# Edifices: Building-like spaces associated to linear algebraic groups

Michael Bate  Department of Mathematics, University of York, York YO10 5DD,
United Kingdom<EMAIL_ADDRESS>,  Benjamin Martin  Department of
Mathematics, University of Aberdeen, King’s College, Fraser Noble Building,
Aberdeen AB24 3UE, United Kingdom<EMAIL_ADDRESS>and  Gerhard Röhrle
Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
<EMAIL_ADDRESS>
(Date: May 19, 2023.)

###### Abstract.

Given a semisimple linear algebraic $k$-group $G$, one has a spherical
building $\Delta_{G}$, and one can interpret the geometric realisation
$\Delta_{G}({\mathbb{R}})$ of $\Delta_{G}$ in terms of cocharacters of $G$.
The aim of this paper is to extend this construction to the case when $G$ is
an arbitrary connected linear algebraic group; we call the resulting object
$\Delta_{G}({\mathbb{R}})$ the _spherical edifice_ of $G$. We also define an
object $V_{G}({\mathbb{R}})$ which is an analogue of the vector building for a
semisimple group; we call $V_{G}({\mathbb{R}})$ the _vector edifice_. The
notions of a linear map and an isomorphism between edifices are introduced; we
construct some linear maps arising from natural group-theoretic operations. We
also devise a family of metrics on $V_{G}({\mathbb{R}})$ and show they are all
bi-Lipschitz equivalent to each other; with this extra structure,
$V_{G}({\mathbb{R}})$ becomes a complete metric space. Finally, we present
some motivation in terms of geometric invariant theory and variations on the
Tits Centre Conjecture.

###### Key words and phrases:

Spherical buildings, Edifices, Tits Centre Conjecture, Geometric invariant
theory

###### 2020 Mathematics Subject Classification:

51E24, 20E42, 20G15

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Edifices
  4. 4 Linear maps of vector edifices
  5. 5 Linear maps arising from base change
  6. 6 Admissible metrics
  7. 7 The Tits Centre Conjecture and geometric invariant theory

##  1\. Introduction

Jacques Tits showed in his seminal 1974 monograph [33] how to associate a
spherical building $\Delta_{G}$ to an isotropic semisimple linear algebraic
group $G$ over a field $k$: the simplices of the building correspond to the
$k$-parabolic subgroups of $G$, ordered by reverse inclusion. This has led to
a rich and fruitful interaction between the theory of algebraic groups and the
theory of buildings; for example, Serre has formulated his theory of
$G$-completely reducible subgroups of $G$ in building-theoretic language [30],
[31]. Tits’s celebrated Centre Conjecture for spherical buildings—whose proof
is the culmination of a series of papers involving several authors [23], [21],
[26]—has consequences for algebraic groups: for instance, it implies the
Borel-Tits Theorem [13, §3]. The authors in previous and current work have
studied the interactions among the Centre Conjecture, $G$-complete
reducibility and geometric invariant theory (GIT), e.g., see [2], [4], [5],
[7], [8, §5], [10, §5.4], and [9]. The link with geometric invariant theory
comes via a rational version of the Hilbert-Mumford Theorem, used to study
orbits of an action of $G$ on an affine variety, see [3], [10, §3].

The geometric realisation $\Delta_{G}({\mathbb{R}})$ of $\Delta_{G}$ has a
striking interpretation in terms of cocharacters of $G$. Given a cocharacter
$\lambda$ of $G$, one can associate to it a parabolic subgroup $P_{\lambda}$
of $G$ in a standard way, see [27, §2], [32, §8.4]. Roughly speaking, points
in $\Delta_{G}({\mathbb{R}})$ correspond to formal ${\mathbb{R}}$-linear
combinations of cocharacters, modulo a natural equivalence relation. If
$x\in\Delta_{G}({\mathbb{R}})$ is the point corresponding to $\lambda$ then
$x$ belongs to the geometric realisation of the simplex corresponding to
$P_{\lambda}$. This construction is based on ideas of Tits and Chevalley and
is described in [16], [9] and below. See [24, Ch. 2, Sec. 2] for further
discussion (note that $\Delta_{G}({\mathbb{Q}})$ is called the _rational flag
complex_ in _loc. cit._). For applications to geometric invariant theory, one
must work with this geometric realisation rather than the simplicial building,
since in the Hilbert-Mumford Theorem one is concerned with the behaviour of
individual cocharacters; see Section 7.

Our purpose in this paper is to generalise the construction of a building via
cocharacters to the case when $G$ is an arbitrary connected linear algebraic
$k$-group, not necessarily semisimple. This was achieved for reductive groups
in [16]; in that case one can still form a spherical building from the
$k$-parabolic subgroups of $G$, which is canonically isomorphic to the
spherical building of the semisimple group $G/Z(G)^{0}$, but one sees a
difference at the level of cocharacters as $G$ may admit non-trivial
cocharacters with image in $Z(G)^{0}$. We define spaces
$\Delta_{G}({\mathbb{R}})$ and $\Delta_{G}$ for arbitrary $G$ in Section 3,
which coincide with the spaces described above when $G$ is semisimple or
reductive. We call $\Delta_{G}({\mathbb{R}})$ the _spherical edifice_
associated to $G$ and $\Delta_{G}$ the _combinatorial edifice_ associated to
$G$. We also define a space $V_{G}({\mathbb{R}})$, which we call the _vector
edifice_ associated to $G$; this is our generalisation for arbitrary $G$ of
the vector building for a semisimple group defined in [28]. In fact, we
concentrate on vector edifices rather than spherical edifices as the former
have some technical advantages (see Remark 4.30).

In general $\Delta_{G}({\mathbb{R}})$ is not the geometric realisation of
$\Delta_{G}$ (or of any spherical building); nonetheless it enjoys some of the
same properties. For instance, $G(k)$ acts on $\Delta_{G}({\mathbb{R}})$ and
we may endow $\Delta_{G}({\mathbb{R}})$ with a $G(k)$-invariant complete
metric: see Section 6. We prove the following key property (Proposition 4.7).
Let $f\colon G\to H$ be a homomorphism of $k$-groups. Then $f$ gives rise in a
natural way to a map $\kappa_{f}\colon V_{G}({\mathbb{R}})\to
V_{H}({\mathbb{R}})$, and $\kappa_{f}$ is continuous with respect to the
metric topologies (Corollary 6.24). For example, if $G$ is reductive and $P$
is a parabolic subgroup of $G$ then $\kappa_{i}\colon V_{P}({\mathbb{R}})\to
V_{G}({\mathbb{R}})$ is a bijection, where $i\colon P\to G$ is inclusion (see
Example 4.14). We also obtain an induced map from $\Delta_{G}({\mathbb{R}})$
to $\Delta_{H}({\mathbb{R}})$. Even when $G$ and $H$ are reductive, $f$ need
not map parabolic subgroups of $G$ to parabolic subgroups of $H$, so $f$ does
not give rise to a map $\Delta_{G}\to\Delta_{H}$ of simplicial buildings (see
Remark 4.27). Our constructions therefore provide extra flexibility: maps
between the geometric objects exist even when maps between the corresponding
combinatorial objects do not. We rely on this heavily in our forthcoming work
[9].

While edifices are worth studying as objects in their own right, we have two
further motivations for investigating them. First, assume $G$ is reductive.
Let $P$ be a parabolic subgroup of $G$ and let $L$ be a Levi subgroup of $P$.
For reasons discussed in Section 7, we wish to construct a map with certain
properties from $V_{G}({\mathbb{R}})$ to $V_{L}({\mathbb{R}})$. It is well
known that such a map exists at the level of the simplicial building: this is
the so-called projection map from $\Delta_{G}$ to $\Delta_{L}$, see Section
4.6. We need a map on the geometric realisations. Our constructions give us
what we want: we define the map to be $\kappa_{\pi}\circ\kappa_{i}^{-1}$,
where $i$ is the inclusion of $P$ in $G$ from above and $\pi\colon P\to L$ is
the canonical projection.

Second, edifices give us a framework for studying $G$-complete reducibility
and geometric invariant theory for non-reductive $G$: they play the role that
the geometric realisation of the spherical building plays when $G$ is
semisimple. We mention here in particular the case of pseudo-reductive $G$. As
explained in [25], one can associate a spherical Tits system to $G$ [15, Thm.
4.1.7], and one can then construct a spherical building $\Delta_{G}$ using the
pseudo-parabolic subgroups of $G$; however, there does not seem to be an
analogue of the “geometric realisation by cocharacters” previously known here.
Moreover, when working with pseudo-reductive $G$, one often needs to extend
scalars to a purely inseparable field extension $k^{\prime}/k$; if the base
change $G_{k^{\prime}}$ has non-trivial unipotent radical over $k^{\prime}$
then $\Delta_{G}$ need no longer be a spherical building, but the formalism of
edifices is flexible enough to handle this situation. The authors and their
collaborators will pursue these ideas in forthcoming work.

The paper is set out as follows. In Section 2 we review some background
material on algebraic groups and cocharacters, and we define the key notion of
an R-parabolic subgroup. We define the vector edifice $V_{G}({\mathbb{K}})$,
the spherical edifice $\Delta_{G}({\mathbb{K}})$ and the combinatorial edifice
$\Delta_{G}$ in Section 3, where ${\mathbb{K}}={\mathbb{R}}$ or
${\mathbb{Q}}$, and introduce some related notions: apartments (certain
subsets of the vector edifice), opposites, addition, and convexity. One
substantial difference from the reductive group case is that the “common
apartment property” can fail: not every pair of points in
$V_{G}({\mathbb{K}})$ need belong to a common apartment. In Section 4 we
introduce the notions of linear maps and isomorphisms between vector edifices.
We show that an algebraic group homomorphism $f\colon G\to H$ gives rise to a
linear map $\kappa_{f}\colon V_{G}({\mathbb{K}})\to V_{H}({\mathbb{K}})$, and
that $\kappa_{f}$ is injective if $\ker(f)$ is finite. In Section 5 we
construct linear maps of vector edifices arising from base change of the
ground field $k$.

We turn to metrics in Section 6. We define the notion of an _admissible
metric_ on $V_{G}({\mathbb{K}})$. The idea is to do this first for reductive
groups, where we can work inside individual apartments; for an arbitrary $G$,
we can embed $G$ inside some reductive $G^{\prime}$, which gives an embedding
of $V_{G}({\mathbb{K}})$ inside $V_{G^{\prime}}({\mathbb{K}})$, then pull back
a metric from $V_{G^{\prime}}({\mathbb{K}})$ to $V_{G}({\mathbb{K}})$.
Admissible metrics on $V_{G}({\mathbb{K}})$ are not unique, but we prove that
any two are bi-Lipschitz equivalent (Corollary 6.23); this takes some work. In
the final section (Section 7), we discuss applications of our formalism to the
Tits Centre Conjecture and to geometric invariant theory.

##  2\. Preliminaries

Throughout $k$ denotes a field with separable closure $k_{s}$ and algebraic
closure $\overline{k}$. We work with affine (linear) algebraic $k$-groups.
Formally, we view such a group $G$ as a functor from $k$-algebras to groups,
represented by an affine $k$-algebra $k[G]$, the coordinate algebra of $G$. If
$X$ is a $k$-variety and $k^{\prime}/k$ is a field extension, then
$X_{k^{\prime}}$ denotes the $k^{\prime}$-variety given by the base change of
$X$ to $k^{\prime}$, with coordinate algebra
$k^{\prime}[X_{k^{\prime}}]:=k[X]\otimes_{k}k^{\prime}$; if $f\colon X\to Y$
is a morphism of $k$-varieties then $f_{k^{\prime}}\colon X_{k^{\prime}}\to
Y_{k^{\prime}}$ is the morphism obtained by base extension. Unless specified
otherwise, we use the phrase “let $G$ be a $k$-group” as shorthand for “let
$G$ be a smooth affine algebraic $k$-group”. We write $\mathfrak{g}$ for the
Lie algebra of $G$. For a $k$-group $G$ and a $k$-algebra $A$, we let $G(A)$
denote the group of $A$-points of $G$. We adopt the framework of [14]: so by a
subgroup of $G$ we mean a $k$-defined subgroup, by a homomorphism we mean a
$k$-homomorphism and by a cocharacter we mean a $k$-defined cocharacter. We
assume, however, that subgroups are smooth; in cases when smoothness is not
evident, we use the terminology subgroup scheme. By the kernel of a
homomorphism we mean the scheme-theoretic kernel (note that this need not be
smooth). Likewise, by the centraliser $C_{G}(H)$ of a subgroup $H$ we mean the
scheme-theoretic centraliser and by the intersection of subgroups we mean the
scheme-theoretic intersection. Note that $C_{G}(H)$ is smooth if $H$ is a
torus. By an embedding of varieties we mean a closed embedding. For background
on linear algebraic groups, see [14], [11], [32]. We record one crucial result
here [14, Thm. C.2.3], which is used frequently in the sequel without further
comment: the maximal split $k$-tori in a $k$-group $G$ are all conjugate under
the action of $G(k)$. By the rank of $G$ we mean the $k$-rank: that is, the
dimension of a maximal split torus of $G$. If $T$ is a maximal split torus of
$G$ then $W_{k}:=(N_{G}(T)/C_{G}(T))(k)$ is the _relative Weyl group_ of $G$.
If $g\in G(k)$ then $\operatorname{Inn}_{g}\colon G\to G$ is conjugation by
$g$.

For an algebraic $k$-scheme $X$ we let $X_{\operatorname{red}}$ denote the
unique reduced subscheme of $X$ with the same underlying topological space as
$X$ [22, A.30]. When $k$ is perfect
$(X_{\operatorname{red}})_{\bar{k}}=(X_{\bar{k}})_{\operatorname{red}}$ is
reduced, and so when $X$ is a group scheme over a perfect field $k$,
$X_{\operatorname{red}}$ is smooth [22, Prop. 1.26, Cor. 1.39].

Our convention in this paper is that reductive groups are connected. If $G$ is
a $k$-group then we write $R_{u,k}(G)$ for the $k$-unipotent radical of $G$
(the largest connected normal unipotent $k$-subgroup). If
$R_{u}(G_{\overline{k}})$ descends to a subgroup of $G$ then we write
$R_{u}(G)$ for $R_{u,k}(G)$. We denote by $Y_{G}$ the set of cocharacters
(one-parameter subgroups) of $G$ — that is, an element $\lambda\in Y_{G}$ is a
homomorphism $\lambda$ from the multiplicative group $\mathbb{G}_{m}$ to $G$.
If $H$ is a subgroup of $G$ then we may regard $Y_{H}$ as a subset of $Y_{G}$
via the inclusion $H\subseteq G$. We say that $\lambda\in Y_{G}$ _evaluates in
$H$_ if $\lambda$ arises from an element of $Y_{H}$ in this way. We say that
two cocharacters _commute_ if their images commute. Given $\lambda\in Y_{H}$
and $g\in G(k)$, we define $g\cdot\lambda\in Y_{gHg^{-1}}$ by
$(g\cdot\lambda)(a)=g\lambda(a)g^{-1}$ for any $k$-algebra $A$ and any
$a\in{\mathbb{G}}_{m}(A)=A^{\times}$; this gives an action of $G(k)$ on
$Y_{G}$.

Now suppose $T$ is a split torus, and let $X_{T}$ denote the group of
characters of $T$: that is, $\chi\in X_{T}$ is a homomorphism
$T\to\mathbb{G}_{m}$. There is the usual pairing between $Y_{T}$ and $X_{T}$:
given $\lambda\in Y_{T}$ and $\chi\in X_{T}$, the map
$\chi\circ\lambda\colon\mathbb{G}_{m}\to\mathbb{G}_{m}$ is an endomorphism of
$\mathbb{G}_{m}$, and hence is given by raising elements to a power; we denote
this power by $\langle\lambda,\chi\rangle$. If now $T$ is a subtorus of some
$k$-group $G$, then for $g\in G(k)$ and $\chi\in X_{T}$, we get a character
$g\cdot\chi\in X_{gTg^{-1}}$ via the formula $(g\cdot\chi)(gtg^{-1})=\chi(t)$
for any $k$-algebra $A$ and $t\in T(A)$. It is easy to check that $\langle
g\cdot\lambda,g\cdot\chi\rangle=\langle\lambda,\chi\rangle$ for all $g\in
G(k)$.

Throughout, the symbol ${\mathbb{K}}$ denotes ${\mathbb{R}}$ or
${\mathbb{Q}}$. We write ${\mathbb{K}}^{+}$ for ${\mathbb{K}}\cap(0,\infty)$.
If $T$ is a torus then $Y_{T}$ with the usual addition of cocharacters is a
free ${\mathbb{Z}}$-module of finite rank. We define
$Y_{T}({\mathbb{K}}):=Y_{T}\otimes_{\mathbb{Z}}{\mathbb{K}}$, a finite-
dimensional ${\mathbb{K}}$-vector space, and identify $Y_{T}$ with the subset
of $Y_{T}({\mathbb{K}})$ consisting of elements of the form $\lambda\otimes 1$
for $\lambda\in Y_{T}$ in the usual way. We give $Y_{T}({\mathbb{K}})$ the
usual topology arising from a positive-definite bilinear form (this topology
does not depend on the form). We may similarly define
$X_{T}({\mathbb{K}}):=X_{T}\otimes_{\mathbb{Z}}{\mathbb{K}}$, and then the
pairing $Y_{T}\times X_{T}\to{\mathbb{Z}}$ extends to give a
${\mathbb{K}}$-bilinear map $Y_{T}({\mathbb{K}})\times
X_{T}({\mathbb{K}})\to{\mathbb{K}}$.

Below $G$ denotes a connected $k$-group. At various points we make the extra
assumption that $G$ is reductive. For $f\colon G\to H$ a homomorphism of
$k$-groups, we say that $f$ is surjective provided $f(\overline{k})\colon
G(\overline{k})\to H(\overline{k})$ is surjective. An _isogeny_ is a
surjective homomorphism of $k$-groups with finite kernel.

###  2.1. Limits and R-parabolic subgroups

For full details on the following construction, the reader is referred to [14,
Sec. 2.1]. Let $V$ be an affine $k$-scheme equipped with an action of
$\mathbb{G}_{m}$. Given $v\in V(A)$ for some $k$-algebra $A$, we say that
$\lim_{a\to 0}a\cdot v$ _exists_ if the orbit map $(\mathbb{G}_{m})_{A}\to
V_{A},a\mapsto a\cdot v$ over $A$ extends to an $A$-morphism
$F:\mathbb{A}^{1}_{A}\to V_{A}$, and in this case we write $\lim_{a\to
0}a\cdot v$ for the limit $F(0)$ (which is necessarily uniquely defined). In
[14, Lem. 2.1.4], it is shown that there is a closed subscheme $V^{\prime}$ of
$V$ such that for any $k$-algebra $A$, $V^{\prime}(A)$ consists of the $v\in
V(A)$ such that the limit $\lim_{a\to 0}a\cdot v$ exists. If $f\colon V\to W$
is an equivariant map of $\mathbb{G}_{m}$-varieties, $v\in V(A)$ and
$\lim_{a\to 0}a\cdot v$ exists then $\lim_{a\to 0}a\cdot f(v)$ exists, and the
converse holds is $f$ is an embedding.

The construction in the previous paragraph is of central importance in this
paper when $V=G$ is a $k$-group, and the action of $\mathbb{G}_{m}$ on $G$ is
the conjugation action via a cocharacter $\lambda\colon\mathbb{G}_{m}\to G$.
In this case, for each $\lambda\in Y_{G}$, one obtains a closed subgroup
$P_{\lambda}(G)$ of $G$: for each $k$-algebra $A$,

$P_{\lambda}(G)(A):=\left\\{g\in G(A)\ \left|\ \lim_{a\to
0}\lambda(a)g\lambda(a)^{-1}\textrm{ exists}\right\\}\right.$

(that $P_{\lambda}(G)$ is a smooth $k$-subgroup scheme is in [14, Lem.
2.1.5]). We also write $\lim_{a\to 0}\lambda(a)\cdot g$ as a shorthand for the
limit.

We call the subgroups of $G$ of the form $P_{\lambda}(G)$ for $\lambda\in
Y_{G}$ the _R-parabolic subgroups_ of $G$.111The “R” here is in honour of
Roger Richardson, who made extensive use of the link between cocharacters and
parabolic subgroups in reductive groups, in the context of geometric invariant
theory (see [27]). We usually suppress the $G$ in $P_{\lambda}(G)$ unless
there is some ambiguity — this occurs, for instance, when $H$ is a subgroup of
$G$ and we wish to view $\lambda\in Y_{H}$ as a cocharacter of both $G$ and
$H$, with two corresponding subgroups $P_{\lambda}(H)\subseteq
P_{\lambda}(G)$. In case $G$ is pseudo-reductive the subgroups
$P_{\lambda}(G)$ are the so-called pseudo-parabolic subgroups of $G$, but this
is not the case in general: a pseudo-parabolic subgroup of a general connected
$k$-group $G$ must also contain $R_{u,k}(G)$ but $P_{\lambda}$ need not; see
[14, Sec. 2.2].

