+1.
\]
\end{enumerate}
\end{theo}


\subsection{Realization of the Generic representations of \texorpdfstring{$\G_{n}$}{Gn}}


In this section, we fix an irreducible generic representation $(\pi,V)$ of $\G_{n}$ and we make the following assumption:

\begin{enonce}[remark]{Assumption}\label{ass521}
$\pi$ is non-spherical, that is the space of $\Gamma_{0}(\pf_{\F}^{0})$-fixed vectors
\[
V^{\Gamma_{0}(\pf_{\F}^{0})}:=\left\{v\in V \,\middle|\, \forall g\in\Gamma_{0}(\pf_{\F}^{0}),~\pi(g)v=v\right\}
\]
is zero.
\end{enonce}

In the following, our aim is to prove that the representation $\pi$ can be realized as a quotient of the cohomology space $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ and if moreover $\pi$ is cuspidal then in fact it can be realized as a subrepresentation of this cohomology space. Furthermore, as in Theorem~(5.3.2) of~\cite{Br09}, we obtain a multiplicity one result for cuspidals but in a more simpler way. The proofs of the results below are similar to those given in~\cite[\S(3.2)]{Br09}. Let us recall that for every vertex $s$ (resp. edge $a$) of $\tilde{\X}_{c(\pi)}$, $\Gamma_{s}$ (resp. $\Gamma_{a}$) denotes the stabilizer in $\G_{n}$ of $s$ (resp. $a$). We recall that
\[
\Gamma_{s_{0}}=\Gamma_{0}(\pf_{\F}^{c(\pi)})\quad\text{and}\quad\Gamma_{a_{0}}=\Gamma_{0}(\pf_{\F}^{c(\pi)+1}),
\]
where $s_{0}$ (resp. $a_{0}$) is the standard vertex (resp. edge) of $\tilde{\X}_{c(\pi)}$.

\begin{lemm}\label{lem23}\ 
\begin{enumerate}\romanenumi
\item \label{23i} For every edge $a$ of $\tilde{\X}_{c(\pi)}$, $ V^{\Gamma_{a}}$ is of dimension one.
\item \label{23ii} Let $a$ be an edge of $\tilde{\X}_{c(\pi)}$ and $s$ be a vertex of $a$. Then for every $v\in V^{\Gamma_{a}}$ we have
\[
\sum_{g\in \Gamma_{s}/\Gamma_{a}}\pi(g)v=0.
\]
\end{enumerate}
\end{lemm}

\begin{proof}
\begin{proof}[\meqref{23i}]
Since $G_{n}$ acts transitively on the set of edges of $\tilde{\X}_{c(\pi)}$ then the subgroup $\Gamma_{a}$ is conjugate to $\Gamma_{a_{0}}$ which gives the result.
\let\qed\relax
\end{proof}

\begin{proof}[\meqref{23ii}]
Clearly the vector
\[
v_{0}:=\sum_{g\in \Gamma_{s}/\Gamma_{a}}\pi(g)v
\]
is fixed by $\Gamma_{s}$. But by transitivity of the action of $G_{n}$ on the set of vertices of $\tilde{\X}_{c(\pi)}$, the subgroup $\Gamma_{s}$ is conjugate to $\Gamma_{s_{0}}$. So Theorem~\ref{th21} implies that $v_{0}=0$.
\end{proof}
\let\qed\relax
\end{proof}

We define a map
\[
\Psi^{\vee}_{\pi}:V^{\vee}\longrightarrow C^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right)
\]
as follows. Let us fix a non-zero vector $v_{0}\in V^{\Gamma_{a_{0}}}$. For every edge $a$ of $\tilde{\X}_{c(\pi)}$, we put
\begin{equation}\label{522}
v_{a}=\pi(g).v_{0},\quad\text{where } a=g.a_{0}
\end{equation}
This definition is well defined since $\G_{n}$ acts transitively on $\tilde{\X}^{1}_{c(\pi)}$ and it does not depend on the choice of $g\in\G_{n}$ such that $v_{a}=g.v_{0}$ as $v_{0}$ is fixed by $\Gamma_{a_{0}}$. The map $\Psi^{\vee}$ is then defined by
\[
\Psi^{\vee}(\varphi)(a)=\varphi(v_{a})
\]
for every $\varphi\in V^{\vee}$ and $a\in\tilde{\X}^{1}_{c(\pi)}$. From \eqref{522} the map $\Psi^{\vee}$ is $G_{n}$-equivariant.

