###  5.3 Lie algebras over real closed fields

Let $\mathbb{K}$ be a subfield of $\mathbb{R}$ and $\mathbb{F}$ be a real
closed fields with $\mathbb{K}\subseteq\mathbb{F}$. In this section we
additionally assume that $\mathbb{K}$ is real closed. Let
$\mathfrak{g}_{\mathbb{K}}\subseteq\mathbb{K}^{n\times n}$ be the
$\mathbb{K}$-points of the Lie algebra of a connected semisimple algebraic
group $G$ defined over $\mathbb{K}$. Let $\mathfrak{g}_{\mathbb{R}}$ and
$\mathfrak{g}_{\mathbb{F}}$ be the semialgebraic extensions. The definitions
of Section 5.2 apply also to $\mathfrak{g}_{\mathbb{F}}$:

An abelian subalgebra $\mathfrak{h}\subseteq\mathfrak{g}_{\mathbb{F}}$ is
called _Cartan subalgebra_ if
$\mathfrak{h}=\operatorname{Nor}_{\mathfrak{g}_{\mathbb{F}}}(\mathfrak{h})$.
An abelian subalgebra $\mathfrak{a}\subseteq\mathfrak{g}_{\mathbb{F}}$ is
called _$\mathbb{F}$ -split_ if
$\operatorname{ad}(X)\colon\mathfrak{g}_{\mathbb{F}}\to\mathfrak{g}_{\mathbb{F}}$
is diagonalizable over $\mathbb{F}$ for every $X\in\mathfrak{a}$. Let
$r_{\mathbb{F}}(\mathfrak{g}_{\mathbb{F}})$ be the maximal dimension of such
an $\mathbb{F}$-split abelian subalgebra and denote by
$V(\mathfrak{g}_{\mathbb{F}})$ the set of all $\mathbb{F}$-split abelian
subalgebras with
$\dim(\mathfrak{a})=r_{\mathbb{F}}(\mathfrak{g}_{\mathbb{F}})$. A _maximally
$\mathbb{F}$-split Cartan subalgebra_ is a Cartan subalgebra containing an
element of $V(\mathfrak{g}_{\mathbb{F}})$ as a subset. We denote by
$\mathcal{C}(\mathfrak{g}_{\mathbb{F}})$ the set of all maximally
$\mathbb{F}$-split Cartan subalgebras. We will now use the transfer principle
to relate $\mathbb{F}$-split algebras to the real subalgebras studied in the
previous section.

######  Lemma 5.15.

Whenever $\mathbb{F}$ and $\mathbb{K}$ are two real closed fields with
$\mathbb{K}\subseteq\mathbb{F}$ and $\mathbb{K}\subseteq\mathbb{R}$, then

  * (1)

$r_{\mathbb{K}}(\mathfrak{g}_{\mathbb{K}})=r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})=r_{\mathbb{F}}(\mathfrak{g}_{\mathbb{F}})$.

  * (2)

$G_{\mathbb{K}}$ acts transitively on $\mathcal{C}(\mathfrak{g}_{\mathbb{K}})$
and $V(\mathfrak{g}_{\mathbb{K}})$ and those two sets are non-empty.

  * (3)

$\mathcal{C}(\mathfrak{g}_{\mathbb{K}})$ contains a $\theta$-stable element,
i.e. there is $\mathfrak{h}\in\mathcal{C}(\mathfrak{g}_{\mathbb{K}})$ such
that $\theta(\mathfrak{h})=\mathfrak{h}$.

  * (4)

Let $\mathfrak{a}$ and $\mathfrak{h}$ be subalgebras of
$\mathfrak{g}_{\mathbb{K}}$. Then

$\displaystyle\mathfrak{a}\in V(\mathfrak{g}_{\mathbb{K}})$
$\displaystyle\iff\mathfrak{a}_{\mathbb{R}}\in
V(\mathfrak{g}_{\mathbb{R}})\iff\mathfrak{a}_{\mathbb{F}}\in
V(\mathfrak{g}_{\mathbb{F}})$
$\displaystyle\mathfrak{h}\in\mathcal{C}(\mathfrak{g}_{\mathbb{K}})$
$\displaystyle\iff\mathfrak{h}_{\mathbb{R}}\in\mathcal{C}(\mathfrak{g}_{\mathbb{R}})\iff\mathfrak{h}_{\mathbb{F}}\in\mathcal{C}(\mathfrak{g}_{\mathbb{F}}).$

###### Proof.

Our first goal is to see $V(\mathfrak{g}_{\mathbb{K}})$ as a semi-algebraic
set. Let $k$ be any field of characteristic $0$ containing $\mathbb{K}$. Let
$\mathcal{L}$ be a semisimple $k$-algebra for any field $k$, such that
$\mathcal{L}$ is also a semi-algebraic set defined over $\mathbb{K}$. Let
$r\leq n:=\operatorname{dim}_{k}(\mathcal{L})$. The set
$\operatorname{Grass}_{r}(\mathcal{L})$ can be identified with an algebraic
subset of $k^{n\times n}$, namely

$\varphi\colon\operatorname{Grass}_{r}(\mathcal{L})\to\left\\{A\in k^{n\times
n}\colon A{{}^{\mathsf{T}}\\!}=A,A^{2}=A,\operatorname{Tr}(A)=r\right\\}$

sends the $k$-dimensional subspace $V$ to the orthogonal projection
$\mathcal{L}\to V$, see [BCR98, Theorem 3.4.4]. Moreover, for any
$A\in\varphi(\operatorname{Grass}_{r}(\mathcal{L}))$ we have the description

$\varphi^{-1}(A)=\\{v\in\mathcal{L}\colon Av=v\\}.$

An abelian subalgebra $\mathfrak{a}\subseteq\mathcal{L}$ is $k$-split exactly
when $\operatorname{ad}(x)$ is diagonalizable over $k$ for all
$x\in\mathfrak{a}$. It is enough to require that for a basis
$\\{v_{1},\ldots,v_{r}\\}$ of $\mathfrak{a}$, the maps
$\operatorname{ad}(v_{i})\colon\mathcal{L}\to\mathcal{L}$ are diagonalizable
over $k$. A linear map with matrix $M=(M_{ij})$ is diagonalizable over $k$ if
and only if its characteristic polynomial decomposes into linear factors,
which can be written as a first-order formula:

$f(M)=\exists x_{1},\ldots,x_{n}\colon\forall
X\colon\det(M_{ij}-X\delta_{ij})=\prod_{i=1}^{n}(X-x_{i}).$

We can write the statement “$\mathfrak{a}=\varphi^{-1}(A)$ is a $k$-split
abelian subalgebra” as a first-order formula

$\displaystyle\exists v_{1},\ldots,v_{n}\in\mathcal{L}\colon\left(\forall
a_{1},\ldots,a_{n}\colon\sum_{i=1}^{n}a_{i}v_{i}=0\to
a_{1}=0\wedge\ldots\wedge a_{n}=0\right)\wedge$
$\displaystyle\bigwedge_{i=1}^{r}Av_{i}=v_{i}\
\wedge\bigwedge_{i,j=1}^{r}[v_{i},v_{j}]=0\ \wedge$
$\displaystyle\bigwedge_{\ell=1}^{r}\left(\exists
M_{11},M_{12},\ldots,M_{nn}\colon\bigwedge_{i=1}^{n}[v_{\ell},v_{i}]=\sum_{j=1}^{n}M_{ij}e_{j}\,\rightarrow
f(M)\right),$

in words: there exists a basis of $\mathcal{L}$ whose first $r$ vectors form a
basis of $\mathfrak{a}=\varphi^{-1}(A)$, such that $\mathfrak{a}$ is abelian
and for every $\ell\in\\{1,\ldots,r\\}$ $\operatorname{ad}(v_{\ell})$ with
matrix $M$ in this basis is diagonalizable over $k$.

By quantifier elimination we get an equivalent first-order statement $g(A)$
without quantifiers and we can write

$\varphi(V(\mathcal{L})_{r})=\\{A\in\varphi(\operatorname{Grass}_{r}(\mathcal{L}))\colon
g(A)\\}$

as a semi-algebraic set defined by polynomials with coefficients in $k$, where
$V(\mathcal{L})_{r}$ denotes the set of all $k$-split abelian subalgebras of
dimension $r$.