We define $L_{\lambda}(G)$ (or just $L_{\lambda}$) to be $C_{G}({\rm
Im}(\lambda))=P_{\lambda}\cap P_{-\lambda}$ and we define a subgroup
$U_{\lambda}=U_{\lambda}(G)$ functorially by

$U_{\lambda}(G)(A):=\left\\{g\in G(A)\ \left|\ \lim_{a\to
0}\lambda(a)g\lambda(a)^{-1}=1\right\\}\right.,$

for each $k$-algebra $A$. It follows from [14, Lem. 2.1.5, Prop. 2.1.8] that
$P_{\lambda}$, $L_{\lambda}$ and $U_{\lambda}$ are smooth connected subgroup
schemes of $G$, $U_{\lambda}$ is a normal unipotent subgroup of $P_{\lambda}$,
and $P_{\lambda}=L_{\lambda}\ltimes U_{\lambda}$. If $P$ is an R-parabolic
subgroup of $G$ then we call any subgroup $L_{\lambda}$ for $\lambda\in Y_{G}$
such that $P=P_{\lambda}$ an _R-Levi subgroup_ of $P$. Given $\lambda\in
Y_{G}$, we have a surjective homomorphism $c_{\lambda}\colon P_{\lambda}\to
L_{\lambda}$ given by $c_{\lambda}(g)=\lim_{a\to 0}\lambda(a)g\lambda(a)^{-1}$
for each $k$-algebra $A$ and each $g\in P_{\lambda}(A)$.

We collect some basic properties of R-parabolic subgroups. First off, it
follows from the definition of limit in this case that for any subgroup $H$ of
$G$ and a cocharacter $\lambda\in Y_{H}\subseteq Y_{G}$, we have

$P_{\lambda}(H)=P_{\lambda}(G)\cap H,\ L_{\lambda}(H)=L_{\lambda}(G)\cap H,\
U_{\lambda}(H)=U_{\lambda}(G)\cap H;$

see [14, Lem. 2.1.5] and the paragraph following the proof of [14, Lem.
2.1.4].

Note that every R-parabolic subgroup of $G$ contains a maximal split torus of
$G$ — for $P=P_{\lambda}$ for some $\lambda$, and if $T$ is a maximal split
torus of $G$ such that $\lambda\in Y_{T}$, then $T\subseteq
L_{\lambda}\subseteq P_{\lambda}$. It is clear that if $g\in G(k)$ and
$\lambda\in Y_{G}$, then $P_{g\cdot\lambda}=gP_{\lambda}g^{-1}$,
$L_{g\cdot\lambda}=gL_{\lambda}g^{-1}$ and
$U_{g\cdot\lambda}=gU_{\lambda}g^{-1}$; in particular, if $g\in
P_{\lambda}(k)$, then $P_{g\cdot\lambda}=P_{\lambda}$. If $k^{\prime}/k$ is a
field extension then $P_{\lambda_{k^{\prime}}}=(P_{\lambda})_{k^{\prime}}$,
$L_{\lambda_{k^{\prime}}}=(L_{\lambda})_{k^{\prime}}$ and
$U_{\lambda_{k^{\prime}}}=(U_{\lambda})_{k^{\prime}}$.

We say the R-parabolic subgroups $P$ and $Q$ are _opposite_ if there exists
$\lambda\in Y_{G}$ such that $P=P_{\lambda}$ and $Q=P_{-\lambda}$: in this
case, $P_{\lambda}\cap P_{-\lambda}=L_{\lambda}$.

The next result follows from [14, Prop. 2.1.8, Prop. 2.1.12].

######  Lemma 2.1.

Let $G$ be a $k$-group and let $\lambda\in Y_{G}$. Then the multiplication map
$P_{\lambda}\times U_{-\lambda}\to G$ is an open immersion. If $G$ is
connected and solvable then this map is an isomorphism.

######  Lemma 2.2.

Let $\lambda\in Y_{G}$ and let $\mu\in Y_{P_{\lambda}}$. Then there exist
$u\in U_{\lambda}(k)$ and a maximal split torus $T$ of $P_{\lambda}$ such that
$\lambda$ and $u\cdot\mu$ belong to $Y_{T}$.

###### Proof.

Choose maximal split tori $T_{0}$ and $T_{1}$ of $P_{\lambda}$ such that
$\lambda\in Y_{T_{0}}$ and $\mu\in Y_{T_{1}}$. By conjugacy of maximal split
tori, there exists $g\in P_{\lambda}(k)$ such that $gT_{1}g^{-1}=T_{0}$. Write
$g=lu$ with $l\in L_{\lambda}(k)$ and $u\in U_{\lambda}(k)$. Then
$\lambda,u\cdot\mu\in Y_{T}$, where $T:=l^{-1}T_{0}l$. ∎

######  Lemma 2.3.

Let $T$ be a maximal split torus of $G$, and suppose $\lambda\in Y_{T}$. We
have $N_{P_{\lambda}(k)}(T)=N_{L_{\lambda}(k)}(T)$ and
$N_{P_{\lambda}(k)}(L_{\lambda})=L_{\lambda}(k)$.

###### Proof.

Suppose we have $g\in P_{\lambda}(k)$ such that $gTg^{-1}=T$. Now write $g=lu$
with $l\in L_{\lambda}(k)$ and $u\in U_{\lambda}(k)$. Note that $\lim_{a\to
0}\lambda(a)\cdot g$ normalises $T$, because $\lambda(a)$ and $g$ do, thus so
does $l=\lim_{a\to 0}\lambda(a)\cdot g$, and hence $u$ normalises $T$ also.
Now the morphism $\mathbb{G}_{m}\to U_{\lambda},a\mapsto\lambda(a)\cdot u$
extends to a morphism from all of $\mathbb{A}^{1}$ (by definition of the
limit), and this morphism has connected image contained in the normaliser of
$T$, containing the points $1=\lim_{a\to 0}\lambda(a)\cdot u$ and
$u=\lambda(1)\cdot u$. Rigidity of tori implies that the identity component of
$N_{G}(T)$ is the identity component of $C_{G}(T)$, [11, 8.10, Cor. 2], and
since $\lambda$ evaluates in $T$, $C_{G}(T)\subseteq L_{\lambda}$. But we also
have $u\in U_{\lambda}$, so this implies $u=1$. Thus $g\in L_{\lambda}(k)$,
which proves the first statement.

Now suppose $h\in P_{\lambda}(k)$ normalises $L_{\lambda}$. Then $hTh^{-1}$ is
another maximal split torus of $L_{\lambda}$, and hence is conjugate to $T$ by
some element of $L_{\lambda}(k)$. But this says that we may multiply $h$ by an
element of $L_{\lambda}(k)$ to obtain an element of
$N_{P_{\lambda}(k)}(T)\subseteq L_{\lambda}(k)$, so we’re done. ∎

######  Lemma 2.4.

Let $P=P_{\lambda}$ be an R-parabolic subgroup of $G$, and let $T$ be a
maximal split torus of $G$ such that $\lambda\in Y_{T}$.

  * (i)

Given any other maximal split torus $T^{\prime}$ of $G$ contained in $P$,
there is a $\mu\in Y_{T^{\prime}}$ with $P_{\mu}=P$ and $U_{\mu}=U_{\lambda}$.
In fact, we can take $\mu=u\cdot\lambda$ for some $u\in U_{\lambda}(k)$.

  * (ii)

The R-Levi subgroup $L_{\lambda}$ is the unique $P_{\lambda}(k)$-conjugate of
$L_{\lambda}$ containing $T$.

  * (iii)

Let $\mu\in Y_{G}$ such that $P_{\mu}=P_{\lambda}$ and $L_{\mu}=L_{\lambda}$.
Then $U_{\mu}=U_{\lambda}$.

###### Proof.

(i). Since $T$ and $T^{\prime}$ are maximal split tori of the smooth connected
$k$-group $P$, they are $P(k)$-conjugate. Let $g\in P(k)$ be such that
$T^{\prime}=gTg^{-1}$. Write $g=ul$ with $u\in U_{\lambda}(k)$ and $l\in
L_{\lambda}(k)$. Then $u\cdot\lambda=g\cdot\lambda\in Y_{T^{\prime}}$,
$P_{u\cdot\lambda}=uP_{\lambda}u^{-1}=P_{\lambda}$, and
$U_{u\cdot\lambda}=uU_{\lambda}u^{-1}=U_{\lambda}$, as required.

(ii). Suppose we have $g\in P(k)$ such that $T\subseteq gL_{\lambda}g^{-1}$.
Then $T$ and $g^{-1}Tg$ are maximal split tori of $L_{\lambda}$, so are
conjugate by an element of $L_{\lambda}(k)$. Hence, by adjusting $g$ with an
element of $L_{\lambda}(k)$ if necessary, we may assume that $g$ normalises
$T$. Now Lemma 2.3 implies that $g\in L_{\lambda}(k)$, as required.

(iii). Since ${\operatorname{Im}}(\mu)\subseteq Z(L_{\lambda})^{0}\subseteq
T\subseteq L_{\lambda}$, $\mu$ belongs to $Y_{T}$. It follows easily that
$c_{\mu}\circ c_{\lambda}=c_{\lambda}\circ c_{\mu}$. So let $v\in
U_{\mu}(\overline{k})$. We can write $v=lu$ for some $l\in
L_{\lambda}(\overline{k})$ and some $u\in U_{\lambda}(\overline{k})$. Then

$1=c_{\lambda}(c_{\mu}(v))=c_{\lambda}(c_{\mu}(lu))=c_{\lambda}(c_{\mu}(l))c_{\lambda}(c_{\mu}(u))=lc_{\mu}(c_{\lambda}(u))=l,$

so $v=u$. This shows that $U_{\mu}\subseteq U_{\lambda}$, and the reverse
inclusion follows by symmetry. ∎

If $G$ is reductive then the R-parabolic subgroups of $G$ are precisely the
parabolic subgroups, and the R-Levi subgroups of a parabolic subgroup are
precisely its Levi subgroups; furthermore, we have
$U_{\lambda}=R_{u}(P_{\lambda})$. These facts mean that the result above has a
rather simpler formulation and proof for reductive $G$: the content of parts
(i) and (ii) of Lemma 2.4 for reductive $G$ are the well-known facts that each
maximal torus of a parabolic subgroup is contained in a unique Levi subgroup,
and the unipotent radical of the parabolic subgroup acts simply transitively
on its Levi subgroups.

If $G$ is not reductive then things are more complicated. We have
$U_{\lambda}\subseteq R_{u,k}(P_{\lambda})$ since $U_{\lambda}$ is a connected
normal unipotent subgroup of $P_{\lambda}$, but the inclusion can be proper.
We can have cocharacters $\lambda,\mu\in Y_{G}$ such that
$P_{\lambda}\subseteq P_{\mu}$ but $U_{\lambda}\not\supseteq U_{\mu}$. In
fact, there can exist $\lambda,\mu\in Y_{G}$ such that $P_{\lambda}=P_{\mu}$
but $U_{\lambda}\not\supseteq U_{\mu}$ and $L_{\mu}$ is not conjugate to
$L_{\lambda}$. For instance, take $k=\overline{k}$ and let
$G=P_{\mu}(G^{\prime})$ be a Borel subgroup of a non-trivial connected
semisimple $k$-group $G^{\prime}$. Choose a maximal torus $T$ of $G$ such that
$\mu\in Y_{T}$. Let $\lambda=0\in Y_{G}$. Then
$P_{\lambda}(G)=P_{\mu}(G)=G=L_{\lambda}(G)\neq L_{\mu}(G)=T$ but
$R_{u}(P_{\mu})=U_{\mu}(G)\nsubseteq U_{\lambda}(G)=1$.

We do, however, have the following result.

######  Lemma 2.5.

Let $P$ be an R-parabolic subgroup of $G$. Suppose $P=P_{\lambda}=P_{\mu}$ and
$U_{\lambda}\subseteq U_{\mu}$ for some $\lambda,\mu\in Y_{G}$. Let $H$ be a
subgroup of $P$ such that $P\cong H\ltimes U_{\mu}$. Then:

  * (i)

$H$ is $U_{\mu}(k)$-conjugate to $L_{\mu}$.

  * (ii)

$H$ is $U_{\mu}(k)$-conjugate to a subgroup of $L_{\lambda}$. In particular,
$L_{\mu}$ is conjugate to a subgroup of $L_{\lambda}$.

###### Proof.

Since $H$ is a complement to $U_{\mu}$ in $P_{\mu}$, $H$ is isomorphic to
$L_{\mu}$ under the (restriction of the) projection $P_{\mu}\to L_{\mu}$, and
hence $H$ contains a maximal split torus $T$ of $P_{\mu}$. Using Lemma 2.4(i),
by conjugating $H$ with some element in $U_{\mu}(k)$, we may assume that $\mu$
evaluates in $T$ as well. Now $H=P_{\mu}(H)=L_{\mu}(H)\ltimes U_{\mu}(H)$, but
$U_{\mu}(H)=U_{\mu}\cap H$ is trivial, so $H=L_{\mu}(H)$. This implies that
$H\subseteq L_{\mu}$, thus $H=L_{\mu}$ since $H$ is isomorphic to $L_{\mu}$
and both are smooth and connected, proving (i).

The proof of (ii) is similar: by Lemma 2.4(i) and the fact that
$U_{\lambda}\subseteq U_{\mu}$, we may also conjugate $H$ by an element of
$U_{\mu}(k)$ so that the maximal split torus $T$ of $H$ contains the image of
$\lambda$. Then $H=P_{\lambda}(H)=L_{\lambda}(H)\ltimes U_{\lambda}(H)$.
Again, since $U_{\lambda}\subseteq U_{\mu}$ and $U_{\mu}\cap H=1$, we have
$H=L_{\lambda}(H)$, so $H$ is contained in $L_{\lambda}$. ∎

######  Corollary 2.6.

Let $\lambda,\mu\in Y_{G}$ such that $P_{\lambda}=P_{\mu}$ and
$U_{\lambda}=U_{\mu}$. Then there is a unique $u\in U_{\lambda}(k)$ such that
$L_{\mu}=uL_{\lambda}u^{-1}$.

###### Proof.

The existence of $u$ follows from Lemma 2.5(i) with $H=L_{\lambda}$. If
$u_{1},u_{2}\in U_{\lambda}(k)$ and
$u_{1}L_{\lambda}u_{1}^{-1}=u_{2}L_{\lambda}u_{2}^{-1}$ then $u_{1}^{-1}u_{2}$
normalises $L_{\lambda}$, so $u_{1}=u_{2}$ by Lemma 2.3. This proves
uniqueness. ∎

######  Remark 2.7.

We need some notation. Let $V$ be a rational $G$-module and let $T$ be a
maximal split torus of $G$. Let $\Phi\subseteq X_{T}$ be the set of weights of
$T$ on $V$. Since $T$ is split, $V$ splits into a direct sum of weight spaces
$V_{\chi}$ for $T$. Given $v\in V(\overline{k})$, we can write $v$ uniquely as
$v=\sum_{\chi\in\Phi}v_{\chi}$, where each $v_{\chi}\in
V_{\chi}(\overline{k})$. We define $\operatorname{supp}(v)=\\{\chi\in\Phi\mid
v_{\chi}\neq 0\\}$. If $g\in G(k)$ then the weights of $gTg^{-1}$ on $V$ are
the characters of the form $g\cdot\chi$, where $\chi$ runs over the set of
weights of $T$ on $V$, and we have $V_{g\cdot\chi}=g\cdot V_{\chi}$ for each
$\chi\in\Phi$. Let $\lambda\in Y_{T}$. Set
$\Phi_{\lambda,\epsilon}=\\{\chi\in\Phi\mid\operatorname{sgn}(\langle\lambda,\chi\rangle)=\epsilon\\}$,
where $\epsilon\in\\{+,-,0\\}$. We define $V_{\lambda,\epsilon}$ to be the sum
of the weight spaces $V_{\chi}$ for $\chi\in\Phi_{\lambda,\epsilon}$, and
$V_{\lambda,\geq 0}:=V_{\lambda,0}\oplus V_{\lambda,+}$ (as in [10, Rem.
2.8]).

Now we consider an important special case of this set-up. By [20, Lem.
1.1(a)], there is a $G$-equivariant embedding $\rho$ of $G$ (as a $G$-variety
under the conjugation action) into a rational $G$-module $V$. Let $T$, etc.,
be as above. For $v\in V(A)$, $\lim_{a\to 0}\lambda(a)\cdot v$ exists if and
only if $v\in V_{\lambda,\geq 0}(A)$ [14, Ex. 2.1.1]. We see that
$P_{\lambda}=\rho^{-1}(V_{\lambda,\geq 0})$, and it follows that
$P_{n\lambda}=P_{\lambda}$ for every $n\in{\mathbb{N}}$. By similar arguments,
$L_{\lambda}=\rho^{-1}(V_{\lambda,0})$ and
$U_{\lambda}=\rho^{-1}(V_{\lambda,+})$.

######  Remark 2.8.

There are only finitely many R-parabolic subgroups $P$ containing a fixed
maximal torus $S$ of $G$. To see this, choose a $G$-equivariant embedding
$\rho$ of $G$ into a rational $G$-module $V$, as in Remark 2.7. Let $T$ be the
unique maximal split subtorus of $S$. By Lemma 2.4(i), we can assume that
$P=P_{\lambda}$ for some $\lambda\in Y_{T}$. We have
$P_{\lambda}=\rho^{-1}(V_{\lambda,\geq 0})$. Since $\Phi$ is finite, there are
only finitely many possibilities for $\Phi_{\lambda,\geq 0}$ and
$V_{\lambda,\geq 0}$ and hence only finitely many possibilities for $P$.

######  Lemma 2.9.

Let $T$ be a maximal split torus of $G$ and let $\lambda,\mu\in Y_{T}$.
Suppose $P_{\lambda}=P_{\mu}$. Then $P_{\lambda+\mu}=P_{\lambda}=P_{\mu}$.

###### Proof.

Set $P=P_{\lambda}=P_{\mu}$. We prove first that $P_{\lambda+\mu}\supseteq P$.
Choose a $G$-equivariant embedding $\rho$ of $G$ into a rational $G$-module
$V$, and fix a maximal split torus $T$ of $G$. For any $\nu\in Y_{T}$ and any
$g\in G(\overline{k})$, we have $g\in P_{\nu}(\overline{k})$ if and only if

$\min\\{\langle\nu,\chi\rangle\mid\chi\in\operatorname{supp}(\rho(g))\\}\geq
0.$

So let $g\in P(\overline{k})$. Then

$\min\\{\langle\lambda,\chi\rangle\mid\chi\in\operatorname{supp}(\rho(g))\\}\geq
0$

and

$\min\\{\langle\mu,\chi\rangle\mid\chi\in\operatorname{supp}(\rho(g))\\}\geq
0,$

so

$\min\\{\langle\lambda+\mu,\chi\rangle\mid\chi\in\operatorname{supp}(\rho(g))\\}\geq
0.$

Hence $g\in P_{\lambda+\mu}(\overline{k})$. As $P$ and $P_{\lambda+\mu}$ are
smooth, this shows that $P_{\lambda+\mu}\supseteq P$.

To complete the proof, it is enough to show that
$\operatorname{Lie}(P_{\lambda+\mu})=\operatorname{Lie}(P)$. Let $\Phi$ be the
set of roots of $G$ with respect to $T$. By [14, Prop. 2.1.8], if $\nu\in
Y_{T}$ then $\operatorname{Lie}(P_{\nu})=\mathfrak{g}_{\nu,\geq 0}$. Since
$\operatorname{Lie}(P_{\lambda})=\operatorname{Lie}(P_{\mu})$, we deduce that
$\Phi_{\lambda,\geq 0}=\Phi_{\mu,\geq 0}$, so we have $\Phi_{\lambda+\mu,\geq
0}=\Phi_{\lambda,\geq 0}=\Phi_{\mu,\geq 0}$ and
$\mathfrak{g}_{\lambda+\mu,\geq 0}=\mathfrak{g}_{\lambda,\geq
0}=\mathfrak{g}_{\mu,\geq 0}$. The result follows. ∎

######  Lemma 2.10.

Let $P$ be an R-parabolic subgroup of $G$ and let $T$ be a maximal torus of
$P$. Then there exists $\lambda\in Y_{T}$ such that $P_{\lambda}=P$.

###### Proof.

Choose a finite Galois extension $k^{\prime}/k$ such that $T$ is
$k^{\prime}$-split and set $\Gamma=\operatorname{Gal}(k^{\prime}/k)$. By Lemma
2.4(i) there exists $\lambda^{\prime}\in Y_{T_{k^{\prime}}}$ such that
$P_{\lambda^{\prime}}=P_{k^{\prime}}$. Now $\gamma\cdot\lambda^{\prime}$
belongs to $Y_{T_{k^{\prime}}}$ for each $\gamma\in\Gamma$ and
$\sum_{\gamma\in\Gamma}\gamma\cdot\lambda^{\prime}$ is $\Gamma$-stable, so it
descends to an element $\lambda$ of $Y_{T}$. It is easily seen that
$P_{\gamma\cdot\lambda^{\prime}}=\gamma(P_{\lambda^{\prime}})=\gamma(P)=P$ for
all $\gamma\in\Gamma$. It follows from Lemma 2.9 that $P_{\lambda}=P$, so we
are done. ∎

######  Lemma 2.11.

Let $P$ and $Q$ be R-parabolic subgroups of $G$. Suppose $P\cap Q$ contains a
maximal split torus $T$ of $G$. Then $P\cap Q$ is smooth.

