Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings

Kiumars Kaveh
Department of Mathematics, University of Pittsburgh,
Pittsburgh, PA, USA.

Christopher Manon
Department of Mathematics, University of Kentucky, Lexington, KY, USA

Boris Tsvelikhovskiy
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA

The first author is partially supported by National Science Foundation Grant DMS-2101843 and a Simons Collaboration Grant (award number 714052). The second author is partially supported by National Science Foundation Grant DMS-2101911 and a Simons Collaboration Grant (award number 587209).

We give a classification of rank $r$ torus equivariant vector bundles $\E$ on a toric scheme $\X$ over a discrete valuation ring, in terms of piecewise affine maps $\Phi$ from the polyhedral complex of $\X$ to the  extended Bruhat-Tits building of $\GL(r)$. This is an extension of Klyachko's classification of torus equivariant vector bundles on a toric variety over a field.
We also give a simple criterion for equivariant splitting of $\E$ into a sum of toric line bundles in terms of its piecewise affine map.

Comments are welcome!

§ INTRODUCTION

Let $T$ be an $n$-dimensional split algebraic torus over a field $K$ with cocharacter lattice $N \cong \Z^n$. Let $X = X_\Sigma$ be a $T$-toric variety associated to a fan $\Sigma$ in $N_\R = N \otimes \R \cong \R^n$.  A torus equivariant vector bundle (or toric vector bundle for short) on $X$ is a vector bundle $\E$ on $X$ together with a linear action of $T$ that lifts that of $X$. Toric vector bundles have been classified by Kaneyama <cit.> (in terms of certain data of cocycles) and by Klyachko <cit.> (in terms of certain data of compatible filtrations, see Section <ref>). In this paper we extend Klyachko's classification to toric schemes over a discrete valuation ring $\O$.
Our classification, on the other hand, extends the known classification of toric line bundles on toric schemes (see <cit.>, <cit.>). We introduce the notion of a piecewise affine map from the polyhedral complex, defining the toric scheme, to the extended Bruhat-Tits building of a general linear group. Our main result states that the isomorphism classes of toric vector bundles over a toric scheme are in one-to-one correspondence with such piecewise affine maps.

The present paper can be considered as a continuation of ideas in <cit.> where classifying torus equivariant principal bundles (or toric principal bundles for short) on toric varieties over a field $K$ is connected with the theory of Tits buildings of algebraic groups. The main result in <cit.> states that for a reductive group $G$, toric principal $G$-bundles on $X_\Sigma$ are classified by piecewise linear maps from the fan $\Sigma$ to the cone over the Tits building of $G$.

We begin with recalling some of the main concepts involved. Throughout $K$ denotes a discretely valued field with valuation $\val: K \to \overline{\Z}$. We denote by $\O$ the corresponding valuation ring $\{x \mid \val(x) \geq 0\}$. Also $\varpi$ denotes a uniformizer, that is, a generator of the unique maximal ideal $\m = \{ x \mid \val(x) > 0\}$.
The scheme $\Spec(\O)$ has two points: the generic point $\eta$ corresponding to the prime ideal $\{0\}$ and the special point $o$ corresponding to the maximal ideal $\mathfrak{m}$. For a scheme $\X$ over $\Spec(\O)$, we call $\X_\eta$ (the fiber over $\eta$) the generic fiber, and $\X_o$ (the fiber over $o$) the special fiber.

Let $T$ be a split torus over $\Spec(\O)$. A toric scheme over $\Spec(\O)$ is a normal integral separated scheme of finite type $\X$ over $\Spec(\O)$ equipped with a dense open embedding $T_K=T_\eta \hookrightarrow \X_\eta$ and an action of $T$ on $\X$ over $\Spec(\O)$ which  extends the translation action of $T_K$ on itself.
While toric varieties (over a field) are classified by fans in $N_\R \cong \R^n$, toric schemes (over $\Spec(\O)$) are classified by fans $\Sigma$ in $N_\R \times \R_{\geq 0}$ (see Section <ref>). For a fan in $N_\R \times \R_{\geq 0}$ we denote the corresponding toric scheme by $\X_\Sigma$ . By intersecting cones in $\Sigma$ with $N_\R \times \{1\}$ one obtains a polyhedral complex $\Sigma_1$ in the affine space $N_\R \times \{1\}$. The generic fiber $\X_\eta$ is a usual toric variety over $K$ with corresponding fan $\Sigma_0$ obtained by intersecting the cones in $\Sigma$ with $N_\R \times \{0\}$. For the rest of the paper, we denote the identity element in $T_\eta \hookrightarrow \X$ by $x_0$.

The insight from <cit.> is that the right gadgets for classification of toric principal bundles are buildings of algebraic groups.
Buildings are special kinds of simplicial complexes arising in the classification theory of algebraic groups over arbitrary fields. Abstractly speaking, a building is an (infinite) simplicial complex together with certain distinguished subcomplexes called apartments that satisfy a list of axioms (see Definition <ref>).

There are two important kinds of buildings: spherical buildings and affine buildings. In a spherical building each apartment is a triangulation of a sphere while in an affine building each apartment is a triangulation of an affine (or Euclidean) space. To a reductive algebraic group over a field one associates its Tits building which is a typical example of a spherical building. Moreover, to a reductive algebraic group over a discretely valued field $K$ one associates its Bruhat-Tits building which is a typical example of an affine building.

Let $E \cong K^r$ be an $r$-dimensional $K$-vector space.
To state the main results of the paper we need to briefly review the construction of the Bruhat-Tits building $\B_\aff(E)$ as well as the extended Bruhat-Tits building $\tilde{\B}_\aff(E)$ of the group $\GL(E)$. Roughly speaking, the Bruhat-Tits building $\B_\aff(E)$ is obtained by gluing infinitely many $(r-1)$-dimensional real affine spaces along some simplices. We have one affine space for each direct sum decomposition of $E$ into $1$-dimensional subspaces. The extended building $\tilde{\B}_\aff(E)$ is just an extension of $\B_\aff(E)$ by a direct sum with $\R$. Below we describe the Bruhat-Tits building of the group $\GL(E)$ in two ways (see Section <ref> for details):

(i) as a simplicial complex which we denote by $\Delta_\aff(E)$,

(ii) as a topological space which we denote by $\B_\aff(E)$.

The topological space $\B_\aff(E)$ is a geometric realization of the abstract simplicial complex $\Delta_\aff(E)$, that is, each simplex in $\Delta_\aff(E)$ can be identified with a subset of $\B_\aff(E)$ homeomorphic to a standard simplex and these simplices intersect along their common subsimplices. By abuse of terminology we refer to both as the Bruhat-Tits building of $\GL(E)$.

We recall that a lattice $\Lambda$ (also called an $\O$-lattice) in $E$ is a full rank $\O$-submodule. In other words, a lattice is a finitely generated submodule of $E$ that spans the latter as a vector space. Two lattices $\Lambda$ and $\Lambda'$ are said to be equivalent if there is nonzero $c \in K^\times$ such that $c\Lambda = \Lambda'$. The vertices in $\Delta_\aff(E)$ correspond to equivalence classes $[\Lambda]$ of lattices $\Lambda \subset E$. Two equivalence classes $[\Lambda]$, $[\Lambda']$ are connected to each other if there are representatives $\Lambda$, $\Lambda'$ such that $\varpi \Lambda \subsetneqq \Lambda'\subsetneqq \Lambda$. The $(k-1)$-dimensional simplices in $\Delta_\aff(E)$ then correspond to $k$ mutually connected vertices $[\Lambda_1], \ldots, [\Lambda_{k}]$, in other words, chains of lattices such that $\varpi \Lambda_k \subsetneqq \Lambda_1 \subsetneqq \cdots \subsetneqq \Lambda_k$.
A $K$-vector space basis $B = \{b_1, \ldots, b_r\}$ for $E$ defines a distinguished subcomplex, namely an apartment, in $\Delta_\aff(E)$. The vertices in the apartment of $B$ consist of the equivalence classes of lattices $\Lambda$ of the form $\Lambda = \sum_{i=1}^r \varpi^{a_i} \O b_i$, for any choice of $a_i \in \Z$. Hence the vertices in the apartment can be identified with the points in $\Z^r / \langle (1, \ldots, 1) \rangle \cong \Z^{r-1}$.

Alternatively, the set of lattices in $E$ can be identified with the set of integer-valued additive norms (see Section <ref>). An additive norm on $E$ is a function $v: E \to \overline{\R} = \R \cup \{\infty\}$ satisfying the following axioms:

(1) $v(\lambda e) = \val(\lambda) + v(e)$, for all $e \in E$ and $\lambda \in K$.

(2)(Non-Archimedean property) $v(e_1+e_2) \geq \min\{v(e_1), v(e_2)\}$, for all $e_1, e_2 \in E$.

(3) $v(e) = \infty$ if and only if $e = 0$.

Two additive norms are said to be equivalent if their difference is a constant. Given a basis $B = \{b_1, \ldots, b_r\}$, an additive norm $v$ is said to be adapted to $B$ if the following holds. For any $e = \sum_i c_i b_i \in E$ we have $v(e) = \min\{ \val(c_i) + v(b_i) \mid i=1, \ldots, r \}$.

The set of equivalence classes of additive norms on $E$ is the (geometric realization of the) Bruhat-Tits building of $\GL(E)$ which we denote by $\B_\aff(E)$. For a basis $B$, the set of equivalence classes of additive norms adapted to it is the (geometric realization of the) apartment associated to $B$ which we denote by $A_\aff(B)$.

Furthermore, we denote the set of all additive norms on $E$ by $\tilde{\B}_\aff(E)$ and call it the extended Bruhat-Tits building of $E$. For a basis $B$, we denote the set of all additive norms adapted to it by $\tilde{A}_\aff(B)$ and call it the extended apartment of $B$.

[Piecewise affine map to an affine building]
Let $\Sigma_1$ be a polyhedral complex in $N_\R \times \{1\}$. We say that a map $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$ is a piecewise affine map if the following holds:

(a) For every polyhedron $\Delta \in \Sigma_1$, there is an apartment $\tilde{A}_\aff(\Delta)$ in $\tilde{\B}_\aff(E)$ such that $\Phi(\Delta)$ lands in $\tilde{A}_\aff(\Delta)$.

(b) The map $\Phi_{|\Delta}: \Delta \to \tilde{A}_\aff(\Delta)$ is the restriction of an affine map from $N_\R \times \{1\}$ to the affine space $\tilde{A}_\aff(\Delta)$.

We say that $\Phi$ is integral if, for every $\Delta \in \Sigma_1$, $\Phi_{|\Delta}$ is the restriction of an integral affine map from $N_\R \times \{1\}$ to $\tilde{A}_\aff(\Delta)$.

Recall that given a piecewise affine function $\phi: |\Sigma_1| \to \R$, one can define $\phi_0: |\Sigma_0| \to \R$, the linear part of $\phi_0$. Here $\Sigma_0$ is the collection of recession cones of the polyhedra in $\Sigma_1$. More generally, for a piecewise affine map $\Phi$, we can define $\Phi_0: |\Sigma_0| \to \tilde{\B}_\sph(E)$, the linear part of $\Phi$. Here $\tilde{\B}_\sph(E)$ is the cone over the spherical building of $\GL(E)$ which can be identified with the collection of prevaluations on $E$ (see Section <ref>).

There is a natural notion of morphism of piecewise affine maps (Definition <ref>) and the collection of piecewise affine maps to extended Bruhat-Tits buildings of finite dimensional vector spaces $E$ form a category.

Let $\X_\Sigma$ be a toric scheme over $\Spec(\O)$ associated to a fan $\Sigma$ in $N_\R \times \R_{\geq 0}$ with corresponding polyhedral complex $\Sigma_1$ in $N_\R \times \{1\}$. Throughout, we will assume that $\Sigma_1$ determines $\Sigma$. That is, all the cones in $\Sigma$ are cones over the polyhedra in $\Sigma_1$ or faces of such cones. For example, this is the case if the support of $\Sigma$ is the whole $N_\R \times \R_{\geq 0}$, or equivalently, if the support of $\Sigma_1$ is the whole $N_\R \times \{1\}$ (this means that the toric scheme $\X_\Sigma$ is proper over $\Spec(\O)$, see also <cit.>).
Given a toric vector bundle $\E$ over $\X_\Sigma$, we denote the fiber $\E_{x_0}$, over the distinguished point $x_0$, by $E$. Our main result is the following.
[Main theorem, first part]
With notation as above, there is a bijection between the isomorphism classes of toric vector bundles on $\X_\Sigma$ and the piecewise affine maps from $|\Sigma_1|$ to $\tilde{\B}_\aff(E)$. Moreover, this bijection extends to an equivalence of the corresponding categories.

The moral of Theorem <ref> is that the simplicial complex $\tilde{\B}_\aff(E)$ can be considered as an analogue of the classifying space of $\GL(E)$ for rank $r$ toric vector bundles over toric schemes.

[Main theorem, second part]
Let $\E$ be a toric vector bundle over a toric scheme $\X_\Sigma$ with the corresponding piecewise affine map $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$. Then the generic fiber $\E_\eta$, which is a (usual) toric vector bundle over the (usual) toric variety $\X_\eta = X_{\Sigma_0}$, corresponds to $\Phi_0: |\Sigma_0| \to \tilde{\B}_\sph(E)$, the linear part of $\Phi_0$, which is a piecewise linear map.

Theorems <ref>
and <ref> extend the well-known results on classification of torus equivariant line bundles on a toric scheme (<cit.>).

We made the assumption that the fan $\Sigma$ (in $N_\R \times \R_{\geq 0}$) is determined by its corresponding polyhedral complex $\Sigma_1$ (in $N_\R \times \{1\}$). One can modify the statement of Theorem <ref> so that this assumption is not necessary. To this end, one uses the notion of a piecewise linear map from the fan $|\Sigma|$ to the “completed” extended building $\tilde{\B}_\aff(E) \cup \tilde{\B}_\sph(E)$.

We expect that Theorem <ref> can be used to give a classification of equivariant vector bundles on normal complexity-one $T$-varieties, in terms of piecewise affine maps, extending the classification of equivariant line bundles on these varieties.

In this regard the results in <cit.> might be relevant.

We say that a toric vector bundle $\E$ over a toric scheme splits equivariantly if it is equivariantly isomorphic to a direct sum of toric line bundles. We give the following criterion for equivariant splitting (Theorem <ref>).
[Criterion for equivariant splitting]
A toric vector bundle $\E$ over a toric scheme $\X_\Sigma$ splits equivariantly if and only if the image of the corresponding piecewise affine map $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$ lands in one extended apartment.

We use the above criterion to give an example of a toric vector bundle over $\mathbb{P}^1_\O$ which does not split equivariantly. This is in contrast with the case of toric vector bundles over a field $K$: every toric vector bundle over $\mathbb{P}^1_K$ splits equivariantly (see <cit.>).

We end this introduction with a question and a conjecture. An important key step in the proof of Theorem <ref> is the local equivariant triviality of toric vector bundles over affine toric schemes (Proposition <ref>). We ask whether this result extends toric principal bundles over toric schemes. More precisely, let $G$ be a reductive algebraic group over $\Spec(\O)$. A toric principal $G$-bundle over a toric scheme $\X$ is a principal $G$-bundle on $\X$ together with an action of $T$ lifting that of $\X$ such that the $T$-action and the $G$-action commute.

Is a toric principal bundle $\mathcal{P}$ over an affine toric scheme $\X$ equivariantly trivial? That is, is there a $(G \times T)$-equivariant isomorphism between $\mathcal{P}$ and $\X \times G \to \X$, where $G$ acts on $\X \times G$ by multiplication on the second factor and $T$ acts on $\X \times G$ diagonally via a homomorphism $T \to G$?

Provided that the answer to the above question is positive, it is reasonable to make the following conjecture. This is an analogue of the main result in <cit.>.

The toric principal bundles over a toric scheme $\X_\Sigma$ are classified by piecewise affine maps from $\Sigma$ to the (extended) Bruhat-Tits building of $G$.

We end the introduction with a question/conjecture motivated by equivariant splitting of toric vector bundles.

Does Helly's theorem for the Bruhat-Tits building of $\GL(2, K)$ (see Example <ref> and <cit.>) hold for other buildings?

The structure of the paper is as follows: Section <ref> reviews preliminary material about Klyachko classification of toric vector bundles over toric varieties, as well as a review of toric schemes. Section <ref> recalls Tits and Bruhat-Tits buildings associated to the general linear group. Section <ref> contains the statements and proofs of our main results. Finally Section <ref> gives a simple criterion for equivariant splitting of toric vector bundles.

Acknowledgement: We would like to thank Huayi Chen, Martin Sombra, Eric Katz and specially Sam Payne for very useful email correspondence. The first two authors are supported by an NSF collaborative research grant as well as Simons Collaboration Grants for Mathematicians.

* $K$, a discretely valued field with valuation $\val: K \to \overline{\Z}$

* $\O$, the valuation ring of $(K, \val)$

* $\mathfrak{m}$, the maximal ideal in $\O$

* $\varpi$, a uniformizer for $\O$ i.e. a generator of $\mathfrak{m}$

* $\k = \O/\mathfrak{m}$, the residue field of $\O$

* $\Spec(\O)$, the affine scheme associated to $\O$

* $\eta$, the generic point of $\Spec$, i.e. the point associated to the prime ideal $\{0\}$

* $o$, the special point of $\Spec$, i.e. the point associated to the maximal ideal $\mathfrak{m}$

* For a scheme $\X$ over $\Spec$, we denote the fibers of $\X$ over $\eta$ and $o$ by $\X_\eta$ and $\X_o$ respectively.