Now for $k=\mathbb{R}$ we know that
$V(\mathfrak{g}_{\mathbb{R}})=V(\mathfrak{g}_{\mathbb{R}})_{r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})}$
is non-empty, but $V(\mathfrak{g}_{\mathbb{R}})_{r}$ is empty for any
$r>r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})$. We consider the first-order
formula

$\exists A\in\varphi(V(\mathfrak{g}_{k})_{r}),$

which is defined over $\mathbb{K}$ since $\mathfrak{g}_{\mathbb{R}}$ and hence
the brackets are defined over $\mathbb{K}$. Since the formula is satisfied for
$k=\mathbb{R}$ and $r=r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})$, we conclude
by the transfer principle that it also holds for $k=\mathbb{K}$ and
$k=\mathbb{F}$, i.e.
$V(\mathfrak{g}_{\mathbb{K}})=V(\mathfrak{g})_{r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})}\neq\emptyset$
and
$V(\mathfrak{g}_{\mathbb{F}})=V(\mathfrak{g}_{\mathbb{F}})_{r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})}\neq\emptyset$.
For any $r>r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})$, we know that the
formula is not satisfied for $k=\mathbb{R}$, therefore by the transfer
principle it is also not satisfied for $k=\mathbb{K}$ and $k=\mathbb{F}$, i.e.
$V(\mathfrak{g}_{\mathbb{K}})_{r}=\emptyset$ and
$V(\mathfrak{g}_{\mathbb{F}})_{r}=\emptyset$ for any
$r>r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})$. It follows that
$r_{\mathbb{F}}(\mathfrak{g}_{\mathbb{F}})=r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})=r_{\mathbb{K}}(\mathfrak{g}_{\mathbb{K}})$,
which is statement (1) of the Lemma we are proving.

We describe

$\displaystyle\mathcal{C}(\mathfrak{g}_{k})=\left\\{\mathfrak{h}\in\operatorname{Grass}_{r_{k}(\mathfrak{g}_{k})}(\mathfrak{g}_{k})\colon\begin{matrix}[\mathfrak{h},\mathfrak{h}]=0,\
\operatorname{Nor}_{\mathfrak{g}_{k}}(\mathfrak{h})=\mathfrak{h},\\\
\exists\mathfrak{a}\in
V(\mathfrak{g}_{k})\colon\mathfrak{a}\subseteq\mathfrak{h}\end{matrix}\right\\}$

similarly. Let $v_{1},\ldots,v_{n}$ again be a basis of a semisimple
$k$-algebra $\mathcal{L}$ such that $v_{1},\ldots,v_{r}$ is a basis of a
subalgebra $\mathfrak{a}\subseteq\mathcal{L}$. We have
$\operatorname{Nor}_{\mathcal{L}}(\mathfrak{a})=\mathfrak{a}$ whenever
$[v_{i},v_{j}]\in\mathfrak{a}\ \forall
i\in\\{1,\ldots,n\\},j\in\\{1,\ldots,r\\}$, i.e. given
$\mathfrak{a}=\varphi^{-1}(A)$ we have

$\displaystyle h(A)\colon\quad$
$\displaystyle\bigwedge_{i=1}^{n}\bigwedge_{j=1}^{r}A[v_{i},v_{j}]=[v_{i},v_{j}].$

We have that $\varphi^{-1}(A)\in\mathcal{C}(\mathcal{L})$ if and only if the
following first-order formula holds

$\displaystyle\exists v_{1},\ldots,v_{n}\in\mathcal{L}\colon\left(\forall
a_{1},\ldots,a_{n}\colon\sum_{i=1}^{n}a_{i}v_{i}=0\to\bigwedge_{i=1}^{n}a_{i}=0\right)\wedge$
$\displaystyle\bigwedge_{i=1}^{r}Av_{i}=v_{i}\ \ \wedge\ \
\bigwedge_{i,j=1}^{r}[v_{i},v_{j}]=0\ \ \wedge\ \ h(A)\ \ \wedge$
$\displaystyle\exists B\in V(\mathcal{L})\colon\forall v\in\mathcal{L}\colon
Bv=v\to Av=v.$

Again we use quantifier elimination to to get an equivalent statement $s(A)$
without quantifiers. Thus, $\mathcal{C}(\mathcal{L})$ can be identified with
the semialgebraic set

$\varphi(\mathcal{C}(\mathcal{L}))=\\{A\in\varphi(\operatorname{Grass}_{r_{k}(\mathcal{L})}(\mathcal{L}))\colon
s(A)\\}.$

From the theory of real Lie groups we know by Lemma 5.14(3) that
$\mathcal{C}(\mathfrak{g}_{\mathbb{R}})$ is non-empty and since $\exists
A\in\varphi(\mathcal{C}(\mathfrak{g}_{k}))$ is a first-order statement,
defined over $\mathbb{K}$, we can use the transfer principle to conclude that
$\mathcal{C}(\mathfrak{g}_{\mathbb{K}})$ and
$\mathcal{C}(\mathfrak{g}_{\mathbb{F}})$ are also non-empty. Statement (4)
follows from the semi-algebraic description of the sets
$V(\mathfrak{g}_{\mathbb{K}})$ and $\mathcal{C}(\mathfrak{g}_{\mathbb{K}})$
and the transfer principle.

The group $G_{k}$ acts by conjugation on
$\operatorname{Grass}_{r}(\mathfrak{g}_{k})$. The corresponding action on
$\varphi(\operatorname{Grass}_{r}(\mathfrak{g}_{k}))$ is given by
$g.A=A\circ\operatorname{Ad}(g^{-1})$, where $g\in G_{k}$,
${A\in\varphi(\operatorname{Grass}_{r}(\mathfrak{g}_{k}))}$ and
$\operatorname{Ad}(g)\colon\mathfrak{g}_{k}\to\mathfrak{g}_{k},X\mapsto
gXg^{-1}$.

We know that this action is transitive on $V(\mathfrak{g}_{\mathbb{R}})$ and
$\mathcal{C}(\mathfrak{g}_{\mathbb{R}})$ by Lemma 5.14(2). As all involved
sets are semi-algebraic, we can formulate transitivity as a first-order
formula,

$\displaystyle\forall A,B\in\varphi(V(\mathfrak{g}_{k}))\ \exists g\in
G\colon\forall v\in\mathfrak{g}_{k}\colon A(g^{-1}vg)=Bv.$

and conclude that $G_{\mathbb{K}}$ acts transitively on
$V(\mathfrak{g}_{\mathbb{K}})$ and $\mathcal{C}(\mathfrak{g}_{\mathbb{K}})$,
concluding the proof of (2).

Finally, for
$\theta\colon\mathfrak{g}_{k}\to\mathfrak{g}_{k},X\mapsto-X{{}^{\mathsf{T}}\\!}$
and $A\in\varphi(\mathcal{C}(\mathfrak{g}_{k}))$, we note that
$v\in\theta\left(\varphi^{-1}(A)\right)$ if and only if
$A(\theta(v))=\theta(v)$, that is $\theta A\theta v=v$. The condition
$\theta(\mathfrak{a})=\mathfrak{a}$ therefore corresponds to $A=\theta
A\theta$. We know by Lemma 5.14(3) that the first-order formula

$\exists A\in\varphi(\mathcal{C}(\mathfrak{g}_{k}))\colon\forall
v\in\mathfrak{g}_{k}\colon Av=\theta A\theta v$

is true for $k=\mathbb{R}$ and conclude that it therefore is also true for
$k=\mathbb{K}$, proving (3). ∎

###  5.4 Split tori of algebraic groups over real closed fields

Let $G$ be a semi-simple connected algebraic group which is invariant under
transposition. Let $\mathbb{K}$ and $\mathbb{F}$ be real closed fields with
$\mathbb{K}\subseteq\mathbb{R}\cap\mathbb{F}$. Let
$\mathfrak{g}_{\mathbb{K}}\subseteq\mathbb{K}^{n\times n}$ be the
$\mathbb{K}$-points of the Lie algebra. Let $\mathfrak{g}_{\mathbb{R}}$ and
$\mathfrak{g}_{\mathbb{F}}$ be the semialgebraic extensions, then
$\mathfrak{g}_{\mathbb{R}}$ is also the Lie algebra of $G_{\mathbb{R}}$.