###### Proof.

We can write $P=P_{\lambda}$ and $Q=P_{\mu}$ for some $\lambda,\mu\in Y_{T}$,
by Lemma 2.4(i). Since $\mu$ evaluates in $P$, it follows that $P\cap
Q=P_{\lambda}\cap P_{\mu}=P_{\mu}(P_{\lambda})$. Since $P_{\lambda}$ is smooth
and connected, this means that the intersection is an R-parabolic subgroup of
$P_{\lambda}$, and hence is also smooth and connected. (A direct proof using
the Lie algebras is also possible, like the one in [14, Prop. 2.1.8].) ∎

######  Lemma 2.12.

Let $P$ and $Q$ be R-parabolic subgroups of $G$. The following are equivalent.

  * (i)

$P\cap Q$ contains a maximal split torus of $G$.

  * (ii)

$P\cap Q$ contains a maximal torus of $G$.

  * (iii)

$P_{\overline{k}}\cap Q_{\overline{k}}$ contains a maximal torus of
$G_{\overline{k}}$.

###### Proof.

Clearly (ii) implies (iii). If $P_{\overline{k}}\cap Q_{\overline{k}}$
contains a maximal torus of $G_{\overline{k}}$ then $P_{\overline{k}}\cap
Q_{\overline{k}}$ is smooth by Lemma 2.11, so $P\cap Q$ is smooth; then $P\cap
Q$ contains a maximal torus and this is a maximal torus of $G$. This shows
that (iii) implies (ii). Now suppose $S$ is a maximal split torus of $G$
contained in $P\cap Q$. We can choose $\lambda,\mu\in Y_{S}$ such that
$P=P_{\lambda}$ and $Q=P_{\mu}$. Let $T$ be a maximal torus of $G$ that
contains $\operatorname{Im}(\lambda)$ and $\operatorname{Im}(\mu)$; then
$T\subseteq L_{\lambda}\cap L_{\mu}\subseteq P_{\lambda}\cap P_{\mu}=P\cap Q$.
Conversely, suppose $T$ is a maximal torus of $G$ contained in $P\cap Q$. By
Lemma 2.10, there exist $\lambda,\mu\in Y_{T}$ such that $P_{\lambda}=P$ and
$P_{\mu}=Q$. Then $\operatorname{Im}(\lambda)\operatorname{Im}(\mu)$ is a
split torus, so it is contained in a maximal split torus $S$ of $G$. We have
$S\subseteq L_{\lambda}\cap L_{\mu}\subseteq P_{\lambda}\cap P_{\mu}=P\cap Q$.
This shows that (i) and (ii) are equivalent. ∎

######  Lemma 2.13.

Suppose $P$ and $Q$ are two R-parabolic subgroups of a $k$-group $G$. Identify
$Y_{P}$ and $Y_{Q}$ as subsets of $Y_{G}$ in the usual way. Then $P\subseteq
Q$ if and only if $Y_{P}\subseteq Y_{Q}$.

###### Proof.

That $Y_{P}\subseteq Y_{Q}$ when $P\subseteq Q$ is obvious. For the other
implication, first observe that for any $\lambda\in Y_{G}$ such that
$Q=P_{\lambda}$, and any $1\neq u\in U_{-\lambda}(k)$, $u\cdot\lambda$ does
not evaluate in $Q$. To see this, note that since the image of $\lambda$
normalises $U_{-\lambda}$, for every $a\in k_{s}$ we can write

$u\lambda(a)u^{-1}=\lambda(a)(\lambda(a)^{-1}u\lambda(a))u^{-1}=\lambda(a)u_{a}$

for some $u_{a}\in U_{-\lambda}(k_{s})$. Since $1\neq u\in U_{-\lambda}$, $u$
does not centralise $\rm Im(\lambda)$, and hence there exists $a\in k_{s}$
such that $u_{a}\neq 1$, which means that $\lambda(a)u_{a}\not\in
P_{\lambda}$, and we’re done.

Now if we assume that $Y_{P}\subseteq Y_{Q}$, then we can find a maximal split
torus $T$ in the intersection $P\cap Q$: for any split torus of $P$ is
generated by the images of its cocharacters, and these also lie in $Y_{Q}$.
Choose cocharacters $\lambda,\mu\in Y_{T}$ such that $P=P_{\lambda}$ and
$Q=P_{\mu}$. Suppose $P$ is not contained in $Q$. Now $U_{-\mu}(P)$ is a non-
trivial unipotent subgroup of $P$ by Lemma 2.1. Since the image of $\mu$ acts
non-trivially on this subgroup, it cannot be wound, by [14, Prop. B4.4]. In
particular, $U_{-\mu}(P)$ has a non-identity $k$-point $u$. Now $\mu$ is a
cocharacter of $Q$ evaluating in $P$, so it is a cocharacter of $P$, so
$u\cdot\mu\in Y_{P}$. The first paragraph of the proof shows that
$u\cdot\mu\not\in Y_{Q}$, which gives a contradiction. ∎

###  2.2. Spherical buildings

As general references on buildings we refer to [1] and [33]. We regard a
spherical building — as opposed to its geometric realisation — as a
combinatorial object. It is a simplicial complex with a distinguished set of
subcomplexes called apartments. If $G$ is reductive then we can form the
spherical building $\Delta_{G}$ as follows: the simplices of $\Delta_{G}$ are
the parabolic subgroups ordered by reverse inclusion, and an apartment of
$\Delta_{G}$ is the subcomplex consisting of all parabolic subgroups
containing a fixed maximal split torus $T$ of $G$. Two simplices are opposite
if the corresponding parabolic subgroups are opposite. There is a notion of
convexity for subcomplexes [31, Prop. 3.1]. An automorphism of a spherical
building $\Delta$ is an automorphism of simplicial complexes that maps
apartments bijectively to apartments. We write $\operatorname{Aut}(\Delta)$
for the group of automorphisms of $\Delta$. For instance, any algebraic group
automorphism of $G$ — e.g., an inner automorphism — gives rise to an
automorphism of $\Delta_{G}$. There are also automorphisms of $\Delta_{G}$
arising from automorphisms of the ground field $k$. Both these kinds of
automorphism have counterparts for vector edifices: see Section 4.2 and
Section 5.2.

##  3\. Edifices

In order to define vector edifices, we need to understand how to glue together
the ${\mathbb{K}}$-vector spaces $Y_{T}({\mathbb{K}})$ for the maximal split
tori of a $k$-group $G$ in a suitable way, where ${\mathbb{K}}={\mathbb{R}}$
or ${\mathbb{Q}}$. This first requires us to attach an R-parabolic subgroup to
an arbitrary element of $Y_{T}({\mathbb{K}})$ (above, we have defined
$P_{\lambda}$ just for $\lambda\in Y_{T}$). This work is achieved in the next
few subsections.

First we clarify some notation. When we write, say, $\lambda\in
Y_{T}({\mathbb{K}})$ then it is understood that we are working with a fixed
choice of $T$. If $\mu$ is an actual cocharacter — that is, if $\mu\in Y_{G}$
— then there can exist many different maximal split tori $T$ such that $\mu\in
Y_{T}$, so there is potential for misunderstanding. This does not cause
problems, however, for we show below that our constructions when applied to
such a $\mu$ don’t depend on the choice of $T$: see the paragraph following
Definition 3.7.

###  3.1. Linear maps induced by homomorphisms of tori

Suppose $f\colon T\to T^{\prime}$ is a homomorphism of split $k$-tori. Then we
get an induced map $Y_{T}\to Y_{T^{\prime}}$ via $\lambda\mapsto
f\circ\lambda$, and this extends to a ${\mathbb{K}}$-linear map $f_{*}\colon
Y_{T}({\mathbb{K}})\to Y_{T^{\prime}}({\mathbb{K}})$. This map is injective if
$f$ has finite kernel. In particular, if $T\subseteq G$ for some $k$-group $G$
and $g\in G(k)$ then the conjugation map $\operatorname{Inn}_{g}$ induces a
linear isomorphism from $Y_{T}({\mathbb{K}})$ to $Y_{gTg^{-1}}({\mathbb{K}})$;
we denote the image of $\lambda\in Y_{T}({\mathbb{K}})$ by $g\cdot\lambda$. If
in fact $\lambda\in Y_{T}$, then $g\cdot\lambda\in Y_{gTg^{-1}}$ as just
defined agrees with the usual definition of $g\cdot\lambda$ given in Section
2. We also have functoriality: if $f^{\prime}\colon T^{\prime}\to
T^{\prime\prime}$ is another homomorphism of split $k$-tori then
$(f^{\prime}\circ f)_{*}=(f^{\prime})_{*}\circ f_{*}$. In particular, if
$f\colon G\to H$ is a homomorphism of $k$-groups then, given split $k$-tori
$T\subseteq G$ and $S\subseteq H$ such that $f(T)\subseteq S$, we have a map
$h_{*}\colon Y_{T}({\mathbb{K}})\to Y_{S}({\mathbb{K}})$, where $h\colon T\to
S$ is the restriction of $f$. Often we abuse notation slightly and write
$f_{*}\colon Y_{T}({\mathbb{K}})\to Y_{S}({\mathbb{K}})$ for this map.

We return to the subject of linear maps induced by group homomorphisms in
Section 4.2 below.

###  3.2. More on R-parabolic subgroups

Now we can define $P_{\lambda}$, $L_{\lambda}$ and $U_{\lambda}$ for
$\lambda\in Y_{T}({\mathbb{K}})$, where $T\subseteq G$ is a split torus. We
may as well assume that $T$ is a maximal split torus of $G$ and $\lambda\in
Y_{T}({\mathbb{K}})$. Then we may choose a $G$-equivariant embedding $\rho$ of
$G$ into a rational $G$-module $V$. Using the notation from Remark 2.7, we
define a subspace $W$ of $Y_{T}({\mathbb{K}})$ by

$W=\\{\mu\in Y_{T}({\mathbb{K}})\mid\langle\mu,\chi\rangle=0\textrm{ for all
}\chi\in\Phi_{\lambda,0}\\}.$

Clearly, $\lambda\in W$, by the definition of $\Phi_{\lambda,0}$. Since
$\Phi_{\lambda,0}\subseteq X_{T}$, $W$ is defined by equations with
coefficients in ${\mathbb{Z}}$ and hence is defined over ${\mathbb{Q}}$.
Therefore, we get that $\dim_{\mathbb{Q}}(W\cap
Y_{T}({\mathbb{Q}}))=\dim_{\mathbb{K}}W$, which implies that $W\cap
Y_{T}({\mathbb{Q}})$ is dense in $W$. Thus there exists a sequence
$(\lambda_{n})$ in $W\cap Y_{T}({\mathbb{Q}})$ converging to $\lambda$. For
all $n\in{\mathbb{N}}$ and for $\lambda^{\prime}:=\lambda_{n}$, it follows
from the definition of $W$ that

(3.1)
$\operatorname{sgn}(\langle\lambda^{\prime},\chi\rangle)=\operatorname{sgn}(\langle\lambda,\chi\rangle)$

holds for all $\chi\in\Phi_{\lambda,0}$, and for $n$ large enough (3.1) even
holds for all $\chi\in\Phi_{\lambda,\pm}$, because the linear functional
$\langle-,\chi\rangle$ is continuous. An element $\lambda^{\prime}\in
Y_{T}({\mathbb{Q}})$ satisfying (3.1) is called a _rational cocharacter
approximation to $\lambda$_ (_with respect to $V$_) (_in
$Y_{T}({\mathbb{K}})$_). Multiplying a rational cocharacter approximation by a
suitable positive integer, we may obtain an element $\lambda^{\prime}\in
Y_{T}$ satisfying (3.1), which we call a _cocharacter approximation to
$\lambda$_ (_with respect to $V$_). If $\lambda^{\prime}\in
Y_{T}({\mathbb{Q}})$ then we set $P_{\lambda^{\prime}}=P_{n\lambda^{\prime}}$,
$L_{\lambda^{\prime}}=L_{n\lambda^{\prime}}$ and
$U_{\lambda^{\prime}}=U_{n\lambda^{\prime}}$ for any $n\in{\mathbb{N}}$ such
that $n\lambda^{\prime}\in Y_{G}$: clearly this is well-defined. We call a
sequence $(\lambda_{n})$ of rational cocharacter approximations to $\lambda$
such that $\lambda_{n}\to\lambda$ a _rational approximating sequence to
$\lambda$_ (_with respect to $V$_) (_in $Y_{T}({\mathbb{K}})$_).

######  Definition 3.2.

Given $\lambda\in Y_{T}({\mathbb{K}})$ and a rational cocharacter
approximation $\lambda^{\prime}$ to $\lambda$, we define
$P_{\lambda}=P_{\lambda^{\prime}}$, $L_{\lambda}=L_{\lambda^{\prime}}$, and
$U_{\lambda}=U_{\lambda^{\prime}}$.

######  Lemma 3.3.

With the notation as above, we have the following.

  * (i)

The subgroups $P_{\lambda}$, $L_{\lambda}$ and $U_{\lambda}$ do not depend on
the choice of $\lambda^{\prime}$ or the choice of $\rho$.

  * (ii)

For any $g\in G(k)$, if $\lambda^{\prime}\in Y_{T}$ is a cocharacter
approximation to $\lambda$ then $g\cdot\lambda^{\prime}\in Y_{gTg^{-1}}$ is a
cocharacter approximation to $g\cdot\lambda$. Consequently, for any $g\in
G(k)$, we have $P_{g\cdot\lambda}=gP_{\lambda}g^{-1}$,
$L_{g\cdot\lambda}=gL_{\lambda}g^{-1}$ and
$U_{g\cdot\lambda}=gU_{\lambda}g^{-1}$.

###### Proof.

(i). We have $P_{\lambda^{\prime}}=\rho^{-1}(V_{\lambda^{\prime},\geq 0})$,
$L_{\lambda^{\prime}}=\rho^{-1}(V_{\lambda^{\prime},0})$ and
$U_{\lambda^{\prime}}=\rho^{-1}(V_{\lambda^{\prime},>0})$ (Remark 2.7), so for
a given $\rho$ the fact that these subgroups do not depend on the choice of
$\lambda^{\prime}$ is clear. Now let $\rho_{i}\colon G\to V_{i}$ be
$G$-equivariant embeddings with $\Phi_{i}$ the sets of weights of $T$ on
$V_{i}$ for $i=1,2$. Let $\rho\colon G\to V_{1}\oplus V_{2}$ be the diagonal
embedding, and let $\Phi=\Phi_{1}\cup\Phi_{2}$ be the associated set of
weights. Choose $\lambda^{\prime}\in Y_{T}$ such that $\lambda^{\prime}$ is a
cocharacter approximation to $\lambda$ with respect to $V_{1}\oplus V_{2}$.
Then
$\operatorname{sgn}(\langle\lambda^{\prime},\chi\rangle)=\operatorname{sgn}(\langle\lambda,\chi\rangle)$
for all $\chi\in\Phi_{1}$ and for all $\chi\in\Phi_{2}$, so $\lambda^{\prime}$
is a cocharacter approximation to $\lambda$ with respect to $V_{1}$ and with
respect to $V_{2}$. This proves part (i).

(ii). The first statement follows from the fact that the pairing
$\langle-,-\rangle$ is $G(k)$-invariant, and the fact that the set of weights
of $gTg{{}^{-1}}$ on $V$ is $g\cdot\Phi$, where $\Phi$ is the set of weights
of $T$. The second statement now follows. ∎

######  Remarks 3.4.

(i). Let $S$ be any split torus of $G$, let $\lambda\in Y_{S}({\mathbb{K}})$,
and let $T$ be any maximal split torus of $G$ containing $S$. We claim that
there exists a cocharacter approximation $\lambda^{\prime}$ to $\lambda$
contained in $Y_{S}\subseteq Y_{T}$. To see this, note that in the
construction above, the $S$-weights on $V$ are given by the restrictions of
the $T$-weights on $V$ to $S$, because $S\subseteq T$. We may therefore
perform the construction of $\lambda^{\prime}$ above entirely within
$Y_{S}({\mathbb{K}})$ just using these restrictions, and we obtain a
cocharacter approximation $\lambda^{\prime}\in Y_{S}$ satisfying (3.1).
Analogous conclusions hold for rational cocharacter approximations and
rational approximating sequences in $Y_{S}({\mathbb{K}})$.

(ii). Now suppose $T_{1}$ and $T_{2}$ are two maximal split tori of $G$ that
contain $S$ and let $i_{1},i_{2}$ be the inclusions of $S$ in $T_{1},T_{2}$
respectively. Let $\lambda^{\prime}$ be the cocharacter approximation
contained in $Y_{S}\subseteq Y_{T_{1}}\cap Y_{T_{2}}$ from part (i). It is
immediate from the construction that
$L_{(i_{j})_{*}(\lambda)}=L_{i_{j}\circ\lambda^{\prime}}$ and
$P_{(i_{j})_{*}(\lambda)}=P_{i_{j}\circ\lambda^{\prime}}$ for $j=1,2$, so
$L_{(i_{1})_{*}(\lambda)}=L_{(i_{2})_{*}(\lambda)}=L_{\lambda}$ and
$P_{(i_{1})_{*}(\lambda)}=P_{(i_{2})_{*}(\lambda)}=P_{\lambda}$. In
particular, both $T_{1}$ and $T_{2}$ are contained in $L_{\lambda}$, and there
is no dependence of our construction on the choice of a maximal split torus
containing $S$.

(iii). Let $T$ be a maximal split torus of $G$. It is clear from the
construction that if $\lambda\in Y_{T}({\mathbb{K}})$ then $T\subseteq
L_{\lambda}\subseteq P_{\lambda}$. It is also clear that
$P_{\lambda}=\rho^{-1}(V_{\lambda,\geq 0})$,
$L_{\lambda}=\rho^{-1}(V_{\lambda,0})$ and
$U_{\lambda}=\rho^{-1}(V_{\lambda,>0})$ for any $\lambda\in
Y_{T}({\mathbb{K}})$.

The next result is an extension to all of $Y_{T}({\mathbb{K}})$ of the obvious
fact that for a cocharacter $\lambda\in Y_{T}$ we have $L_{\lambda}=G$ if and
only if $\lambda$ evaluates in the centre of $G$.

######  Lemma 3.5.

Let $T$ be a maximal split torus of $G$ and let $\lambda\in
Y_{T}({\mathbb{K}})$. Let $Z_{\lambda}$ be the unique maximal split torus of
$Z(L_{\lambda})^{0}$ and let $Z$ be the unique maximal split torus of
$Z(G)^{0}$. Then:

  * (i)

$\lambda\in Y_{Z_{\lambda}}({\mathbb{K}})$.

  * (ii)

$L_{\lambda}=G$ if and only if $\lambda\in Y_{Z}({\mathbb{K}})$.

###### Proof.

(i). Let $(\lambda_{n})$ be a rational approximating sequence to $\lambda$ in
$Y_{T}({\mathbb{K}})$. Choose $a_{n}\in{\mathbb{N}}$ such that
$a_{n}\lambda_{n}\in Y_{T}$ for each $n\in{\mathbb{N}}$. Then for all
$n\in{\mathbb{N}}$,
$L_{\lambda_{n}}=L_{a_{n}\lambda_{n}}=C_{G}(a_{n}\lambda_{n})$, so
$a_{n}\lambda_{n}\in Y_{Z_{\lambda_{n}}}$, and so $\lambda_{n}\in
Y_{Z_{\lambda_{n}}}({\mathbb{K}})$. But $L_{\lambda_{n}}=L_{\lambda}$ for $n$
sufficiently large, and $Y_{Z_{\lambda}}({\mathbb{K}})$ is closed in
$Y_{T}({\mathbb{K}})$, so $\lambda\in Y_{Z_{\lambda}}({\mathbb{K}})$, as
required.

(ii). The forward implication follows from part (i). The reverse implication
follows from Remark 3.4(i) applied to the split torus $Z$ — we may find a
cocharacter approximation $\lambda^{\prime}\in Y_{Z}$ and then we have
$L_{\lambda}=L_{\lambda^{\prime}}=G$. ∎

###  3.3. The vector edifice

Now we can define $Y_{G}({\mathbb{K}})$ and $V_{G}({\mathbb{K}})$. We work
with the disjoint union $\bigsqcup_{T}Y_{T}({\mathbb{K}})$, where $T$ ranges
over the maximal split tori of $G$. It is convenient to write elements of
$\bigsqcup_{T}Y_{T}({\mathbb{K}})$ explicitly as pairs $(T,\lambda)$, where
$T$ is a maximal split torus of $G$ and $\lambda\in Y_{T}({\mathbb{K}})$. The
group $G(k)$ acts on $\bigsqcup_{T}Y_{T}({\mathbb{K}})$ by
$g\cdot(T,\lambda)=(gTg^{-1},g\cdot\lambda)$.