* $T \cong \mathbb{G}_m^n$, a split torus over $\O$. We have $T_\eta \cong (K^\times)^n$ and $T_o \cong (\k^\times)^n$.

* $N$ and $M$, cocharacter and character lattices of $T$, with $N_\R = N \otimes \R$ and $M_\R = M \otimes \R$

* $\tilde{N} = N \times \Z$ and $\tilde{M} = M \times \Z$

For a cone $\sigma \subset N_\R \times \R_{\geq 0}$, we let $\Delta =  \sigma \cap (N_\R \times \{1\})$. Conversely, for a polyhedron $\Delta \subset N_\R \times \{1\}$ we let $\sigma \subset N_\R \times \R_{\geq 0}$ be the cone over $\Delta$.

* $\Sigma$, a fan in $N_\R \times \R_{\geq 0}$

* $\Sigma_1$, a polyhedral complex in $N_\R \times \{1\}$ obtained by intersecting the cones in $\Sigma$ with $N_\R \times \{1\}$

* $\X = \X_{\Sigma}$, toric scheme associated to a fan $\Sigma$.

* $\B_\aff(E)$, the collection of all additive norms on a $K$-vector space $E$ modulo homotheties. It can be considered as (the geometric realization of) the Bruhat-Tits building of $\GL(E)$ (as well as $\SL(E)$ or $\PGL(E)$).

* $\tilde{\B}_\aff(E)$, the collection of all additive norms on a $K$-vector space $E$. It is called the extended Bruhat-Tits building of $\GL(E)$.

§ PRELIMINARIES ON TORIC VARIETIES AND TORIC SCHEMES

§.§ Klyachko's classification of toric vector bundles over a field

In this section we review Klyachko's classification of toric vector bundles <cit.>.
We mainly follow the exposition in <cit.>. The first classification of toric vector bundles goes back to <cit.>. We refer the reader to <cit.> for a nice brief history of the subject.

Let $T \cong \mathbb{G}_m^n$ denote an $n$-dimensional algebraic torus over a field $\k$. We let $M$ and $N$ denote its character and cocharacter lattices respectively. We also denote by $M_$ and $N_$ the $$-vector spaces spanned by $M$ and $N$. For cone $σ∈N_$ let $M_σ$ be the quotient lattice:
$$M_\sigma = M / (\sigma^\perp \cap M).$$
Let $Σ$ be a (finite rational polyhedral) fan in $N_$ and let $X_Σ$ be the corresponding toric variety. Also $X_σ$ denotes the invariant affine open subset in $X_Σ$ corresponding to a cone $σ∈Σ$. We denote the support of $Σ$, that is the union of all the cones in $Σ$, by $|Σ|$. For each $i$, $Σ(i)$ denotes the subset of $i$-dimensional cones in $Σ$. In particular, $Σ(1)$ is the set of rays in $Σ$. For each ray $ρ∈Σ(1)$ we let $v_ρ$ be the primitive vector along $ρ$, i.e. $v_ρ$ is the unique vector on $ρ$ whose integral length is equal to $1$.

We say that $$ is a {\it toric vector bundle} on $X_Σ$ if $$ is a vector bundle on $X_Σ$ equipped with a $T$-linearization. This means that there is an action of $T$ on $$ which lifts the $T$-action on $X_Σ$ such that the action map $_x →_t ·x$ for any $t ∈T$, $x ∈X_Σ$ is linear.

We fix a point $x_0 ∈X_0 ⊂X_Σ$ in the dense orbit $X_0$. We often identify $X_0$ with $T$ and think of $x_0$ as the identity element in $T$. We let $E = _x_0$ denote the fiber of $$ over $x_0$. It is an $r$-dimensional vector space where $r = ()$.

For each cone $σ∈Σ$ we have an invariant open subset $X_σ⊂X_Σ$. The space of sections $H^0(X_σ, )$ is a $T$-module. We let  $H^0(X_σ, )_u ⊆H^0(X_σ, )$ be the weight space corresponding to a weight $u ∈M$; these spaces define the weight decomposition:

$$H^0(X_\sigma, \E) = \bigoplus_{u \in M} H^0(X_\sigma, \E)_u.$$ Every section in $H^0(X_σ, )_u$ is determined by its value at $x_0$.  Thus, by restricting sections to $E = _x_0$, we get an embedding $H^0(X_σ, )_u ↪E$. Let us denote the image of $H^0(X_σ, )_u$ in $E$ by $E_u^σ$. Note that if $u' ∈σ^∨∩M$ then multiplication by the character  $χ^u'$ gives an injection $H^0(X_σ, )_u ↪H^0(X_σ, )_u-u'$. Moreover, the multiplication map by $χ^u'$ commutes with the evaluation at $x_0$ and hence induces an inclusion $E_u^σ⊂E_u-u'^σ$. If $u' ∈σ^⊥$ then these maps are isomorphisms and thus $E_u^σ$ depends only on the class $[u] ∈M_σ= M/(σ^⊥∩M)$. For a ray $ρ∈Σ(1)$ we write $$E^\rho_j = E_u^\rho,$$ for any $u ∈M$ with $⟨u, v_ρ⟩= j$ (all such $u$ define the same class in $M_ρ$). Equivalently, one can define $E^ρ_u$ as follows (see \cite[\S 0.1]{Klyachko}). Pick a point $x_ρ$ in the orbit $O_ρ$ and let:
$$E^\rho_u = \{ e \in E \mid \lim_{t \cdot x_0 \to x_\rho} \chi^u(t)^{-1}(t \cdot e) \textup{ exists in } \E \},$$
where $t$ varies in $T$ in such a way that $t ·x_0$ approaches $x_ρ$. One checks that $E^ρ_u$ does not depend on the choice of $x_ρ$ and only depends on $j = ⟨u, v_ρ⟩$.

We thus have a decreasing filtration of $E$:
\begin{equation}  \label{equ-filt-E-rho}
\cdots \supset E^\rho_{j-1} \supset E^\rho_j \supset E^\rho_{j+1} \supset \cdots
\end{equation}
%Let $\sigma \in \Sigma$ be a cone with the corresponding invariant open subset $X_\sigma$.

An important step in the classification of toric vector bundles is that a toric vector bundle over an affine toric variety is {\it equivariantly trivial}. That is, it decomposes $T$-equivariantly as a sum of trivial line bundles. Let $σ$ be a strictly convex rational polyhedral cone with corresponding affine toric variety $X_σ$. Given $u ∈M$, let $Ł_u$ be the trivial line bundle $X_σ×𝔸^1$ on $X_σ$ where $T$ acts on $𝔸^1$ via the character $-u$. One observes that the toric line bundle $Ł_u$ in fact only depends on the class $[u] ∈M_σ$. Hence we also denote this line bundle by $Ł_[u]$. One has the following (\cite[Proposition 2.1.1]{Klyachko}):
\begin{proposition}   \label{prop-toric-vb-over-affine-equiv-trivial}
Let $\E$ be a toric vector bundle of rank $r$ on an affine toric variety $X_\sigma$. Then $\E$ splits equivariantly into a sum of line bundles $\L_u$:
$$\E = \bigoplus_{i=1}^r \L_{[u_i]}$$
where $[u_i] \in M_\sigma$.
\end{proposition}
%Without loss of generality we can assume that $\sigma$ is a full dimensional cone. Let $x_\sigma$ denote the unique $T$-fixed point in $X_\sigma$. One knows that every vector bundle over an affine variety is globally generated \cite[Example II 5.16.2]{Hartshorne}. Thus the restriction map $H^0(X_\sigma, \E) \to \E_{x_\sigma}$ is a surjective $T$-equivariant map. Hence we can find weight sections $s_1, \ldots, s_r \in H^0(X_\sigma, \E)$ such that their restrictions form a basis for $E_{x_\sigma}$. Now the set of $x \in X_\sigma$ where the set $\{s_1(x), \ldots, s_r(x)\}$ is linearly dependent is a closed $T$-invariant subset which does not contain $x_\sigma$, hence it must be the empty set. The weight sections $s_i$ then provide an equivariant trivialization of $\E$.

We denote the multiset ${ [u_1], …, [u_r]} ⊂M_σ$ by $u(σ)$. The above shows that the filtrations $(E^ρ_i)_i ∈$, $ρ∈Σ(1)$, satisfy the following compatibility condition:
There is a decomposition $E = ⊕_j=1^r L_j$ of $E$ into a direct sum of $1$-dimensional subspaces $L_j$ and a multiset $u(σ) = { [u_1], …, [u_r]} ⊂M_σ$ such that for any ray $ρ∈σ(1)$ we have:
\begin{equation}  \label{equ-Klyachko-comp-condition}
E^\rho_i = \sum_{\langle u_j, v_\rho \rangle \geq i}  L_j.
\end{equation}

%{\color{red} Say something about the morphisms in this setting (or a forward reference to where this is discussed later) in order to give meaning to the stated equivalence of categories.}
{We call a collection of decreasing $\Z$-filtrations $\{(E_i^\rho) \mid \rho \in \Sigma(1)\}$ a \emph{compatible collection of filtrations} if for any $\sigma \in \Sigma$ there is a direct sum decomposition $E=\bigoplus_{j=1}^r L_j$ of $E$ into $1$-dimensional subspaces and a multiset $\{ [u_1], \ldots, [u_r]\} \subset M_\sigma$ such that \eqref{equ-Klyachko-comp-condition} holds. We also need the notion of a morphism between compatible filtrations. Let $E$, $E'$ be finite dimensional vector spaces with compatible collections of filtrations $\{(E^\rho_i) \mid \rho \in \Sigma(1)\}$ and
$\{({E'}^\rho_i) \mid \rho \in \Sigma(1)\}$ respectively. A \emph{morphism} between these compatible collections is a linear map $F: E \to E'$ such that $F(E_i^\rho) \subset {E'}_i^\rho$, for all $i \in \Z$ and $\rho \in \Sigma(1)$.}

%The following theorem of Klyachko (Theorem \ref{th-Klyachko}) states that the category of toric principal bundles on $X_\Sigma$ is naturally equivalent to the category of compatible filtrations on $E$.

%Before we state Klyachko's result on classification of toric vector bundles, which is stated as an equivalence of categories, we need to define notions of morphism in our setting.

The following is Klyachko's result on classification of toric vector bundles (see \cite[Theorem 2.2.1]{Klyachko}).
\begin{theorem}[Klyachko]   \label{th-Klyachko}
The category of toric vector bundles $\E$ on $X_\Sigma$ is naturally equivalent to the category of compatible collections of filtrations on finite dimensional vector spaces $E$.
\end{theorem}

\subsection{Toric schemes over a discrete valuation ring}   \label{subsec-toric-schemes}
%Review of classification of toric schemes over a DVR using fans in $N_\R \times \R$.
Let $Ø$ be a discrete valuation ring with field of fractions $K$. We let $ϖ$ denote a uniformizer for $Ø$, namely an ideal generator for $𝔪$. We let $=̨ Ø/ 𝔪$ be the residue field of $Ø$. Also $o$ and $η$ denote the \emph{special} and \emph{generic} points in $(Ø)$ corresponding to the maximal ideal $𝔪$ and the prime ideal ${0}$ respectively. For a scheme $$ over $(Ø)$ we let $_η= ×_(Ø) (K)$ and $_o = ×_(Ø) ()̨$ be the \emph{generic fiber} and \emph{special fiber} of $$ respectively.

We let $N$ and $M$ be dual free abelian groups of rank $n$, $Ñ = N ×$ and $M̃ = M ×$.
We also let $T$ denote the split torus over $(Ø)$ with cocharacter lattice $N$. For $m ∈M$ we denote the corresponding character of $T$ by $χ^m$. The generic fiber $T_η$ is the split torus over $K$ isomorphic to $(K^×)^n$. The coordinate ring
$K[T_η]$ is isomorphic to the Laurent polynomial algebra with coefficients in $K$ in $n$ indeterminates.

A \textit{toric scheme} $$ is an irreducible normal scheme of finite type over $(Ø)$ with a $T$-action with a choice of an open embedding of $T_η$ in $$ such that the action of $T$ on $$ extends the translation action of $T_η$ on itself. We denote the image of the identity point in $T_η$ in $$ with $x_0$.

For a rational polyhedral cone $σ⊂N_×_≥0$ we let $Δ$ be the polyhedron obtained by intersecting $σ$ and the plane $N_×{1}$ (it may be empty). Conversely, for a rational polyhedron $Δ⊂N_×{1}$. We let $σ⊂N_×_≥0$ denote the cone over $Δ$. Then $σ↔Δ$ gives a one-to-one correspondence between rational polyhedra in $N_×{1}$ and rational polyhedral cones in $N_×_≥0$ that intersect $N_×{1}$.

\begin{definition}  \label{def-sigma-Delta}
Let $\Sigma$ be a fan in $N_\R \times \R_{\geq 0}$. We denote by $\Sigma_1$ the polyhedral complex in $N_\R \times \{1\}$ obtained by intersecting all the cones in $\Sigma$ by $N_\R \times \{1\}$. Conversely, for a polyhedral complex $\Sigma_1$ in $N_\R \times \{1\}$ we let $\Sigma$ be the fan consisting of the cones over the polyhedra in $\Sigma$ as well as the faces of these cones.
\end{definition}

We recall that the \emph{recession cone} $(Δ)$ of a polyhedron $Δ⊂N_$ is defined as:
$$\recc(\Delta) = \{ x \in N_\R \mid x+\Delta \subset \Delta \}.$$

\begin{remark}   \label{rem-rec-fan}
Let $\Sigma_1$ be a complete strongly convex rational polyhedral complex in $N_\R \times \{1\}$ (for precise definitions see \cite[Section 2.1]{BPS}). Then the collection $\Sigma$ of cones over the polyhedra in $\Sigma_1$ is indeed a fan in $\N_\R \times \R_{\geq 0}$. In particular, the collection $\Sigma_0$ of recession cones of the polyhedra in $\Sigma$ is a complete fan in $N_\R$. On the other hand, as is shown in \cite[Example 3.1]{BS}, this is not the case if $\Sigma_1$ is not a complete strongly convex rational polyhedral complex.
\end{remark}

For a strictly convex rational polyhedral cone $σ⊂N_×_≥0$ let $S_σ ⊂K[T_η]$ be the subring defined by:
$$S_{\sigma} = \O[\chi^u \varpi^k \mid (u, k) \in \sigma^\vee \cap \tilde{M}],$$
and let $_σ = (S_σ)$.
When $σ$ is the cone over a polyhedron $Δ$ in $N_×{1}$ we also denote $S_σ$ by $S_Δ$.

By gluing the $_σ$, $σ∈Σ$, one can construct a toric scheme $_Σ$. When the fan $Σ$ is obtained from a polyhedral complex $Σ_1$ in $N_×{1}$ we also denote the toric scheme $_Σ$ by $_Σ_1$.
%{\color{red} Can we have toric schemes corresponding to fans that do not come from a polyhedral complex i.e. have parts lying in the zero level?}
%{\color{red} Mention we are assuming that $\Sigma$ is complete.}

One has the following (\cite[Chap IV, Sec. 3, p. 191]{Kempf}).
\begin{theorem}
\begin{itemize}
\item[(a)] $\sigma \mapsto \X_{\sigma}$ gives rise to an equivalence of categories between strictly convex rational polyhedral cones in $N_\R \times \R_{\geq 0}$ and affine toric $T$-schemes over $S$.
\item[(b)] $\Sigma \mapsto \X_{\Sigma}$ gives an equivalence of categories between fans in $N_\R \times \R_{\geq 0}$ and toric $T$-schemes over $S$.
\item[(c)] There is a bijection between the orbits of $T_\eta$ in $\X_\eta$ and the cones in $\Sigma_0$, that is the cones in $\Sigma$ that lie in $N_\R \times \{0\}$. Moreover, there is bijection between the orbits of $T_o$ in $\X_o$ and the cones of $\Sigma$ that intersect $N_\R \times \{1\}$, or in other words, the polyhedra in the polyhedral complex $\Sigma_1$.
\item[(d)] In particular, the irreducible components of the special fiber $(\X_\Sigma)_o$ are in bijection with the vertices of $\Sigma$. Let
$\rho$ be a vertex in $\Sigma$ and let $(v_\rho, \ell_\rho)$ be the primitive vector on the ray generated by $\rho$. Then $\ell_\rho$ is the order of vanishing of the uniformizer $\varpi$ along the irreducible component of $\X_o$ corresponding to $\rho$.
\end{itemize}
\end{theorem}

\begin{example}  \label{ex-toric-scheme-P1}
Let $n=1$ and $K = \C((t))$ be the field of formal Laurent series in one indeterminate $t$.
Consider the polyhedral complex $\Sigma_1$ in $N_\R \times \{1\}$ consisting of the
polyhedra $(-\infty, 0]$, $[0, 1]$, $[1, \infty)$ (Figure \ref{fig-toric-scheme-poly-P1}). The corresponding fan $\Sigma$ is depicted in Figure \ref{fig-toric-scheme-fan-P1}. Intuitively, the toric scheme $\X_{\Sigma}$ is the family over $S$ whose generic fiber is a projective line $\mathbb{P}^1$ and the special fiber is a union of two copies of $\mathbb{P}^1$ glued at a fixed point (Figure \ref{fig-toric-scheme-P1}).
\begin{figure}[h]
\centering
\begin{tikzpicture}[>=stealth]
\begin{axis}[xmin=-2.5, xmax=2.5, ymin=-0.5, ymax=2.3,
axis lines*=middle,
\addplot[name path=F,domain={-3:3}] {1.5};
\plot[name path=I,domain={-3:3}, opacity=0] {2};
\addplot[name path=G,domain={-0.5:2.3}] {\x};
\addplot[name path=H,domain={-2:2}] {0} node [pos=0.97, below] {$\Sigma_0$};
\addplot[pattern=north east lines, pattern color=blue]fill between[of=I and H, soft clip={domain=-2:0}];
\addplot[pattern=horizontal lines, pattern color=red]fill between[of=I and G, soft clip={domain=0:2}];
\addplot[pattern=vertical lines, pattern color=green]fill between[of=H and G, soft clip={domain=0:2}];
\node at (axis cs:0,0) [circle, scale=0.4, draw=black!80,fill=black!80] {};
\node at (axis cs:0,1.5) [circle, scale=0.4, draw=black!80,fill=black!80] {};
\node at (axis cs:1.5,1.5) [circle, scale=0.4, draw=black!80,fill=black!80] {};
\node at (axis cs:2.3,2) [] {$\widetilde{\Sigma}$};
\end{axis}
\end{tikzpicture}