All subfields $\mathbb{K}$ of $\mathbb{R}$ are dense in $\mathbb{R}$, in the
sense that $\overline{\mathbb{K}}=\mathbb{R}$ where $\overline{\mathbb{K}}$ is
the closure of $\mathbb{K}$ in the usual topology of $\mathbb{R}$. The
following generalization of this fact will be used in the proof of Theorem
5.17, the main result of this section.

######  Lemma 5.16.

Let $A\subseteq\mathbb{R}^{n}$ be an algebraic set defined over $\mathbb{K}$.
Then we have $\overline{A_{\mathbb{K}}}=A_{\mathbb{R}}$ in the usual
$\mathbb{R}^{n}$ topology.

###### Proof.

We first note that the algebraic set $A_{\mathbb{R}}$ is Zariski-closed and
therefore also closed in the usual topology of $\mathbb{R}^{n}$,
$\overline{A_{\mathbb{R}}}=A_{\mathbb{R}}$. We know that
$A_{\mathbb{K}}\subseteq A_{\mathbb{R}}$ and therefore
$\overline{A_{\mathbb{K}}}\subseteq\overline{A_{\mathbb{R}}}=A_{\mathbb{R}}$.

On the other hand, let $x\in A_{\mathbb{R}}$. Since
$\overline{\mathbb{K}}=\mathbb{R}$, there are $y_{k}\in\mathbb{K}^{n}$ and
$\varepsilon_{k}>0$ with $|y_{k}-x|<\varepsilon_{k}$ and $\varepsilon_{k}\to
0$ as $k\to\infty$. Now we have the following first-order formula

$\varphi(y,\varepsilon)\colon\quad\exists z\in A\colon\lVert
z-y\rVert<\varepsilon.$

For every $k\in\mathbb{N}$, the formula $\varphi(y_{k},\varepsilon_{k})$ is
true over $\mathbb{R}$, we can just take $z=x$ for every $k$. By the transfer
principle it is also true over $\mathbb{K}$, which means that we have
$z_{k}\in A_{\mathbb{K}}\colon\lVert z_{k}-y_{k}\rVert<\varepsilon_{k}.$ We
conclude

$\lVert z_{k}-x\rVert\leq\lVert z_{k}-y_{k}\rVert+\lVert
y_{k}-x\rVert<2\varepsilon_{k}\to 0$

as $k\to\infty$, i.e. $x\in\overline{A_{\mathbb{K}}}$. ∎

By Subsection 4.2.1, a torus may be split with respect to some field, while
not being split in a subfield. We now prove that this does not happen as long
as all involved fields are real closed.

######  Theorem 5.17.

Any maximal $\mathbb{K}$-split torus $S<G$ is maximal $\mathbb{F}$-split.
Moreover, there is a maximal $\mathbb{F}$-split torus $S$ so that
$g{{}^{\mathsf{T}}\\!}=g$ for all $g\in S$.

###### Proof.

We first find a maximal $\mathbb{K}$-split torus $T^{s}$ with
$g{{}^{\mathsf{T}}\\!}=g$ for all $g\in T^{s}$.

Lemma 5.15(3) states that there is a maximally $\mathbb{K}$-split Cartan
subalgebra $\mathfrak{h}\in\mathcal{C}(\mathfrak{g}_{\mathbb{K}})$ such that
$\theta(\mathfrak{h})=\mathfrak{h}$. We consider the complexification
$\mathfrak{h}_{\mathbb{C}}=\mathbb{C}\otimes_{\mathbb{K}}\mathfrak{h}=\mathfrak{h}_{\mathbb{R}}\oplus
i\mathfrak{h}_{\mathbb{R}}\subseteq\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}_{\mathbb{R}}\oplus
i\mathfrak{g}_{\mathbb{R}}$, which by Lemma 5.10(ii) is also a Cartan
subalgebra, in the sense that $\mathfrak{h}_{\mathbb{C}}$ is abelian and
$\operatorname{Nor}_{\mathfrak{g}_{\mathbb{C}}}(\mathfrak{h}_{\mathbb{C}})=\mathfrak{h}_{\mathbb{C}}$.
Then

$T_{\mathbb{C}}:=\operatorname{Nor}_{G_{\mathbb{C}}}(\mathfrak{h}_{\mathbb{C}})^{\circ}=\\{g\in
G_{\mathbb{C}}\colon\operatorname{Ad}(g)(\mathfrak{h}_{\mathbb{C}})=\mathfrak{h}_{\mathbb{C}}\\}^{\circ}.$

forms a Zariski-connected closed subgroup of $G_{\mathbb{C}}$ with Lie algebra
$\operatorname{Nor}_{\mathfrak{g}_{\mathbb{C}}}(\mathfrak{h}_{\mathbb{C}})=\mathfrak{h}_{\mathbb{C}}$.
By [Bor91, Lemma 18.5], $T_{\mathbb{C}}$ is defined over $\mathbb{K}$. Since
$\mathfrak{h}_{\mathbb{C}}$ is finite dimensional, $T_{\mathbb{C}}$ can be
written as the zero-set of finitely many polynomials and is therefore the
$\mathbb{C}$-points of an algebraic group $T$.

Since $\mathfrak{g}_{\mathbb{C}}$ is semisimple and
$\mathfrak{h}_{\mathbb{C}}$ is a Cartan subalgebra, $\operatorname{ad}(H)$ is
simultaneously diagonalizable for all $H\in\mathfrak{h}_{\mathbb{C}}$ Lemma
5.10(v). For $H\in\mathfrak{h}_{\mathbb{C}}$, $\exp(H)\in T_{\mathbb{C}}$ and
$\operatorname{Ad}(\exp(H))=\exp(\operatorname{ad}(H))$ is diagonalizable.
Possibly not all elements in $T_{\mathbb{C}}$ are of the form $\exp(H)$ with
$H\in\mathfrak{h}_{\mathbb{C}}$, but $\exp(\mathfrak{h}_{\mathbb{C}})$ is an
open neighborhood of the identity and thus generates $T_{\mathbb{C}}$. For any
$g\in T_{\mathbb{C}}$ there are
$H_{1},\ldots,H_{n}\in\mathfrak{h}_{\mathbb{C}}$ such that

$\displaystyle\operatorname{Ad}(g)$
$\displaystyle=\operatorname{Ad}\left(\prod_{i=1}^{n}\exp(H_{i})\right)=\operatorname{Ad}\left(\exp\left(\sum_{i=1}^{n}H_{i}\right)\right)$
$\displaystyle=\exp\left(\operatorname{ad}\left(\sum_{i=1}^{n}H_{i}\right)\right)=\exp\left(\sum_{i=1}^{n}\operatorname{ad}\left(H_{i}\right)\right),$

where we used that $\mathfrak{h}_{\mathbb{C}}$ is abelian. Since
$\operatorname{ad}(\mathfrak{h}_{\mathbb{C}})$ is simultaneously
diagonalizable, $\operatorname{Ad}(g)$ is diagonalizable. Similarly we have
$\operatorname{Ad}(g)\operatorname{Ad}(h)=\operatorname{Ad}(h)\operatorname{Ad}(g)$
for all $g,h\in T_{\mathbb{C}}$. Therefore $\operatorname{Ad}(T_{\mathbb{C}})$
is simultaneously diagonalizable, meaning $T_{\mathbb{C}}$ is diagonalizable
by [Bor91, Proposition 8.4]. Since $T_{\mathbb{C}}$ is connected, $T$ is a
torus by [Bor91, Proposition 8.5].

If we restrict to symmetric matrices

$T^{s}=\\{g\in T\colon\sigma(g)=g^{-1}\\}$

we see that
$\operatorname{Lie}(T^{s}_{\mathbb{R}})=\operatorname{Lie}(T_{\mathbb{R}})\cap\mathfrak{p}=\mathfrak{h}_{\mathbb{R}}\cap\mathfrak{p}$,
where $\mathfrak{g}_{\mathbb{R}}=\mathfrak{p}\oplus\mathfrak{k}$ is the Cartan
decomposition corresponding to the standard Cartan involution $\theta$.