We define two relations on $\bigsqcup_{T}Y_{T}({\mathbb{K}})$ by

$(T_{1},\lambda_{1})\sim(T_{2},\lambda_{2})\ \mbox{if there exists $l\in
L_{\lambda_{1}}(k)$ such that $T_{2}=lT_{1}l^{-1}$ and
$\lambda_{2}=l\cdot\lambda_{1}$}$

and

$(T_{1},\lambda_{1})\approx(T_{2},\lambda_{2})\ \mbox{if there exists $g\in
P_{\lambda_{1}}(k)$ such that $T_{2}=gT_{1}g^{-1}$ and
$\lambda_{2}=g\cdot\lambda_{1}$}.$

######  Lemma 3.6.

With the notation as above, we have the following.

  * (i)

If $(T_{1},\lambda_{1})\sim(T_{2},\lambda_{2})$ then
$L_{\lambda_{1}}=L_{\lambda_{2}}$.

  * (ii)

If $(T_{1},\lambda_{1})\approx(T_{2},\lambda_{2})$ then
$P_{\lambda_{1}}=P_{\lambda_{2}}$.

  * (iii)

The relations $\sim$ and $\approx$ are equivalence relations.

###### Proof.

Parts (i) and (ii) are a consequence of Lemma 3.3(ii), and (iii) follows from
(i) and (ii). ∎

######  Definition 3.7.

We define

$Y_{G}({\mathbb{K}}):=\bigsqcup_{T}Y_{T}({\mathbb{K}})/\sim$

and

$V_{G}({\mathbb{K}}):=\bigsqcup_{T}Y_{T}({\mathbb{K}})/\approx,$

and we call $V_{G}({\mathbb{K}})$ the _vector edifice of $G$ (over
${\mathbb{K}}$)_. When we refer to “a vector edifice $V$” below, it is with
the understanding that $V=V_{G}({\mathbb{K}})$ for some $k$-group $G$ and for
${\mathbb{K}}={\mathbb{Q}}$ or ${\mathbb{R}}$.

We can regard $V_{G}({\mathbb{K}})$ as a quotient of $Y_{G}({\mathbb{K}})$,
since $\approx$ is coarser than $\sim$, and we denote the corresponding
projection by

$\phi_{G}\colon Y_{G}({\mathbb{K}})\to V_{G}({\mathbb{K}}).$

We write $\varpi_{G}\colon\bigsqcup_{T}Y_{T}({\mathbb{K}})\to
Y_{G}({\mathbb{K}})$ for the canonical projection and we define

$\omega_{G}:=\phi_{G}\circ\varpi_{G}\colon\bigsqcup_{T}Y_{T}({\mathbb{K}})\to
V_{G}({\mathbb{K}}).$

Henceforth, we often abuse notation and denote elements of
$Y_{G}({\mathbb{K}})$ simply by $\lambda$. Note that we may regard $Y_{G}$ as
a subset of $Y_{G}({\mathbb{K}})$: for given $\lambda\in Y_{G}$, the
equivalence relation $\sim$ does the job of identifying all pairs
$(T,\lambda)$ where $T$ is a maximal split torus of $G$ containing the image
of $\lambda$, because all such tori are conjugate by elements of
$L_{\lambda}(k)$. More generally, if $S$ is a subtorus of two maximal split
tori $T_{1}$ and $T_{2}$, then $\sim$ formally matches up the “common
subspace” $Y_{S}({\mathbb{K}})$ inside $Y_{T_{1}}({\mathbb{K}})$ and
$Y_{T_{2}}({\mathbb{K}})$.

######  Remark 3.8.

We observe that Lemma 2.2 holds not only for elements of $Y_{G}$ but also for
elements of $Y_{G}({\mathbb{K}})$: If $\lambda\in Y_{G}({\mathbb{K}})$ and
$\mu\in Y_{P_{\lambda}}({\mathbb{K}})$ then there exist a maximal split torus
$T$ of $G$ and $u\in U_{\lambda}(k)$ such that $\lambda,u\cdot\mu$ belong to
$Y_{T}({\mathbb{K}})$. To see this, repeat the proof of Lemma 2.2 with
$Y_{T_{0}}$, $Y_{T_{1}}$, and $Y_{T}$ replaced by $Y_{T_{0}}({\mathbb{K}})$,
$Y_{T_{1}}({\mathbb{K}})$, and $Y_{T}({\mathbb{K}})$.

######  Remark 3.9.

Lemmas 2.3, 2.4 and 2.5 and Corollary 2.6 extend to $Y_{G}({\mathbb{K}})$:
just apply each original result to suitable cocharacter approximations.

######  Remarks 3.10.

(i). One checks easily that the action of $G(k)$ on
$\bigsqcup_{T}Y_{T}({\mathbb{K}})$ descends to give actions of $G(k)$ on
$Y_{G}({\mathbb{K}})$ and $V_{G}({\mathbb{K}})$. We sketch the argument for
$V_{G}({\mathbb{K}})$. Suppose we have a pair $(T,\lambda)$ consisting of a
maximal split torus $T$ and $\lambda\in Y_{T}({\mathbb{K}})$, and elements
$p\in P_{\lambda}(k)$ and $g\in G(k)$. We wish to show that

$(gTg^{-1},g\cdot\lambda)\approx(g(pTp^{-1})g^{-1},g\cdot(p\cdot\lambda))=(gpT(gp)^{-1},(gp)\cdot\lambda).$

Since both $gTg^{-1}$ and $gpT(gp)^{-1}$ are split tori of
$gP_{\lambda}g^{-1}=P_{g\cdot\lambda}$, there is a $p^{\prime}\in
P_{g\cdot\lambda}(k)$ such that $p^{\prime}$ conjugates $gTg^{-1}$ to
$gpT(gp)^{-1}$. Writing $p^{\prime}=gqg^{-1}$ for some $q\in P_{\lambda}(k)$,
we see that $q^{-1}p$ belongs to $N_{P_{\lambda}(k)}(T)$, and this group
coincides with $N_{L_{\lambda}(k)}(T)$, by the extension of Lemma 2.3 to
$Y_{G}({\mathbb{K}})$ (Remark 3.9). Consequently, $q^{-1}p$ fixes $\lambda$,
so

$p^{\prime}\cdot(g\cdot\lambda)=(gq)\cdot\lambda=(gp)\cdot\lambda,$

and we are done.

(ii). In a similar vein, if $(T,\lambda)\approx(T,\mu)$ then $\lambda=\mu$.
For suppose $g\in P_{\lambda}(k)$ is such that $gTg^{-1}=T$ and
$g\cdot\lambda=\mu$. Then, by appealing to the extension of Lemma 2.3 to
$Y_{G}({\mathbb{K}})$ as in (i), we see that $g\in L_{\lambda}(k)$, and hence
$\lambda=\mu$.

######  Definition 3.11.

Given $x\in V_{G}({\mathbb{K}})$, we set $P_{x}:=P_{\lambda}$ for any
$\lambda\in Y_{G}({\mathbb{K}})$ such that $\phi_{G}(\lambda)=x$. Since the
relation $\approx$ only identifies pairs $(T_{1},\lambda_{1})$ and
$(T_{2},\lambda_{2})$ for which $P_{\lambda_{1}}=P_{\lambda_{2}}$ (Lemma
3.6(ii)), this is well-defined.

######  Remark 3.12.

For now we regard $V_{G}({\mathbb{K}})$ just as a set. In Section 6 we
topologise it by endowing it with a metric.

######  Example 3.13.

The definition of $P_{\lambda}$, $L_{\lambda}$ and $U_{\lambda}$ for
$\lambda\in Y_{G}({\mathbb{K}})$ agree with those of [7, Sec. 2.2] when $G$ is
reductive (where the definitions are given in terms of pairings with roots).
To see this, we can assume by Remark 3.4(iii) that $k$ is separably closed and
choose a maximal torus $T$ of $G$. Choose an equivariant embedding $i$ of $G$
in a rational $G$-module $V$. Let $\lambda\in Y_{T}({\mathbb{K}})$ and let
$\lambda^{\prime}\in Y_{T}$ be a cocharacter approximation to $\lambda$. The
derivative of $i$ gives a $G$-equivariant embedding of $\mathfrak{g}$ in
$T_{0}(V)$, and $T_{0}(V)$ is isomorphic to $V$ as a $G$-module. Hence all the
roots of $G$ with respect to $T$ appear in the set of weights of $T$ on $V$.
It follows that
$\operatorname{sgn}\langle\lambda^{\prime},\alpha\rangle=\operatorname{sgn}\langle\lambda,\alpha\rangle$
for every root $\alpha$. Now $P_{\lambda}$ is equal to $P_{\lambda^{\prime}}$,
which is generated by $T$ together with the root groups $U_{\alpha}$ for the
roots $\alpha$ such that $\langle\lambda^{\prime},\alpha\rangle\geq 0$. This
shows that $P_{\lambda}$ coincides with the subgroup defined in [7, Sec. 2.2],
since $\langle\lambda,\alpha\rangle\geq 0$ if and only if
$\langle\lambda^{\prime},\alpha\rangle\geq 0$ for each root $\alpha$. The
argument for $L_{\lambda}$ and $U_{\lambda}$ is similar: here
$L_{\lambda^{\prime}}$ is generated by $T$ together with the root groups
$U_{\alpha}$ for the roots $\alpha$ such that
$\langle\lambda^{\prime},\alpha\rangle=0$, and $U_{\lambda^{\prime}}$ is
generated by $T$ together with the root groups $U_{\alpha}$ for the roots
$\alpha$ such that $\langle\lambda^{\prime},\alpha\rangle>0$.

######  Lemma 3.14.

Let $f\colon G\to H$ be a homomorphism of connected $k$-groups. Let $T$ be a
maximal split torus of $G$ and let $S$ be a maximal split torus of $H$ such
that $f(T)\subseteq S$. Let $\lambda\in Y_{T}({\mathbb{K}})$. Then there exist
a $G$-equivariant embedding $\nu\colon G\to V$, an $H$-equivariant embedding
$\psi\colon H\to W$, and $\lambda^{\prime}\in Y_{T}$ such that
$\lambda^{\prime}$ is a cocharacter approximation to $\lambda$ with respect to
$V$ and $f_{*}(\lambda^{\prime})$ is a cocharacter approximation to
$f_{*}(\lambda)$ with respect to $W$.

###### Proof.

Pick a $G$-equivariant embedding $\phi\colon G\to V_{1}$ for some rational
$G$-module $V_{1}$, and an $H$-equivariant embedding $\psi\colon H\to W$ for a
rational $H$-module $W$. Define $\nu\colon G\to V:=V_{1}\oplus W$ by
$\nu(g)=(\phi(g),\psi(f(g)))$ for $g\in G(A)$ for any $k$-algebra $A$. Then
$\nu$ is a $G$-equivariant embedding of $G$ in $V$, and it follows from the
construction that if $\lambda^{\prime}$ is a cocharacter approximation to
$\lambda$ with respect to $V$ then $f_{*}(\lambda^{\prime})$ is a cocharacter
approximation to $f_{*}(\lambda)$ with respect to $W$. Here we used the fact
that $\langle\lambda,\chi\circ f\rangle=\langle f\circ\lambda,\chi\rangle$,
where $\chi$ is any weight of $S$ on $W$ and $\lambda\in Y_{T}$, and that this
formula extends to all $\lambda\in Y_{T}({\mathbb{K}})$ (replacing
$f\circ\lambda$ with $f_{*}(\lambda)$). ∎

######  Remark 3.15.

The proof of Lemma 3.14 actually gives a stronger conclusion: in the notation
of the lemma there exist equivariant embeddings such that for any $\lambda\in
Y_{T}({\mathbb{K}})$ and _any_ cocharacter approximation $\lambda^{\prime}$ to
$\lambda$ with respect to $V$, $f_{*}(\lambda^{\prime})$ is also a cocharacter
approximation to $f_{*}(\lambda)$ with respect to $W$. It follows that if
$(\lambda_{n})$ is a rational approximating sequence in $Y_{T}({\mathbb{K}})$
to $\lambda$ then $(f(\lambda_{n}))$ is a rational approximating sequence in
$Y_{S}({\mathbb{K}})$ to $f_{*}(\lambda)$.

######  Corollary 3.16.

Suppose $G$ is a subgroup of a connected $k$-group $H$. Then for any
$\lambda\in Y_{G}({\mathbb{K}})$ we have $P_{\lambda}=P_{\lambda}(H)\cap G$,
$L_{\lambda}=L_{\lambda}(H)\cap G$ and $U_{\lambda}=U_{\lambda}(H)\cap G$.

###### Proof.

Let $i\colon G\to H$ be the inclusion. By Lemma 3.14, there exists
$\lambda^{\prime}\in Y_{G}$ such that we have the following equalities:
$P_{\lambda}=P_{\lambda^{\prime}}$; $L_{\lambda}=L_{\lambda^{\prime}}$;
$U_{\lambda}=U_{\lambda^{\prime}}$;
$P_{i^{*}(\lambda)}(H)=P_{i^{*}(\lambda^{\prime})}(H)$;
$L_{i^{*}(\lambda)}(H)=L_{i^{*}(\lambda^{\prime})}(H)$; and
$U_{i^{*}(\lambda)}(H)=U_{i^{*}(\lambda^{\prime})}(H)$. But
$P_{i^{*}(\lambda^{\prime})}(H)=P_{\lambda^{\prime}}(H)$, because
$i^{*}(\lambda^{\prime})$ is just $\lambda^{\prime}$ viewed as an element of
$Y_{H}$ via the inclusion $Y_{G}\subseteq Y_{H}$, and similarly
$L_{i^{*}(\lambda^{\prime})}(H)=L_{\lambda^{\prime}}(H)$ and
$U_{i^{*}(\lambda^{\prime})}(H)=U_{\lambda^{\prime}}(H)$. The result follows
since $P_{\lambda^{\prime}}=P_{\lambda^{\prime}}(H)\cap G$,
$L_{\lambda^{\prime}}=L_{\lambda^{\prime}}(H)\cap G$ and
$U_{\lambda^{\prime}}=U_{\lambda^{\prime}}(H)\cap G$. ∎

######  Lemma 3.17.

Let $f\colon G\to H$ be a homomorphism of $k$-groups, let $T$ be a maximal
split torus of $G$ and let $\lambda,\mu\in Y_{T}({\mathbb{K}})$. Let $S$ be a
maximal split torus of $H$ such that $f(T)\subseteq S$.

  * (i)

We have $f(P_{\lambda})\subseteq P_{f_{*}(\lambda)}$ and
$f(L_{\lambda})\subseteq L_{f_{*}(\lambda)}$, with equality if $f$ is
surjective.

  * (ii)

If $f$ has finite kernel, then $P_{f_{*}(\lambda)}=P_{f_{*}(\mu)}$ implies
$P_{\lambda}=P_{\mu}$, and similarly $L_{f_{*}(\lambda)}=L_{f_{*}(\mu)}$
implies $L_{\lambda}=L_{\mu}$. Hence, if $f$ is surjective and has finite
kernel, then for $\lambda,\mu\in Y_{T}({\mathbb{K}})$ we have
$P_{\lambda}=P_{\mu}$ if and only if $P_{f_{*}(\lambda)}=P_{f_{*}(\mu)}$, and
$L_{\lambda}=L_{\mu}$ if and only if $L_{f_{*}(\lambda)}=L_{f_{*}(\mu)}$.

###### Proof.

(i). For any $\lambda^{\prime}\in Y_{T}$, it is immediate from the definitions
that $f(P_{\lambda^{\prime}})\subseteq P_{f_{*}(\lambda^{\prime})}$ and
$f(L_{\lambda^{\prime}})\subseteq L_{f_{*}(\lambda^{\prime})}$, and we have
equality by [14, Cor. 2.1.9] if $f$ is surjective (note that the hypotheses of
_loc. cit._ hold because a surjective map of smooth affine group schemes is
faithfully flat by [22, Prop. 1.70]). By Lemma 3.14 there exists
$\lambda^{\prime}\in Y_{T}$ such that $\lambda^{\prime}$ is a cocharacter
approximation to $\lambda$ and $f_{*}(\lambda^{\prime})$ is a cocharacter
approximation to $f_{*}(\lambda)$. So
$f(P_{\lambda})=f(P_{\lambda^{\prime}})\subseteq
P_{f_{*}(\lambda^{\prime})}=P_{f_{*}(\lambda)}$, with equality if $f$ is
surjective, and similarly for $L_{\lambda}$.

(ii). As in part (i), it is enough by Lemma 3.14 to prove the result when
$\lambda\in Y_{G}$. Let $P=P_{f_{*}(\lambda)}$. Since $f$ has finite kernel,
the preimage $f^{-1}(P)$ is a finite extension of the smooth connected group
$P_{\lambda}$. Hence, we may conclude that
$((f_{\bar{k}})^{-1}(P_{\bar{k}}))_{\operatorname{red}}$ and
$(P_{\lambda})_{\bar{k}}$ have the same identity component, which is
$(P_{\lambda})_{\bar{k}}$ itself. Setting $Q=P_{f_{*}(\mu)}$, we deduce that

$\displaystyle P=Q$ $\displaystyle\implies
f^{-1}(P)=f^{-1}(Q)\implies((f_{\bar{k}})^{-1}(P_{\bar{k}}))_{\operatorname{red}}=((f_{\bar{k}})^{-1}(Q_{\bar{k}}))_{\operatorname{red}}$
$\displaystyle\implies(P_{\lambda})_{\bar{k}}=(P_{\mu})_{\bar{k}}\implies
P_{\lambda}=P_{\mu}.$

A similar argument works for $L_{\lambda}$ and $L_{\mu}$. The final assertion
now follows from part (i). ∎

###  3.4. Apartments

It follows from Remark 3.10 that the restriction of $\phi_{G}$ gives rise to a
bijection from $Y_{T}({\mathbb{K}})$ onto its image in $V_{G}({\mathbb{K}})$;
we denote this image by $V_{T}({\mathbb{K}})$ and we call this an _apartment_
of $V_{G}({\mathbb{K}})$. The set $V_{T}({\mathbb{K}})$ inherits the structure
of a ${\mathbb{K}}$-vector space from $Y_{T}({\mathbb{K}})$. We denote the
common zero of all the apartments by $0$.

######  Remark 3.18.

Our new notation has introduced a potential inconsistency — given a split
torus $T$ in $G$, we have two objects labelled $V_{T}({\mathbb{K}})$: the
apartment of $V_{G}({\mathbb{K}})$ corresponding to $T$, and the vector
edifice of $T$ as a $k$-group in its own right. It is not hard to see that
these two objects are essentially the same, but we leave a proper discussion
of this until we have introduced linear maps below; see Remark 4.12.

######  Lemma 3.19.

The map $T\mapsto V_{T}({\mathbb{K}})$ gives a bijection between the set of
maximal split tori in $G$ and the set of apartments in $V_{G}({\mathbb{K}})$.

###### Proof.

The given map is surjective by definition, so we just need to show that it is
injective. Suppose $T_{1}$ and $T_{2}$ are maximal split tori in $G$, and
suppose that $V_{T_{1}}({\mathbb{K}})=V_{T_{2}}({\mathbb{K}})$. Then for all
$\lambda\in Y_{T_{1}}$ there exists $\mu\in Y_{T_{2}}$ such that
$(T_{1},\lambda)\approx(T_{2},\mu)$. This means that $T_{2}\subseteq
P_{\lambda}$ for every $\lambda\in Y_{T_{1}}$, and hence $T_{2}\subseteq
P_{\lambda}\cap P_{-\lambda}=L_{\lambda}$ for every $\lambda\in Y_{T_{1}}$.
Hence $T_{2}$ commutes with every cocharacter of $T_{1}$, which implies that
$T_{2}$ commutes with $T_{1}$, and we conclude that $T_{2}=T_{1}$, by
maximality of $T_{1}$ and $T_{2}$. ∎

###  3.5. Common apartments, addition, opposites

As detailed in the introduction, the structures we have just defined are
generalisations to arbitrary $k$-groups of constructions which already appear
in the literature for reductive groups, where they give rise to (vector and
spherical) buildings [16], [24, Ch. 2, Sec. 2], [7, Sec. 2.4]. However, we
cannot import all of the structure/properties we see in the reductive case to
the case of arbitrary $G$. One fundamental problem is that two points of
$V_{G}({\mathbb{K}})$ need not lie in a common apartment, as Example 3.22
below shows.222For another example of building-like structures where the same
phenomenon occurs, see the theory of hovels [29], [18]. Before giving that
example, we give a characterisation of when two points _do_ lie in a common
apartment.