\begin{lemm}\label{lem24}
The map $\Psi^{\vee}$ is injective and its image is contained in $\mathcal{H}_{\infty}(\tilde{\X}_{c(\pi)},\mathbb{C})$.
\end{lemm}

\begin{proof}
The $G_{n}$-equivariant map $\Psi^{\vee}$ is injective as it is nonzero and as the representation $\pi$ is irreducible. Let $\varphi\in V^{\vee}$. Let us prove that for every vertex $s$ of $\tilde{\X}^{1}_{c(\pi)}$,
\[
\sum_{a\in\tilde{\X}^{1}_{c(\pi)}}[a:s]\varphi(v_{a})=0.
\]
Let $s$ be a vertex of $\tilde{\X}^{1}_{c(\pi)}$. By Proposition~\ref{prop17}, the stabilizer $\Gamma_{s}$ acts transitively on the two sets
\[
 \mathcal{V}^{-}(s)=\left\{a\in\tilde{\X}_{c(\pi)}^{1}\,\middle|\,a^{-}=s\right\}\quad\text{and}\quad\mathcal{V}^{+}(s)=\left\{a\in\tilde{\X}_{c(\pi)}^{1}\,\middle|\,a^{+}=s\right\}.
\]
Let us fix $a_{s}^{+}\in\mathcal{V}^{+}(s)$ and $a_{s}^{-}\in\mathcal{V}^{-}(s)$. We have then
\begin{align*}
\sum_{a\in\tilde{\X}^{1}_{c(\pi)}}[a:s]\varphi(v_{a})&=\varphi\left(\sum_{a\in \mathcal{V}^{+}(s)}v_{a}-\sum_{a\in \mathcal{V}^{-}(s)}v_{a}\right)
\\
&=\varphi\left(\sum_{g\in\Gamma_{s}/\Gamma_{a_{s}^{+}}}\pi(g).v_{a_{s}^{+}}-\sum_{g\in\Gamma_{s}/\Gamma_{a_{s}^{-}}}\pi(g).v_{a_{s}^{-}}\right)=0
\end{align*}
by Lemma~\ref{lem23}. Consequently, $\Imop(\Psi^{\vee})$ is contained in $\mathcal{H}(\tilde{\X}_{c(\pi)},\mathbb{C})$ which implies that it is contained in $\mathcal{H}_{\infty}(\tilde{\X}_{c(\pi)},\mathbb{C})$.
\end{proof}

By Lemma~\ref{lem1} we have the isomorphism of smooth $\G_{n}$-module
\[
\mathcal{H}_{\infty}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right)\simeq H_{c}^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right)^{\vee}.
\]
So applying contragredients to the operator $\Psi^{\vee}_{\pi}:V^{\vee}\longrightarrow \mathcal{H}_{\infty}(\tilde{\X}_{c(\pi)},\mathbb{C})$ we obtain an intertwining operator
\[
\Psi^{\vee\vee}_{\pi}:H_{c}^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right)^{\vee\vee}\longrightarrow V^{\vee\vee}.
\]
It is well known that a smooth $\G_{n}$-module $W$ have a canonical injection in the contragredient of its contragredient $W^{\vee\vee}$. So the smooth $\G_{n}$-module $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ canonically injects in $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})^{\vee\vee}$. Moreover the representation $\pi$ is irreducible and hence admissible then $V$ and $V^{\vee\vee}$ are canonically isomorphic. In the following, if $\omega\in C_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ we write $\bar{\omega}$ for its image in $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$.

\begin{theo}\label{th25}
The restriction of $\Psi^{\vee\vee}_{\pi}$ to the space $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ define a nonzero intertwining operator
\[
\Psi_{\pi}:H_{c}^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right)\longrightarrow V
\]
given by
\[
\Psi_{\pi}(\bar{\omega})=\sum_{a\in\tilde{\X}_{c(\pi)}^{1}}\omega(a)v_{a}
\]
In particular, $(\pi,V)$ is isomorphic to a quotient of $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$. Moreover, if $(\pi,V)$ is cuspidal then it is isomorphic to a subrepresentation of $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$.
\end{theo}