Let $\mathfrak{a}\in V(\mathfrak{g}_{\mathbb{K}})$ be a maximal abelian
$\mathbb{K}$-split subalgebra of $\mathfrak{g}_{\mathbb{K}}$ with
$\mathfrak{a}\subseteq\mathfrak{h}$. By Lemma 5.15(4), we have
$\mathfrak{h}_{\mathbb{R}}\in\mathcal{C}(\mathfrak{g}_{\mathbb{R}})$ and
$\mathfrak{a}_{\mathbb{R}}\in V(\mathfrak{a}_{\mathbb{R}})$ with
$\mathfrak{a}_{\mathbb{R}}\subseteq\mathfrak{h}_{\mathbb{R}}$. By Proposition
5.13, $\mathfrak{h}_{\mathbb{R}}$ is a maximal $\mathbb{R}$-split Cartan
subalgebra and
$\mathfrak{a}_{\mathbb{R}}=\mathfrak{h}_{\mathbb{R}}\cap\mathfrak{p}$.

Thus
$\operatorname{Lie}(T^{s}_{\mathbb{R}})=\mathfrak{h}_{\mathbb{R}}\cap\mathfrak{p}$
is maximal $\mathbb{R}$-split (or maximally noncompact in the terminology of
Section 5.2). We conclude that $T^{s}_{\mathbb{R}}$ is a maximal
$\mathbb{R}$-split torus. Note that $T^{s}$ is defined over $\mathbb{K}$.

We now want to show that $T^{s}$ is in fact $\mathbb{K}$-split. We consider
the relative root system $\vphantom{\Phi}{}_{\mathbb{R}}\Phi:=\Phi(G,T^{s})$.
For any root $\alpha\colon T^{s}\to\mathbb{G}_{m}$,
$\alpha\in\leavevmode\nobreak\ \\!\\!\vphantom{\Phi}_{\mathbb{R}}\Phi$, we
know that
$\alpha_{\mathbb{R}}(T^{s}_{\mathbb{K}})\subseteq\mathbb{K}^{\times}$, since
$\alpha$ is defined by the property that
$\operatorname{Ad}(g)X=\alpha_{\mathbb{R}}(g)X$ for $g\in
T^{s}_{\mathbb{K}},X\in\mathfrak{g}_{\mathbb{K}}$. The graph of $\alpha$ is an
algebraic set

$\operatorname{graph}(\alpha)=\\{(x,z)\in T^{s}\times\mathbb{G}_{m}\colon
z=\alpha(x)\\}$

defined over $\mathbb{R}$, our goal is to show that it is actually defined
over $\mathbb{K}$.

Now $(x,z)\in\operatorname{graph}(\alpha)_{\mathbb{R}}$ if and only if $x\in
T^{s}_{\mathbb{R}}$ and $z=\alpha_{\mathbb{R}}(x)$. In view Lemma 5.16,
$T^{s}_{\mathbb{R}}=\overline{T^{s}_{\mathbb{K}}}$, so $x\in
T^{s}_{\mathbb{R}}$ is equivalent to saying that there is a sequence of
$x_{n}\in T^{s}_{\mathbb{K}}$ such that $x_{n}\to x$ as $n\to\infty$. Since
$\alpha_{\mathbb{R}}$ is continuous, we have $\alpha_{\mathbb{R}}(x_{n})\to z$
as $n\to\infty$. We conclude that
$(x,z)\in\operatorname{graph}(\alpha)_{\mathbb{R}}$ if and only if there is a
sequence
$(x_{n},\alpha_{\mathbb{R}}(x_{n}))\in\operatorname{graph}(\alpha)_{\mathbb{K}}$
with $(x_{n},\alpha_{\mathbb{R}}(x_{n}))\to(x,z)$ as $n\to\infty$, i.e.

$\overline{\operatorname{graph}(\alpha)_{\mathbb{K}}}=\operatorname{graph}(\alpha)_{\mathbb{R}},$

which means that $\operatorname{graph}(\alpha)_{\mathbb{K}}$ is dense in
$\operatorname{graph}(\alpha)_{\mathbb{R}}$, in particular Zariski-dense
(Zariski-open sets are also open in the usual topology and a set is dense if
every open set intersects it). Viewing $\operatorname{graph}(\alpha)$ as an
algebraic group, we can use [Zim84, Proposition 3.1.8] to conclude that
$\operatorname{graph}(\alpha)$ is defined over $\mathbb{K}$. Hence every
$\alpha\in\vphantom{\Phi}_{\mathbb{R}}\Phi$ is defined over $\mathbb{K}$,
indeed every multiplicative character is defined over $\mathbb{K}$. By [Bor91,
Corollary 8.2], this means that $T^{s}$ is $\mathbb{K}$-split. We have thus
found a maximal $\mathbb{K}$-split torus which satisfies
$g{{}^{\mathsf{T}}\\!}=g$ for all $g\in T^{s}$.

Next we will prove that
$\operatorname{rank}_{\mathbb{F}}(G)=\operatorname{rank}_{\mathbb{K}}(G)$ as
follows.

$\displaystyle\operatorname{rank}_{\mathbb{F}}(G)$ $\displaystyle\leq
r_{\mathbb{F}}(\mathfrak{g}_{\mathbb{F}})=r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})=\operatorname{rank}_{\mathbb{K}}(G)\leq\operatorname{rank}_{\mathbb{F}}(G)$

We recall that $\operatorname{rank}_{\mathbb{F}}(G)$ is the dimension of a
maximal $\mathbb{F}$-split torus and
$r_{\mathbb{F}}(\mathfrak{g}_{\mathbb{F}})$ is the dimension of an element in
$V(\mathfrak{g}_{\mathbb{F}})$. Indeed, the Lie algebra of any maximal
$\mathbb{F}$-split torus is abelian and $\mathbb{F}$-split, i.e. contained in
an element of $V(\mathfrak{g}_{\mathbb{F}})$. The first equality is Lemma
5.15(1). The second equality is what we did in this proof: we found the
maximal $\mathbb{K}$-split torus $T^{s}$ with
$\operatorname{Lie}(T^{s}_{\mathbb{R}})=\mathfrak{h}_{\mathbb{R}}\cap\mathfrak{p}\in
V(\mathfrak{g}_{\mathbb{R}})$, with dimension
$r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})$. The last inequality holds because
every $\mathbb{K}$-split torus is also $\mathbb{F}$-split.

Let $S$ be a maximal $\mathbb{K}$-split torus. Then $S$ is also
$\mathbb{F}$-split (but possibly not maximal). Let $T$ be a maximal
$\mathbb{F}$-split torus with $S\subseteq T$. Then
$\dim(S)=\operatorname{rank}_{\mathbb{K}}(G)=\operatorname{rank}_{\mathbb{F}}(G)=\dim(T)$
and since tori are connected $S=T$. Thus every maximal $\mathbb{K}$-split
torus $S$ is also maximal $\mathbb{F}$-split. ∎

######  Corollary 5.18.

Let $S<G$ be a maximal $\mathbb{K}$-split torus. Then the set
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ of $\mathbb{K}$-roots of $S$ in $G$
coincides with $\vphantom{\Phi}{}_{\mathbb{F}}\Phi$ and hence the set of
standard parabolic $\mathbb{K}$-subgroups coincides with the set of standard
parabolic $\mathbb{F}$-subgroups. In particular any $\mathbb{F}$-parabolic
subgroup is $G_{\mathbb{F}}$-conjugate to a parabolic $\mathbb{K}$-subgroup.

###### Proof.

The set of $\mathbb{K}$-roots is defined as
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi:=\Phi(S,G)$, where $S$ is a maximal
$\mathbb{K}$-split torus of $G$ [Bor91, 21.1]. Since $S$ is also
$\mathbb{F}$-split by Theorem 5.17, we have
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi:=\Phi(S,G)=\vphantom{\Phi}_{\mathbb{F}}\Phi$.