######  Lemma 3.20.

Given $x,y\in V_{G}({\mathbb{K}})$, the apartments containing both $x$ and $y$
are precisely those of the form $V_{T}({\mathbb{K}})$ where $T$ is a maximal
split torus of $G$ contained in $P_{x}\cap P_{y}$. In particular, $x$ and $y$
are contained in a common apartment if and only if $P_{x}\cap P_{y}$ contains
a maximal split torus of $G$.

###### Proof.

If $x$ and $y$ are contained in $V_{T}({\mathbb{K}})$ for some maximal split
torus $T$, then we may write $x=\phi_{G}(\lambda)$, $y=\phi_{G}(\mu)$ for
$\lambda,\mu\in Y_{T}({\mathbb{K}})$. But then $T\subseteq P_{\lambda}\cap
P_{\mu}=P_{x}\cap P_{y}$ by Remark 3.4(iii). Conversely, let $T$ be a maximal
split torus of $G$ contained in $P_{x}\cap P_{y}$. Choose $\lambda,\mu\in
Y_{G}({\mathbb{K}})$ such that $x=\phi_{G}(\lambda)$ and $y=\phi_{G}(\mu)$. By
Lemma 2.4(i) and Remark 3.9, there exist $u\in U_{\lambda}(k)$ and $v\in
U_{\mu}(k)$ such that $u\cdot\lambda,v\cdot\mu\in Y_{T}({\mathbb{K}})$. Then
$x=\phi_{G}(u\cdot\lambda)\in V_{T}({\mathbb{K}})$ and
$y=\phi_{G}(v\cdot\mu)\in V_{T}({\mathbb{K}})$. ∎

######  Remark 3.21.

In particular, taking $x=y$ in Lemma 3.20 we see that for any $x\in
V_{G}({\mathbb{K}})$, $x$ belongs to $V_{T}({\mathbb{K}})$ if and only if $T$
is contained in $P_{x}$. This also shows that if $T^{\prime}$ is another
maximal split torus of $G$ such that $x\in V_{T^{\prime}}({\mathbb{K}})$, then
there is $g\in P_{x}(k)$ such that $T^{\prime}=gTg^{-1}$.

######  Example 3.22.

Let $B$ be a Borel subgroup of a reductive group $G$ with $G$ not a torus, and
write $B=P_{\lambda}$ for some $\lambda\in Y_{G}$. Then $\phi_{B}(-\lambda)$
and $\phi_{B}(-u\cdot\lambda)$ do not lie in a common apartment of
$V_{B}({\mathbb{K}})$ for any $1\neq u\in R_{u}(B)(k)$, because
$P_{-\lambda}(B)=T$ and $P_{-u\cdot\lambda}(B)=uTu^{-1}$ are distinct maximal
tori of $B$, and $V_{T}({\mathbb{K}})$ (resp., $V_{uTu^{-1}}({\mathbb{K}})$)
is the unique apartment containing $\phi_{B}(-\lambda)$ (resp.,
$\phi_{B}(-u\cdot\lambda)$), by Lemma 3.20.

The preceding discussion motivates the following definition.

######  Definition 3.23.

We say that $V_{G}({\mathbb{K}})$ has the _common apartment property_ if every
$x,y\in V_{G}({\mathbb{K}})$ lie in a common apartment of
$V_{G}({\mathbb{K}})$.

It follows from Lemma 3.20 that the common apartment property holds for $G$ if
and only if the intersection of any two R-parabolic subgroups contains a
maximal split torus of $G$ — this is the case for reductive groups, and more
generally for pseudo-reductive groups, where the R-parabolic subgroups are the
same thing as the _pseudo-parabolic subgroups_ , see [14, Def. 2.2.1, Ex.
2.2.2]. In fact, [14, Prop. 3.5.12(1)] shows that the intersection of two
pseudo-parabolic subgroups in any $k$-group $G$ contains a maximal split
torus, but working with pseudo-parabolic subgroups rather than R-parabolic
subgroups does not give the structure we need for applications below.
Returning to the example of the Borel subgroup $B$ in Example 3.22, the only
pseudo-parabolic subgroup of $B$ is $B$ itself; in contrast, we show below
(Example 4.14) that there is a naturally arising _bijection_ between the
vector edifice $V_{B}({\mathbb{K}})$ and the vector edifice
$V_{G}({\mathbb{K}})$.

An immediate consequence of this lack of common apartment is that there is no
obvious addition law on $V_{G}({\mathbb{K}})$ in general, and the notions of
convexity and opposition become more complicated. One can, however, define
addition when the two points one considers live in a common apartment.

######  Lemma 3.24.

  * (i)

Let $T$ be a maximal split torus of $G$. Suppose $x,y\in V_{G}({\mathbb{K}})$
are contained in the common apartment $V_{T}({\mathbb{K}})$. Since
$V_{T}({\mathbb{K}})$ is a vector space, there is an element $x+y\in
V_{T}({\mathbb{K}})$. The element $x+y$ is independent of the choice of
apartment containing $x$ and $y$.

  * (ii)

Let $x\in V_{G}({\mathbb{K}})$ and let $a\in{\mathbb{K}}^{+}$. The element
$ax$ is independent of the choice of apartment containing $x$.

###### Proof.

(i). The common apartments containing $x$ and $y$ correspond to the maximal
split tori of $G$ contained in the intersection $P_{x}\cap P_{y}$ (Lemma
3.20). By hypothesis, there is at least one such $T$. Let $T^{\prime}$ be
another. By Lemma 2.11, $P_{x}\cap P_{y}$ is smooth, so there exists $g\in
P_{x}(k)\cap P_{y}(k)$ such that $gTg^{-1}=T^{\prime}$. Choose $\lambda,\mu\in
Y_{T}({\mathbb{K}})$ such that $x=\phi_{G}(\lambda)$ and $y=\phi_{G}(\mu)$.
Then $x=\phi_{G}(g\cdot\lambda)$ and $y=\phi_{G}(g\cdot\mu)$. Fix a
$G$-equivariant embedding of $G$ in a rational $G$-module $V$, and let
$\lambda^{\prime}$ and $\mu^{\prime}$ be cocharacter approximations (with
respect to $V$) in $Y_{T}$ to $\lambda$ and $\mu$, respectively. Then
$\lambda^{\prime}+\mu^{\prime}\in Y_{T}$ is a cocharacter approximation to
$\lambda+\mu$, and so $P_{\lambda+\mu}=P_{\lambda^{\prime}+\mu^{\prime}}$.
Since $P_{x}=P_{\lambda}=P_{\lambda^{\prime}}$ and
$P_{y}=P_{\mu}=P_{\mu^{\prime}}$, we see that $\lim_{a\to
0}(\lambda^{\prime}+\mu^{\prime})(a)\cdot g$ exists, so $g\in
P_{\lambda^{\prime}+\mu^{\prime}}(k)=P_{\lambda+\mu}(k)$. Hence
$\phi_{G}(g\cdot\lambda+g\cdot\mu)=\phi_{G}(g\cdot(\lambda+\mu))=\phi_{G}(\lambda+\mu)$.
The result follows.

(ii). Let $T,T^{\prime}$ be maximal split tori of $G$ such that $x\in
V_{T}({\mathbb{K}})\cap V_{T^{\prime}}({\mathbb{K}})$. There exists $g\in
P_{x}$ such that $gTg^{-1}=T^{\prime}$. Pick $\lambda\in Y_{T}({\mathbb{K}})$
such that $x=\phi_{G}(\lambda)$. Then
$\phi_{G}(a(g\cdot\lambda))=\phi_{G}(g\cdot(a\lambda))=\phi_{G}(a\lambda)$
since $P_{a\lambda}=P_{\lambda}$. The result follows. ∎

######  Definition 3.25.

We say that $x$ and $y$ are _opposite_ in $V_{G}({\mathbb{K}})$ if there is an
apartment containing $x$ and $y$ and such that $x+y=0$ in that apartment. It
follows from Lemma 3.24 that if $x$ and $y$ are opposite, then $x+y=0$ in
every apartment containing $x$ and $y$.

######  Remark 3.26.

Note that part (ii) of Lemma 3.24 is false if $a=-1$ (or, more generally, if
$a<0$): for $x$ has many different opposites, one for each apartment
containing $x$, and each is of the form $-x$ inside the corresponding
apartment. See Remark 3.28.

Here is a criterion for points to be opposite in terms of cocharacters.

######  Lemma 3.27.

Suppose $x$ and $y$ are opposite points of $V_{G}({\mathbb{K}})$. Then $P_{x}$
and $P_{y}$ are opposite R-parabolic subgroups of $G$, and there is a unique
$\lambda\in Y_{G}({\mathbb{K}})$ such that $x=\phi_{G}(\lambda)$ and
$y=\phi_{G}(-\lambda)$.

###### Proof.

First choose an apartment $V_{T}({\mathbb{K}})$ such that $x,y\in
V_{T}({\mathbb{K}})$. Then since $\phi_{G}$ restricts to a linear isomorphism
$Y_{T}({\mathbb{K}})\to V_{T}({\mathbb{K}})$, we have a unique $\lambda\in
Y_{T}({\mathbb{K}})$ such that $x=\phi_{G}(\lambda)$ and
$y=\phi_{G}(-\lambda)$. But then $P_{x}=P_{\lambda}$ and $P_{y}=P_{-\lambda}$,
hence $P_{x}$ and $P_{y}$ are opposite. Any maximal split torus $T^{\prime}$
of $G$ such that $x,y\in V_{T^{\prime}}({\mathbb{K}})$ must lie in $P_{x}\cap
P_{y}=L_{\lambda}$ by Lemma 3.20. Since $L_{\lambda}$ is smooth, any two such
tori are conjugate by an element of $L_{\lambda}(k)$. But elements of
$L_{\lambda}(k)$ fix $\lambda$ and $-\lambda$, which shows that $\lambda$ is
indeed unique as an element of $Y_{G}({\mathbb{K}})$. ∎

######  Remark 3.28.

Note that for a given $\lambda\in Y_{G}({\mathbb{K}})$, there are in general
many opposite points to $\phi_{G}(\lambda)$ in $V_{G}({\mathbb{K}})$. To see
this, note that any of the cocharacters $\mu=u\cdot(-\lambda)$ has this
property, where $u\in U_{\lambda}(k)$ — the point here is that for such a $u$,
$\phi_{G}(u\cdot\lambda)=\phi_{G}(\lambda)$, but
$\phi_{G}(u\cdot(-\lambda))\neq\phi_{G}(-\lambda)$ in general.

######  Definition 3.29.

Let $x,y\in V_{G}({\mathbb{K}})$. If there is an apartment in
$V_{G}({\mathbb{K}})$ containing $x$ and $y$, then we let

$[x,y]:=\\{ax+(1-a)y\mid a\in[0,1]\cap{\mathbb{K}}\\}$

denote the _geodesic_ between $x$ and $y$ in $V_{G}({\mathbb{K}})$ — again,
Lemma 3.24 implies that this geodesic (when it exists at all) is independent
of the apartment we choose to draw it in. We say a subset $S\subseteq
V_{G}({\mathbb{K}})$ is _convex_ if for all $x,y\in S$ such that $x$ and $y$
lie in a common apartment, $[x,y]\subseteq S$.

The failure of the common apartment property makes it harder to metrise
$V_{G}({\mathbb{K}})$. We return to this in Section 6.

######  Definition 3.30.

We say that $x,y\in V_{G}({\mathbb{K}})$ _have the same type_ if $y=g\cdot x$
for some $g\in G(k)$.

######  Remark 3.31.

Suppose $x,y\in V_{G}({\mathbb{K}})$ have the same type and $x$ and $y$ belong
to a common apartment $V_{T}({\mathbb{K}})$. Choose $\lambda,\mu\in
Y_{T}({\mathbb{K}})$ such that $x=\phi_{G}(\lambda)$ and $y=\phi_{G}(\mu)$.
Since $x$ and $y$ have the same type, there exists $g\in G(k)$ such that
$\mu=g\cdot\lambda$. Then $\mu\in Y_{T^{\prime}}({\mathbb{K}})$, where
$T^{\prime}:=gTg^{-1}$. So $T,T^{\prime}$ are maximal split tori of $L_{\mu}$,
so they are $L_{\mu}(k)$-conjugate. Now $L_{\mu}(k)$ fixes $\mu$, so after
multiplying $g$ on the left by an element of $L_{\mu}(k)$, we can assume that
$g$ normalises $T$. This shows that $x$ and $y$ are $W_{k}$-conjugate, where
$W_{k}$ is the relative Weyl group. We conclude that for a fixed maximal split
torus $T$ of $G$ and fixed $x\in V_{T}({\mathbb{K}})$, the set $\\{z\in
V_{T}({\mathbb{K}})\mid\mbox{$x$ and $z$ have the same type}\\}$ is finite.

For example, suppose $G=\operatorname{SL}_{2}$. If $x,y\in
V_{G}({\mathbb{K}})$ are distinct elements having the same type then $x$ and
$y$ are opposite, because each apartment $V_{T}({\mathbb{K}})$ is
1-dimensional and the non-trivial element of the relative Weyl group $W_{k}$
acts by $z\mapsto-z$.

###  3.6. The spherical edifice

Recall from Lemma 3.24(ii) that we have a well-defined operation of scalar
multiplication by elements of ${\mathbb{K}}^{+}$ on the vector edifice
$V_{G}({\mathbb{K}})$, allowing the following definition.

######  Definition 3.32.

The _spherical edifice_ $\Delta_{G}({\mathbb{K}})$ is defined to be the set of
${\mathbb{K}}^{+}$-orbits on $V_{G}({\mathbb{K}})\setminus\\{0\\}$. We denote
the natural map from $V_{G}({\mathbb{K}})\setminus\\{0\\}$ to
$\Delta_{G}({\mathbb{K}})$ by

$\zeta_{G}:V_{G}({\mathbb{K}})\setminus\\{0\\}\to\Delta_{G}({\mathbb{K}}).$

######  Remark 3.33.

The obvious inclusion of $V_{G}({\mathbb{Q}})$ in $V_{G}({\mathbb{R}})$ gives
an inclusion of $\Delta_{G}({\mathbb{Q}})$ in $\Delta_{G}({\mathbb{R}})$.

######  Remark 3.34.

The notions of apartment, opposite points and convexity make sense for
$\Delta_{G}({\mathbb{K}})$ as well. We define an _apartment_ of
$\Delta_{G}({\mathbb{K}})$ to be a subset of the form
$\zeta_{G}(V_{T}({\mathbb{K}})\backslash\\{0\\})$ for some maximal split torus
$T$ of $G$. If $y_{1},y_{2}\in\Delta_{G}({\mathbb{K}})$ then we say that
$y_{1}$ and $y_{2}$ are _opposite_ if there exists a maximal split torus $T$
of $G$ and a point $x\in V_{T}({\mathbb{K}})$ such that $y_{1}=\zeta_{G}(x)$
and $y_{2}=\zeta_{G}(-x)$. If $y_{1}=\zeta_{G}(x_{1})$ and
$y_{2}=\zeta_{G}(x_{2})$ are not opposite then we define the _geodesic_
$[y_{1},y_{2}]$ in $\Delta_{G}({\mathbb{K}})$ to be
$\zeta_{G}([x_{1},x_{2}])$; this does not depend on the choice of $x_{1}$ and
$x_{2}$. We say a subset $\Sigma$ of $\Delta_{G}({\mathbb{K}})$ is _convex_ if
for all $y_{1},y_{2}\in\Sigma$ such that $[y_{1},y_{2}]$ exists,
$[y_{1},y_{2}]\subseteq\Sigma$.

###  3.7. The combinatorial edifice

######  Definition 3.35.

We define the _combinatorial edifice_ $\Delta_{G}$ to be the poset formed by
the R-parabolic subgroups of $G$ under reverse inclusion. Given $\lambda\in
Y_{G}({\mathbb{K}})$, $P=P_{\lambda}$ and $x=\phi_{G}(\lambda)\in
V_{G}({\mathbb{K}})$, we denote the corresponding element of $\Delta_{G}$ by
$\sigma_{P}$, $\sigma_{\lambda}$ or $\sigma_{x}$, and we write $\leq$ for the
partial order on $\Delta_{G}$: i.e., $\sigma_{Q}\leq\sigma_{P}$ if and only if
$P\subseteq Q$. We let $\varnothing$ denote the element in this poset
corresponding to $G$ itself.

The elements of the combinatorial edifice can be realised geometrically inside
$V_{G}({\mathbb{K}})$: given an R-parabolic subgroup $P$ of $G$, consider the
subset

$\widetilde{\sigma}_{P}=\\{x\in V_{G}({\mathbb{K}})\mid P_{x}=P\\}.$

(We also define
$\widetilde{\sigma}_{\lambda}:=\widetilde{\sigma}_{P_{\lambda}}$ and
$\widetilde{\sigma}_{x}:=\widetilde{\sigma}_{P_{x}}$.) By choosing a maximal
split torus $T$ of $P$, we may realise $\widetilde{\sigma}_{P}$ as a subset of
$V_{T}({\mathbb{K}})$ (Lemma 2.10), so it makes sense to take linear
combinations of elements in $\widetilde{\sigma}_{P}$.

######  Lemma 3.36.

Let $x,y\in\widetilde{\sigma}_{P}$. Then $ax+by\in\widetilde{\sigma}_{P}$ for
all $a,b\in{\mathbb{K}}^{+}$.

###### Proof.

Fix a maximal split torus $T$ of $P$. We can choose $\lambda,\mu\in
Y_{T}({\mathbb{K}})$ such that $x=\phi_{G}(\lambda)$ and $y=\phi_{G}(\mu)$;
then $P_{\lambda}=P_{\mu}=P$. Recall that $P_{\nu}=\rho^{-1}(V_{\nu,\geq 0})$
for any $\nu\in Y_{T}({\mathbb{K}})$ (Remark 3.4(iii)). It follows that
$P_{a\lambda}=P_{\lambda}$ and $P_{b\mu}=P_{\mu}$ for any $a,b>0$. So it is
enough to show that $P_{\lambda+\mu}=P$.

Fix a $G$-equivariant embedding $\rho^{\prime}$ of $G$ in a rational
$G$-module $V^{\prime}$. Let $V$ be the rational $G$-module
$V^{\prime}\oplus\mathfrak{g}$ and let $\rho$ be $\rho^{\prime}$ followed by
the obvious inclusion of $V^{\prime}$ in $V$. Since
$P_{\nu}=\rho^{-1}(V_{\nu,\geq 0})$ for any $\nu\in Y_{T}({\mathbb{K}})$, the
argument of the first part of the proof of Lemma 2.9 shows that
$P_{\lambda+\mu}\supseteq P$.

Let $\nu\in Y_{T}({\mathbb{K}})$ and let $\nu^{\prime}\in Y_{T}$ be a
cocharacter approximation to $\nu$. By our choice of $\rho$ we have
$\mathfrak{g}_{\nu,\geq 0}=\mathfrak{g}_{\nu^{\prime},\geq 0}$, and we deduce
that

$\operatorname{Lie}(P_{\nu})=\operatorname{Lie}(P_{\nu^{\prime}})=\mathfrak{g}_{\nu^{\prime},\geq
0}=\mathfrak{g}_{\nu,\geq 0},$

where the middle equality is from [14, Prop. 2.1.8]. The argument of the
second part of the proof of Lemma 2.9 shows that
$\operatorname{Lie}(P_{\lambda+\mu})=\operatorname{Lie}(P)$, and we deduce
that $P_{\lambda+\mu}=P$, as required. ∎

It is well known that if $G$ is semisimple then $\Delta_{G}$ is a spherical
building; in particular, it is a simplicial complex. The same is true for
reductive $G$ because the parabolic subgroups of $G$ are in bijective
correspondence with the parabolic subgroups of $[G,G]$. The following example
shows that the combinatorial edifice need not be a simplicial complex for
arbitrary $G$, and the partial order on $\Delta_{G}$ need not be realised
geometrically by relationships between closures of the corresponding subsets
in $V_{G}({\mathbb{K}})$.