\caption{A fan $\Sigma$ defining a toric scheme with $\mathbb{P}^1$ as general fiber}
\label{fig-toric-scheme-fan-P1}
\begin{tikzpicture}
\draw[-] (-3,0)--(3.2,0);
\draw (-3,1.5)--(3.5,1.5);
\draw[decorate,decoration={coil,segment length=4pt}] (-3,1.5)--(-0.2,1.5);
\draw[decorate,decoration={coil,segment length=4pt}] (0.2,1.5)--(1.3,1.5);
\draw[decorate,decoration={coil,segment length=4pt}] (1.7,1.5)--(3.5,1.5);
\draw[-] (0,-0.5)--(0,2.5);
\node at  (0.4,2.5){$\R_{\geq 0}$};
\node at  (0,1.5){$\bullet$};
\node at  (1.5,1.5){$\bullet$};
\node at  (3.5,0){$N_{\R}$};

\end{tikzpicture}
\label{fig-toric-scheme-poly-P1}
\caption{The polyhedral complex $\Sigma$ associated to the fan $\Sigma$}
\begin{tikzpicture}
\draw[thick](-1.7,1.3) circle (0.4);
\draw[thick] (-1.65,0.9) arc (-23:23:1);
\draw[thick, dashed] (-1.7,1.7) arc (157:203:1);
\draw[thick] (0.95,0.9) arc (-23:23:1);
\draw[thick, dashed] (0.95,1.7) arc (157:203:1);
\draw[thick] (1.75,0.9) arc (-23:23:1);
\draw[thick, dashed] (1.75,1.7) arc (157:203:1);
\draw[thick](0.9,1.3) circle (0.4);
\draw[thick](1.7,1.3) circle (0.4);
\draw[-] (-2.5,0)--(2.5,0);
\draw[dashed, ->] (-0.8,1.3)--(0.1,1.3);
\draw[densely dotted, -] (-1.7,0.7)--(-1.7,0.2);
\draw[densely dotted, -] (1.3,0.7)--(1.3,0.2);
\node at  (-2.1,1.3){\tiny{$\bullet$}};
\node at  (-1.3,1.3){\tiny{$\bullet$}};
\node at  (0.5,1.3){\tiny{$\bullet$}};
\node at  (1.3,1.3){\tiny{$\bullet$}};
\node at  (2.1,1.3){\tiny{$\bullet$}};
\node at  (-1.7,0){\tiny{$\bullet$}};
\node at  (1.3,0){\tiny{$\bullet$}};
\node at  (-1.7,-0.3){$\eta$};
\node at  (1.3,-0.3){$o$};
\node at  (-1.7,2.1){$\X_\eta$};
\node at  (1.3,2.1){$\X_o$};
\end{tikzpicture}
\label{fig-toric-scheme-P1}
\caption{The general and special fibers of the toric scheme associated to the fan $\Sigma$}
\end{figure}
\end{example}

\begin{definition}   \label{def-tlb-over-S}
A \emph{toric line bundle} on $\X$ is a line bundle $\L$ over $\X$ together with a linear action of $T$ on $\L$.
\end{definition}

We also define a \emph{framed} toric line bundle on $$ to be a toric line bundle together with a choice of a nonzero vector $e$ in the fiber $(Ł_η)_x_0$. The choice of $e$ is equivalent to the choice of a $K$-linear isomorphism between $(Ł_η)_x_0$ and $K$.

\begin{remark}    \label{rem-BPS}
Our notion of a framed toric line bundle is the same as the notion of a \emph{toric line bundle with a toric section} in \cite[Definition 3.6.4]{BPS}. We point out that we use the term \emph{toric line bundle} to mean a torus equivariant line bundle.
\end{remark}

Recall that a function $Φ: N_→$ is an \emph{affine function} if $Φ$ is a linear function followed by a translation. In other words,
there is a dual vector $u ∈M_$ and $b ∈$ such that for all $x ∈N_$ we have:
$$\Phi(x) = \langle u, x \rangle + b.$$
We say that $Φ$ is an \emph{integral} affine function if $Φ(N) ⊂$. This means that $u ∈M$ and $b ∈$.

Let $Σ$ be a polyhedral complex. We recall that $|Σ|$ denotes the \emph{support} of $Σ$, the union of all the polyhedra in $Σ$.
\begin{definition}[Piecewise affine function]
We say that a function $\phi:|\Sigma| \to \R$ is \emph{piecewise affine}, if for every polyhedron $\sigma \in \Sigma$, the function $\phi_{|\sigma}$ is the restriction of an affine function on $N_\R$ to $\sigma$. We say that $\phi$ is an integral piecewise affine function, if for every $\sigma \in \Sigma$, the linear $\phi_{|\sigma}$ is the restriction of an integral affine function on $N_\R$ to $\sigma$, that is, it maps $N$ to $\Z$.\end{definition}

One has the following (it is essentially the same as \cite[Chap IV, Section 3, item (h)]{Kempf}).
\begin{theorem}    \label{th-tlb-over-S}
There is a bijection between the set of (framed) toric line bundles $\mathcal{L}$ over $X_\Sigma$ and integral piecewise affine functions $\phi: \Sigma \to \R$.
\end{theorem}

Our Theorem \ref{th-main} extends the above (Theorem \ref{th-tlb-over-S}) to toric vector bundles over $S$.

\section{Preliminaries on buildings} \label{sec-prelim-bldg}
%\subsection{General notion of a building}
A \emph{building} is an (abstract) simplicial complex together with a collection of distinguished subcomplexes called \emph{apartments} that satisfy certain axioms (see Definition \ref{def-building}). One can think of the notion of building as a ``discretization'' of the notion of symmetric space from Lie theory and differential geometry.   %While symmetric spaces can be defined for real and complex groups, buildings can be defined for any linear algebraic group over a (not necessarily algebraically closed) field.

There are two important kinds of buildings: \emph{spherical buildings} and \emph{affine buildings}.
\begin{itemize}
\item In a spherical building, every apartment is a triangulation of a sphere. To an algebraic group $G$ over a field there corresponds a spherical building which is usually referred to as
the \emph{Tits building} of $G$. As a simplicial complex, it encodes the relative position of \emph{parabolic subgroups} of $G$.
\item In an affine building every apartment is a triangulation of an affine (Euclidean) space. To an algebraic group $G$ over a discretely valued field there corresponds an affine building which is usually referred to as
the \emph{Bruhat-Tits building} of $G$. As a simplicial complex, it encodes the relative position of \emph{parahoric subgroups} of $G$.
\end{itemize}

The main results of the paper are concerned with the Bruhat-Tits building of a general linear group $(r, K)$ over a discretely valued field $K$. The Tits building of $(r, K)$ also makes an appearance in our main results as the \emph{boundary at infinity} of the Bruhat-Tits building. We discuss the usual descriptions of these buildings in Sections \ref{subsec-BT-bldg} and \ref{subsec-Tits-bldg}. In Section
\ref{subsec-bldg-at-infinity} we explain how the Tits building can be regarded as the boundary at infinity of the Bruhat-Tits building.

The notion of a building was invented by Jacques Tits (see the introduction in \cite{AB} and references therein) in order to describe semisimple algebraic groups over an arbitrary field. Given a linear algebraic group $G$, Tits showed how to associate a simplicial complex $Δ(G)$ together with an action of $G$ (the Tits building of $G$). The action of $G$ imposes very strong combinatorial regularity conditions on the complex $Δ$. Taking these conditions as axioms for a class of simplicial complexes, Tits gave his first definition of a building. A part of the data defining a building $Δ$ is a Coxeter group $W$. As a simplicial complex, each apartment in a building is the \emph{Coxeter complex} of $W$. The building $Δ$ is glued together from multiple copies of the Coxeter complex. When W is a finite Coxeter group, the action of this group divides the Euclidean space into a finite number of cones and the Coxeter complex is the triangulation of sphere obtained by intersecting these chambers with the unit sphere. Hence, for a finite Coxeter group, (the geometric realization of) its Coxeter complex is a topological sphere, and the corresponding buildings are spherical buildings. When $W$ is an affine Weyl group, the Coxeter complex is a triangulation of the affine space into (identical) simplices. In this case, (the geometric realization of) the Coxeter complex is a Euclidean space and the corresponding buildings are affine buildings. Iwahori–Matsumoto, Borel-Tits and Bruhat-Tits showed that in analogy with Tits's construction of Tits buildings, affine buildings can also be constructed from algebraic groups $G$ over a discretely valued field (local non-Archimedean field). This is the Bruhat-Tits buildings of $G$.

As mentioned before, in this paper, we are only concerned with the Bruhat-Tits building of a general linear group $(r, K)$ and we do not discuss the technical construction of the Bruhat-Tits building of an arbitrary linear algebraic group. For the general construction we refer the reader to \cite{Bruhat-Tits1} or or \cite[Section 3]{RTW}. In the appendix (Section \ref{sec-appendix}) we give a brief review of buildings of algebraic groups. For the general theory of buildings we refer the reader to the monographs \cite{AB} and \cite{Garrett}.
%Buildings come in two flavors: spherical buildings and affine buildings. Main examples of buildings come from reductive algebraic groups.

\subsection{The Bruhat-Tits building of $\GL(r, K)$}     \label{subsec-BT-bldg}
%{\color{red} Call $\tilde{\B}(E)$ the extended Bruhat-Tits building of $\GL(E)$, call $\tilde{A}_\aff(B)$, the extended apartments. Mention Goldman-Iwahori space.}

In this section we review the descriptions of the Bruhat-Tits building of the general group in terms of lattices and additive norms. The reader can also consult \cite[Section 6.9]{AB} and \cite[Section 1.2.1]{RTW}.
%{\color{red} Read Mumford for definition of affine building of $G$.}

%{\color{red} After introducing spherical building, define and introduce notation for the geometric realization and cone over the geometric realization of the building.}

Let $E ≅K^r$ be an $r$-dimensional vector space over $K$. We first describe the Bruhat-Tits building of $(E)$ as an abstract simplicial complex which we denote by $Δ_(E)$. This description is in terms of of \emph{lattices} in $E$. Then we describe the geometric realization of the simplicial complex $Δ_(E)$ as a topological space which we denote by $_(E)$. This geometric realization is in terms of \emph{additive norms} on $E$. Every simplex in $Δ_(E)$ corresponds to a subset of $_(E)$ homeomorphic to a standard simplex. By abuse of terminology we refer to both $Δ_(E)$ and $_(E)$ as the Bruhat-Tits building of $(E)$.

\subsubsection{Lattices}
A \emph{lattice} $Λ$ in $E$ is a rank $r$ free $Ø$-submodule in $E$.
Two lattices $Λ_1$, $Λ_2$ are said to be \emph{equivalent} if for some nonzero $c ∈K^×$ we have $cΛ_1 = Λ_2$. We denote the equivalence class of a lattice $Λ$ by $[Λ]$.

The vertices in the Bruhat-Tits building $Δ_(E)$ correspond to the equivalence classes of lattices in $E$. Two vertices $[Λ_1]$ and $[Λ_2]$ are connected if for some representatives $Λ_1$, $Λ_2$ we have $ϖΛ_1 ⫋Λ_2 ⫋Λ_1$. A $(k-1)$-simplex in $Δ_(E)$ corresponds to $k$ vertices $[Λ_1], …, [Λ_k]$ such that:
\begin{equation}  \label{equ-periodic-lattice-chain}
\varpi \Lambda_k \subsetneqq \Lambda_1 \subsetneqq \cdots \subsetneqq \Lambda_k.
\end{equation}
We call a chain of lattices as in \eqref{equ-periodic-lattice-chain} a \emph{periodic lattice chain}.

A \emph{full periodic lattice} chain is a chain of lattices

\begin{equation}
\varpi\Lambda_r\subsetneqq \Lambda_1\subsetneqq \hdots \subsetneqq \Lambda_r.
\end{equation}
Then each subsequent quotient $Λ_i+1/Λ_i$ is a locally free $𝒪$-module of rank $1$. The full periodic lattice chains correspond to simplices of maximum dimension (also called \emph{chambers}).

Let ${b_1, …, b_r}$ be a $K$-vector space basis for $E$ and let $Λ_i⊂E$ be the free
$𝒪$-module (lattice) generated by
${b_1,…, b_i,ϖb_i+1, …,ϖb_r}$. The chain of lattices
\begin{equation}   \label{equ-full-lattice-chain}
\bigoplus_{i=1}^r \varpi\,\O\, b_i = \varpi \Lambda_r \subsetneqq\Lambda_{1}\subsetneqq \hdots \subsetneqq \Lambda_r= \bigoplus_{i=1}^r \O\, b_i
\end{equation}
is a full periodic lattice chain. Moreover, every full periodic lattice chain has the above form \eqref{equ-full-lattice-chain} for some choice of a basis $B$.

A \emph{frame} $L={L_1, …, L_r}$ is a direct sum decomposition $E = ⊕_i=1^r L_i$ of $E$ into $1$-dimensional subspaces $L_i$. One thinks of frames as equivalence classes of vector space bases up to multiplication of basis elements by nonzero scalars.
To a frame $L$ there corresponds a subcomplex of $Δ_(E)$ called the \emph{apartment} associated to $L$ as follows.
Let $B={b_1, …, b_r}$ be a basis of $E$ with $b_i ∈L_i$.
The vertices in the apartment are equivalence classes $[Λ]$ where $Λ$ is of the form
\begin{equation} \label{equ-affine-apt}
\Lambda = \bigoplus_{i=1}^r \O \varpi^{a_i} b_i,
\end{equation}
for any choice of $a_i ∈$. Note that a different choice of a basis $B$ in the same frame $L$ gives rise to the same collection of vertices but possibly with different coordinates $(a_1, …, a_r)$. More precisely, let ${b'_1, …, b'_r}$ be another basis with $b'_i = λ_i b_i$, $0 ≠λ_i ∈K$. Then $Λ$ from \eqref{equ-affine-apt} can be written as $$\Lambda = \bigoplus_{i=1}^r \O \varpi^{a_i-\val(\lambda_i)} b'_i.$$
Clearly, if $(a_1, …, a_r)$ runs over all the points in $^r$ then $(a_1-(λ_1), …, a_r-(c_r))$ also runs over all the points in $^r$. %This is the sense in which the apartment corresponding to $L$ is an affine space.
The set of vertices in the apartment is a copy of $^r$ but without a specified point as the origin. A choice of a basis $B$, with the frame $L$, distinguishes a point in the apartment as the origin.

%The Bruhat-Tits building of $\SL(r, K)$ (which is the same as that of $\GL(r, K)$ or $\PGL(r, K)$) is an abstract simplicial complex whose vertices correspond to parahoric subgroups of $\GL(r, K)$ (with reverse inclusion). The apartments correspond to split maximal tori $H$ and consist of all parahoric subgroups containing $H$.
\begin{remark}[Iwahori and parahoric subgroups] \label{rem-Iwahori-subgp}
%\begin{definition}[Iwahori and parahoric subgroups]
The $\GL(E)$-stabilizer of a full periodic lattice chain  is called an \emph{Iwahori subgroup}. Let us fix a basis $B$ and identify $E$ with $K^r$. Then the Iwahori subgroup that is the stabilizer of a full lattice chain as in \eqref{equ-full-lattice-chain} is equal to the preimage of a Borel subgroup $B \subset \GL(r, \k)$ under the evaluation map $p: \GL(r, \mathcal{O})\rightarrow \GL(r, \k)$ sending $\varpi$ to $0$. The stabilizer of a periodic lattice chain is called a \emph{parahoric subgroup}. It is the preimage, under $p$, of a parabolic subgroup in $\GL(r, \Bbbk)$.

Each frame $L$ in $E$ determines a split maximal torus $H_L \subset \GL(E)$. The simplices in the apartment of $L$ correspond to the parahoric subgroups in $\GL(E)$ that contain the maximal torus $H_L$.
\end{remark}

\subsubsection{Additive norms}
Next we describe the geometric realization $_(E)$ of the simplicial complex $Δ_(E)$ in terms of \emph{additive norms} on $E$. This realization is due to Goldman and Iwahori and precedes the work of Bruhat and Tits on general theory of Bruhat-Tits buildings of algebraic groups (see \cite{GI}).

\begin{definition}[Additive norm]   \label{def-add-norm}
We call a function $v: E \to \overline{\R} = \R \cup \{\infty\}$ an {\it additive norm} if the following hold:
\begin{itemize}
\item[(1)] For all $e \in E$ and $\lambda \in K$ we have $v(\lambda e) = \val(\lambda) + v(e)$.
\item[(2)] For all $e_1, e_2 \in E$, the non-Archimedean inequality $v(e_1+e_2) \geq \min\{v(e_1), v(e_2)\}$ holds.
\item[(3)] $v(e) = \infty$ if and only if $e=0$.
\end{itemize}
%It is convenient to define $v(0) = \infty$ and consider $v$ as a function from $E$ to $\R \cup \{\infty\}$.
An additive norm is \emph{integral} if it attains values in $\overline{\Z} = \Z \cup \{\infty\}$.
\end{definition}

To an integral additive norm $v$ one assigns the lattice $Λ_v$ given by:
\begin{equation}   \label{equ-Lambda-v}
\Lambda_v = \{ e \in E \mid v(e) \geq 0\}.
\end{equation}
Conversely, a lattice $Λ$ gives rise to an additive norm $v_Λ: E →$ by:
$$v_\Lambda(e) = \max\{ k \in \Z \mid \varpi^{-k} e \in \Lambda \}.$$
One verifies that the maps
$v ↦Λ_v$ and $Λ↦v_Λ$ are mutually inverse and provide a one-to-one correspondence between integral additive norms and lattices.