Following [Bor91, 21.11], we choose an ordering on
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ and fix the minimal parabolic
$\mathbb{K}$-subgroup $P$ associated to the positive roots in
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$. Any parabolic $\mathbb{K}$-subgroup
containing $P$ is called _standard parabolic_. The standard parabolic
$\mathbb{K}$-subgroups are in one-to-one correspondence with the subsets
$I\subseteq\vphantom{\Delta}_{\mathbb{K}}\Delta$ of the simple roots
$\vphantom{\Delta}{}_{\mathbb{K}}\Delta$ of
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ [Bor91, Proposition 21.12]. Since
$\vphantom{\Delta}{}_{\mathbb{F}}\Delta=\vphantom{\Delta}_{\mathbb{K}}\Delta$,
the standard parabolic $\mathbb{K}$-groups coincide with the standard
parabolic $\mathbb{F}$-groups. By [Bor91, Proposition 21.12], any parabolic
$\mathbb{F}$-group is conjugate to one and only one standard parabolic, by an
element in $G_{\mathbb{F}}$. In particular every $\mathbb{F}$-parabolic
subgroup is $G_{\mathbb{F}}$-conjugate to a $\mathbb{K}$-parabolic subgroup. ∎

##  6 Semialgebraic groups

Real Lie groups admit more decompositions than semisimple algebraic groups. In
this section we extend some of the Lie group decompositions to algebraic
groups over real closed fields. This includes the Iwasawa-decomposition $KAU$,
Cartan-decomposition $KAK$, Bruhat-decomposition $BWB$ as well as Kostant-
convexity, the Jacobson-Morozov Lemma and the study of rank 1 subgroups. We
believe that these results are of independent interest.

Our application of these results lies in the construction of an affine
$\Lambda$-building in Section 7 as well as the verification of its axioms in
Section 8. The Cartan-decomposition is crucial in the definition of the
building as well as in the verification of axiom (A3). The Iwasawa-
decomposition together with Kostant convexity allows us to verify the triangle
inequality. The Bruhat-decomposition is used in the verification of axioms
(A2), (A4) and (EC). Jacobson-Morozov shows up in the verification of axioms
(A2) and (EC).

Let $\mathbb{K}$ and $\mathbb{F}$ be real closed fields such that
$\mathbb{K}\subseteq\mathbb{R}\cap\mathbb{F}$. Let $G$ be a semisimple
connected self-adjoint algebraic $\mathbb{K}$-group777The whole theory could
also be formulated for a group $\tilde{G}$ that is defined as a semialgebraic
set, and that lies between the semialgebraic connected component of the
identity $G_{\mathbb{F}}^{\circ}$ of $G_{\mathbb{F}}$ and $G_{\mathbb{F}}$
itself, so $G_{\mathbb{F}}^{\circ}<\tilde{G}<G_{\mathbb{F}}$, see [BIPP23].
and $S$ a maximal $\mathbb{K}$-split torus, which we may assume to be self-
adjoint by Theorem 5.17. We consider the $\mathbb{K}$-points
$\mathfrak{g}_{\mathbb{K}}\subseteq\mathbb{K}^{n\times n}$ of the Lie algebra
$T_{e}G$. Let $\mathfrak{g}_{\mathbb{R}}$ be the semialgebraic extension of
$\mathfrak{g}_{\mathbb{K}}$. Then $\mathfrak{g}_{\mathbb{R}}$ is the Lie
algebra of the real Lie group $G_{\mathbb{R}}$. Since $G_{\mathbb{R}}$ is
self-adjoint, so is $\mathfrak{g}_{\mathbb{R}}$. The differential of the Lie
group homomorphism $G_{\mathbb{R}}\to
G_{\mathbb{R}},g\mapsto(g^{-1}){{}^{\mathsf{T}}\\!}$ is the standard Cartan-
involution
$\theta\colon\mathfrak{g}_{\mathbb{R}}\to\mathfrak{g}_{\mathbb{R}},X\mapsto-X^{T}$
and leads to the Cartan decomposition into two eigenspaces
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}\oplus\mathfrak{p}$ as in Section 5.1.
We define the algebraic $\mathbb{K}$-group

$K=G\cap\operatorname{SO}_{n}$

whose real points $K_{\mathbb{R}}$ is a real Lie group with Lie algebra

$\operatorname{Lie}(K_{\mathbb{R}})=\operatorname{Lie}(G_{\mathbb{R}}\cap\operatorname{SO}_{n}(\mathbb{R}))=\operatorname{Lie}(G_{\mathbb{R}})\cap\operatorname{Lie}(\operatorname{SO}_{n}(\mathbb{R}))=\mathfrak{g}_{\mathbb{R}}\cap\mathfrak{so}_{n}=\mathfrak{k}.$

Let $A_{\mathbb{K}}=(S_{\mathbb{K}})^{\circ}$ be the semialgebraic connected
component of $S_{\mathbb{K}}$ containing the identity. Let $A_{\mathbb{R}}$ be
the semialgebraic extension of $A_{\mathbb{K}}$. We denote the Lie algebra of
$A_{\mathbb{R}}$ by $\mathfrak{a}$. Note that unlike for $G$ and $K$, there
may not be an algebraic group $A$.

We note that $\mathfrak{a}\subseteq\mathfrak{p}$. By Theorem 5.17, $S$ is
maximal $\mathbb{K}$-split as well as maximal $\mathbb{R}$-split and hence
$\mathfrak{a}$ is maximal abelian in $\mathfrak{p}$. As in Section 5.1, we can
now define a root space decomposition

$\mathfrak{g}_{\mathbb{R}}=\mathfrak{g}_{0}\oplus\bigoplus_{\alpha\in\Sigma}\mathfrak{g}_{\alpha}$

where $\Sigma=\\{\alpha\in\mathfrak{a}^{\star}\colon\alpha\neq
0,\mathfrak{g}_{\alpha}\neq 0\\}$ is a root system.

We note that

$N_{\mathbb{R}}:=\operatorname{Nor}_{K_{\mathbb{R}}}(\mathfrak{a})=\left\\{k\in
K_{\mathbb{R}}\colon kXk^{-1}\in\mathfrak{a}\text{ for all
}X\in\mathfrak{a}\right\\}$

and

$M_{\mathbb{R}}:=\operatorname{Cen}_{K_{\mathbb{R}}}(\mathfrak{a})=\left\\{k\in
K_{\mathbb{R}}\colon kXk^{-1}=X\text{ for all }X\in\mathfrak{a}\right\\},$

are semialgebraic groups, since it suffices to verify the conditions for $X$
on a basis of $\mathfrak{a}$.

A choice of ordered basis $\Delta\subseteq\Sigma$ gives a total order on
$\Sigma$. We consider the Lie algebra

$\mathfrak{n}=\bigoplus_{\alpha>0}\mathfrak{g}_{\alpha}$

which is nilpotent by Lemma 5.6. Thus the exponential map and the logarithm
are polynomials and we can define the Lie group

$U_{\mathbb{R}}=\\{g\in G_{\mathbb{R}}\colon\log(g)\in\mathfrak{n}\\}$

which is the group of $\mathbb{R}$-points of an algebraic $\mathbb{K}$-group
$U$ defined the same way888In the theory of Lie groups, this group is often
denoted by $N$, while $U$ is more common in the setting of algebraic groups..