######  Example 3.37.

Let $k$ be algebraically closed, and set $G=\operatorname{GL}_{2}\ltimes V$,
where $V$ is the natural two-dimensional module for $\operatorname{GL}_{2}$.
Let $e_{1}$ and $e_{2}$ be the usual standard basis vectors in $V$, and let
$V_{1}$ and $V_{2}$ be the corresponding one-dimensional subspaces of $V$. Let
$B^{+}$ denote the upper triangular Borel subgroup in $\operatorname{GL}_{2}$
and $B^{-}$ the lower triangular Borel subgroup. Let $T$ be the diagonal
maximal torus in $\operatorname{GL}_{2}$. Then an element of
$Y_{T}({\mathbb{K}})$ can be identified with a pair
$(a,b)\in{\mathbb{K}}^{2}$, and the corresponding R-parabolic subgroup has the
form $P\ltimes W$ for some parabolic subgroup $P$ of $\operatorname{GL}_{2}$
and some subspace $W$ of $V$ as in the following table:

Conditions on $(a,b)$ | $P\ltimes W$ | Conditions on $(a,b)$ | $P\ltimes W$
---|---|---|---
$a=b\geq 0$ | $G$ | $0>a=b$ | $\operatorname{GL}_{2}$
$a>b\geq 0$ | $B^{+}\ltimes V$ | $0>b>a$ | $B^{-}$
$a\geq 0>b$ | $B^{+}\ltimes V_{1}$ | $b\geq 0>a$ | $B^{-}\ltimes V_{2}$
$0>a>b$ | $B^{+}$ | $b>a\geq 0$ | $B^{-}\ltimes V$

In an (abstract) simplicial complex, we can recognise any element of the
complex by the vertices (minimal non-empty elements) it contains. In this
case, the minimal elements in the poset $\Delta_{G}$ correspond to the maximal
proper R-parabolic subgroups: thus, here we get (the conjugates in $G$ of)
$\operatorname{GL}_{2}$, $B^{+}\ltimes V$ and $B^{-}\ltimes V$. Consider
$\sigma=\sigma_{B^{+}\ltimes V_{1}}\in\Delta_{G}$. Any
$\varnothing\neq\tau\in\Delta_{G}$ with $\tau\lneq\sigma$ corresponds to an
R-parabolic subgroup $P$ of $G$ with $P\supsetneq B^{+}\ltimes V_{1}$. Such a
$P$ must contain $T$, so $P=B^{+}\ltimes V$ is the only option. Thus
$\tau=\sigma_{B^{+}\ltimes V}$ and $\sigma$ contain the same minimal elements,
and $\Delta_{G}$ is not a simplicial complex.

We leave it as an exercise for the reader to sketch the regions of the plane
${\mathbb{R}}^{2}$ that correspond to each R-parabolic subgroup and show that
(for example) the region $a\geq 0>b$ is not in the closure of the region
$a>b\geq 0$, even though we have a containment $B^{+}\ltimes V_{1}\subseteq
B^{+}\ltimes V$.

##  4\. Linear maps of vector edifices

In this section we define the notion of a linear map between vector edifices,
and show how homomorphisms between $k$-groups give rise to such maps. For some
of the later work (especially in Section 5) it is important to consider maps
between edifices for algebraic groups defined over possibly different fields,
so that is how our definitions are framed in Section 4.1.

###  4.1. Definition and first properties

######  Definition 4.1.

Suppose $V_{1}=V_{G}({\mathbb{K}})$ and $V_{2}=V_{H}({\mathbb{K}})$ are two
vector edifices, where $G$ is a $k$-group and $H$ is a $k^{\prime}$-group for
two (possibly different) fields $k$ and $k^{\prime}$. A function $\kappa\colon
V_{1}\to V_{2}$ is called a _linear map of vector edifices_ if for every
apartment $A_{1}$ of $V_{1}$, there exists an apartment $A_{2}$ of $V_{2}$
such that $\kappa(A_{1})\subseteq A_{2}$ and $\kappa|_{A_{1}}\colon A_{1}\to
A_{2}$ is a ${\mathbb{K}}$-linear map.

It is immediate that the composition of linear maps is a linear map.

######  Remark 4.2.

The reader is warned that the inverse of a bijective linear map of vector
edifices is _not_ necessarily itself a linear map. The reason for this is that
even if $V_{1}$ and $V_{2}$ are in bijection with each other, it might not be
the case that their systems of apartments are in bijection with each other.
See Example 4.14 below.

######  Definition 4.3.

Suppose $V_{1}=V_{G}({\mathbb{K}})$ and $V_{2}=V_{H}({\mathbb{K}})$ are two
vector edifices as above. A linear map $\kappa\colon V_{1}\to V_{2}$ is an
_isomorphism of vector edifices_ if it is bijective and the inverse map is
also a linear map of vector edifices. If $V_{1}=V_{2}$ then we call $\kappa$
an _automorphism of vector edifices_. We write $\operatorname{Aut}(V)$ for the
group of automorphisms of a vector edifice $V$.

In practice, we often wish to construct linear maps of vector edifices from
corresponding maps on cocharacters. The following result shows that this
process can be reversed.

######  Lemma 4.4.

Let $G$ be a $k$-group and $H$ a $k^{\prime}$-group for two fields $k$ and
$k^{\prime}$. Let $\kappa\colon V_{G}({\mathbb{K}})\to V_{H}({\mathbb{K}})$ be
a linear map. Then $\kappa$ lifts uniquely to a map

$\widehat{\kappa}\colon Y_{G}({\mathbb{K}})\to Y_{H}({\mathbb{K}})$

with the following property: for any maximal split torus $T$ of $G$ and for
any maximal split torus $S$ of $H$ such that
$\kappa(V_{T}({\mathbb{K}}))\subseteq V_{S}({\mathbb{K}})$, we have
$\widehat{\kappa}(Y_{T}({\mathbb{K}}))\subseteq Y_{S}({\mathbb{K}})$, and the
following diagram commutes:

$\textstyle{Y_{T}({\mathbb{K}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{G}}$$\scriptstyle{\widehat{\kappa}}$$\textstyle{Y_{S}({\mathbb{K}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{H}}$$\textstyle{V_{T}({\mathbb{K}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{V_{S}({\mathbb{K}})}$

###### Proof.

First off, it follows from Remark 3.10(ii) that $\widehat{\kappa}$ is unique
if it exists.

Let $\lambda\in Y_{G}({\mathbb{K}})$, and set $x=\phi_{G}(\lambda)$ and
$y=\phi_{G}(-\lambda)$. Since $\kappa$ is linear on apartments,
$\kappa(x)+\kappa(y)=\kappa(x+y)=\kappa(0)=0$, so $\kappa(x)$ and $\kappa(y)$
are opposite points in (the corresponding apartment of) $V_{H}({\mathbb{K}})$.
By Lemma 3.27 we can find a unique $\mu\in Y_{H}({\mathbb{K}})$ such that
$\phi_{H}(\mu)=\kappa(x)$, $\phi_{H}(-\mu)=\kappa(y)$. We set
$\widehat{\kappa}(\lambda):=\mu$.

Now let $T$ be a maximal split torus of $G$ such that $\lambda\in
Y_{T}({\mathbb{K}})$ and let $S$ be any maximal split torus of $H$ such that
$\kappa(V_{T}({\mathbb{K}}))\subseteq V_{S}({\mathbb{K}})$. Set
$x=\phi_{G}(\lambda)$ and $y=\phi_{G}(-\lambda)$. Since
$\kappa(x),\kappa(y)\in V_{S}({\mathbb{K}})$ by construction and
$\kappa(x)+\kappa(y)=0$, there exists a unique $\nu\in Y_{S}({\mathbb{K}})$
such that $\phi_{H}(\nu)=\kappa(x)$ and $\phi_{H}(-\nu)=\kappa(y)$. We see
that $\widehat{\kappa}(\lambda)=\mu=\nu\in Y_{S}({\mathbb{K}})$, so that
$\widehat{\kappa}(Y_{T}({\mathbb{K}}))\subseteq Y_{S}({\mathbb{K}})$, and the
commutativity of the given diagram follows. ∎

######  Remark 4.5.

It is immediate from the construction that $\widehat{\kappa}$ restricts to a
linear map from each $Y_{T}({\mathbb{K}})$ to the corresponding
$Y_{S}({\mathbb{K}})$. We have obvious functoriality: $\widehat{\rm id}={\rm
id}$ and $\widehat{\tau\circ\kappa}=\widehat{\tau}\circ\widehat{\kappa}$.

###  4.2. Linear maps induced by group homomorphisms

Let $f\colon G\to H$ be a homomorphism of connected $k$-groups. We show that
$f$ gives rise to a linear map $\kappa_{f}\colon V_{G}({\mathbb{K}})\to
V_{H}({\mathbb{K}})$.

######  Lemma 4.6.

Let $T$ be a maximal split torus of $G$ and let $S_{1}$ and $S_{2}$ be maximal
split tori of $H$ such that $f(T)\subseteq S_{1}\cap S_{2}$. Let $f_{1}\colon
T\to S_{1}$ and $f_{2}\colon T\to S_{2}$ be the maps induced by $f$. Let
$\lambda\in Y_{T}({\mathbb{K}})$ and set $\lambda_{1}=(f_{1})_{*}(\lambda)$,
$\lambda_{2}=(f_{2})_{*}(\lambda)$. Then
$(S_{1},\lambda_{1})\sim(S_{2},\lambda_{2})$.

###### Proof.

By Remark 3.4(ii), $S_{1}$ and $S_{2}$ are maximal split tori of
$L_{\lambda_{1}}$, and hence $lS_{1}l^{-1}=S_{2}$ for some $l\in
L_{\lambda_{1}}(k)$. The conjugation map $\operatorname{Inn}_{l}$ gives rise
to a linear map $\omega$ from $Y_{S_{1}}({\mathbb{K}})$ to
$Y_{S_{2}}({\mathbb{K}})$, and we have $\omega\circ(f_{1})_{*}=(f_{2})_{*}$ by
functoriality (Section 3.1). The result follows. ∎

For each maximal split torus $T$ of $G$, choose a maximal split torus $S_{T}$
of $H$ such that $f(T)\subseteq S_{T}$. The maps $f_{*}\colon
Y_{T}({\mathbb{K}})\to Y_{S_{T}}({\mathbb{K}})$ for each $T$ give rise to a
map from $\bigsqcup_{T}Y_{T}({\mathbb{K}})$ to
$\bigsqcup_{S}Y_{S}({\mathbb{K}})$. Consider the compositions
$\bigsqcup_{T}Y_{T}({\mathbb{K}})\to\bigsqcup_{S}Y_{S}({\mathbb{K}})\stackrel{{\scriptstyle\varpi_{H}}}{{\to}}Y_{H}({\mathbb{K}})$
and
$\bigsqcup_{T}V_{T}({\mathbb{K}})\to\bigsqcup_{S}V_{S}({\mathbb{K}})\stackrel{{\scriptstyle\omega_{H}}}{{\to}}V_{H}({\mathbb{K}})$.
By Lemma 4.6, these maps do not depend on the choices of the maximal split
tori $S_{T}$ for each $T$.

######  Proposition 4.7.

The maps above descend to give well-defined maps $Y_{G}({\mathbb{K}})\to
Y_{H}({\mathbb{K}})$ and

$\kappa_{f}\colon V_{G}({\mathbb{K}})\to V_{H}({\mathbb{K}}).$

Moreover, $\kappa_{f}$ is a linear map of vector edifices.

###### Proof.

We give the proof for $\kappa_{f}$. Suppose
$(T_{1},\lambda_{1})\approx(T_{2},\lambda_{2})$ for
$(T_{1},\lambda_{1}),(T_{2},\lambda_{2})\in\bigsqcup_{T}Y_{T}({\mathbb{K}})$.
Then $T_{2}=gT_{1}g^{-1}$ and $g\cdot\lambda_{1}=\lambda_{2}$ for some $g\in
P_{\lambda_{1}}(k)$. Let $f_{i}\colon T_{i}\to S_{T_{i}}$ be the map induced
by $f$. It is enough to prove that
$(S_{T_{1}},(f_{1})_{*}(\lambda_{1}))\approx(S_{T_{2}},(f_{2})_{*}(\lambda_{2}))$.
By Lemma 4.6, there is no harm in taking $S_{T_{2}}$ to be
$f(g)S_{T_{1}}f(g)^{-1}$. Observe that this means that
$f_{2}\circ\operatorname{Inn}_{g}=\operatorname{Inn}_{f(g)}\circ f_{1}$. It
follows from functoriality that

$f(g)\cdot(f_{1})_{*}(\lambda_{1})=(f_{2})_{*}(g\cdot\lambda_{1})=(f_{2})_{*}(\lambda_{2}).$

Since $f(g)$ belongs to $P_{(f_{1})_{*}(\lambda_{1})(H)}(k)$ by Lemma 3.17(i),
we obtain the desired result. It is also clear that $\kappa_{f}$ is linear —
for each maximal split torus $T$ of $G$, the corresponding map $f_{*}\colon
Y_{T}({\mathbb{K}})\to Y_{S_{T}}({\mathbb{K}})$ is ${\mathbb{K}}$-linear.

The proof for the map $Y_{G}({\mathbb{K}})\to Y_{H}({\mathbb{K}})$ is very
similar: it just uses a conjugating element $l\in L_{\lambda_{1}}(k)$ instead.
∎

######  Remarks 4.8.

(i). If $\lambda\in Y_{G}$ then $\widehat{\kappa_{f}}(\lambda)=f\circ\lambda$.
It is clear that $\kappa_{{\rm id}_{G}}={\rm id}_{V_{G}({\mathbb{K}})}$ and if
$f\colon G\to H$ and $l\colon H\to K$ are homomorphisms of connected
$k$-groups then $\kappa_{l\circ f}=\kappa_{l}\circ\kappa_{f}$. In particular,
if $f\in\operatorname{Aut}(G)$ then $\kappa_{f}$ belongs to
$\operatorname{Aut}(V_{G}({\mathbb{K}}))$ and has inverse $\kappa_{f^{-1}}$.

(ii). Let $f\colon G\to H$ be a surjective homomorphism of connected
$k$-groups and let $x\in V_{G}({\mathbb{K}})$. It follows from Lemma 3.17(i)
that $P_{\kappa_{f}(x)}=f(P_{x})$.

######  Example 4.9.

Let $g\in G(k)$. Untangling the definitions, we see that
$\kappa_{\operatorname{Inn}_{g}}(x)$ is just $g\cdot x$ in the sense of
Section 3.3. By Remark 4.8(i), $\kappa_{\operatorname{Inn}_{g}}$ is invertible
with inverse $\kappa_{\operatorname{Inn}_{g^{-1}}}$. Remark 4.8(ii) implies
that $P_{g\cdot
x}=P_{\kappa_{\operatorname{Inn}_{g}}(x)}=\operatorname{Inn}_{g}(P_{x})=gP_{x}g^{-1}$.

######  Remark 4.10.

Let $f\colon G\to H$ be a homomorphism. Then $\kappa_{f}$ is equivariant in
the following sense: if $x\in V_{G}({\mathbb{K}})$ and $g\in G(k)$ then
$\kappa_{f}(g\cdot x)=f(g)\cdot\kappa_{f}(x)$. To see this, note that if
$\lambda\in Y_{G}$ and $g\in G(k)$ then
$f\circ(g\cdot\lambda)=f(g)\cdot(f\circ\lambda)$; the result follows from
linearity. This shows that $\kappa_{f}$ is type-preserving (see Definition
3.30).

###  4.3. Homomorphisms which induce injective linear maps

We can now show that inclusions of $k$-groups give inclusions of vector
edifices. In fact, we can do better.

######  Proposition 4.11.

Let $f\colon G\to H$ be a homomorphism of connected $k$-groups with finite
kernel. Then the maps $Y_{G}({\mathbb{K}})\to Y_{H}({\mathbb{K}})$ and
$\kappa_{f}\colon V_{G}({\mathbb{K}})\to V_{H}({\mathbb{K}})$ are injective.

###### Proof.

Again, we give the proof for $\kappa_{f}$; the other proof is very similar.
Let
$(T_{1},\lambda_{1}),(T_{2},\lambda_{2})\in\bigsqcup_{T}Y_{T}({\mathbb{K}})$
be such that
$(S_{T_{1}},f_{*}(\lambda_{1}))\approx(S_{T_{2}},f_{*}(\lambda_{2}))$. Let
$f_{i}\colon T_{i}\to S_{T_{i}}$ be the map induced by $f$ for each $i$. Then
$P_{(f_{1})_{*}(\lambda_{1})}(H)=P_{(f_{2})_{*}(\lambda_{2})}(H)$, so
$P_{(f_{1})_{*}(\lambda_{1})}(f(G))=P_{(f_{2})_{*}(\lambda_{2})}(f(G))$ by
Corollary 3.16, and hence by Lemma 3.17(i) we have
$f(P_{\lambda_{1}})=f(P_{\lambda_{2}})$. It follows from Lemma 3.17(ii)
(applied to the surjective homomorphism $G\to f(G)$) that
$P_{\lambda_{1}}=P_{\lambda_{2}}$. Thus $T_{1}$ and $T_{2}$ are maximal split
tori of $P_{\lambda_{1}}$, which means we can find $g\in P_{\lambda_{1}}(k)$
such that $gT_{2}g^{-1}=T_{1}$. Since $f(g)\in
P_{(f_{1})_{*}(\lambda_{1})}(H)(k)=P_{(f_{2})_{*}(\lambda_{2})}(H)(k)$, we
have
$P_{(f_{2})_{*}(g\cdot\lambda_{2})}(H)=P_{f(g)\cdot(f_{2})_{*}(\lambda_{2})}(H)=f(g)P_{(f_{2})_{*}(\lambda_{2})}(H)f(g)^{-1}=P_{(f_{2})_{*}(\lambda_{2})}(H)$
by Remark 4.10 and Lemma 3.3(ii). Hence, replacing $\lambda_{2}$ with
$g\cdot\lambda_{2}$, we may assume that $T_{1}=T_{2}$. Now, since $f$ has
finite kernel, the map $(f_{1})_{*}$ is injective on
$Y_{T_{1}}({\mathbb{K}})$, as observed in Section 3.1. Since we have reduced
to the case that $\lambda_{1},\lambda_{2}\in Y_{T_{1}}({\mathbb{K}})$ with
$(f_{1})_{*}(\lambda_{1})=(f_{1})_{*}(\lambda_{2})$, we conclude that
$\lambda_{1}=\lambda_{2}$. ∎

######  Remark 4.12.

We can now resolve the potential ambiguity noted in Remark 3.18 above. If $T$
is a maximal split torus of $G$ then the notation $V_{T}({\mathbb{K}})$ could
mean the corresponding apartment of $V_{G}({\mathbb{K}})$ or it could mean the
vector edifice associated to $T$ in its own right. But we see that if $i\colon
T\to G$ is the inclusion then $\kappa_{i}$ gives a linear isomorphism from the
latter object to the former, so the abuse of notation is harmless. Indeed,
this observation works for any subgroup $H$ of $G$ — we may identify the
vector edifice $V_{H}({\mathbb{K}})$ as a subobject of the vector edifice
$V_{G}({\mathbb{K}})$ via the injective linear map $\kappa_{i}\colon
V_{H}({\mathbb{K}})\to V_{G}({\mathbb{K}})$ arising from the inclusion
$i\colon H\to G$. We often do this without further comment in the rest of the
paper.

######  Remark 4.13.

If $\lambda\in Y_{G}({\mathbb{K}})$ then we may regard $\lambda$ as an element
of $Y_{L_{\lambda}}({\mathbb{K}})$ and $x:=\phi_{G}(\lambda)$ as an element of
$V_{L_{\lambda}}({\mathbb{K}})$ and $V_{P_{\lambda}}({\mathbb{K}})$ in the
obvious way. Since all maximal split tori in $L_{\lambda}$ are
$L_{\lambda}(k)$-conjugate, it follows that if $T$ is a maximal split torus of
$L_{\lambda}$ then $x\in V_{T}({\mathbb{K}})$.

######  Example 4.14.

Suppose that $G$ is such that $V_{G}({\mathbb{K}})$ has the common apartment
property, and $P$ is an R-parabolic subgroup of $G$. Then the inclusion
$i\colon P\to G$ actually induces a _bijection_ $\kappa_{i}\colon
V_{P}({\mathbb{K}})\to V_{G}({\mathbb{K}})$. To see this, note that for every
maximal split torus $T$ of $G$ and every $\lambda\in Y_{T}({\mathbb{K}})$,
there is a maximal split torus $S_{T,\lambda}$ contained in $P_{\lambda}\cap
P$, because of the common apartment property, and hence
$(T,\lambda)\approx(S_{T,\lambda},\mu)$, where $\mu=g\cdot\lambda$ for $g\in
P_{\lambda}(k)$ conjugating $T$ to $S_{T,\lambda}$. Therefore, if we let
$x=\phi_{P}(\mu)\in V_{P}({\mathbb{K}})$ and $y=\phi_{G}(\lambda)\in
V_{G}({\mathbb{K}})$, we have $\kappa_{i}(x)=y$. Thus $\kappa_{i}$ is
surjective, and we already know it is injective by Proposition 4.11. Note
that, even though $\kappa_{i}$ is a bijective linear map of vector edifices,
it is _not_ an isomorphism in general: e.g., in the setting of Example 3.22 we
have points $x_{1},x_{2}\in V_{P}({\mathbb{K}})$ such that $x_{1}$ and $x_{2}$
do not lie in a common apartment of $V_{P}({\mathbb{K}})$, but
$\kappa_{i}(x_{1})$ and $\kappa_{i}(x_{2})$ do lie in a common apartment of
$V_{G}({\mathbb{K}})$, because $G$ is reductive in that case (and hence
$V_{G}({\mathbb{K}})$ has the common apartment property).