%The set of all additive norms on $E$ can be identified with the geometric realization $\B_\aff(\GL(E))$ of the Bruhat-Tits building of $\GL(E)$. On the other hand, the set of all the equivalence classes of additive norms on $E$ (with respect to multiplication with nonzero scalars) can be identified with the geometric realization $\B_\aff(\SL(E))$, of the Bruhat-Tits building of $\SL(E)$. We note that $\B_\aff(\SL(E)$ and $\B_\aff(\PGL(E)$ coincide and $\PGL(E)$ is the semisimple quotient of $\GL(E)$.

%This is usually known as the Goldman-Iwahori realization of the \emph{Bruhat-Tits building} of $\GL(E)$. The set $\\tilde{\B}_\aff(K^r)$ is simply the collection of all additive norms on $E$.

We say that an additive norm $v: E →$ is \emph{adapted} to a frame $L={L_1, …, L_r}$ for $E$ if the following holds. For any $e = ∑_i e_i$, $e_i ∈L_i$, we have:
$$v(e) =  \min\{ v(e_i) \mid i=1, \ldots, r\}.$$
In other words, if $B = {b_1, …, b_r}$ is a basis with $b_i ∈L_i$, then for any $e = ∑_i λ_i b_i$ we have: $$v(e) = \min\{ \val(\lambda_i) + v(b_i) \mid i=1, \ldots, r\}.$$ That is, the additive norm $v$ is determined by its values on the basis $B$. In particular, the additive norm $v_Λ$ associated to a lattice $Λ= ⊕_i=1^r Øϖ^a_i b_i$, adapted to $B$, is given by $v_Λ(e) = min{ (λ_i) + a_i |i = 1, …, r}$.

Note that if $v$ is an additive norm then for any $a ∈$, $v_a$ defined by $v_a(e) = v(e) + a$ is also an additive norm. Two additive norms $v$, $v'$ are said to be \emph{equivalent} if their difference is a constant.%\color{red}{I might show some lack of education here:) As far as I know, two norms $\|\|_1$ and $\|\|_2$ are called equivalent if there exist two constants $C_1,C_2 \in K^\times$, s.t. $C_1\|v, w\|_1\leq \|v, w\|_2\leq C_2\|v, w\|_1$ for any $v, w \in E$. Do you mean that for additive norms one uses addition instead of multiplication and the constant is $C_2-C_1$??}\color{black}{}

\begin{definition}[Geometric realization of Bruhat-Tits building]
We define $\B_\aff(E)$, the \emph{(geometric realization of) the Bruhat-Tits building} of $\GL(E)$ to be the set of equivalence classes of additive norms on $E$. For a frame $L$, the \emph{(geometric realization of) the apartment} associated to $L$, denoted by $A_\aff(L)$, is the set of equivalence classes of all additive norms adapted to $L$.
\end{definition}

\begin{definition}[Extended Bruhat-Tits building]
We let $\tilde{\B}_\aff(E)$ to be the set of all additive norms on $E$ (without taking the equivalence classes). It is sometimes called the \emph{extended} Bruhat-Tits building of $\GL(E)$ (see \cite[Remark 1.23]{RTW}). For a frame $L$, we denote the set of all additive norms adapted to $L$ by $\tilde{A}_\aff(L)$ and call it the \emph{extended apartment} associated to $L$.
\end{definition}

%the corresponding apartment in $\B_\aff(E)$ consists of the additive norms $v$ that are adapted to $L$, that is, for any $e \in E$ if we write $e = \sum_i e_i$, $e_i \in L_i$ then we have:
%$$v(e) =  \min\{ v(e_i) \mid i=1, \ldots, r\}.$$
%One checks that an integral additive norm $v$ is adapted to $L$ if its corresponding lattice $\Lambda_v$ lies in the apartment of $L$.
\begin{remark}
%Fix a basis $B=\{b_1, \ldots, b_r\}$ in $E$. It gives rise to a base lattice $\Lambda_0 = \O b_1 + \ldots + \O b_r$. We can then identify the space of lattices in $E$ with the affine Grassmannian $\GL(r, K) / \GL(r, \O)$.
The lattices in $E=K^r$ correspond to closed points of the affine Grassmannian $\GL(r, K) / \GL(r, \O)$.
%The space of all lattices in $E$ is usually referred to as the \emph{affine Grassmannian} of $\GL(E)$. After a choice of a basis, it can be identified with the homogeneous space $\GL(r, K) / \GL(r, \O)$.
\end{remark}

%{\color{red} Bruhat-Tits building of $\GL(r)$ vs that of $\SL(r)$.}

\begin{figure}
\begin{tikzpicture}
\draw[-] (-3,0)--(2,0);
\node at  (-2,0){$\bullet$};
\node at  (-1,0){$\bullet$};
\node at  (0,0){$\bullet$};
\node at  (1,0){$\bullet$};

\end{tikzpicture}
\vspace{.5cm}
\caption{An apartment in the Bruhat-Tits/affine building of $\GL(2)$, a subdivision of the affine line into identical segments. An extended apartment is the Cartesian product of this picture with a copy of $\R$}

\vspace{.5cm}
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\node at (axis cs:-0.75,-0.5) [circle, scale=0.5, draw=black!80,fill=black!80] {};
\end{axis}
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\caption{An apartment in the Bruhat-Tits/affine building of $\GL(3)$, a triangulation of the affine plane. An extended apartment is the Cartesian product of this picture with a copy of $\R$}
\end{figure}

%In Example \ref{ex-GL2} we see that the Bruhat-Tits building of $\GL(2, \Q_p)$, where $\Q_p$ denotes the field of $p$-adic numbers, is an $(p+1)$-regular infinite tree (that is, the degree of every vertex is equal to $p+1$). The geometric realization $\B_\aff(2, \Q_p)$ is the underlying topological space of the tree. On the other hand, the geometric realization $\B_\aff(\GL(2, \Q_p))$ is the Cartesian product of this tree and a copy of $\R$. %\textcolor{red}{Boris: is it $\R$ or $\Q_p^\times=\Q_p\setminus{0}$ ($\det$ is an element of $\Q_p^\times$)?}

%\begin{example}[Bruhat-Tits building of $\GL(2, K)$]  \label{ex-GL2}
\subsubsection{Bruhat-Tits building of $\GL(2, K)$} \label{subsubsec-GL2}
%Case of $\GL(2)$. Description of simplices in an apartment. A vertex lying in two apartments.
%Let us consider $G = \GL(2)$.
Let $B = {b_1, b_2}$ be a basis for $E = K^2$. Consider the lattice $$\Lambda = \O b_1 \oplus \O b_2.$$ The equivalence class $[Λ]$ is comprised of the lattices $Øϖ^a b_1 ⊕Øϖ^a b_2$, $a ∈$.
%Let $A_\aff(B)$ denote the apartment corresponding to the basis $B$.
The vertices in the apartment $A_(B)$ consist of all the equivalence classes of lattices of the form:
$$\Lambda_{a_1, a_2} = \O \varpi^{a_1} b_1 \oplus \O \varpi^{a_2} b_2.$$ Thus the set of vertices in $A_(B)$ is ${[Λ_a, 0] |a ∈}$ and $A_(B)$ is a $1$-dimensional affine space.

A vertex $[Λ']$ is connected to the vertex $[Λ]$ if for some representative $Λ'$ we have:
$$\varpi \Lambda \subsetneqq \Lambda' \subsetneqq \Lambda.$$
Let $Λ'$ be the image of $Λ'$ in the $$̨-vector space $\Lambda/\varpi \Lambda \cong \k^2$.
One sees that $\Lambda' \mapsto \overline{\Lambda'}$ gives a one-to-one correspondence between the lattices $\Lambda'$, $\varpi \Lambda \subsetneqq \Lambda' \subsetneqq \Lambda$, and the $\k$-vector subspaces $L$, ${0} ⫋L ⫋Λ/ϖΛ$. It follows that the set of vertices $[Λ']$ connected to $[Λ]$ is in one-to-one correspondence with the points on the projective line $ℙ(Λ/ϖΛ) ≅ℙ^1$. In particular, if the residue field $$̨ is a finite field with $q$ elements, then every vertex is connected to $q+1$ other vertices. It can also be shown that the graph obtained by connecting the vertices as above is a tree, that is, there are no cycles. Moreover, if the base field $K$ is complete then
any one-sided infinite path lies in an apartment
(see <cit.> and <cit.>). An example is $K = \Q_p$, the field of $p$-adic numbers (Figure <ref>).

grow cyclic,
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child foreach in 1,...,2
child foreach in 1,...,2

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The Bruhat-Tits buildings of $\GL(2, \Q_2)$ and $\GL(2, \Q_3)$

Let us determine for what other bases $B' = \{b'_1, b'_2\}$ the vertex $[\Lambda]$ lies in the corresponding apartment $A_{B'}$. If $[\Lambda]$ lies in $A_{B'}$ then we can find $\gamma_1$, $\gamma_2 \in \Z$ such that:

\begin{equation}   \label{equ-Lambda-B-B'}
\O b_1 \oplus \O b_2 = \Lambda = \O \varpi^{\gamma_1} b'_1 \oplus \O \varpi^{\gamma_2} b'_2.
\end{equation}

Let $C = [c_{ij}]$, $1 \leq i, j \leq 2$, be the change of coordinates matrix from $B$ to $B'$. That is, $b'_i = \sum\limits_{j=1}^2 c_{ij} b_j$. The equation (<ref>) is then equivalent to

$$C D \in \GL(2, \O),$$
where $D = \operatorname{diag}(\varpi^{\gamma_1}, \varpi^{\gamma_2})$.
For instance, we can take $\gamma_1 = \gamma_2 = 0$ and $C = \begin{bmatrix} 1 &  \varpi^\delta \\ 0 & 1 \\ \end{bmatrix}$ with $\delta \geq 0$. Then $C D = C \in \GL(2, \O)$ and thus

$$\O b_1 \oplus \O b_2 = \Lambda = \O b_1 \oplus \O (\varpi^\delta b_1 + b_2).$$
Hence $\Lambda$ belongs to the apartment $A_{B'}$ where $B' = \{b_1, \varpi^\delta b_1 + b_2 \}$.

To demonstrate how apartments can overlap we consider the apartments associated to the bases $B = \{b_1, b_2\}$ and $B' = \{b_1+b_2, b_2\}$. The vertices in the apartments $A_B$ and $A_{B'}$ correspond respectively to the chain of lattices:

$$\cdots \subset \O b_1 \oplus \O\varpi b_2 \subset \O b_1 \oplus \O b_2 \subset \O b_1 \oplus \O \varpi^{-1} b_2 \subset \cdots $$

$$ \cdots \subset \O (b_1+b_2) \oplus \O\varpi b_2 \subset \O (b_1+b_2) \oplus \O b_2 \subset \O (b_1+b_2) \oplus \O \varpi^{-1} b_2 \subset \cdots.$$

One verifies that for any $k \geq 0$, $\O b_1 \oplus \O\varpi^{-k} b_2 = \O (b_1+b_2) \oplus \O\varpi^{-k} b_2$. Thus the two chains of lattices overlap on the half-infinite path with vertices $[\O b_1 \oplus \O\varpi^{-k} b_2] = [\O (b_1+b_2) \oplus \O\varpi^{-k} b_2]$, $k \geq 0$, but become separate afterwards (Figure <ref>).

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at (axis cs:0,0) [circle, scale=0.4, draw=black!80,fill=black!80] ;
[black,right] at (axis cs:0,0)$\mathcal{O}b_1\oplus\mathcal{O}b_2$;
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Two apartments (infinite paths) corresponding to the bases $B=\{b_1, b_2\}$ and $B' = \{b_1+b_2, b_2\}$ overlapping on a half-infinite path.

§.§.§ Floor, ceiling and pull-back of an additive norm

In this section we discuss some basic operations on additive norms, namely the floor and ceiling functions and pull-back, and prove some facts about them for future use in Section <ref>.

One has a natural partial order $\preceq$ on the set $\tilde{\B}(E)$ of all additive norms.
Let $v$, $v'$ be additive norms on $E$. We say that $v \preceq v'$ if $v(e) \leq v'(e)$, for all $e \in E$.

Recall that for $r \in \R$, the floor $\lfloor r \rfloor$ (respectively the ceiling $\lceil r \rceil$) is the largest integer less than or equal to $r$ (respectively the smallest integer greater than or equal to $r$).

For an additive norm $v: E \to \overline{\R}$ we define its floor $\lfloor v \rfloor$ and  ceiling $\lceil v \rceil: E \to \overline{\Z}$ to be functions from $E$ to $\overline{\Z}$ given by:

$$\lfloor v \rfloor(e) = \lfloor v(e) \rfloor,$$

$$\lceil v \rceil(e) = \lceil v(e) \rceil, \quad \forall e \in E.$$

$\lfloor v \rfloor$ and $\lceil v \rceil$ are additive norms.

We write the proof for the ceiling, the proof for the floor is similar. We know that there exists a basis $B = \{b_1, \ldots, b_r\}$ such that $v$ is adapted to $B$. Thus:

$$v\left(\sum_i \lambda_i b_i \right) = \min\{ \val(\lambda_i)+v(b_i) \mid i=1, \ldots, r\}.$$

\begin{align*}
\lceil v \rceil \left(\sum_i \lambda_i b_i\right) &= \lceil v(\sum_i \lambda_i b_i) \rceil \\
&= \lceil \min\{ \val(\lambda_i)+v(b_i) \mid i=1, \ldots, r\} \rceil \\ &= \min\{ \val(\lambda_i)+\lceil v(b_i) \rceil \mid i=1, \ldots, r\}.\\
\end{align*}

But any expression $\min\{ \val(\lambda_i)+a_i \mid i=1, \ldots, r\}$, $a_i \in \R$, defines an additive norm. This proves that $\lceil v \rceil$ is indeed an additive norm.

The floor (respectively ceiling) of an additive norm $v$ is the greatest integral additive norm that is smaller than or equal to $v$ (respectively smallest integral additive norm that is greater than or equal to $v$).

Let $F: E \to E'$ be a linear map between $K$-vector spaces $E$ and $E'$. Given an additive norm $v': E \to \overline{\R}$ one can consider its pull-back $F^*(v')$ given by
$$F^*(v')(e) = v'(F(e)), \quad \forall e \in E.$$

We note that $F^*(v')$ may not be an additive norm on $E$ for the following reason: $F$ may have nonzero kernel in which case $F^*(v')$ assigns value $\infty$ to nonzero elements in the kernel. We will need the following in Section <ref>.

Let $F: E \to E'$ be a linear map between $K$-vector spaces $E$ and $E'$. Let $v$ and $v'$ be integral additive norms on $E$ and $E'$ respectively. Then, for all $e \in E$,
$$v(e) \leq F^*(v')(e)$$
if and only if
$$F(\Lambda_v) \subset \Lambda_{v'}.$$
Recall that $\Lambda_v$, $\Lambda_{v'}$ denote the lattices corresponding to the integral additive norms $v$, $v'$ respectively (see (<ref>)).

Suppose $v(e) \leq F^*(v')(e)$, for all $e \in E$, but $F(\Lambda_v) \not\subset \Lambda_{v'}$. Then there exists $e \in \Lambda_v$ with $F(e) \notin \Lambda_{v'}$. Since $e \in \Lambda_v$ we have $v(e) \geq 0$ and since $F(e) \notin \Lambda_{v'}$ we have $v'(F(e)) < 0$, the contradiction proves the claim. Conversely, suppose $F(\Lambda_v) \subset \Lambda_{v'}$ then for any $e \in E$ we have:

\begin{align*}
v(e) &= \max\{ k \mid \varpi^{-k} e \in \Lambda_v\}, \\
& \leq \max\{ k \mid \varpi^{-k} F(e) \in F(\Lambda_v) \}, \\
&\leq \max\{ k \mid \varpi^{-k} F(e) \in \Lambda_{v'} \},\\
&= v'(F(e)).\\
\end{align*}

§.§ The Tits building of $\GL(r, K)$

We now turn to the discussion of Tits building of $\GL(E)$, which is a spherical building associated to the group $\GL(E)$. As in the case of Burhat-Tits building, we describe the Tits building of $\GL(E)$ in two ways:

(i) as a simplicial complex $\Delta_\sph(E)$ and,

(ii) its geometric realization $\B_\sph(E)$ as a topological space.

The abstract simplicial complex, $\Delta_\sph(E)$, can be described as follows. Its vertices are the proper nonzero vector subspaces of $E$. Two subspaces $F_1$ and $F_2$ are connected if one of them is a subset of the other. The $(k-1)$-simplices of $\Delta_\sph(E)$ are formed by sets of $k$ mutually connected vertices, hence $(k-1)$-simplices correspond to partial flags

$$F_\bullet = (\{0\} \subsetneqq F_1 \subsetneqq \cdots \subsetneqq F_{k} \subsetneqq E).$$
Maximal simplices, i.e. chambers, correspond to complete flags.