###  6.1 Examples

For the algebraic group $G=\operatorname{SL}_{n}$ with maximal
$\mathbb{K}$-split torus

$S=\left\\{\begin{pmatrix}\star&&\\\ &\ddots&\\\
&&\star\end{pmatrix}\in\operatorname{SL}_{n}\right\\}.$

we have

$\displaystyle K$ $\displaystyle=\operatorname{SO}_{n}$ $\displaystyle
A_{\mathbb{F}}$ $\displaystyle=\left\\{a=(a_{ij})\in S_{\mathbb{F}}\colon
a_{ii}>0\right\\}$ $\displaystyle U_{\mathbb{F}}$
$\displaystyle=\left\\{g=(g_{ij})\in\operatorname{SL}_{n}(\mathbb{F})\colon
g_{ii}=1,\,g_{ij}=0\text{ for }i>j\right\\}$ $\displaystyle N_{\mathbb{F}}$
$\displaystyle=\left\\{\text{ permutation matrices with entries in }\pm
1\right\\}$ $\displaystyle M_{\mathbb{F}}$
$\displaystyle=\left\\{a=(a_{ij})\in S_{\mathbb{F}}\colon a_{ii}\in\\{\pm
1\\}\right\\}$ $\displaystyle B_{\mathbb{F}}$
$\displaystyle=\left\\{g=(g_{ij})\in\operatorname{SL}_{n}(\mathbb{F})\colon
g_{ij}=0\text{ for }i>j\right\\}.$

###  6.2 Compatibility of the root systems

In Chapter 4.4 about algebraic groups, we defined the root system
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ relative to a $\mathbb{K}$-split torus
$S$ consisting of those $\alpha\in\hat{S}\setminus\\{1\\}$ for which

$\mathfrak{g}_{\alpha}^{(S)}=\\{X\in\mathfrak{g}\colon\operatorname{Ad}(s)X=\alpha(s)\cdot
X\text{ for all }s\in S\\}$

is non-zero. By Theorem 5.17, we have
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi=\vphantom{\Phi}_{\mathbb{R}}\Phi=\vphantom{\Phi}_{\mathbb{F}}\Phi$.
In this chapter we defined a root system $\Sigma$ consisting of those
$\alpha\in\mathfrak{a}^{\star}\setminus\\{0\\}$ for which

$\mathfrak{g}_{\alpha}=\\{X\in\mathfrak{g}_{\mathbb{R}}\colon\operatorname{ad}(H)X=\alpha(H)\cdot
X\text{ for all }H\in\mathfrak{a}\\}$

is non-empty. In this section we show that all these notions of root systems
and the various notions of spherical Weyl groups coincide. Let us first see
how to construct an algebraic character in
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ from a root in $\Sigma$.

######  Lemma 6.1.

Let $\alpha\in\Sigma$. Then the homomorphism

$\displaystyle\chi_{\alpha}\colon A_{\mathbb{R}}$
$\displaystyle\to\mathbb{R}_{>0}$ $\displaystyle\exp(H)$ $\displaystyle\mapsto
e^{\alpha(H)}$

is the restriction to $A_{\mathbb{R}}$ of an algebraic character in $\hat{S}$.

###### Proof.

We choose a basis of $\mathfrak{g}$ consistent with the root decomposition

$\mathfrak{g}=\mathfrak{g}_{0}\oplus\bigoplus_{\alpha\in\Sigma}\mathfrak{g}_{\alpha}.$

In this basis, the matrices in $\operatorname{Ad}(S_{\mathbb{R}})$ and
$\operatorname{ad}(\mathfrak{a})$ are diagonal. In fact any diagonal
entry999Note that $\mathfrak{g}_{\alpha}$ may not be one-dimensional, but then
all the diagonal entries of $\operatorname{ad}(H)$ corresponding to basis
vectors in $\mathfrak{g}_{\alpha}$ are equal, so it makes sense to talk about
$(\operatorname{ad}(H))_{\alpha\alpha}$. corresponding to the root
$\alpha\in\Sigma$ satisfies

$\displaystyle(\operatorname{ad}(H))_{\alpha\alpha}$ $\displaystyle=\alpha(H)$

for all $H\in\mathfrak{a}$. We define a character

$\displaystyle\chi(a)$
$\displaystyle:=(\operatorname{Ad}(a))_{\alpha\alpha}\quad\quad\text{for }a\in
S$

which is algebraic by its definition. Let $\chi_{\mathbb{R}}\colon
S_{\mathbb{R}}\to\mathbb{R}$ be the $\mathbb{R}$-points of $\chi$ and
$\chi_{\mathbb{R}}|_{A_{\mathbb{R}}}$ its restriction to $A_{\mathbb{R}}$. We
now claim that $\chi_{\alpha}=\chi_{\mathbb{R}}|_{A_{\mathbb{R}}}$: when
$a=\exp(H)\in A_{\mathbb{R}}$ for $H\in\mathfrak{a}$, we have

$\displaystyle\chi_{\mathbb{R}}(a)$
$\displaystyle=(\operatorname{Ad}(a))_{\alpha\alpha}=(\operatorname{Ad}(\exp(H)))_{\alpha\alpha}$
$\displaystyle=(\exp(\operatorname{ad}(H)))_{\alpha\alpha}=e^{\operatorname{ad}(H)_{\alpha\alpha}}=e^{\alpha(H)}=\chi_{\alpha}(a).$

∎

######  Lemma 6.2.

Let $\alpha\in\Sigma$. Then there is an $x_{\alpha}\in\mathfrak{a}$ such that
the homomorphism

$\displaystyle t_{\alpha}^{\mathbb{R}}\colon\mathbb{R}_{>0}$ $\displaystyle\to
A_{\mathbb{R}}$ $\displaystyle e^{s}$ $\displaystyle\mapsto\exp(sx_{\alpha})$

is the restriction of an algebraic one-parameter group $t_{\alpha}$ in
$X_{\star}(S)$, defined over $\mathbb{K}$. The non-degenerate bilinear form in
Proposition 4.6 is then given by

$b(\chi_{\alpha},t_{\beta})=\frac{2\langle\alpha,\beta\rangle}{\langle\beta,\beta\rangle}\in\mathbb{Z}.$

###### Proof.

Recall that the Killing form gives rise to a scalar product $B_{\theta}$ on
$\mathfrak{a}=\operatorname{Lie}(A_{\mathbb{R}})$, inducing an isomorphism
$\mathfrak{a}^{\star}\cong\mathfrak{a},\gamma\mapsto H_{\gamma}$ with the
defining property $B_{\theta}(H_{\gamma},H)=\gamma(H)$ for all
$H\in\mathfrak{a}$. The coroot

$\alpha^{\vee}=\frac{2}{B_{\theta}(H_{\alpha},H_{\alpha})}\alpha$

can be used to define

$x_{\alpha}:=H_{\alpha^{\vee}}=\frac{2}{B_{\theta}(H_{\alpha},H_{\alpha})}H_{\alpha}.$

We now define $t_{\alpha}^{\mathbb{R}}(e^{s})=\exp(sx_{\alpha})$ and note that

$\chi_{\alpha}^{\mathbb{R}}(t_{\alpha}^{\mathbb{R}}(e^{s}))=e^{s\alpha(x_{\alpha})}=(e^{s})^{2}.$

We have to show that $t_{\alpha}^{\mathbb{R}}$ is algebraic. Let
$\mathcal{A}(H_{\alpha})$ be the smallest algebraic subgroup whose Lie algebra
contains $H_{\alpha}$, [Bor91, 7.1]. Since $H_{\alpha}\in\mathbb{K}^{n\times
n}$, $\mathcal{A}(H_{\alpha})$ is defined over $\mathbb{K}$, [Bor91, 2.1(b)].
Note that for every $r\in\mathbb{G}_{m}$ there is a unique element
$a\in\mathcal{A}(H_{\alpha})$ with $\chi_{\alpha}(a)=r^{2}$. We define the map
$t_{\alpha}\colon\mathbb{G}_{m}\to S$ by its graph

$\operatorname{graph}(t_{\alpha})=\left\\{(r,a)\in\mathbb{G}_{m}\times S\colon
a\in\mathcal{A}(H_{\alpha}),\chi_{\alpha}(a)=r^{2}\right\\}.$

By Proposition 4.2, $t_{\alpha}$ is algebraic and defined over $\mathbb{K}$.
By the defining property of the bilinear form $b\colon\hat{S}\times
X_{\star}(S)\to\mathbb{Z}$ from Proposition 4.6, we have
$\chi_{\alpha}(t_{\beta}(r))=r^{b(\chi_{\alpha},t_{\beta})}$ for all
$\alpha,\beta\in\Sigma$. Over $\mathbb{R}$ we have for
$r=e^{s}\in\mathbb{R}_{>0}$

$\displaystyle\chi_{\alpha}^{\mathbb{R}}(t_{\beta}^{\mathbb{R}}(r))=\chi_{\alpha}^{\mathbb{R}}(\exp(sx_{\alpha}))=e^{s\alpha(x_{\beta})}=r^{\frac{2\langle\alpha,\beta\rangle}{\langle\beta,\beta\rangle}},$

as claimed. ∎

######  Proposition 6.3.

The root systems $\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ and $\Sigma$ are
isomorphic. The spherical Weyl groups $\vphantom{W}{}_{\mathbb{K}}W$ and
$N_{\mathbb{R}}/M_{\mathbb{R}}$ are isomorphic.