###  4.4. Homomorphisms which induce surjective linear maps

Next we investigate the conditions under which a homomorphism $f\colon G\to H$
gives rise to a surjective linear map. It turns out that the obvious condition
(that $f$ is surjective) is not sufficient; this situation is made more
complicated by, for example, the existence over imperfect fields of
inseparable isogenies. This motivates the following definition. See also
Remark 4.17(i) below.

######  Definition 4.15.

Suppose $f\colon G\to H$ is a homomorphism of connected $k$-groups. We say $f$
is _apte_ if every maximal split torus $S$ of $H$ has the form $S=f(T)$ for a
maximal split torus $T$ of $G$.

With the key definition in hand, the following result is easy.

######  Lemma 4.16.

Let $f\colon G\to H$ be an apte homomorphism of connected $k$-groups. Then
$\kappa_{f}$ is surjective.

###### Proof.

Let $S$ be a maximal split torus of $H$. The assumption that $f$ is apte means
we can find a maximal split torus $T$ of $G$ with $S=f(T)$. Let $h\colon T\to
S$ be restriction of $f$. Since the map of character groups $h^{*}\colon
X_{S}\to X_{T}$ is injective, the dual map $h_{*}\colon Y_{T}\to Y_{S}$ has
finite cokernel. This implies that $f_{*}\colon Y_{T}({\mathbb{K}})\to
Y_{S}({\mathbb{K}})$ is surjective, and the result follows. ∎

######  Remarks 4.17.

(i). The lemma does not hold without the assumption that $f$ is apte. For a
concrete example, let $G=H=\operatorname{SL}_{2}$ over an imperfect field $k$
in characteristic $2$, and let $f\colon G\to G$ be the standard Frobenius map
which squares the entries of $2\times 2$ matrices. That $k$ is imperfect means
that there is some $a\in\overline{k}\setminus k$ such that $a^{2}\in k$. Now
let

$u=\left(\begin{array}[]{cc}1&a\\\ 0&1\end{array}\right)\in G(\overline{k}),$

and let $T$ be the standard (split) diagonal torus in $G$. The torus
$uT_{\overline{k}}u^{-1}$ is not a $k$-subgroup, hence in particular is not
$k$-split. However, its image
$f_{\overline{k}}(uT_{\overline{k}}u^{-1})=f(u)T_{\overline{k}}f(u)^{-1}$
descends to a split $k$-torus $S=f(u)Tf(u)^{-1}$, since $f(u)\in G(k)$. This
shows that $f$ is not apte, and it is also clear that not every element of
$V_{S}({\mathbb{K}})$ belongs to $\kappa_{f}(V_{G}({\mathbb{K}}))$.

(ii). Following [11, 22.11], we call a surjective homomorphism $f\colon G\to
H$ a _central isogeny_ if it has finite kernel which is contained in the
_scheme-theoretic_ centre of $G$ — that is, for every $k$-algebra $A$ and
every $A$-algebra $A^{\prime}$, the kernel of $f(A)\colon G(A)\to H(A)$
centralises $G(A^{\prime})$.333For $G$ reductive, Borel [11, 22.3] gives
another definition of central isogeny, but this is shown to be equivalent to
the scheme-theoretic one just given in [11, 22.15 Prop.]. If (a) $f$ is
surjective and $k$ is perfect or (b) $f$ is a central isogeny, then $f$ is
apte: this follows from [11, 22.6 Thm.(ii)]. Neither of these hypotheses holds
for the example in (i).

(iii). Let $f$ be a surjective homomorphism. Suppose $f$ is smooth (this is
equivalent to requiring $\ker f$ to be smooth, by [22, Prop. 1.63]). Then $f$
is apte. To see this, let $S$ be a maximal split torus of $H$ and let $S_{1}$
be a maximal torus of $H$ containing $S$. Then the subgroup scheme
$f^{-1}(S_{1})$ of $G$ is smooth. Let $T_{1}$ be a maximal torus of
$f^{-1}(S_{1})$ and let $T$ be the unique maximal split torus of $T_{1}$. By
[14, A.2.8], $f(T_{1})=S_{1}$, so (the restriction of) $f$ is a surjective map
of $k$-tori. It follows from [11, 8.15 Prop.(3)] and surjectivity that
$f(T)=S$.

(iv). Example 4.14 shows that $\kappa_{f}$ can be surjective even when $f$ is
not apte.

We can now prove the key result of this subsection:

######  Proposition 4.18.

Suppose $f\colon G\to H$ is an apte homomorphism with finite kernel. Then
$\kappa_{f}$ is an isomorphism of vector edifices.

###### Proof.

It follows from Proposition 4.11 and Lemma 4.16 that $\kappa_{f}$ is a
bijective linear map. Since $f$ is apte, for each maximal split torus $S$ of
$H$ there is a maximal split torus $T$ of $G$ such that $f(T)=S$, and since
$f$ has finite kernel, this $T$ is unique. Therefore, $\kappa_{f}$ pairs up
the apartments of $V_{G}({\mathbb{K}})$ and $V_{H}({\mathbb{K}})$, and hence
$\kappa_{f}^{-1}$ is also a linear map of vector edifices. ∎

We finish with some results related to the common apartment property.

######  Lemma 4.19.

Let $N$ be a connected normal unipotent subgroup of $G$ and let $\pi\colon
G\to G/N$ be the canonical projection. Let $x_{1},x_{2}\in
V_{G}({\mathbb{K}})$. Then $\kappa_{\pi}(x_{1})=\kappa_{\pi}(x_{2})$ if and
only if $x_{2}=n\cdot x_{1}$ for some $n\in N(k)$.

###### Proof.

If $n\in N(k)$ and $y=n\cdot x$ then $\kappa_{\pi}(y)=\kappa_{\pi}(n\cdot
x)=\pi(n)\cdot\kappa_{\pi}(x)=\kappa_{\pi}(x)$ where the second equality is
from Remark 4.10. Conversely, suppose
$\kappa_{\pi}(x_{1})=\kappa_{\pi}(x_{2})$; call this common value $y$. For
each $i$, let $G_{i}=P_{x_{i}}N$, let $Q_{i}=\pi(G_{i})$ and let
$\pi_{i}\colon G_{i}\to Q_{i}$ be the restriction of $\pi$. Then $\pi_{i}$ is
surjective, so $Q_{i}=\pi(P_{x_{i}})=P_{y}(G/N)$ by Remark 4.8(ii).

Choose a maximal split torus $T$ of $G/N$ such that $y\in
V_{T}({\mathbb{K}})$. Then $T$ is a maximal split torus of $P_{y}(G/N)$ by
Remark 3.21. For each $i$, since $\pi_{i}$ is smooth, $\pi_{i}$ is apte
(Remark 4.17(iii)), so there is a maximal split torus $T_{i}$ of $G_{i}$ such
that $\pi(T_{i})=T$. So $T_{1}$ and $T_{2}$ are maximal split tori of
$H:=T_{1}N\cong T_{1}\ltimes N$. Hence there exists $n\in N(k)$ such that
$T_{2}=nT_{1}n^{-1}$. Now $x_{i}$ belongs to $V_{T_{i}}({\mathbb{K}})$ for
each $i$, so $n\cdot x_{1}$ and $x_{2}$ belong to $V_{T_{2}}({\mathbb{K}})$.
We have $\kappa_{\pi}(n\cdot x_{1})=\kappa_{\pi}(x_{2})=y$ by Remark 4.10. But
$\pi$ induces an isomorphism from $T_{2}$ onto $T$ since $N$ is unipotent, so
$\kappa_{\pi}$ induces an isomorphism from $V_{T_{2}}({\mathbb{K}})$ to
$V_{T}({\mathbb{K}})$. We conclude that $x_{2}=n\cdot x_{1}$, as required. ∎

######  Lemma 4.20.

Let $x,y\in V_{G}({\mathbb{K}})$. Then there exists $u\in R_{u,k}(G)(k)$ such
that $u\cdot x$ and $y$ lie in a common apartment.

###### Proof.

Let $H=G/R_{u,k}(G)$ and let $\pi\colon G\to H$ be the canonical projection.
Since $H$ is pseudo-reductive, there is a maximal split torus $S$ of $H$ such
that $\kappa_{\pi}(x),\kappa_{\pi}(y)\in V_{S}({\mathbb{K}})$ (see the
discussion following Definition 3.23). By Remark 4.17(iii), as $\pi$ is smooth
there is a maximal split torus $T$ of $G$ such that $\pi(T)=S$. The map
$\kappa_{\pi}$ gives an isomorphism from $V_{T}({\mathbb{K}})$ to
$V_{S}({\mathbb{K}})$, so there exists $x^{\prime},y^{\prime}\in
V_{T}({\mathbb{K}})$ such that $\kappa_{\pi}(x^{\prime})=\kappa_{\pi}(x)$ and
$\kappa_{\pi}(y^{\prime})=\kappa_{\pi}(y)$. By Lemma 4.19, there exist $v,w\in
R_{u,k}(G)(k)$ such that $v\cdot x=x^{\prime}$ and $w\cdot y=y^{\prime}$. Set
$u:=w^{-1}v$. Then $u\cdot x=w^{-1}\cdot x^{\prime}$ and $y=w^{-1}\cdot
y^{\prime}$ lie in the common apartment $V_{w^{-1}Tw}({\mathbb{K}})$, as
required. ∎

######  Remark 4.21.

Even though the common apartment property may fail to hold, we can show the
following: if $x,y\in V_{G}({\mathbb{K}})$ then there exists $z\in
V_{G}({\mathbb{K}})$ such that $x$ and $z$ lie in a common apartment, and $y$
and $z$ lie in a common apartment. For instance, we can take $z=0$, although
this is slightly unsatisfactory as 0 does not yield a point in
$\Delta_{G}({\mathbb{K}})$. More generally, we can take $z$ to be any element
of $V_{Z(G)^{0}}({\mathbb{K}})$. Here is a construction which does not involve
$Z(G)^{0}$.

Let $U=R_{u,k}(G)$. By Lemma 4.20, there exist a maximal split torus $T$ of
$G$ and $u\in U(k)$ such that $u\cdot x,y\in V_{T}({\mathbb{K}})$. We can pick
$\lambda\in Y_{u^{-1}Tu}({\mathbb{K}})$ such that $x=\phi_{G}(\lambda)$ and
$u\cdot x=\phi_{G}(u\cdot\lambda)$. Note that $x$ and $\phi_{G}(-\lambda)$ lie
in the common apartment $V_{u^{-1}Tu}({\mathbb{K}})$, and $u\cdot\lambda$ and
$-u\cdot\lambda$ both belong to $Y_{T}({\mathbb{K}})$. By [32, 13.4.4 Cor.],
there exist $v\in P_{\lambda}(U)(k)$ and $w\in U_{-\lambda}(U)(k)$ such that
$u=vw$. Let $z=v\cdot\phi_{G}(-\lambda)$. Then $x$ and $z$ lie in the common
apartment $V_{vu^{-1}Tuv^{-1}}({\mathbb{K}})$ since $v\cdot x=x$. Now
$w\cdot\phi_{G}(-\lambda)=\phi_{G}(-\lambda)$, so
$z=vw\cdot\phi_{G}(-\lambda)=u\cdot\phi_{G}(-\lambda)=\phi_{G}(-u\cdot\lambda)$
belongs to $V_{T}({\mathbb{K}})$. Hence $z$ and $y$ lie in a common apartment,
as required.

###  4.5. Isomorphisms of vector edifices induce isomorphisms of combinatorial
edifices

Our next aim is to prove that isomorphisms between vector edifices behave
nicely with respect to the structure coming from the poset of R-parabolic
subgroups under reverse inclusion. Now that we have shown that for a subgroup
$M$ of $G$ we can identify the sets $Y_{M}({\mathbb{K}})$ and
$V_{M}({\mathbb{K}})$ inside $Y_{G}({\mathbb{K}})$ and $V_{G}({\mathbb{K}})$,
we can extend Lemma 2.13, which is the equivalence (i) $\iff$ (ii) in the
following.

######  Lemma 4.22.

Let $P,Q$ be R-parabolic subgroups of $G$. Then the following are equivalent:

  * (i)

$P\subseteq Q$;

  * (ii)

$Y_{P}\subseteq Y_{Q}$;

  * (iii)

$Y_{P}({\mathbb{K}})\subseteq Y_{Q}({\mathbb{K}})$.

###### Proof.

(i)$\iff$(ii) follows from Lemma 2.13 and that (iii)$\implies$(ii) is easy to
see, basically by construction of these objects. Suppose (ii) holds. Let $T$
be a maximal split torus of $P$. Because of the equivalence of (i) and (ii),
we have $T\subseteq Q$, so $Y_{T}({\mathbb{K}})\subseteq Y_{Q}({\mathbb{K}})$.
This shows that (ii)$\implies$(iii). ∎

######  Lemma 4.23.

Let $\kappa\colon V_{G}({\mathbb{K}})\to V_{H}({\mathbb{K}})$ be a linear map,
and let $\widehat{\kappa}\colon Y_{G}({\mathbb{K}})\to Y_{H}({\mathbb{K}})$ be
the corresponding lift from Lemma 4.4. Then for any $\lambda\in
Y_{G}({\mathbb{K}})$, we have

$\widehat{\kappa}(Y_{L_{\lambda}}({\mathbb{K}}))\subseteq
Y_{L_{\widehat{\kappa}(\lambda)}}({\mathbb{K}})\quad\textrm{ and
}\quad\widehat{\kappa}(Y_{P_{\lambda}}({\mathbb{K}}))\subseteq
Y_{P_{\widehat{\kappa}(\lambda)}}({\mathbb{K}}).$

Hence also $\kappa(V_{L_{\lambda}}({\mathbb{K}}))\subseteq
V_{L_{\widehat{\kappa}(\lambda)}}({\mathbb{K}})$ and
$\kappa(V_{P_{\lambda}}({\mathbb{K}}))\subseteq
V_{P_{\widehat{\kappa}(\lambda)}}({\mathbb{K}})$. Thus, if we set
$x=\phi_{G}(\lambda)$, we have $\kappa(V_{P_{x}}({\mathbb{K}}))\subseteq
V_{P_{\kappa(x)}}({\mathbb{K}})$.

###### Proof.

Choose any $z\in V_{L_{\lambda}}({\mathbb{K}})$: say,
$z=\phi_{L_{\lambda}}(\sigma)$ for some $\sigma\in
Y_{L_{\lambda}}({\mathbb{K}})$. Choose any maximal split torus $T$ of
$L_{\lambda}$ such that $\sigma\in Y_{T}({\mathbb{K}})$. Then $\lambda\in
Y_{T}({\mathbb{K}})$, by Remark 4.13, and $T$ is a maximal split torus of $G$.
Let $S$ be any maximal split torus of $H$ such that
$\kappa(V_{T}({\mathbb{K}}))\subseteq V_{S}({\mathbb{K}})$. Then
$\widehat{\kappa}(\lambda)$ and $\widehat{\kappa}(\sigma)$ belong to
$Y_{S}({\mathbb{K}})$. Thus $S\subseteq L_{\widehat{\kappa}(\lambda)}$ and so
$Y_{S}({\mathbb{K}})\subseteq
Y_{L_{\widehat{\kappa}(\lambda)}}({\mathbb{K}})$. This shows that
$\widehat{\kappa}(Y_{L_{\lambda}}({\mathbb{K}}))\subseteq
Y_{L_{\widehat{\kappa}(\lambda)}}({\mathbb{K}})$, by Lemma 4.22.

For the second statement, let $w\in V_{P_{\lambda}}({\mathbb{K}})$: say,
$w=\phi_{P_{\lambda}}(\tau)$ for some $\tau\in Y_{P_{\lambda}}({\mathbb{K}})$.
Set $x=\phi_{G}(\lambda)$. Choose a maximal split torus $T$ of $P_{\lambda}$
such that $\tau\in Y_{T}({\mathbb{K}})$. By Lemma 2.4(i) and Remark 3.9, there
exists $u\in U_{\lambda}(k)$ such that $uTu^{-1}\subseteq L_{\lambda}$. Then
$\lambda\in Y_{uTu^{-1}}({\mathbb{K}})$, again by Remark 4.13, so
$u^{-1}\cdot\lambda\in Y_{T}({\mathbb{K}})$, so $T\subseteq
L_{u^{-1}\cdot\lambda}$ by Remark 3.4(iii), so $\tau\in
Y_{L_{u^{-1}\cdot\lambda}}({\mathbb{K}})$. By the previous paragraph, we can
conclude that $\widehat{\kappa}(\tau)\in
Y_{L_{\widehat{\kappa}(u^{-1}\cdot\lambda)}}({\mathbb{K}})\subseteq
Y_{P_{\widehat{\kappa}(u^{-1}\cdot\lambda)}}({\mathbb{K}})$. But
$P_{\widehat{\kappa}(u^{-1}\cdot\lambda)}=P_{\kappa(u^{-1}\cdot
x)}=P_{\kappa(x)}=P_{\widehat{\kappa}(\lambda)}$ by construction, so
$\widehat{\kappa}(\tau)\in Y_{P_{\widehat{\kappa}(\lambda)}}({\mathbb{K}})$,
as required.

Now the corresponding statements on the level of vector edifices follow
immediately. ∎

######  Lemma 4.24.

Let $G$ and $H$ be connected $k$-groups and let $\kappa\colon
V_{G}({\mathbb{K}})\to V_{H}({\mathbb{K}})$ be an isomorphism of vector
edifices. Then $\kappa$ induces a poset isomorphism
$\sigma_{x}\mapsto\sigma_{\kappa(x)}$ between the corresponding combinatorial
edifices.

###### Proof.

Let $\tau\in Y_{G}({\mathbb{K}})$. Then
$\widehat{\kappa}(Y_{P_{\tau}}({\mathbb{K}}))\subseteq
Y_{P_{\widehat{\kappa}(\tau)}}({\mathbb{K}})$ by Lemma 4.23. On the other
hand,
$\widehat{\kappa}^{-1}(Y_{P_{\widehat{\kappa}(\tau)}}({\mathbb{K}}))\subseteq
Y_{P_{\widehat{\kappa}^{-1}(\widehat{\kappa}(\tau))}}({\mathbb{K}})=Y_{P_{\tau}}({\mathbb{K}})$
using Lemma 4.23 again, so we deduce that

(4.25)
$\widehat{\kappa}(Y_{P_{\tau}}({\mathbb{K}}))=Y_{P_{\widehat{\kappa}(\tau)}}({\mathbb{K}})$

for all $\tau\in Y_{G}({\mathbb{K}})$.

Now let $\lambda,\mu\in Y_{G}({\mathbb{K}})$ such that $P_{\lambda}\subseteq
P_{\mu}$. Then $Y_{P_{\lambda}}({\mathbb{K}})\subseteq
Y_{P_{\mu}}({\mathbb{K}})$ by Lemma 4.22, so
$Y_{P_{\widehat{\kappa}(\lambda)}}({\mathbb{K}})=\widehat{\kappa}(Y_{P_{\lambda}}({\mathbb{K}}))\subseteq\widehat{\kappa}(Y_{P_{\mu}}({\mathbb{K}}))=Y_{P_{\widehat{\kappa}(\mu)}}({\mathbb{K}})$
using (4.25). We deduce from Lemma 4.22 that

(4.26) $P_{\widehat{\kappa}(\lambda)}\subseteq P_{\widehat{\kappa}(\mu)}.$

In particular, if $P_{\lambda}=P_{\mu}$ then
$P_{\widehat{\kappa}(\lambda)}=P_{\widehat{\kappa}(\mu)}$, so the map
$\sigma_{x}\mapsto\sigma_{\kappa(x)}$ is well-defined. Clearly, this map is
invertible (with inverse $\sigma_{w}\mapsto\sigma_{\kappa^{-1}(w)}$), and
(4.26) implies that it is a poset isomorphism. ∎

######  Remark 4.27.

If $\kappa\colon V_{G}({\mathbb{K}})\to V_{H}({\mathbb{K}})$ is an arbitrary
linear map of vector edifices then the prescription
$\sigma_{x}\mapsto\sigma_{\kappa(x)}$ need not give a well-defined map, even
when $G$ and $H$ are reductive. For instance, if $f$ is the inclusion of a
torus $G$ in a non-commutative reductive group $H$ then $P_{x}(G)=G$ for every
$x\in V_{G}({\mathbb{K}})$ but there is more than one possible value for
$P_{\kappa_{f}(x)}(H)$. So there are linear maps of vector buildings/vector
edifices that do not correspond to any maps at the combinatorial level. We
will exploit the extra flexibility yielded by this geometric approach in our
forthcoming work [9].

Even if $\kappa$ is bijective, the conclusion of Lemma 4.24 can fail. Let $B$
be a Borel subgroup in a non-commutative split reductive group $G$ and let
$f\colon B\to G$ be inclusion. Then Example 4.14 shows that $\kappa_{f}\colon
V_{B}({\mathbb{K}})\to V_{G}({\mathbb{K}})$ is a bijection. We can choose
$x=0\in V_{B}({\mathbb{K}})$, $y\in V_{B}({\mathbb{K}})$ such that
$G=P_{\kappa_{f}(x)}(G)\neq P_{\kappa_{f}(y)}(G)=B$; but
$P_{x}(B)=P_{y}(B)=B$.