The apartments in $\Delta_\sph(E)$, correspond to frames in $E$. Recall that
a frame in $E$ is a direct sum decomposition $E = \bigoplus_{i=1}^r L_i$ into one-dimensional subspaces $L_i$.
A flag $F_\bullet$ is said to be adapted to a frame $L = \{L_1, \ldots, L_r\}$ if all the subspaces in the flag are spanned by a subset of the $L_i$. The apartment corresponding to a frame $L$ is the collection of all flags adapted to $L$.

A convenient way to describe the geometric realization of the spherical building of $\GL(E)$ is as the space of prevaluations on $E$.
(see <cit.>). This is an analogue of the Goldman-Iwahori realization of the Bruhat-Tits building of $\GL(E)$ as the space of equivalence classes of additive norms on $E$ (see Section <ref>). This description of Tits building of the general linear group appears in <cit.> but the authors are not aware of another reference for this.

We call a function $v: E \to \overline{\R}$ a prevaluation if the following hold:

(1) For all $e \in E$ and $0 \neq \lambda \in K$ we have $v(\lambda e) = v(e)$.

(2) For all $e_1, e_2 \in E$, the non-Archimedean inequality $v(e_1+e_2) \geq \min\{v(e_1), v(e_2)\}$ holds.

(3) $v(e) = \infty$ if and only if $e = 0$.

We note that the only difference with the definition of an additive norm (Definition <ref>) is item (1).

We call a prevaluation $v$ integral if it attains values in $\overline{\Z}$. In short, a prevaluations is a valuation on $E$ extending the trivial valuation on $K$, and an additive norm is a valuation on $E$ extending the valuation $\val$ on $K$.

The term prevaluation is taken from the paper <cit.>.  Prevaluation is a standard commutative algebra notion although in most of the literature the term valuation on a vector space is used. We use the term prevaluation to distinguish prevaluations from valuations on rings.

For a prevaluation $v$, consider the decreasing filtration $\{E_{v \geq a}\}_{a \in \R}$, where  $E_{v \geq a} = \{ e \in E \mid v(e) \geq a\}$. Let

$$\{0\} \subsetneqq F_{1} \subsetneqq \cdots \subsetneqq F_{k} \subsetneqq E,$$

be all the distinct subspaces among the $E_{v \geq a}$. They determine the flag associated to the simplex that contains $v$.

An integral prevaluation $v$ on $E$ gives a $\Z$-filtration $\{E_{v \geq a}\}_{a \in \Z}$. One verifies that this gives a one-to-one correspondence between integral prevaluations and $\Z$-filtrations $\{E_a\}_{a \in \Z}$ such that $\underset{a \in \Z}{\bigcap} E_a = \{0\}$ and $\underset{a \in \Z}{\bigcup} E_a = E$.

We say that a prevaluation $v$ is adapted to a frame $L=\{L_1, \ldots, L_r\}$ if the values of $v$ can be obtained by taking a minimum of the values $v(L_i)$. More precisely, for any $e \in E$ let us write $e = \sum_i e_i$, $e_i \in L_i$. Then:

$$v(e) = \min\{v(e_i) \mid i=1, \ldots, r\}.$$

This is equivalent to requiring that, for any $a \in \R$, the subspace

\begin{equation}  \label{equ-F-a}
F_{a} = \{ e \in E \mid v(e) \geq a\}
\end{equation}

is spanned by some of the $L_i$.

Two prevaluations $v_1$, $v_2$ are said to be equivalent if there is $\lambda > 0$ such that $\lambda v_1 = v_2$. The (geometric realization of the) Tits building $\B_\sph(E)$ can be identified with the set of all equivalence classes of prevaluations on $E$. The collection of all equivalence classes of prevaluations that are adapted to a given frame $L$ is the (geometric realization of the) apartment associated to $L$. Each such prevaluation $v$ is uniquely determined by the values $\{v(L_1), \ldots, v(L_r)\}$. Hence this space can be identified with the $(r-1)$-dimensional sphere $\R^r / \R_{> 0} \cong S^{r-1}$ (since we are taking equivalence classes of prevaluations).

We denote the set of all prevaluations on $E$ by $\tilde{\B}_\sph(E)$ and call it the cone over the Tits building of $\GL(E)$. For a frame $L$, we denote the set of all prevaluations on $E$ that are adapted to $L$ by $\tilde{A}_\sph(L)$ and call it the cone over the apartment associated to $L$.

[-, thick] (-6,0)–(6,0);
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[thick](0,0) circle (5);

at (0,0) $\bullet$;
at (5,0) $\bullet$;
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at (2.2,-4.4) $\bullet$;
at (-2.2,-4.4) $\bullet$;

An apartment in the Tits building of $\SL(3)$, a subdivision of circle into $6$ arcs

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] (1,0,0) arc (0:180:1);

] (1,0,0) arc (0:180:1);

] (0.92,-0.3,0) arc (0:132:1.1);

] (0.92,-0.3,0) arc (360:335:1.25);

] (0.92,-0.3,0) arc (0:132:1.1);

] (0.92,-0.3,0) arc (360:335:1.25);

[thick,-] (-0.83,-0.57,0) – (0.9,0.4,0) node;
[thick,-] (-0.9,0.4,0) – (0.83,-0.57,0) node;

An apartment in the Tits building of $\SL(4)$, a triangulation of $2$-sphere

§.§ Building at infinity

As usual let $E \cong K^r$ be a finite dimensional vector space over a discretely valued field $K$.
We now explain that $\tilde{\B}_\sph(E)$, the space of prevaluations on $E$, can be regarded as the 'boundary at infinity' of $\tilde{\B}_\aff(E)$, the space of additive norms on $E$. This is related to a general construction for affine buildings. Given an affine building $\B_\aff$ one can consider the parallelism equivalence classes of rays in the buildings (that is, each equivalence class consists of all the rays parallel to each other). This set of equivalence classes naturally has a structure of a spherical building (see Section <ref>).

Take a basis $B = \{b_1, \ldots, b_r\}$ for $E$. Recall that the extended apartment $\tilde{A}_\aff(B)$ can be identified with the set of all additive norms $v: E \to \overline{\R}$ that are adapted to the basis $B$, namely, for any $e = \sum_i \lambda_i b_i$ we have:

$$v(e) = \min\{ \val(\lambda_i) + v(b_i) \mid i=1, \ldots, r \}.$$

In $\tilde{A}_\aff(B)$, a ray is given by $\rho: t \mapsto v_t$ where $v_t$ is the additive norm adapted to $B$ given by

$$v_t(e) = \min\{ \val(\lambda_i) + a_i + tv_i \mid i=1, \ldots, r \},$$
where $a=(a_1, \ldots, a_r)$, $v=(v_1, \ldots, v_r) \in \R^r$. Then an equivalence class of parallel rays in the apartment $\tilde{A}_\aff(B)$ is obtained by fixing $v$ and varying $a$. Let us consider $v_\infty: E \to \overline{\R}$ defined by:
$$v_\infty(e) = \lim_{t \to
\infty}  \frac{v_t(e)}{t}.$$
We have:

\begin{align*}
v_\infty(e) &= \lim_{t \to \infty} \min\{\val(\lambda_i)/t + a_i / t + v_i \mid i=1, \ldots, r\}  \\ &= \min\{ v_i \mid \lambda_i \neq 0 \}.
\end{align*}

Thus $v_\infty: E \to \overline{\R}$ is in fact a prevaluation, adapted to the basis $B$, which represents the parallelism equivalence class of a ray $\rho$ in $\tilde{\B}_\aff(E)$. We recall that the cone over the spherical building $\tilde{\B}_\sph(E)$ can be viewed as the space of all prevaluations on $E$.
This explains that $\tilde{\B}_\sph(E)$ can be viewed as the boundary at infinity of $\tilde{\B}_\aff(E)$.

Recall that, for a basis $B$, we denote by $\tilde{A}_\sph(B)$ the set of all prevaluations adapted to the basis $B$, that is, the cone over the apartment associated to $B$. The above shows that $\tilde{A}_\sph(B)$ can be thought of as the boundary at infinity of the extended affine apartment $\tilde{A}_\aff(B)$.

§ TORIC VECTOR BUNDLES OVER A DISCRETE VALUATION RING

As before let $\X = \X_\Sigma$ be a toric scheme over $\Spec(\O)$. Here $\Sigma$ is a fan in $N_\R \times \R_{\geq 0}$ with the corresponding polyhedral complex $\Sigma_1$ in $N_\R \times \{1\}$. We recall that $\X$ comes with a distinguished point $x_0$ in the open chart $\X_0$ giving an identification of this open set with the torus $T_\eta = T_K$.

A toric vector bundle over $\X$ is a $T$-linearized vector bundle $\E$ over $\X$.

A toric vector bundle is the same as a $T$-linearized locally free sheaf of constant rank $\O_\X$-modules on $\X$. By a morphism of toric vector bundles over $\X_\Sigma$ we mean a torus equivariant morphism of vector bundles. Toric vector bundles over $\X_\Sigma$ and their morphisms form a category.

We refer the reader to <cit.>
for the definition of a $T$-linearization of a vector bundle on a $T$-scheme (the definition in the above mentioned reference is for line bundles but the same works for vector bundles as well). In short, a $T$-linearization of $\E$ is a lifting of action of $T$ on $\X$ to an action of $T$ on $\E$ by morphisms of vector bundles.

We assume that the fan $\Sigma$ is determined by the polyhedral complex $\Sigma_1$. That is, each cone in $\Sigma$ is a cone over a polyhedron in $\Sigma_1$ or a face of such cone.
Geometrically, the assumption says that every $T_K$-orbit in the generic fiber $\X_\eta$ extends to the special fiber $\X_o$, that is, its closure intersects $\X_o$. In particular, if $\Sigma$ is a complete polyhedral complex this condition is satisfied.

Under the above assumption (Assumption <ref>) in this section we give a classification of toric vector bundles over $\X$ in terms of piecewise affine maps from $\Sigma_1$ to $\tilde{\B}_\aff(E)$, the extended Bruhat-Tits buildings of $\GL(E)$, for finite dimensional $K$-vector spaces $E$.

§.§ Local equivariant triviality

Similar to the case of toric vector bundles over a field, the following is a key step in the classification. The proof is similar to the one for toric vector bundles over a field (see <cit.>).

Let $\E$ be a toric vector bundle on an affine toric scheme $\X_{\sigma}$. Then $\E$ is equivariantly trivial. That is, there exist characters $u_1, \ldots, u_r \in M$ such that $H^0(\X_\sigma, \E) = \bigoplus_{i=1}^r S_{\sigma, i}$ where $T$ acts on $S_{\sigma, i} \cong S_\sigma$ via character $u_i$.

It suffices to show there are $T$-weight sections $s_1, \ldots, s_r \in H^0(\X_{\sigma}, \E)$ that freely generate $H^0(\X_{\sigma}, \E)$ as an $S_{\sigma}$-module.
If the cone $\sigma$ lies in $N_\R \times \{0\}$, there is no special fiber $\X_{\sigma, o}$ and we are in the case of a toric vector bundle over an affine toric variety $\X_{\sigma, \eta}$. Thus, we can assume that $\sigma$ does not lie in $N_\R \times \{0\}$. In this case,
$\sigma$ corresponds to the unique closed $T_o$-orbit $O_\sigma$ in the special fiber $\X_o$. Recall that $T_o = T_\k$ is the torus over the residue field $=̨ Ø/ $. Since $O_σ$ is an orbit, the vector bundle $_|O_σ$ is equivariantly trivial. That is, we can find a basis of $T$-weight sections $s'_1, …, s'_r ∈H^0(O_σ, _|O_σ)$. By general properties of coherent sheaves on affine schemes, one knows that the $S_σ$-module morphism:
$$j^*: H^0(\X_{\sigma}, \E) \to H^0(O_\sigma, \E_{|O_\sigma}),$$ induced by the inclusion morphism $j: O_σ↪_σ$ is surjective. Note that the $S_σ$-module structure on $H^0(O_σ, _|O_σ)$ comes from the homomorphism $S_σ →[̨O_σ]$ determined by the embedding $O_σ↪_σ$. Moreover, %since $O_\sigma$ is $T_o$-invariant,
$j^*$ is $T$-equivariant. In other words, $j^*$ is $M$-grading preserving. Thus we can find $T$-weight sections $s_1, …, s_r ∈H^0(, )$ such that $j^*(s_i) = s'_i$, for all $i$. We claim that these $s_i$ freely generate $H^0(, )$ as an $S_σ$-module. First we claim that for any $x ∈$, the images of the $s_i$ in the fiber $_x$ are linearly independent. This is because the subscheme of $_σ$ where these images are linearly dependent is a closed $T$-invariant subscheme and hence, if non-empty, should contain $O_σ$. But since the $s'_i$ are a basis for $H^0(O_σ, _|O_σ)$, this subscheme does not contain $O_σ$ as a subscheme, and thus is empty which proves the claim. Now from the assumption that $$ is locally free we conclude that for all $x ∈$ the images of $s_i$ in $_x$ form a basis. It follows that the $s_i$ are an $S_σ$-basis for $H^0(_σ, )$ as required.
\end{proof}

\subsection{Piecewise affine maps}
%Recall that the set of vertices of the Bruhat-Tits building $\B_\aff(\GL(r))$ is the set of all $\O$-lattices in $K^r$.

%For each basis $B=\{b_1, \ldots, b_r\}$ for $K^r$ the corresponding apartment $A_\aff(B)$ contains the following lattices:
%$$ L_a = \{ \sum_{i=1}^r \O \varpi^{a_i} b_i \mid a=(a_1, \ldots, a_r) \in \Z^r \}.$$

%Let $\Sigma$ be a fan in $\tilde{N}_\R = N_\R \times \R$. Let $\Sigma_1 = \Sigma \cap (\N_\R \times \{1\})$.
Recall from Section \ref{subsec-BT-bldg} that
$_(E)$ denotes the extended Bruhat-Tits building of the reductive group $(E)$. We think of $_(E)$ as the set of all additive norms on $E$ (while the Bruhat-Tits building $_(E)$ is the set of additive norms up to addition by constants). The set of vertices in $_(E)$ consists of all the lattices in $E$ (while the set of vertices in $_(E)$ consists of all the lattices in $E$ up to scalar multiplication). Each apartment in $_(E)$ is a copy of $^r$, $r=(E)$ (while each apartment in $_(E)$ is a copy of $^r-1$).

We also recall that a function $Φ: N_→^r$ is an \emph{affine map} if $Φ$ is a linear map followed by a translation. In other words,
there is a linear map $Φ_0: N_→^r$ and $b ∈^r$ such that for all $x ∈N_$ we have
$$\Phi(x) = \Phi_0(x) + b.$$
The linear map $Φ_0$ is called the \emph{linear part} of $Φ$.
We say that $Φ$ is an \emph{integral} affine function if $Φ(N) ⊂^r$. This means that the entries of the matrix of $Φ_0$, with respect to a $$-basis for $N$ and the standard basis for $^r$, are integers, and moreover, $b ∈^r$.

%Recall that an affine map between two affine spaces is a composition of a linear transformation with a translation.
As usual, let $Σ$ be a fan in $N_×_≥0$ and $Σ_1$ the polyhedral complex obtained by intersecting the cones in $Σ$ with the affine hyperplane $N_×{1}$. Similarly, $Σ_0$ is the fan obtained by taking the cones in $Σ$ that lie in the hyperplane $N_×{0}$.
%In fact, $\Sigma_0$ is the recession fan of the polyhedral complex $\Sigma_1$, that is, it consists of the recession cones of the polyhedra in $\Sigma_1$ %Alternatively, it is obtained by intersection the fan $\Sigma$ in $\N_\R \times \R_{\geq 0}$ over $\Sigma_1$ with $N_\R \times \{0\}$
%(see Definition \ref{def-sigma-Delta} and Remark \ref{rem-rec-fan}).

\begin{definition}[Piecewise affine map to $\tilde{\B}(E)$]
Let $\Sigma_1$ be a polyhedral complex in the affine space $N_\R \times \{1\}$.
A map $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$ is a \emph{piecewise affine} map if the following holds:
\begin{itemize}
\item[(a)] For any polyhedron $\Delta \in \Sigma_1$, there is an extended apartment $\tilde{A}_\aff(\Delta)$ (not necessarily unique) such that the restriction $\Phi_\Delta := \Phi_{|\Delta}$ maps $\Delta$ into $\tilde{A}_\aff(\Delta)$.
\item[(b)] For any $\Delta \in \Sigma_1$, $\Phi_\Delta$ is the restriction of an affine map from $N_\R \times \{1\}$ to the affine space $\tilde{A}_\aff(\Delta)$.
\end{itemize}
We say that $\Phi$ is an \emph{integral} piecewise affine map if
for each $\Delta \in \Sigma_1$, $\Phi_\Delta$ is the restriction of an integral affine map from $N_\R \times \{1\}$ to $\tilde{A}_\aff(\Delta)$.
\end{definition}

%Recall that $\tilde{\B}(E)$ is the space of all additive norms on $E$.
Let $Φ: |Σ_1| →(E)$ be an integral piecewise affine map. For any polyhedron $Δ∈Σ_1$, the requirement that $Φ_Δ:= Φ_|Δ: Δ→Ã_(Δ)$ is an integral affine map means that there exists a $K$-basis $B_Δ= {b_Δ, 1, …, b_Δ, r}$ for $E$ and ${u_Δ, 1, …, u_Δ, r} ⊂M$ such that for any $x ∈σ$, $Φ_Δ(x)$, as an additive norm on $E$, is given by
\begin{equation}   \label{equ-Phi-sigma}
\Phi_\Delta(x)\left(\sum_i \lambda_i b_{\Delta, i}\right) = \min\{ \val(\lambda_i) + \langle u_{\Delta, i}, x \rangle \mid i=1, \ldots, r\}.
\end{equation}
%\Phi_\sigma(x){\langle u_{\sigma, i}, v \rangle} b_{\sigma, i}.$$
%Let $\B_\sph(\GL(E))$ (respectively $\B(\GL(E))$) be the spherical building of $\GL(E)$.