###### Proof.

We first find a compatible map $\vphantom{\Phi}{}_{\mathbb{K}}\Phi\to\Sigma$.
Let $\chi\in\vphantom{\Phi}_{\mathbb{K}}\Phi$. Since $S$ is
$\mathbb{K}$-split, $\chi$ is defined over $\mathbb{K}$ [Bor91, Corollary
8.2]. Hence we can consider the Lie group homomorphism
$\chi_{\mathbb{R}}\colon S_{\mathbb{R}}\to\mathbb{R}^{\times}$ and its
derivative
$\operatorname{d}_{e}\\!\chi_{\mathbb{R}}\colon\mathfrak{a}\to\mathbb{R}$,
which we claim to be an element in $\Sigma$. We first claim that
$\operatorname{d}_{e}\\!\chi_{\mathbb{R}}$ is nonzero, since otherwise
$\chi_{\mathbb{R}}$ would be locally constant, hence only take finitely many
values. But since $\chi_{R}(S_{\mathbb{R}})$ is Zariski dense in
$\chi(S)=\mathbb{G}_{m}$ [Bor91, Corollary 18.3], this cannot be the case.
Next, we claim that

$\mathfrak{g}_{\chi}:=\left(\mathfrak{g}_{\chi}^{(S)}\right)_{\mathbb{R}}\subseteq\mathfrak{g}_{\operatorname{d}_{e}\\!\chi_{\mathbb{R}}},$

which shows that $\vphantom{\Phi}{}_{\mathbb{K}}\Phi\to\Sigma$ is well
defined. Let $H\in\mathfrak{a}$, such that $a=\exp(H)\in
A_{\mathbb{R}}\subseteq S_{\mathbb{R}}$. Since $S$ is $\mathbb{K}$-split, we
have

$\operatorname{Ad}(\exp(H))=\operatorname{Id}\cdot\alpha_{\mathbb{R}}(\exp(H))$

as maps $\mathfrak{g}_{\chi}\to\mathfrak{g}_{\chi}$. In fact,
$\operatorname{Ad}\circ\exp=\operatorname{Id}\cdot(\chi_{\mathbb{R}}\circ\exp)$
as maps $\mathfrak{a}\to\operatorname{GL}(\mathfrak{g}_{\chi})$. Taking the
derivative at $0\in\mathfrak{a}$, we get

$\displaystyle\operatorname{ad}$
$\displaystyle=\operatorname{ad}\circ\operatorname{Id}=\operatorname{d}_{e}\\!\operatorname{Ad}\circ\operatorname{d}_{0}\exp=\operatorname{d}_{0}(\operatorname{Ad}\circ\exp)$
$\displaystyle=\operatorname{d}_{0}(\operatorname{Id}\cdot(\chi_{\mathbb{R}}\circ\exp))=\operatorname{Id}\cdot\operatorname{d}_{e}\\!\chi_{\mathbb{R}}\circ\operatorname{d}_{0}\exp=\operatorname{Id}\cdot\operatorname{d}_{e}\\!\chi_{\mathbb{R}}$

as maps $\mathfrak{a}\to\mathfrak{gl}(\mathfrak{g}_{\chi})$. This means that
for all $X\in\mathfrak{g}_{\chi}$ and all $H\in\mathfrak{a}$, we have
$\operatorname{ad}(H)X=\operatorname{d}_{e}\\!\chi_{\mathbb{R}}(H)\cdot X$ and
hence $X\in\mathfrak{g}_{\operatorname{d}_{e}\\!\chi_{\mathbb{R}}}$.

Now, we will show that the map $\alpha\mapsto\chi_{\alpha}$ in Lemma 6.1,
$\chi_{\alpha}$ now viewed as an algebraic character in $\hat{S}$, actually
sends $\Sigma\to\vphantom{\Phi}_{\mathbb{K}}\Phi$. Since

$(\chi_{\alpha})_{\mathbb{R}}(\exp(H))=e^{\alpha(H)},$

we can easily see that $\chi_{\alpha}\neq 1$, when $\alpha\neq 0$. Next we
will show that

$\mathfrak{g}_{\alpha}\subseteq\left(\mathfrak{g}_{\chi_{\alpha}}^{(S)}\right)_{\mathbb{R}}.$

For $a=\exp(H)\in A_{\mathbb{R}}$, we see from the above formula that

$\displaystyle\operatorname{Ad}(a)$
$\displaystyle=\exp(\operatorname{ad}(H))=\exp(\operatorname{Id}\cdot\alpha(H))=\operatorname{Id}\cdot
e^{\alpha(H)}=\operatorname{Id}\cdot\chi_{\alpha}(a)$

as functions on $\mathfrak{g}_{\alpha}$. Since $A_{\mathbb{R}}$ is Zariski
dense in $S$, $\operatorname{Ad}(s)X=\chi_{\alpha}(a)\cdot X$ for all $s\in S$
and $X\in\mathfrak{g}_{\alpha}$.

We note that the two maps between $\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ and
$\Sigma$ are inverses of each other, which follows from the fact that

$\left(\mathfrak{g}_{\alpha}^{(S)}\right)_{\mathbb{R}}\subseteq\mathfrak{g}_{\operatorname{d}_{e}\\!\alpha_{\mathbb{R}}}\subseteq\left(\mathfrak{g}_{\chi_{\operatorname{d}_{e}\\!\alpha_{\mathbb{R}}}}^{(S)}\right)_{\mathbb{R}}\quad\text{
for }\alpha\in\vphantom{\Phi}_{\mathbb{K}}\Phi$

and

$\mathfrak{g}_{\alpha}\subseteq\left(\mathfrak{g}_{\chi_{\alpha}}^{(S)}\right)_{\mathbb{R}}\subseteq\mathfrak{g}_{\operatorname{d}_{e}\\!(\chi_{\alpha})_{\mathbb{R}}}\quad\text{
for }\alpha\in\Sigma.$

We note that since
$[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]\subseteq\mathfrak{g}_{\alpha+\beta}$
the map $\vphantom{\Phi}{}_{\mathbb{K}}\Phi\to\Sigma$ extends to an
isomorphism on their vector spaces. For both
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ and $\Sigma$, the Weyl groups
$\vphantom{W}{}_{\mathbb{K}}W$ and $W_{s}=N_{\mathbb{R}}/M_{\mathbb{R}}$ are
generated by reflections along hyperplanes perpendicular to the roots [Bor91,
Theorem 21.2] and [Kna96, Proposition 7.32]. This implies that the scalar
products of two roots are preserved under the map
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi\to\Sigma$ and hence the two root systems
are isomorphic. ∎

We can also conclude that the various Weyl groups that can be defined
coincide.

######  Proposition 6.4.

The following definitions of Weyl groups are isomorphic.

  * (i)

The Weyl group $W$ generated by reflections in roots of the root system
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$.

  * (ii)

The Weyl group
$\vphantom{W}{}_{\mathbb{R}}W=\operatorname{Nor}_{K}(S)/\operatorname{Cen}_{K}(S)$
from the theory of algebraic groups.

  * (iii)

The Weyl group $W_{s}$ generated by reflections in roots of the root system
$\Sigma$.

  * (iv)

The Weyl group in the Lie groups setting
$W(G_{\mathbb{R}},A_{\mathbb{R}})=N_{\mathbb{R}}/M_{\mathbb{R}}$.

  * (v)

The Weyl group $N_{\mathbb{F}}/M_{\mathbb{F}}$ of the semialgebraic
extensions.