###  4.6. Projection maps

We can now, as promised in the introduction, relate our constructions to the
standard notion of _projection_ for a spherical building. First, recall that
given a simplicial spherical building $\Delta$ and any pair $\sigma,\tau$ of
simplices in $\Delta$ there is a uniquely defined simplex $\sigma\tau$ which
is the projection of $\tau$ onto $\sigma$, [33, 2.30, 3.19], [1, Def. 3.109].
If $\Delta=\Delta_{G}$ is the combinatorial edifice for a reductive $k$-group
$G$ (which is a simplicial spherical building), then this projection has a
natural realisation in terms of the corresponding parabolic subgroups of $G$:
if $\sigma=\sigma_{P}$ and $\tau=\sigma_{Q}$, then $\sigma\tau$ is the simplex
corresponding to $(P\cap Q)R_{u}(P)$, which is also a parabolic subgroup of
$G$ [12, Prop. 4.4] and is contained in $P$. Further, if we choose a maximal
split torus $T$ of $G$ contained in $P\cap Q$ and write $P=P_{\lambda}$,
$Q=P_{\mu}$ for $\lambda,\mu\in Y_{T}$, then $(P\cap
Q)R_{u}(P)=P_{n\lambda+\mu}$ for sufficiently large $n\in{\mathbb{N}}$ [4,
Cor. 6.9].

Now let $L=L_{\lambda}$ be the Levi subgroup of $P$ corresponding to the
choice of $\lambda$ above. Then $L$ is a reductive group and the simplicial
building $\Delta_{L}$ of $L$ can be realised inside $\Delta_{G}$. Again, this
can be seen in terms of parabolic subgroups: for each parabolic subgroup $Q$
of $G$ contained in $P$, the subgroup $Q\cap L$ is a parabolic subgroup of
$L$, all the parabolic subgroups of $L$ arise in this way, and there is an
inverse map $X\mapsto XR_{u}(P)$ mapping parabolic subgroups of $L$ to
parabolic subgroups of $G$ contained in $P$ [12, Prop. 4.4] — in this way we
may identify $\Delta_{L}$ with the _link_ or _star_ of $\sigma_{P}$ [33, 1.1],
[1, Prop. 4.9]; in more modern terminology, $\Delta_{L}$ arises as a _residue_
, see [1, Sec. 5.3, Cor. 5.30]. Combining this with the observations from the
previous paragraph, we obtain a (surjective) map
$\operatorname{proj}_{P,L}:\Delta_{G}\to\Delta_{L}$; this is the same as the
_projection onto a residue_ of [1, Def. 5.35].

The key observation of this section is that these constructions can be
realised in the setting of a vector edifice $V_{G}({\mathbb{K}})$ with the
common apartment property (Definition 3.23). Since a reductive group $G$ gives
rise to such a vector edifice, what follows can be viewed as a generalisation
of the previously described constructions. So now suppose $G$ is any $k$-group
such that $V_{G}({\mathbb{K}})$ has the common apartment property. We have
seen in Example 4.14 above that for such a $G$ and an R-parabolic subgroup
$P=P_{\lambda}$ of $G$, the inclusion $i:P\to G$ gives a bijective linear map
$\kappa_{i}:V_{P}({\mathbb{K}})\to V_{G}({\mathbb{K}})$. It follows from Lemma
4.16 that if we let $L=L_{\lambda}$, the projection $f:P\to L\cong
P/U_{\lambda}$ gives rise to a surjective linear map $V_{P}({\mathbb{K}})\to
V_{L}({\mathbb{K}})$. Hence the composition

$F_{P,L}:=\kappa_{f}\circ\kappa_{i}^{-1}:V_{G}({\mathbb{K}})\to
V_{L}({\mathbb{K}})$

is a surjective map.444The notation $F_{P,L}$ makes sense because
$U_{\lambda}$ does not depend on the choice of $\lambda$ for fixed $P$ and
$L$: see Lemma 2.4(iii). Note that $f$ is apte by Remark 4.17(iii), because
the kernel $U_{\lambda}$ of $f$ is smooth.

######  Lemma 4.28.

With the notation just set up, $F_{P,L}$ induces a well-defined map of
combinatorial edifices $\Delta_{G}\to\Delta_{L}$, which we denote by
$\operatorname{proj}_{P,L}$. If $G$ is reductive, then
$\operatorname{proj}_{P,L}$ is the projection map on the spherical building of
$G$ described above.

###### Proof.

To check that the map on combinatorial edifices induced by $F_{P,L}$ is well-
defined, we need to check that if $\sigma_{\mu}=\sigma_{\nu}$ for $\mu,\nu\in
Y_{G}({\mathbb{K}})$, then $\sigma_{F_{P,L}(\mu)}=\sigma_{F_{P,L}(\nu)}$.
Unwinding the definitions, this amounts to checking that if $\mu,\nu\in
Y_{L}({\mathbb{K}})$ give rise to the same R-parabolic subgroup of $G$ then
they give rise to the same R-parabolic subgroup of $L$, which is true by
Corollary 3.16.

Now to check that the map induced by $F_{P,L}$ corresponds to
$\operatorname{proj}_{P,L}$ in the case that $G$ is reductive, we again
translate into a condition on R-parabolic subgroups. This time, it suffices to
show that given any $\mu\in Y_{L_{\lambda}}({\mathbb{K}})$, we have
$P_{\mu}(L_{\lambda})=P_{n\lambda+\mu}(L_{\lambda})$ for all
$n\in{\mathbb{N}}$. To see this, first note that for any cocharacter
approximation $\lambda^{\prime}$ to $\lambda$ and any $\mu^{\prime}\in
Y_{L_{\lambda}}$ we have
$P_{\mu^{\prime}}(L_{\lambda^{\prime}})=P_{n\lambda^{\prime}+\mu^{\prime}}(L_{\lambda^{\prime}})$
since the image of $\lambda^{\prime}$ is central in
$L_{\lambda^{\prime}}=L_{\lambda}$. Hence, the result follows after choosing
any cocharacter approximation $\mu^{\prime}\in Y_{L_{\lambda}}$ to $\mu$. ∎

######  Remarks 4.29.

(i). The map $F_{P,L}$ is not linear in general. For example, let
$G=\operatorname{SL}_{2}$, let $P$ be the standard upper triangular Borel
subgroup $B$ and let $L$ be the standard diagonal maximal torus $T$. Choose
$\lambda\in Y_{T}$ such that $P_{\lambda}=B$ and set $x=\phi_{G}(-\lambda)$.
Let $y=u\cdot x$, where $1\neq u\in U_{\lambda}(k)$. Now $x$ and $y$ are
opposite by the example in Remark 3.31, so if $F_{P,L}$ is linear then
$F_{P,L}(x)$ and $F_{P,L}(y)$ must be opposite. But this is not the case as
$F_{P,L}(x)=F_{P,L}(y)=x$ by Lemma 4.19.

(ii). Still in the setting of the previous example, let $Q=P_{-\lambda}$ be
the opposite Borel to $B$ with respect to $T$. We can form the map $F_{P,L}$
using the projection $P\to L$ and the map $F_{Q,L}$ using the projection $Q\to
L$. Then $F_{P,L}(y)=F_{P,L}(x)$; but Lemma 4.19 implies that $F_{Q,L}(y)\neq
F_{Q,L}(x)$, as $y$ is not $U_{-\lambda}(k)$-conjugate to $x$. This shows that
$F_{P,L}$ depends on the choice of $P$ for given $L$.

###  4.7. Linear maps of spherical edifices

Let $\kappa\colon V_{G_{1}}({\mathbb{K}})\to V_{G_{2}}({\mathbb{K}})$ be a
linear map of vector edifices. If $\kappa$ is injective then we get an induced
map

$\kappa^{\flat}\colon\Delta_{G_{1}}({\mathbb{K}})\to\Delta_{G_{2}}({\mathbb{K}})$

between the corresponding spherical edifices given by
$\kappa^{\flat}({\mathbb{K}}^{+}\cdot x)={\mathbb{K}}^{+}\cdot\kappa(x)$.
Clearly this map of spherical edifices is also injective. We call a map of the
form $\kappa^{\flat}$ a _linear map_. (To make this definition we need
$\kappa$ to be injective, for otherwise there are nonzero elements of
$V_{G_{1}}({\mathbb{K}})$ mapped to $0\in V_{G_{2}}({\mathbb{K}})$, and then
$\kappa^{\flat}$ cannot be defined on all of $\Delta_{G_{1}}({\mathbb{K}})$.)

######  Remark 4.30.

Note that we get a linear map $\kappa_{f}\colon V_{G}({\mathbb{K}})\to
V_{H}({\mathbb{K}})$ for any homomorphism from $G$ to a connected $k$-group
$H$, but we only get a corresponding map of spherical buildings
$\Delta_{G}({\mathbb{K}})$ to $\Delta_{H}({\mathbb{K}})$ if $\kappa_{f}$ is
injective. This is one of the main advantages of working with vector edifices
rather than with spherical edifices.

######  Definition 4.31.

We define an _automorphism_ of $\Delta_{G}({\mathbb{K}})$ to be a bijective
linear map whose inverse is also a linear map. We write
$\operatorname{Aut}(\Delta_{G}({\mathbb{K}}))$ for the group of automorphisms
of $\Delta_{G}({\mathbb{K}})$.

Clearly, if $\kappa\in\operatorname{Aut}(V_{G}({\mathbb{K}}))$ then
$\kappa^{\flat}\in\operatorname{Aut}(\Delta_{G}({\mathbb{K}}))$. Conversely,
suppose $\kappa,\rho\colon V_{G}({\mathbb{K}})\to V_{G}({\mathbb{K}})$ are
injective linear maps and $\rho^{\flat}=(\kappa^{\flat})^{-1}$. Then
$(\rho\circ\kappa)^{\flat}=\rho^{\flat}\circ\kappa^{\flat}={\rm
id_{\Delta_{G}({\mathbb{K}})}}$, so $\rho\circ\kappa$ maps each ray in
$V_{G}({\mathbb{K}})$ to itself and each apartment in $V_{G}({\mathbb{K}})$ to
itself. Hence $\rho$ is a bijection, and it follows that $\kappa$ is a
bijection. We see that if $T$, $T^{\prime}$ are maximal split tori of $G$ and
$\rho$ maps $V_{T}({\mathbb{K}})$ to $V_{T^{\prime}}({\mathbb{K}})$ then
$\kappa^{-1}$ also maps $V_{T}({\mathbb{K}})$ to
$V_{T^{\prime}}({\mathbb{K}})$, so $\kappa$ belongs to
$\operatorname{Aut}(V_{G}({\mathbb{K}}))$. Hence
$\operatorname{Aut}(\Delta_{G}({\mathbb{K}}))=\\{\kappa^{\flat}\mid\kappa\in\operatorname{Aut}(V_{G}({\mathbb{K}}))\\}$.

##  5\. Linear maps arising from base change

Let $\alpha\colon k\to k^{\prime}$ be a homomorphism of fields — for example,
$\alpha$ could be a field isomorphism, or the inclusion arising from a field
extension $k^{\prime}/k$ — and let $G$ be a $k$-group. Base change along
$\alpha$ gives rise to a $k^{\prime}$-group ${{}^{\alpha}}G$: from the
functorial point of view, given a $k^{\prime}$-algebra $A^{\prime}$, we have
${{}^{\alpha}}G(A^{\prime})=G(A^{\prime}\otimes_{\alpha}k)$, which makes sense
as $A^{\prime}\otimes_{\alpha}k$ is a $k$-algebra via scalar multiplication in
the second factor. Note that when $\alpha\colon k\to k^{\prime}$ is the
inclusion arising from a field extension $k^{\prime}/k$, then
${{}^{\alpha}}G=G_{k^{\prime}}$, but this construction works more generally,
for example when $\alpha\colon k\to k$ is a field automorphism.

Identifying a point $x\in G(k)$ with a morphism $\mathrm{Spec}(k)\to G$, we
obtain by base change along $\alpha$ a morphism
$\mathrm{Spec}(k^{\prime})\to{{}^{\alpha}}G$, which is to say a point
${{}^{\alpha}}x\in{{}^{\alpha}}G(k^{\prime})$. The corresponding map

$\psi_{\alpha}\colon G(k)\to{{}^{\alpha}}G(k^{\prime}),\quad
x\mapsto{{}^{\alpha}}x$

is a homomorphism of abstract groups.

###  5.1. Base change induces an injection of vector edifices

We first wish to show how base change gives rise to a linear map of the
corresponding vector edifices.

Suppose $f\colon H\to G$ is a homomorphism of $k$-groups. Then base change
gives rise to a homomorphism
${{}^{\alpha}}f\colon{{}^{\alpha}}H\to{{}^{\alpha}}G$ of $k^{\prime}$-groups.
Suppose $G=(\mathbb{G}_{m})_{k}$ is the multiplicative group over $k$. Then
${{}^{\alpha}G}\cong(\mathbb{G}_{m})_{k^{\prime}}$ is the multiplicative group
over $k^{\prime}$ (the ring $A^{\prime}\otimes_{\alpha}k$ has the same units
as $A^{\prime}$). Combining this observation with the previous one, we see
that for each cocharacter $\lambda\in Y_{G}$, we obtain a cocharacter
${{}^{\alpha}}\lambda\in Y_{{{}^{\alpha}}G}$. We also see that a split torus
$T$ of $G$ gives rise to a split torus ${{}^{\alpha}}T$ of ${{}^{\alpha}}G$,
and the $k$-rank of $T$ is the same as the $k^{\prime}$-rank of
${{}^{\alpha}}T$.

Now suppose $T$ is a maximal split torus of $G$. Then the map
$\lambda\mapsto{{}^{\alpha}}\lambda$ is an isomomorphism of abelian groups
$Y_{T}\to Y_{{{}^{\alpha}}T}$, and hence induces an isomorphism
$Y_{T}({\mathbb{K}})\to Y_{{{}^{\alpha}}T}({\mathbb{K}})$, which we also
denote by $\lambda\mapsto{{}^{\alpha}}\lambda$. The map
$\chi\mapsto{{}^{\alpha}}\chi$ is an isomomorphism of abelian groups $X_{T}\to
X_{{{}^{\alpha}}T}$, and we have
$\langle{{}^{\alpha}}\lambda,{{}^{\alpha}}\chi\rangle=\langle\lambda,\chi\rangle$
for all $\lambda\in Y_{T}$ and all $\chi\in X_{T}$.

Let $\lambda\in Y_{T}({\mathbb{K}})$ and choose a $G$-equivariant embedding of
$G$ in a rational $G$-module $V$. Base change gives an
${{}^{\alpha}}G$-equivariant embedding of ${{}^{\alpha}}G$ in the rational
${{}^{\alpha}}G$-module ${{}^{\alpha}}V$. The weights of ${{}^{\alpha}}V$ with
respect to ${{}^{\alpha}}T$ are the characters of the form
${{}^{\alpha}}\chi$, where $\chi$ runs over the set of weights of $V$ with
respect to $T$. We see that if $\lambda^{\prime}$ is a cocharacter
approximation to $\lambda$ with respect to $V$ then
${{}^{\alpha}}\lambda^{\prime}$ is a cocharacter approximation to
${{}^{\alpha}}\lambda$ with respect to ${}^{\alpha}V$. We have
$P_{{{}^{\alpha}}\lambda}={{}^{\alpha}}P_{\lambda}$,
$L_{{{}^{\alpha}}\lambda}={{}^{\alpha}}L_{\lambda}$ and
$U_{{{}^{\alpha}}\lambda}={{}^{\alpha}}U_{\lambda}$ for any $\lambda\in
Y_{G}({\mathbb{K}})$; this holds if $\lambda\in Y_{G}$ by [14, Lem. 2.1.5] and
the discussion that precedes it, and for arbitrary $\lambda$ we get the result
by taking cocharacter approximations. It follows that
$\psi_{\alpha}(P_{\lambda}(k))\subseteq P_{{{}^{\alpha}}\lambda}(k^{\prime})$,
$\psi_{\alpha}(L_{\lambda}(k))\subseteq L_{{{}^{\alpha}}\lambda}(k^{\prime})$
and $\psi_{\alpha}(U_{\lambda}(k))\subseteq
U_{{{}^{\alpha}}\lambda}(k^{\prime})$ for all $\lambda\in
Y_{G}({\mathbb{K}})$.

######  Lemma 5.1.

Suppose $\alpha\colon k\to k^{\prime}$ is a homomorphism of fields. Then the
map $\lambda\mapsto{{}^{\alpha}}\lambda$ gives rise to well-defined injective
maps $Y_{G}({\mathbb{K}})\to Y_{{{}^{\alpha}}G}({\mathbb{K}})$ and
$V_{G}({\mathbb{K}})\to V_{{{}^{\alpha}}G}({\mathbb{K}})$, and the latter map
is a linear map of vector edifices.

###### Proof.

Since we have already observed bijectivity and linearity at the level of
apartments above, the essential point is well-definedness. What we need
follows from functoriality of base change: if $T_{1},T_{2}$ are maximal split
tori of $G$ and $g\in G(k)$ is such that $gT_{1}g^{-1}=T_{2}$, then we have
$\psi_{\alpha}(g)({{}^{\alpha}}T_{1})\psi_{\alpha}(g)^{-1}={{}^{\alpha}}T_{2}$.
Similarly, if $g\cdot\lambda_{1}=\lambda_{2}$ for $\lambda_{1}\in
Y_{T_{1}}({\mathbb{K}})$ and $\lambda_{2}\in Y_{T_{2}}({\mathbb{K}})$, then
$\psi_{\alpha}(g)\cdot{{}^{\alpha}}\lambda_{1}={{}^{\alpha}}\lambda_{2}$. We
can now check that the map $\lambda\mapsto{{}^{\alpha}}\lambda$ behaves well
with respect to the relations $\sim$ and $\approx$ before and after base
change. ∎

######  Remark 5.2.

Since it is not necessarily true that every maximal split torus of
${{}^{\alpha}}G$ is of the form ${{}^{\alpha}}T$ for some maximal split torus
$T$ of $G$ (for example, take $\alpha$ a Frobenius endomorphism of a non-
perfect field $k$), the maps in Lemma 5.1 are not necessarily surjective.

######  Definition 5.3.

Suppose $\alpha\colon k\to k^{\prime}$ is a homomorphism of fields, and $G$ is
a $k$-group. We denote the corresponding injective linear map of vector
edifices by

$\nu_{\alpha}:=\nu_{G,\alpha}\colon V_{G}({\mathbb{K}})\to
V_{{{}^{\alpha}}G}({\mathbb{K}}).$

We note in particular that this formalism allows us to identify the vector
edifice $V_{G}({\mathbb{K}})$ as a subset of the vector edifices
$V_{G_{k_{s}}}({\mathbb{K}})$ and $V_{G_{\overline{k}}}({\mathbb{K}})$, which
is helpful for later arguments on metrics.

######  Lemma 5.4.

Suppose $G$ and $H$ are $k$-groups, and let $\alpha\colon k\to k^{\prime}$ be
a field homomorphism. For any homomorphism $f\colon G\to H$, we have
$\kappa_{({{}^{\alpha}}f)}\circ\nu_{G,\alpha}=\nu_{H,\alpha}\circ\kappa_{f}$.

###### Proof.

This is basically just functoriality of base change: it is clear that for
$\lambda\in Y_{G}$ we have
${{}^{\alpha}}(f\circ\lambda)={{}^{\alpha}}f\circ{{}^{\alpha}}\lambda$, and
the result follows by chasing through the definitions. ∎

###  5.2. Field automorphisms and vector edifice automorphisms

First observe that if $\alpha\colon k\to k^{\prime}$ is an isomorphism of
fields and $G$ is a $k$-group, then the linear maps $\nu_{G,\alpha}$ and
$\nu_{{{}^{\alpha}}G,\alpha^{-1}}$ are inverses of each other, and hence
$V_{G}({\mathbb{K}})$ and $V_{{{}^{\alpha}}G}({\mathbb{K}})$ are isomorphic.

We also wish to consider the possibility that ${{}^{\alpha}}G$ and $G$ are
isomorphic as $k$-groups. For ease of exposition, suppose
$\alpha\in\operatorname{Aut}(k)$ — i.e., $\alpha$ is given to us as a field
_automorphism_ — so that $G$ and ${{}^{\alpha}}G$ are both $k$-groups. Then if
there exists an isomorphism $f\colon{{}^{\alpha}}G\to G$, we obtain a linear
automorphism

$\kappa_{\alpha,f}:=\kappa_{f}\circ\nu_{\alpha}$

of $V_{G}({\mathbb{K}})$.

We discuss this further in [9]. Here we give two examples of this
construction, showing how it captures naturally occurring phenomena.

######  Example 5.5.

Suppose $\alpha\in\operatorname{Aut}(k)$ and let $k^{\alpha}$ be the fixed
field of $\alpha$. Suppose that $G$ has a $k^{\alpha}$-descent — that is, that