%Recall that $\\tilde{\B}_\sph(\GL(E))$ (respectively $\B_\aff(\GL(E))$) can be regarded as the spherical building at infinity of $\B_\aff(\GL(E))$ (respectively $\B_\aff(\GL(E))$) (see Section \ref{subsec-sph-bldg-infinity}). %Let also $\\tilde{\B}_\sph(\GL(E))$ denote the cone over this spherical building (see Section \ref{subsec-}).

%Let $\Phi: N_\R \to \R^r$ be an affine map. It can be written as $\Phi(x) = \Phi_0(x) + b$ where $b \in \R^r$ and $\Phi_0: N_\R \to \R^r$ is a linear map. The linear map $\Phi_0$ is called the \emph{linear part} of the affine map $\Phi$.

\begin{definition}[Linear part of a piecewise affine map]
Let $\Phi:|\Sigma_1| \to \tilde{\B}_\aff(E)$ be a piecewise affine map. There is a well-defined piecewise linear map $\Phi_0: |\Sigma_0| \to \tilde{\B}_\sph(E)$ such that for each polyhedron $\Delta \in \Sigma_1$, $\Phi_0$ is the linear part of the affine map $\Phi_{|\Delta}: \Delta \to \tilde{A}_\aff(\Delta)$. The linear part of $\Phi_{|\Delta}$ is a linear map from the recession cone $\recc(\Delta)$ to $\tilde{A}_\sph(\Delta)$, the space of prevaluations adapted to $B$. As explained in Section \ref{subsec-bldg-at-infinity}, $\tilde{A}_\sph(\Delta)$ can be thought of as the boundary at infinity of the extended affine apartment $\tilde{A}_\aff(\Delta)$.
\end{definition}

\begin{remark}   \label{rem-linear-part}
The above definitions extend to arbitrary affine buildings.
\end{remark}

%In this paper we will be interested in piecewise affine maps to the Bruhat-Tits building $\GL(E)$.

\begin{definition}[Morphism of piecewise affine maps]   \label{def-morphism-pa-map}
Let $F: E \to E'$ be a linear transformation between finite dimensional $K$-vector spaces $E$, $E'$ and let $\Phi: |\Sigma_1| \to \tilde{\B}(E)$ and $\Phi': |\Sigma_1| \to \tilde{\B}(E')$ be two piecewise affine maps. We say that $F$ gives a morphism of piecewise affine maps $\Phi \to \Phi'$ if for any $x \in |\Sigma_1|$ we have: $$\Phi(x) \preceq F^*(\Phi'(x)),$$
(see Definition \ref{def-pull-back} and cf. Proposition \ref{prop-F-v-v'}).
\end{definition}

\subsection{Main theorems}   \label{subsec-main}
Recall that $Σ$ denotes a fan in $N_×_≥0$ with $Σ_1$ the corresponding polyhedral complex in $N_= N_×{1}$ and $_Σ$ the corresponding toric scheme over $(Ø)$.

We break our classification of toric vector bundles on $_Σ$ into the following theorems (Theorems \ref{th-main}, \ref{th-main2}, \ref{th-main3}). We present the proofs later in this section. The proofs involve the notion of \emph{ceiling} of an additive norm (Definition \ref{def-ceiling}).

Recall that we fix a point $x_0$ in the open orbit in the generic fiber $_η$.
\begin{theorem}    \label{th-main}
%There is an equivalence of categories between the category of toric vector bundles over the toric scheme $\X=\X_\Sigma$ and the category of integral piecewise affine maps from $|\Sigma_1|$ to $\tilde{\B}_\aff(E)$, the Bruhat-Tits buildings of general linear groups of finite dimensional $K$-vector spaces $E$.
There is a one-to-one correspondence between the isomorphism classes of toric vector bundles over a toric scheme $\X=\X_\Sigma$ and the integral piecewise affine maps from $|\Sigma_1|$ to $\tilde{\B}_\aff(E)$, the extended Bruhat-Tits buildings of general linear groups of finite dimensional $K$-vector spaces $E$.
\end{theorem}

\begin{remark}  \label{rem-plm-sph-bldg}
Let $\Sigma$ be a fan in $N_\R$ and $X=X_\Sigma$ the corresponding toric variety over a field $K$.
The result \cite[Theorem 1]{KM-tpb} applied to $\GL(r, K)$ shows that toric vector bundles over a toric variety $X_\Sigma$ are classified by piecewise linear maps $\Phi: |\Sigma| \to \tilde{\B}_\sph(E)$. This is a reformulation of the Klyachko classification of toric vector bundles (see also \cite[Example 2.8]{KM-tpb}).
\end{remark}

If $$ is a toric vector bundle over a toric scheme $$ then the generic fiber $_η$ is a usual toric variety over the field $K$ and $_|_η$ is a usual toric vector bundle on it. The next theorem describes the piecewise linear map (equivalently, the Klyachko filtrations) associated to the toric vector bundle $_|_η$. %(see Remark \ref{rem-plm-sph-bldg}).
\begin{theorem}  \label{th-main2}
Given a toric vector bundle $\E$ with associated integral piecewise affine map $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$, the toric vector bundle $\E_{|\X_\eta}$ over the generic fiber $\X_\eta$, which is a toric vector bundle over the field $K$, corresponds to the piecewise linear map $\Phi_0: |\Sigma_0| \to \tilde{\B}_\sph(E)$, which is the linear part of $\Phi$ (see \cite[Example 2.8]{KM-tpb}).
\end{theorem}

%Also, we can talk about morphisms of piecewise affine maps. Let $\Phi: |\Sigma_1| \to \tilde{\B}(E)$ and $\Phi': |\Sigma_1| \to \tilde{\B}(E')$, be piecewise affine maps. A morphism from $\Phi$ to $\Phi'$ is a linear map $F: E \to E'$ such that for every $x \in \Sigma_1$ we have $\Phi(x)(e) \leq \Phi'(x)(F(e))$, for all $e \in E$. Piecewise affine maps and their morphisms form a category as well.

\begin{theorem}    \label{th-main3}
The correspondence $\E \mapsto \Phi_\E$ gives
an equivalence of categories between the category of toric vector bundles over the toric scheme $\X=\X_\Sigma$ and the category of integral piecewise affine maps from $|\Sigma_1|$ to $\tilde{\B}_\aff(E)$, for finite dimensional $K$-vector spaces $E$.
\end{theorem}

\begin{remark}
One can modify the statements of Theorems \ref{th-main} and \ref{th-main2} so that
Assumption \ref{ass-fan}  is not necessary. To this end, one should use the notion of a piecewise linear map from the fan $|\Sigma|$ to the ``completed'' extended building $\tilde{\B}_\aff(E) \cup \tilde{\B}_\sph(E)$.
\end{remark}

\begin{proof}[Proofs of Theorems \ref{th-main} and \ref{th-main2}]
Recall that $\Sigma$ is the fan in $N_\R \times \R_{\geq 0}$ consisting of the cones over the polyhedra in $\Sigma_1$ as well as faces of such cones. For each $\sigma \in \Sigma$, we have the affine toric scheme $\X_\sigma = \Spec(S_{\sigma})$ where
\begin{equation}   \label{equ-S-sigma}
S_{\sigma} = \O[\chi^u \varpi^k \mid (u, k) \in \sigma^\vee \cap \tilde{M}] \subset K[T_\eta].
\end{equation}
In particular, $S_0 = K[T_\eta]$ is the coordinate ring of the torus $T_\eta \cong (K^\times)^r$, isomorphic to the Laurent polynomial ring over $K$ in $r$ indeterminates.
The affine toric schemes $\X_{\sigma}$, $\sigma \in \Sigma$, glue together to form the toric scheme $\X = \X_\Sigma$.
%In particular, $S_0 = \O[\chi^u \varpi^k \mid (u, k) \in \tilde{M}]$ is the coordinate ring of the torus $\X_0 = T_\eta$.
The scheme $\X_0 = \Spec(K[T_\eta])$ comes with a distinguished point $x_0 =  1 \in T_\eta$.
%We fix a point $x_0$ in the open orbit in the generic fiber $\X_\eta$. It gives an identification of the open orbit in $\X_\eta$ with $T_K$.

Let $\E$ be a toric vector bundle on $\X_\Sigma$ of rank $r$. First we see how to associate to $\E$ a piecewise affine map $\Phi=\Phi_\E: |\Sigma_1| \to \tilde{\B}_\aff(E)$. The toric vector bundle $\E$  is equivalent to the data of a $T$-linearized sheaf $\F$ of locally free $\O_\X$-modules of rank $r$ on $\X$. Let $E = \E_{x_0}$ which is an $r$-dimensional $K$-vector space. For each cone $\sigma \in \Sigma$, consider the $M$-graded $S_{\sigma}$-module $\F_{\sigma} := H^0(\X_{\sigma}, \E_{|\X_{\sigma}})$.
Since $\E$ is trivial over the open orbit $\X_0 = T_\eta$, we can identify $\E_{|\X_0}$ with $T_\eta \times E$ and hence $\F_0$ with $K[T_\eta] \otimes_K E$.
Then, for each $\sigma \in \Sigma$, the restriction of $\F_\sigma$ to $\X_0$ gives us an embedding $$\F_{\sigma} \hookrightarrow \F_0 = K[T_\eta] \otimes_K E.$$
Consider the decomposition of $\F_\sigma$ into $M$-homogeneous components: $$\F_{\tilde{\sigma}} = \bigoplus_{u \in M} \F_{\sigma, u},$$ where $\F_{\sigma, u} \subset \chi^{u} E \subset K[T_\eta] \otimes_K E$ is the $u$-homogeneous component of $\F_{\sigma}$. Note that the $u$-homogeneous component is the $(-u)$-weight space in $\F_\sigma$). Since the $u$-homogeneous component in $K[T_\eta] \otimes_K E$ is $\chi^{u} E$, we can write:
$$\F_{\sigma, u} = \chi^{u} \Lambda_{\sigma, u},$$ for some $\O$-submodule $\Lambda_{\sigma, u} \subset E$.

On the other hand, by the local equivariant triviality (Proposition \ref{prop-local-equiv-trivial}) the $M$-graded $S_{\sigma}$-module $\F_{\sigma}$ is
free of rank $r$ and we can write:
\begin{equation} \label{equ-F-sigma-free}
\F_{\sigma} = \bigoplus_{i=1}^r S_{\sigma} \cdot s_{\sigma, i},
\end{equation}
where $s_{\sigma, i}$ is an $M$-homogeneous section whose $M$-degree we denote by $u_{\sigma, i}$, that is, the image of $s_{\sigma, i}$ in $K[T_\eta] \otimes_K E$ lands in $\chi^{u_{\sigma, i}}E$.
%Fix a basis for the $K$-vector space $E$.
%As a homogeneous section $s_{\sigma, i} \in \F_\sigma \hookrightarrow E \otimes K[T_\eta]$ can be written as:
Let us write:
$$s_{\sigma, i} = \chi^{u_{\sigma, i}} b_{\sigma, i}$$ for some $b_{\sigma, i} \in E$. Since \eqref{equ-F-sigma-free} is a direct sum, the set of vectors $B_\sigma = \{b_{\sigma, 1}, \ldots, b_{\sigma, r}\}$ forms a $K$-vector space basis for $E$. We thus have:
\begin{equation}  \label{equ-F-sigma}
\F_{\sigma} = \bigoplus_{i=1}^r \bigoplus_{(u',k') \in \sigma^\vee \cap \tilde{M}} \O \cdot\chi^{u'} \varpi^{k'} \, \chi^{u_{\sigma, i}} b_{\sigma, i}.
\end{equation}
To compute the $u$-homogeneous component $\F_{\sigma, u}$ in \eqref{equ-F-sigma} we should sum over all $(u', k')$ such that $u' = u - u_{\sigma, i}$ and $(u', k') \in \sigma^\vee \cap \tilde{M}$. This means that $$\langle (u - u_{\sigma, i}, k'), y \rangle \geq 0,~~ \forall y \in \sigma.$$ That is,
\begin{equation}   \label{equ-Lambda-sigma-u}
\Lambda_{\sigma, u} = \sum_{i=1}^r \sum_{k': (u - u_{\sigma, i}, k') \in \sigma^\vee \cap \tilde{M}} \O\, \varpi^{k'} b_{\sigma, i}.
\end{equation}
We note that the condition $(u - u_{\sigma, i}, k') \in \sigma^\vee$ is equivalent to $(u - u_{\sigma, i}, k') \in \rho^\vee$, for all $\rho \in \sigma(1)$. where $\sigma(1)$ denotes the set of rays in $\sigma$. Thus, from \eqref{equ-Lambda-sigma-u}, we have:   $$\Lambda_{\sigma, u} = \bigcap_{\rho \in \sigma(1)} \Lambda_{\rho, u},$$ Hence we can conclude that the sheaf $\F$ is completely determined by the $\O$-modules $\Lambda_{\rho, u}$, for $\rho \in \Sigma(1)$ and $u \in M$.
%Let us consider the case when $\sigma = \rho$ is a ray. We have two cases:

Let $\rho$ be a ray in $\Sigma$. We consider two cases:

\textbf{Case 1:} The ray $\rho$ does not lie in $N_\R \times \{0\}$. Let $(v, 1)$ be the unique point of intersection of $\rho$ and $N_\R \times \{1\}$. Note that $v$ is a rational point but in general may not be a lattice point. Then:
\begin{align}
(u-u_{\sigma, i}, k') \in \rho^\vee \Leftrightarrow& ~\langle u - u_{\sigma, i}, v  \rangle + k' \geq 0 \\
\Leftrightarrow&~ k' \geq \langle u_{\sigma, i} - u, v \rangle \\
\Leftrightarrow&~ k' \geq \lceil \langle u_{\sigma, i} - u, v \rangle \rceil. \label{equ-k'}
\end{align}
Here $\lceil r \rceil$ denotes the ceiling of a real number $r$, that is, the smallest integer greater than or equal to $r$.
%Let $f_{\rho}(u) = \frac{1}{\ell_{\rho}} \langle u_{\tilde{\sigma}, i} - u, v \rangle$.
From \eqref{equ-k'} we then get:
\begin{equation}  \label{equ-L-rho}
\Lambda_{\rho, u} = \bigoplus_{i=1}^r \O \, \varpi^{\lceil \langle u_{\sigma, i} - u, v \rangle \rceil} b_{\sigma, i},
\end{equation}
%provided that $\langle u_{\sigma, i} - u, v_{\rho} \rangle$ is integer for all $i$, otherwise $\Lambda_{\rho, u} = \{0\}$.
Since $B_\sigma = \{b_{\sigma, 1}, \ldots, b_{\sigma, r}\}$ is a $K$-basis for $E$ we see that, for any $u \in M$, the $\O$-module $\Lambda_{\rho, u} \subset E$ is of full rank, i.e. it is a lattice.

\textbf{Case 2:} The ray $\rho$ lies in $N_\R \times \{0\}$. Let $(v, 0)$ be a primitive vector in $\rho$. Then:
\begin{equation}  \label{equ-k'-2}
(u - u_{\sigma, i}, k')  \in \rho^\vee \Leftrightarrow ~\langle u - u_{\sigma, i}, v  \rangle \geq 0, ~~\forall k' \in \Z.
\end{equation}
Noting that $K = \bigcup_{k' \in \Z} \O \varpi^{k'}$ we get:
\begin{equation}  \label{equ-L-rho-2}
\Lambda_{\rho, u} = \bigoplus_{i: \langle u, v  \rangle \geq \langle u_{\sigma, i}, v  \rangle} K \cdot b_{\sigma, i}.
\end{equation}
%{\color{red} the filtration is decreasing or increasing?}
In other words, $\Lambda_{\rho, u}$ depends only on the integer $\langle u, v \rangle$, that is, on the class of $[u]$ of $u$ in $M_{\rho} := M / (\rho^\perp \cap M) \cong \Z$. Let us write:
\begin{equation} \label{equ-L-rho-3}
E^\rho_j = \Lambda_{\rho, u}
\end{equation}
where $j = \langle -u, v \rangle$ (the reason for $-u$ instead of $u$ is to make the $\{E^\rho_j \}_{j \in \Z}$ a decreasing filtration and hence match with the convention in \cite{Klyachko}). We observe that for rays $\rho$ that lie in $\sigma \cap (N_\R \times \{0\})$, the equation \eqref{equ-L-rho-2} for the filtrations $\{E^\rho_j\}_{j \in \Z}$ is the Klyachko compatibility condition. It follows, exactly as in the classification of toric vector bundles over a field (Theorem \ref{th-Klyachko}) that $\F$ restricted to the generic fiber $\X_\eta$ is determined, up to an equivariant isomorphim, by the decreasing $\Z$-filtrations $\{E^\rho_j\}_{j \in \Z}$, $\rho \in \Sigma_0(1)$.

Now we are ready to construct the piecewise affine map $\Phi$ associated to the toric vector bundle $\E$. Let $\Delta \in \Sigma_1$ be a polyhedron with $\sigma \in \Sigma$ the corresponding cone over it. In this case, we denote the basis $B_\sigma = \{b_{\sigma, i}\}$ by $B_\Delta = \{b_{\Delta, i}\}$. Also, if $v$ is a vertex in $\Delta$ with corresponding ray $\rho \in \sigma(1)$, we denote the $\O$-module $\Lambda_{\rho, u}$ by $\Lambda_{v, u}$.