\begin{proof}
The fact that the restriction of the map $\Psi^{\vee\vee}_{\pi}$ to the space $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ is given exactly by the map $\Psi_{\pi}$ follows by a straightforward computation. Let $\omega_{0}\in C_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ defined on the basis $\tilde{\X}^{1}_{c(\pi)}$ of $C_{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ as follows : for every edge $a$ of $\tilde{\X}_{c(\pi)}$, $\omega_{0}(a)=1$ if $a=a_{0}$ and $\omega_{0}(a)=0$ otherwise. We have
\[
\Psi_{\pi}(\bar{\omega}_{0})=\sum_{a\in\tilde{\X}_{c(\pi)}^{1}}\omega_{0}(a)v_{a}=v_{0}\neq0.
\]
So the map $\Psi_{\pi}$ is nonzero. Hence by irreducibility of $\pi$ the map $\Psi_{\pi}$ is surjective and then as desired $(\pi,V)$ is isomorphic to a quotient of $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$. If the representation $(\pi,V)$ is cuspidal, so in particular generic, then it is isomorphic to a quotient of $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$. But $(\pi,V)$ is cuspidal and then it is projective in the category of smooth complex representation of $\G_{n}$. So we have in fact an embedding of $(\pi,V)$ in $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$.
\end{proof}

\begin{theo}
If the representation $(\pi,V)$ is cuspidal then it have a unique realization in the cohomology space $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$, that is
\[
\dim_{\mathbb{C}}\Hom_{\G_{n}}\left(\pi,H_{c}^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right)\right)=1.
\]
\end{theo}

\begin{proof}
Since $G_{n}$ acts transitively on the set of vertices and edges of $\tilde{\X}_{c(\pi)}$ then the two $G_{n}$-modules $\C_{c}^{0}(\tilde{\X}_{c(\pi)},\mathbb{C})$ and $\C_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ are respectively isomorphic to the following compactly induced representation
\[
\cind_{\Gamma_{0}(\pf_{\F}^{c(\pi)})}^{\G_{n}}1\quad\text{and}\quad\cind_{\Gamma_{0}(\pf_{\F}^{c(\pi)+1})}^{\G_{n}}1
\]
(where 1 denotes the trivial character). The space $H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ is by definition the cokernel of the coboundary map
\[
C_{c}^{0}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right) \overset{d}\longrightarrow C_{c}^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right)
\]
Then we have a surjective map
\[
\varphi:\cind_{\Gamma_{0}(\pf_{\F}^{c(\pi)+1})}^{\G_{n}}1\longrightarrow\Hrm_{c}^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right)
\]
and so we obtain an injective map
\[
\widetilde{\varphi}:\Hom_{\G_{n}}\left(H_{c}^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right),\pi\right)\longrightarrow\Hom_{\G_{n}}\left(\cind_{\Gamma_{0}(\pf_{\F}^{c(\pi)+1})}^{\G_{n}}1,\pi\right)
\]
On the other hand, by Frobenius reciprocity we have
\[
\Hom_{\G_{n}}\left(\cind_{\Gamma_{0}(\pf_{\F}^{c(\pi)+1})}1,\pi\right)\simeq V^{\Gamma_{n}(\pf_{\F}^{c(\pi)+1})}
\]
But by the Theorem~\ref{th21}, the space of fixed vectors $V^{\Gamma_{n}(\pf_{\F}^{c(\pi)+1})}$ is of dimension one. Thus we obtain
\[
\dim_{\mathbb{C}}\Hom_{\G_{n}}\left(H_{c}^{1}\left(\tilde{\X}_{c(\pi)},\mathbb{C}\right),\pi\right)\leqslant1.
\]
On the other hand, since the representation $(\pi,V)$ is cuspidal then it is a projective object of the category of smooth representations of $\G_{n}$. So the two spaces $\Hom_{\G_{n}}(H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C}),\pi)$ and $\Hom_{\G_{n}}(\pi,H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C})$ are in fact isomorphic. But by the previous theorem $\Hom_{\G_{n}}(H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C}),\pi)$ is nonzero. So as desired the space $\Hom_{\G_{n}}(\pi,H_{c}^{1}(\tilde{\X}_{c(\pi)},\mathbb{C}))$ is one dimensional.
\end{proof}

\section*{Acknowledgements}

The author would like to thank the anonymous referee for their careful reading and helpful suggestions which improved an earlier draft.

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