Let $\tilde{A}_\aff(\Delta)$ be the extended apartment in the extended building $\tilde{\B}_\aff(E)$ corresponding to the basis $B_\Delta$. Recall that it consists of additive norms that are adapted to the basis $B_\Delta$ (see Section \ref{subsec-BT-bldg}).
Note that choice of a basis $B_\Delta$ gives an identification of the extended apartment $\tilde{A}_\aff(\Delta)$ with $\R^r$. We define the affine map $\Phi_\Delta: \Delta \to \tilde{A}_\aff(\Delta)$ to be the linear map given by $\Phi_\Delta(x) = (\langle u_{\sigma, 1}, x \rangle, \ldots, \langle u_{\sigma, r}, x \rangle).$
In terms of additive norms, $\Phi_\Delta(x): E \to \overline{\R}$ is the additive norm given by
%As in \eqref{equ-Phi-sigma}, we define the affine map $\Phi_\Delta: \Delta \to \tilde{A}_\aff(\Delta)$ as follows: for $x \in \Delta$, $\Phi_\Delta(x)$ is the additive norm given by:
\begin{equation}   \label{equ-Phi-Delta-1}
\Phi_\Delta(x)\left(\sum_i \lambda_i b_{\Delta, i}\right) = \min\{ \val(\lambda_i) + \langle u_{\sigma, i}, x \rangle \mid i=1, \ldots, r\}.
\end{equation}
For a vertex $v$ of the polyhedron $\Delta$ and $u \in M$, the lattice $\Lambda_{v, u}$ can be recovered from $\Phi_\Delta$ as
\begin{equation}   \label{equ-Phi-Delta}
\Lambda_{v, u} = \bigoplus_{i=1}^r \O \varpi^{\lceil \Phi_\Delta(v)(b_{\Delta, i}) - \langle u, v \rangle \rceil} b_{\Delta, i}.
\end{equation}

%Moreover, from \eqref{equ-L-rho} it follows that $\Lambda_{\rho, u}$ is determined by $v$ and $\Lambda_{\rho, 0}$, namely
%\begin{equation}  \label{equ-Lambda-u-0}
%\Lambda_{\rho, u} = \varpi^{-\langle u, v \rangle} \Lambda_{\rho, 0}.\end{equation}

%Let us see that what information does the linear part $\Phi_{\Delta, 0}$ of $\Phi_\Delta$ encodes.

We also need to check that the affine map $\Phi_\Delta$ contains the data of filtrations $\{E^\rho_j\}_{j \in \Z}$, where the $\rho \in N_\R \times \{0\}$ are horizontal rays in the cone $\sigma$ (see \eqref{equ-L-rho-2} and \eqref{equ-L-rho-3}). We see that the filtrations $\{E^\rho_j\}_{j \in \Z}$ can be recovered from the linear part $\Phi_{\Delta, 0}$ of $\Phi_\Delta$. %Let $(v, 0)$ be the primitive vector for $\rho$. By Assumption \ref{ass-fan}, we know that $\rho$ is the ray in a recession cone of some polyhedron $\Delta \in \Sigma_1$. Then for all $(y, 1) \in \Delta$ and $t>0$, $(y, 1) + t(v, 0) \in \Delta$.
As in Section \ref{subsec-bldg-at-infinity}, for any $(x, 0)$ in the linear span of the recession cone of $\Delta$, we can compute the value of the prevaluation $\Phi_{\Delta, 0}(x, 0)$ on a vector $e = \sum_i \lambda_i b_{\Delta, i}$ by:
\begin{align*}
\Phi_{\Delta, 0}(x, 0)\left(\sum_i \lambda_i b_{\Delta, i}\right) &= \lim_{t \to \infty} \frac{\min\{ \val(\lambda_i) + \langle u_{\Delta, i}, tx \rangle \mid i=1, \ldots, r\}}{t}, \\
&= \min\{\langle u_{\Delta, i}, x \rangle \mid i=1, \ldots, r\}.
\end{align*}
Let $(v, 0)$ be the primitive vector for a ray $\rho$ in $\sigma$, the cone over $\Delta$.
In particular, if we let $x=-v$, the prevaluation $\Phi_{\Delta, 0}(-v, 0)$ is given by:
$$\Phi_{\Delta, 0}(-v, 0)\left(\sum_i \lambda_i b_{\Delta, i}\right) = \min\{\langle u_{\Delta, i}, -v \rangle \mid i=1, \ldots, r \},$$ and its corresponding
filtration $\{E_j\}_{j \in \Z}$ is: $$E_j = \textup{span}_K\{ b_{\Delta, i} \mid j \leq \langle u_{\Delta, i}, -v \rangle \}.$$
%{\color{red} the filtration is decreasing or increasing?}
{But this is the same as \eqref{equ-L-rho-2}, the filtration associated to the ray $\rho \in \Sigma_0$, as required.}
%{\color{red} More argument is needed!}

It remains to show that the $\Phi_\Delta$, $\Delta \in \Sigma_1$, glue together to give a piecewise affine map $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$. To this end, we need to show that for all $\Delta, \Delta' \in \Sigma_1$, $\Phi_\Delta$ and $\Phi_{\Delta'}$ coincide on the intersection $\Delta \cap \Delta'$. This is the content of the next lemma. If the vertices of $\Delta \cap \Delta'$ are lattice points, the lemma is easy to show because we can remove the ceiling function in the expression for $\Lambda_{u, v}$ in \eqref{equ-Phi-Delta}. In general, the presence of the ceiling function adds extra details to the argument.
\begin{lemma}  \label{lem-gluing}
Let $v$ be a vertex of the convex polyhedron $\Delta \cap \Delta'$. Then the additive norms $\Phi_\Delta(v)$ and $\Phi_{\Delta'}(v)$ coincide.
\end{lemma}
\begin{proof}
Let $B_\Delta = \{b_{\Delta, i}\}$ and $B_{\Delta'} = \{b_{\Delta', i}\}$ be the bases corresponding to $\Delta$ and $\Delta'$ respectively. We note that, by \eqref{equ-Phi-Delta}, the additive norm corresponding to $\Lambda_{v, u}$ is:
\begin{equation}   \label{equ-Phi-Phi'-1}
\lceil \Phi_\Delta(v)(\cdot) - \langle u, v \rangle \rceil = \lceil \Phi_{\Delta'}(v)(\cdot) - \langle u, v \rangle \rceil.
\end{equation}
By contradiction, suppose $\Phi_\Delta(v) \neq \Phi_{\Delta'}(v)$. Then for some $e \in E$, $\Phi_\Delta(v)(e) \neq \Phi_{\Delta'}(v)(e)$. Let $e = \sum_i \lambda_i b_{\Delta, i} = \sum_i \lambda'_i b_{\Delta', i}$. From \eqref{equ-Phi-Delta} we see that, after a reordering of basis elements if necessary, there is an index $j$ such that:
$$\Phi_\Delta(e) = \val(\lambda_j) + \langle u_{\sigma, j}, v \rangle \neq  \val(\lambda'_j) + \langle u_{\sigma', j}, v \rangle = \Phi_{\Delta'}(e).$$
%Let $\sigma, \sigma' \in \Sigma$ be cones corresponding to the polyhedra $\Delta, \Delta' \in \Sigma_1$ respectively.
On the other hand, in \eqref{equ-Phi-Phi'-1}, if we let $u=u_{\Delta, j}$ we get:
$$\lceil \langle u_{\Delta', j}, v \rangle - \langle u_{\Delta, j}, v \rangle \rceil = \val(\lambda_j) - \val(\lambda'_j).$$
Also, in \eqref{equ-Phi-Phi'-1}, if we let $u = u_{\Delta', j}$, in the same way we obtain:
$$\lceil \langle u_{\Delta, j}, v \rangle - \langle u_{\Delta', j}, v \rangle \rceil = \val(\lambda'_i) - \val(\lambda_j).$$
It follows that $\langle u_{\Delta, j}, v \rangle - \langle u_{\Delta', j}, v \rangle$ is an integer. Thus, $\Phi_\Delta(v)(e) - \Phi_{\Delta'}(v)(e)$ is an integer. This together with \eqref{equ-Phi-Phi'-1} implies that $\Phi_\Delta(v)(e) = \Phi_{\Delta'}(v)(e)$. The contradiction proves the claim.
\end{proof}

We have shown that $\Phi_\Delta$ and $\Phi_{\Delta'}$ coincide on the vertices of the polyhedron $\Delta \cap \Delta'$. We need to show that the linear parts of $\Phi_\Delta$ and $\Phi_{\Delta'}$ also coincide on the recession cone of $\Delta \cap \Delta'$. Let $\rho \in \Sigma_0$ be a ray in the recession cone of the polyhedron $\Delta \cap \Delta'$ with primitive vector $(v, 0)$. As we saw above, both prevaluations  $\Phi_{\Delta, 0}(v, 0)$ and $\Phi_{\Delta', 0}(v, 0)$ on $E$ correspond to the filtration $\{ E^\rho_j \}$ on $E$ and hence coincide.

Now $\Phi_\Delta$ and $\Phi_{\Delta'}$ are two affine maps on $\Delta \cap \Delta'$ that coincide on the vertices of $\Delta \cap \Delta'$ and also have the same linear parts on $\Delta \cap \Delta'$. This implies that they coincide everywhere on $\Delta \cap \Delta'$ as required. Hence the $\Phi_\Delta$, $\Delta \in \Sigma_1$, glue together to give a piecewise affine map from $|\Sigma_1|$ to the extended building $\tilde{\B}_\aff(E)$.

%We claim that the lattices $\Lambda_{\rho, u}$, $u \in M$, determine the value $\Phi_\Delta(v)$. By \eqref{equ-Phi-Delta}, the information of $\Lambda_{v, u}$ is the same as the information of ${\lceil \Phi_\Delta(v)(b_{\sigma, i}) - \langle u, v \rangle \rceil$. We claim that these determine the value $\langle u_{\sigma, i}, v \rangle$.

Note that the above arguments also show that $\Phi$ is uniquely determined by the vector bundle $\E$. This is because, as shown above, the $\Phi_\Delta$ (and hence $\Phi$) are determined by the lattices $\Lambda_{\rho, u}$, $u \in M$, (for rays $\rho$ in $\Sigma$ that intersects $N_\R \times \{1\}$), and the filtrations $\Lambda_\rho$ (for rays $\rho$ that lie in $N_\R \times \{0\}$).

%Conversely, \eqref{equ-Phi-Delta} shows that the lattices $\Lambda_{x, u}$, for all $u \in M$ and $x \in \Delta$, determine the additive norm $\Phi_\Delta(v)$. It follows that $\Phi_\Delta(v)$ only depends on the vertex $v$ and not on the polyhedron $\Delta$ containing it. We thus conclude that the $\Phi_\Delta$, $\Delta \in \Sigma_1$, glue together to give a piecewise affine map $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$.

%Thus we get  a piecewise linear map $\Phi_0$ which is the linear part of the piecewise affine map $\Phi$.

%We conclude that $\Phi_\sigma$ is an affine map from $\sigma$ to the apartment $\tilde{A}_\sigma$ consisting of all lattices adapted to the basis $\{b_{\sigma, 1}, \ldots, b_{\sigma, r}\}$. Thus $\E$ gives rise to an integral piecewise affine map $\Phi: \Sigma \to \tilde{\B}_\aff(E)$.

In conclusion we have a piecewise affine map $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$ that determines the sheaf $\F$, and its linear part $\Phi_0$ gives the Klyachko's data for the restriction of the sheaf $\F$ to the generic orbit $\X_\eta$.

Conversely, let $\Phi: |\Sigma_1| \to \tilde{\B}_\aff(E)$ be a piecewise affine map. One verifies that reversing the construction gives a toric vector bundle $\E_\Phi$ over the toric scheme $\X_{\Sigma}$. %For each $\sigma \in \Sigma$, the space of sections $\F_\sigma = H^0(\X_\sigma, \E_{|\X_\sigma}\})$ is given by:
\end{proof}

Next we prove the claim about the equivalence of categories. Again, if all the vertices in the the polyhedral complex $Σ_1$ are lattice points, then arguments are easier. In general, we need extra details involving the ceiling function.
\begin{proof}[Proof of Theorem \ref{th-main3}]
A $T$-equivariant morphism of vector bundles is the same as a $T$-equivariant morphism of the corresponding locally free sheaves of modules.
Let $\E$, $\E'$ be toric vector bundles on $\X = \X_\Sigma$ with the corresponding sheaves of $\O_\X$-modules $\F$ and $\F'$ respectively. Let $F: \F \to \F'$ be a $T$-equivariant morphism. %It amounts to a collection of compatible $T$-equivariant $S_{\tilde{\sigma}}$-module homomorphisms $F_{\tilde{\sigma}}: \F_{\tilde{\sigma}} \to \F'_{\tilde{\sigma}}$, for all $\tilde{\sigma} \in \tilde{\Sigma}$.
Since $T_\eta$ is open in $\X$, $F$ is determined by its restriction to $T_\eta$. On the other hand, $F$ is $T$-equivariant and hence $F_{|T_\eta}$ is determined by its restriction to $E = \E_{x_0}$. We note that $F: \E_{x_0} \to \E'_{x_0}$ is a $K$-linear map. We would like to characterize which $K$-linear maps $F: \E_{x_0} \to \E'_{x_0}$ give rise to $T$-equviariant morphisms $F: \F \to \F'$. In fact, one verifies that a linear map $F$ gives rise to a $T$-equivairant morphism if and only if for any $\sigma \in \Sigma$ and $u \in M$ we have:
\begin{equation}    \label{equ-F-Lambda-sigma}
F(\Lambda_{\sigma, u}) \subset \Lambda'_{\sigma, u}.
\end{equation}
We note that since $\Lambda_{\sigma, u} = \bigcap_{\rho \in \sigma(1)} \Lambda_{\rho, u}$, the above is equivalent to
\begin{equation}   \label{equ-F-Lambda-rho}
F(\Lambda_{\rho, u}) \subset \Lambda'_{\rho, u}, \quad \forall \rho \in \Sigma(1), ~\forall u \in M.
\end{equation}
The next lemma shows that the condition in \eqref{equ-F-Lambda-rho} is equivalent to
$\Phi$ being dominated by $F^*(\Phi')$. %Recall that this means that for any $(x, 1) \in \Sigma_1$ we have $\Phi(x, 1) \leq F^*(\Phi')(x, 1)$, as additive norms.
This finishes the proof of Theorem \ref{th-main3}
\end{proof}
%This finishes the proof of Theorems \ref{th-main} and \ref{th-main2}.
%If follows from \eqref{} that for this to hold it suffices that it holds for each vertex $\rho=\sigma \in \Sigma$.
% It is straightforward to verify that $F: \E_{x_0} \to \E'_{x_0}$ gives rise to such a morphism if and only if for any vertex $\rho \in \Sigma$ we have:
%On the other hand,
%by \eqref{equ-Lambda-u-0} we know that each space $\Lambda_{\rho, u}$ is determined by $\Lambda_{\rho, 0}$. Hence $F$ gives rise to a morphism if and only if for each vertex $\rho \in \Sigma$ we have:
%$$F(\Lambda_{\rho, 0}) \subset \Lambda'_{\rho, 0}.$$
%From this the required claim follows.

\begin{lemma}
The following are equivalent:
\begin{itemize}
\item[(a)] 	For all $\rho \in \Sigma$ and $u \in M$,
$$F(\Lambda_{\rho, u}) \subset \Lambda'_{\rho, u},$$ where $\Lambda_{\rho, u}$ is given by \eqref{equ-L-rho} if $\rho$ intersects $N_\R \times \{1\}$ and is given by \eqref{equ-L-rho-2} otherwise.
%For all $x \in |\Sigma|$ and $u \in M$,
%$$F(\Lambda_{x, u}) \subset \Lambda'_{x, u}$$
%if and only if for all $x \in |\Sigma|$ and $e \in E$,
\item[(b)] For all $x \in |\Sigma_1|$, $u \in M$ and $e \in E$:
$$\lceil \Phi(x)(e) - \langle u, v \rangle \rceil  \leq \lceil \Phi'(x)(F(e)) -  \langle u, x \rangle \rceil.$$
\item[(c)] For all $x \in |\Sigma_1|$ and $e \in E$:
$$\Phi(x)(e) \leq F^*(\Phi'(x))(e) := \Phi'(x)(F(e)).$$
\end{itemize}
\end{lemma}
\begin{proof}
(a) $\Rightarrow$ (b): Define $\Psi_u(x)(e) = \lceil \Phi(x)(e) - \langle u, v \rangle \rceil$ and  $\Psi'_u(x)(e) = \lceil \Phi'(x)(F(e)) -  \langle u, x \rangle \rceil$. Then $\Psi_u$ and $\Psi'_u$ are piecewise affine functions on $|\Sigma_1|$. The condition in (a) implies that, for any vertex $v$ in $\Sigma_1$, $\Psi_u(v) \preceq \Psi'(v)$, and moreover the linear part of $\Psi_u$ is less than or equal to the linear part of $\Psi'_u$. These put together imply that $\Psi_u(x) \preceq \Psi'_u(x)$, for all $x \in |\Sigma_1|$.

(b) $\Rightarrow$ (c):
%suppose for all $u \in M$, $x \in |\Sigma|$, $e \in E$, we have $\lceil \Phi(x)(e) - \langle x, u \rangle \rceil \leq \lceil \Phi'(x)(F(e)) - \langle x, u \rangle \rceil$.
%$F(\Lambda_{x, u}) \subset \Lambda'_{x, u}$. By Proposition \ref{prop-F-v-v'} this implies that
%\begin{equation}  \label{equ-Phi-Phi'}
%\lceil \Phi(x)(e) - \langle x, u \rangle \rceil \leq \lceil \Phi'(x)(F(e)) - \langle x, u \rangle \rceil.
%We would like to show that for all $x \in |\Sigma|$ and $e \in E$ we have $\Phi(x)(e) \leq \Phi'(x)(F(e))$.
Suppose by contradiction that there is $x$ and $e$ such that $\Phi(x)(e) > \Phi'(x)(F(e))$.
By the piecewise affineness of $\Phi'$ we know that, for any polyhedron $\Delta \in \Sigma_1$ containing $x$,  there is a basis $B'_{\Delta} = \{b'_{\Delta, 1}, \ldots, b'_{\Delta, r}\}$ for $E'$ and characters $\{u'_{\Delta, 1}, \ldots, u'_{\Delta, r}\} \subset M$ such that for all $e' = \sum_i \lambda_i b'_{\Delta, i} \in E$ we have:
$$\Phi'(x)(e') = \min\{ \val(\lambda_i)+\langle u'_{\Delta, i}, x\rangle \mid i=1, \ldots, r \}.$$
Let $j$ be the index where the minimum above is attained for $e' = F(e)$, thus
\begin{equation}   \label{equ-Phi'-F-e}
\Phi'(x)(F(e)) = \val(\lambda_j) + \langle u'_{\Delta, j}, x \rangle.
\end{equation}
%Also let $\ell > 0$ be the smallest integer such that $x' := \ell x \in N$. Since $x'$ is a primitive vector $\langle M, x' \rangle = \Z$ and thus we can find $u \in M$ such that  $\langle u, x' \rangle = \langle u'_{\Delta, j}, x' \rangle$, that is, $\langle u, x \rangle = \langle u'_{\Delta, j}, x \rangle$.
Letting $u = u'_{\Delta, j}$, we see from \eqref{equ-Phi'-F-e}  that $\Phi'(x)(F(e)) - \langle u, x\rangle \in \Z$. But
$$\Phi(x)(e) - \langle u, x \rangle  > \Phi'(x)(F(e)) - \langle u, x \rangle.$$ Since the right-hand side is an integer we see that
$$\lceil \Phi(x)(e) - \langle u, x \rangle \rceil > \lceil \Phi'(x)(F(e)) - \langle u, x \rangle \rceil,$$ which contradicts the assumption in (b).

(c) $\Rightarrow$ (a):
First suppose for all $x \in |\Sigma|$ and $e \in E$ we have
$\Phi(x)(e) \leq F^*(\Phi'(x))(e) := \Phi'(x)(F(e))$.
Then, for all $u \in M$, $x \in |\Sigma|$, $e \in E$ one has
\begin{align*}
\Phi(x)(e) - \langle x, u \rangle &\leq \Phi'(x)(F(e)) - \langle x, u \rangle, \\
\lceil \Phi(x)(e) - \langle x, u \rangle \rceil &\leq \lceil \Phi'(x)(F(e)) - \langle x, u \rangle \rceil.
\end{align*}
%We recall that $\Lambda_{x, u}$ and $\Lambda'_{x,u}$ are lattices corresponding to the additive norms $\lceil \Phi(x)(e) - \langle x, u \rangle \rceil$ and $\lceil \Phi'(x)(e) - \langle x, u \rangle \rceil$ respectively.
From Proposition \ref{prop-F-v-v'} we now conclude that $F(\Lambda_{\rho, u}) \subset \Lambda'_{\rho, u}$ as required.
\end{proof}

\begin{example}   \label{ex-PA-map-tree}
Let the base field be the $2$-adic field $\Q_2$. Recall that the Bruhat-Tits building $\B_\aff(\Q_2^2)$ of $\GL(2, \Q_2)$ is the $3$-regular infinite tree depicted in Figure \ref{fig-Tree-Q2-Q3}. Each apartment in the Bruhat-Tits building is a two-sided infinite path in the tree. Each extended apartment is the Cartesian product of a two-sided infinite path and $\R$.

Consider the canonical model of $\mathbb{P}^1$ over $\Q_2$. Its fan $\Sigma$ is the Cartesian product of fan of $\mathbb{P}^1$ and $\R_{\geq 0}$. The polyhedral complex $\Sigma_1$ in $N_\R \times \{1\}$ consists of three polyhedra $(-\infty, 0]$, $\{0\}$ and $[0, \infty\}$. The generic fiber is $\mathbb{P}^1$ over $\Q_2$ and the special fiber is $\mathbb{P}^1$ over the residue field $\mathbb{F}_2$.
In Figure \ref{fig-tree-PA-map}, we have marked three one-sided infinite paths $I_1$, $I_2$ and $I'_2$. We let $\tilde{I}_1$, $\tilde{I}_2$ and $\tilde{I'}_2$ denote the Cartesian products of the $I_1$, $I_2$ and $I'_2$ with $\R$ respectively.

%we have marked two edges $I_i$, $i=1, 2$, in the tree and two one-sided infinite paths $I_3$ and $I'_3$. Let the $\tilde{I}_i$ and $\tilde{I'}_3$ denote the Cartesian products of the $I_i$ and $I_3$ with $\R$ respectively.

Consider piecewise affine maps $\Phi_i: \Sigma \to \tilde{\B}_\aff(E)$, $i=1, 2$, where $\Phi_1$ is given by some affine maps ${\Phi_1}_{|(-\infty, 0]} :(-\infty, 0] \to \tilde{I}_1$ and ${\Phi_1}_{|[0, \infty)}: [0, \infty) \to \tilde{I}_2$. Similarly, $\Phi_2$ is given by some affine maps ${\Phi_2}_{|(-\infty, 0]} :[-\infty, 0] \to \tilde{I}_1$ and ${\Phi_2}_{|[0, \infty)}: [0, \infty) \to \tilde{I}'_2$.
The piecewise affine maps $\Phi_i$, $i=1, 2$, give rise to toric vector bundles $\E_{\Phi_i}$, $i=1, 2$, of rank $2$ on the canonical model of $\mathbb{P}^1$. In Example \ref{ex-split-tvb} we will show that $\E_{\Phi_1}$ splits equivariantly into sum of line bundles while $\E_{\Phi_2}$ does not.
%One observe that any piecewise affine map $\Phi$ lands in an apartment. %It follows that any rank $2$ toric vector bundle on a toric model of $\mathbb{P}^1$ splits.
%The image of $\Phi$ does not land in one apartment. In light of Theorem \ref{th-splitting}, it gives a rank $2$ vector bundle on a model of $\P^1$ that does not split
\end{example}

\begin{figure}
\centering
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\hspace{0.5in}
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\caption{One-sided infinite paths in the tree of $\Q_2$}
\label{fig-tree-PA-map}
\end{figure}

%Consider the $1$-dimensional toric scheme$\X = %\X_\Sigma$ associated to the following %$1$-dimensional polyhedral complex $\Sigma$ with %corresponding $2$-dimensional fan $\tilde{\Sigma}$ %as in Figure \ref{}.
%We determine the piecewise affine map associated to %the tangent bundle of $\X$.
%Example of rank $2$ toric vector bundles over $\Q_p$ using the tree structure of Bruhat-Tits building of $\GL(2, \Q_p)$.

\section{Equivariant splitting of toric vector bundles on toric schemes}   \label{sec-splitting}
In this section we give a simple criterion for splitting of a toric vector bundle into a sum of toric vector bundles in terms of its associated piecewise affine map. Similar criterion exists for toric vector bundles over field (see \cite[p. 5, A little application]{KM-tpb}).
\begin{definition} \label{def-equiv-splitting}
We say that a toric vector bundle $\E$ \emph{splits equivariantly} if it is equivariantly isomorphic to a direct sum of toric line bundles.
\end{definition}

\begin{theorem}[Criterion for equivariant splitting]  \label{th-splitting}
A toric vector bundle $\E$ over $\X_\Sigma$ splits equivariantly if and only if the image of the corresponding piecewise affine map $\Phi_\E$ lands in one extended apartment.
\end{theorem}
\begin{proof}
%The proof is rather straightforward.
Let $\E = \underset{i=1}{\overset{r}{\bigoplus}} \L_i$ where the $\L_i$ are toric line bundles on $\X_\Sigma$. As before, the local equivariant triviality (Proposition \ref{prop-local-equiv-trivial}) applied to each toric line bundle $\L_i$ implies that there exists a basis $B=\{b_1, \ldots, b_r\}$ such that for any $\sigma \in \Sigma$, the space of sections $\F_\sigma = H^0(\X_\sigma, \E_{|\X_\sigma})$ is given by (cf. \eqref{equ-F-sigma}):
$$ \F_{\sigma} = \bigoplus_{i=1}^r \bigoplus_{(u',k') \in \sigma^\vee \cap \tilde{M}} \O \cdot\chi^{u'} \varpi^{k'} \, \chi^{u_{\sigma, i}} b_{i}  ~~\subset E \otimes K[T_\eta],$$
for some set of weights $\{u_{\sigma, i} \mid i=1, \ldots, r\}$. The corresponding piecewise affine map $\Phi_\E$ is then given by (cf. \eqref{equ-Phi-Delta-1}):
$$\Phi_\E(x)\left(\sum_i \lambda_i b_i\right) = \min\{ \val(\lambda_i) + \langle u_{\sigma, i}, x \rangle \mid i=1, \ldots, r\},$$
for any $x \in \Delta = \sigma \cap (N_\R \times \{1\})$. This shows that all the additive norms $\Phi_\E(x)$, $x \in |\Sigma_1|$, are adapted to the basis $B$ and hence the image of $\Phi_\E$ lands in the extended apartment $\tilde{A}_\aff(B)$.

Conversely, if there is a basis $B = \{b_i\}$ such that all the additive norms $\Phi_\B(x)$ are adapted to $B$ then, as above, one sees that $\E = \underset{i=1}{\overset{r}{\bigoplus}} \L_i$ where $\L_i$ is the toric line bundle whose sheaf of sections $\F_i$ is given as follows: for any $\sigma \in \Sigma$, $$\F_i(\X_\sigma) = \bigoplus_{(u',k') \in \sigma^\vee \cap \tilde{M}} \O \cdot\chi^{u'} \varpi^{k'} \, \chi^{u_{\sigma, i}} b_{i}.$$
\end{proof}

\begin{example}   \label{ex-split-tvb}
In Example \ref{ex-PA-map-tree} the image of $\Phi_1$ lands in an apartment, that is, an infinite two-sided path, while the image of $\Phi_2$ does not.
It follows from Theorem  \ref{th-splitting} that the toric vector bundle $\E_{\Phi_1}$, associated to the piecewise affine map $\Phi_1$, is equivariantly split while $\E_{\Phi_2}$, associated to $\Phi_2$, is not. This is in contrast to the field case: over a field, any toric vector bundle over $\mathbb{P}^1$ splits equivariantly (see \cite[Theorem 6.1.2]{Klyachko} as well as \cite[p. 5, A little application]{KM-tpb}). The example of $\E_{\Phi_2}$ shows that this is not the case over a discrete valuation ring.
\end{example}

%In the rest of this section we prove an analogue of Klyachko's criterion for splitting of toric vector bundles of sufficiently high rank on projective spaces (\cite[Corollary 6.1.4]{Klyachko}) to toric schemes.

%First we give a slightly different proof of \cite[Theorem 6.1.2]{Klyachko}. It is a statement about the intersection of simplices in the Tits building of the general linear group. Klyachko calls it \emph{an analogue of Helly's theorem} (about intersections of convex subsets in a Euclidean space). Our argument is in the spirit of the argument in Step 3 of the proof of \cite[Theorem 6.2.1]{Klyachko} and is more convenient for establishing a generalization of this result.%\color{red}{ This is a very 'proofy' sentence:) How about we write smth like 'Our argument is in the spirit of the one used in Step 3  \cite[Theorem 6.2.1]{Klyachko} and is more convenient for establishing a generalization of this result.'?} \color{black}{}

%\begin{theorem}		\label{parabolics}
%Let $\{P_{\alpha}\}, \alpha \in A$, be a finite family of parabolic subgroups in $\GL(r, \Bbbk)$. If the intersection of any $r+1$ subgroups in the family contains a maximal torus, then so does the whole family.

%We argue by induction on the number of elements in $A$, which will be denoted by $n+1:=|A|\geq r+1$. Let $\mathbb{P}^n=\Bbbk[e_1:e_2:\hdots:e_{n+1}]$ be a projective space with coordinates corresponding to the parabolic subgroups indexed by $A$. Notice that by inductive assumption the filtrations corresponding to every collection of subgroups strictly contained in $A$ satisfy the compatibility condition. Therefore such collection of filtrations gives rise to an equivariant vector bundle $E$ on $\mathbb{P}^n$. We need to show that this bundle splits. The existence of a maximal torus in the intersection of parabolics implies mutual preservation of the graded components coming from filtrations. One starts with any coordinate $e_{\alpha_1}$, such that the  filtration associated with $\alpha_1$ has at least two terms and proceeds by choosing $e_{\alpha_2}$, which nontrivially refines the filtration (subdivides the graded spaces obtained on the first step), etc. If at some point it is impossible to find $e_{\alpha_i}$ providing further refinement to the filtration, then there must exist a Levi subgroup in the intersection of all parabolics in the family and, thus, a maximal torus (contained in that subgroup). After $k\leq r-1$ iterations the process will terminate with either providing a desired maximal torus or subdiving $E$ into one-dimensional subspaces. In the second case we consider the projective space $\mathbb{P}^{n-k}=\{\Bbbk[e_1:e_2:\hdots:e_{n+1}]~|~e_{\alpha_1}=e_{\alpha_2}=\hdots=e_{\alpha_k}=0\}\subset\mathbb{P}^n$. The stabilizer of a generic point $x\in \mathbb{P}^{n-k}$ is the $k$-dimensional torus $T':=\Bbbk^*e_{\alpha_1}\times\Bbbk^*e_{\alpha_2}\hdots\times\Bbbk^*e_{\alpha_k}$, moreover, by construction the fiber $E\big\vert_x$ has a simple $T'$ spectrum. %Let $\widetilde{T}:=\prod\limits_{i\not\in\{\alpha_1,\hdots\alpha_k\}} \Bbbk^*e_i$, then the action of $\widetilde{T}$ allows to carry the splitting over  to the whole of $\mathbb{P}^{n-k}$.
%We observe that $n-k\geq 2$ and the restriction of $E$ to $ \mathbb{P}^{n-k}$ splits ($E\big\vert_x$ is a sum of one-dimensional irreducible representations of $T'$). Hence, Horrocks' theorem (see \cite{Hor}) implies the splitting of $E$ and the existence of a maximal torus in the intersection of parabolic subgroups indexed by $A$.
%	\end{proof}
%	\begin{remark}
%	In the proof above, we have used that $E$ splits as an ordinary bundle. The ordinary splitting is equivalent to the equivariant one	as can be inferred from  \cite[Theorem 1.2.3]{Klyachko}.
%	\end{remark}

%Next we prove an analogue of Theorem \ref{parabolics} for the intersection of parahoric subgroups, that is, an analogue of Helly's theorem for the Bruhat-Tits building of a general linear group.

%	\begin{proposition}   \label{prop-parahorics}
%		\label{Parahorics} Let $\{P_{\alpha}\}, \alpha \in A$, be a finite family of parahoric subgroups in $\GL(r, \k)$. That is, each $P_\alpha$ is the stabilizer of a lattice chain in $K^r$. Then
%		if the intersection of any $r+1$ subgroups in the family contains a maximal split torus, then so does the whole family.
%	\end{proposition}
%	\begin{proof}
%		We argue by induction on the number of elements in $A$, which will be denoted by $n+1:=|A|\geq r+1$. Let $\mathbb{P}_{\mathcal{O}}^n$ be a projective space over $\Spec(\mathcal{O})$. Notice that by inductive assumption the filtrations corresponding to every collection of subgroups with indices forming a subset strictly contained in $A$ satisfy the compatibility condition. Therefore such collection of filtrations gives rise to an equivariant vector bundle $E$ on $\mathbb{P}_{\mathcal{O}}^n=\Proj(\mathcal{O}[e_1,\hdots,e_n])$. Moreover, \cite[Proposition 1.2]{Bal} asserts that for $n>1$ the splitting of $E$ is equivalent to the splitting of the associated vector bundle $E_{\Bbbk}$ over $\mathbb{P}_{\Bbbk}^n$. The result of \cite{Bal} is for general bundles (with no equivariance assumption). The assertion holds true for toric bundles as well, which is evident from Corollary \ref{equivSplitting} below.
%		As $E_{\Bbbk}$ is given by the filtrations corresponding to the parabolic subgroups  $\{\widetilde{P}_{\alpha}\}\subset \GL(n, \Bbbk)$ and the collection $\{\widetilde{P}_{\alpha}\} \in A$ satisfies the assumptions of Theorem \ref{parabolics}, the statement follows.
%	\end{proof}

%	\begin{corollary}[A Helly's theorem for Bruhat-Tits building of $\GL(E)$]  \label{cor-affine-Helly}
% 	Let $A$ be a finite collection of simplices in a Bruhat-Tits building $\B_\aff(K^r)$, such that any $r+1$ simplices lie in the same apartment, then all simplices in $A$ lie in the same apartment.
%	\end{corollary}
%	\begin{proposition}
%	Let $X$ be a toric variety over a discrete valuation ring, then the Krull-Schmidt theorem holds for the category of equivariant vector bundles on $X$.
%	\end{proposition}
%	\begin{proof}