###### Proof.

By Proposition 6.3, $\vphantom{\Phi}{}_{\mathbb{K}}\Phi\cong\Sigma$, so (i)
and (iii) coincide. By [Bor91, Theorem 21.2], $W=\vphantom{W}_{\mathbb{R}}W$,
so the notions (i) and (ii) agree. By [Kna96, Proposition 7.32],
$W_{s}=W(G_{\mathbb{R}},A_{\mathbb{R}})$, so the notions (iii) and (iv) agree.
Let $W_{s}=\\{w_{1},\ldots,w_{|W_{s}|}\\}$ be a list of the finitely many
elements in $W_{s}$ considered as an abstract group. The first-order formula

$\displaystyle\varphi\colon\quad$ $\displaystyle\exists
n_{1},\ldots,n_{|W_{s}|}\in N\colon\left(\bigwedge_{i,j=0\atop i\neq
j}^{|W_{s}|}\neg\ n_{j}^{-1}n_{i}\in M\right)\wedge$
$\displaystyle\left(\forall n\in N\colon\bigvee_{i=1}^{|W_{s}|}n^{-1}n_{i}\in
M\right)\wedge\left(\bigwedge_{i,j,\ell=0\atop
w_{i}w_{j}=w_{\ell}}^{|W_{s}|}n_{\ell}^{-1}n_{i}n_{j}\in M\right)$

states that there are $|W_{s}|$ many elements in $N$ with distinct
representatives in $N/M$ and there are only $|W_{s}|$ many and they satisfy
the same group multiplication able as $W_{s}$. In short it says that $N/M$ is
isomorphic to $W_{s}$. Now, by (iv) the formula $\varphi$ holds over
$\mathbb{R}$ and we can apply the transfer principle to get the statement over
$\mathbb{F}$, showing that the notion in (v) gives the same Weyl group. ∎

###  6.3 Iwasawa decomposition $G=KAU$

We recall the classical Iwasawa-decomposition, which applies to
$G_{\mathbb{R}}$.

######  Theorem 6.5.

([Kna96, Theorem 6.46]) Let $G_{\mathbb{R}}$ be a semisimple Lie group. Let
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n}$
be an Iwasawa decomposition of its Lie algebra. Let $A_{\mathbb{R}}$ and
$U_{\mathbb{R}}$ be the analytic subgroups with Lie algebras $\mathfrak{a}$
and $\mathfrak{n}$. Then the multiplication map $K_{\mathbb{R}}\times
A_{\mathbb{R}}\times U_{\mathbb{R}}\to G_{\mathbb{R}}$ is a diffeomorphism.
This decomposition is unique.

We note that all the groups $G_{\mathbb{R}},K_{\mathbb{R}},A_{\mathbb{R}}$ and
$U_{\mathbb{R}}$ are semialgebraic defined over $\mathbb{K}$, hence we can
consider $G_{\mathbb{F}},K_{\mathbb{F}},A_{\mathbb{F}}$ and
$U_{\mathbb{F}}\subseteq\mathbb{F}^{n\times n}$ for any real closed field
$\mathbb{F}$ containing $\mathbb{K}$. We use the transfer principle to deduce
the following semialgebraic version of the Iwasawa decomposition. We remark
that by taking inverses, we also have a decomposition $G=UAK$ in addition to
$G=KAU$, both of which will be called the _Iwasawa decomposition_.

######  Theorem 6.6 ($G=KAU$).

For every $g\in G_{\mathbb{F}}$, there are $k\in K_{\mathbb{F}},a\in
A_{\mathbb{F}},u\in U_{\mathbb{F}}$ such that $g=kau$. This decomposition is
unique.

###### Proof.

We write the decomposition as a first-order formula

$\displaystyle\varphi\colon\quad$ $\displaystyle\left(\forall g\in G\ \exists
k\in K,a\in A,u\in U\colon g=kau\right)\ \wedge$ $\displaystyle\left(\forall
k,k^{\prime}\in K,a,a^{\prime}\in A,u,u^{\prime}\in U\colon
kau=k^{\prime}a^{\prime}u^{\prime}\to\left(k=k^{\prime}\wedge
a=a^{\prime}\wedge u=u^{\prime}\right)\right)$

which holds over $\mathbb{R}$ by the classical Iwasawa decomposition of
Theorem 6.5. By the transfer principle, Theorem 2.1, $\varphi$ holds over all
real closed fields $\mathbb{F}$. Note that to apply the classical Iwasawa-
theorem we use that $K_{\mathbb{R}}$ is maximal compact,
$\operatorname{Lie}(A_{\mathbb{R}})=\mathfrak{a}$ and
$\operatorname{Lie}(U_{\mathbb{R}})=\mathfrak{n}$. ∎

###  6.4 Cartan decomposition $G=KAK$

We use the Cartan decomposition for real Lie groups to find an analogue
statement over $\mathbb{F}$.

######  Theorem 6.7.

([Kna96, Theorem 7.39]) Every element $g\in G_{\mathbb{R}}$ has a
decomposition $g=k_{1}ak_{2}$ with $k_{1},k_{2}\in K_{\mathbb{R}}$ and $a\in
A_{\mathbb{R}}$. In this decomposition, $a$ is uniquely determined up to a
conjugation by a member of $W(G_{\mathbb{R}},A_{\mathbb{R}})$.

###### Proof.

By Proposition 6.4,
$\vphantom{W}{}_{\mathbb{R}}W=\operatorname{Nor}_{G}(S)/\operatorname{Cen}_{G}(S)$
is isomorphic to
$W(G_{\mathbb{R}},A_{\mathbb{R}})=\operatorname{Nor}_{K_{\mathbb{R}}}(\mathfrak{a})/\operatorname{Cen}_{K_{\mathbb{R}}}(\mathfrak{a})$.
Then this is the statement of [Kna96, Theorem 7.39]. ∎

######  Theorem 6.8 ($G=KAK$).

Every element $g\in G_{\mathbb{F}}$ has a decomposition $g=k_{1}ak_{2}$ with
$k_{1},k_{2}\in K_{\mathbb{F}}$ and $a\in A_{\mathbb{F}}$. In this
decomposition, $a$ is uniquely determined up to a conjugation by a member of
$N_{\mathbb{F}}/M_{\mathbb{F}}$.

###### Proof.

The existence part of the statement follows from the first-order formula

$\varphi\colon\quad\forall g\in G\ \exists k_{1},k_{2}\in K\ \exists a\in
A\colon g=k_{1}ak_{2},$

which holds over $\mathbb{R}$ by Theorem 6.7 and hence over $\mathbb{F}$ by
the transfer principle. For uniqueness of $a$, we consider the first-order
logic formula

$\displaystyle\psi\colon\quad$ $\displaystyle\forall a,a^{\prime}\in
A,k_{1},k_{1}^{\prime},k_{2},k_{2}^{\prime}\in K\colon
k_{1}ak_{2}=k_{1}^{\prime}a^{\prime}k_{2}^{\prime}\ \to$
$\displaystyle\left(\exists n\in N\colon a=na^{\prime}n^{-1}\ \wedge\
\left(\forall n^{\prime}\in N\colon a=n^{\prime}a^{\prime}(n^{\prime})^{-1}\to
n^{-1}n^{\prime}\in M\right)\right)$

which states that $a$ is determined up to a conjugation by a member of $N$ and
that this member is unique up to an element of $M$. Over $\mathbb{R}$, $\psi$
holds by Theorem 6.7, hence $\psi$ also holds over $\mathbb{F}$ by the
transfer principle, concluding the proof. ∎

###  6.5 Bruhat decomposition $G=BWB$

By [Kna96, page 398],
$B_{\mathbb{R}}:=M_{\mathbb{R}}A_{\mathbb{R}}U_{\mathbb{R}}$ is a closed
subgroup of $G_{\mathbb{R}}$ and we have the following Bruhat decomposition.

######  Theorem 6.9.

([Kna96, Theorem 7.40]) Every element $g\in G_{\mathbb{R}}$ can be written as
$g=b_{1}nb_{2}$ with $b_{1},b_{2}\in B_{\mathbb{R}}$ and $n\in
N_{\mathbb{R}}$. In this decomposition, $n$ is unique up to multiplying by an
element in $M_{\mathbb{R}}$. Since the spherical Weyl group is
$W_{s}=N_{\mathbb{R}}/M_{\mathbb{R}}$, we have a disjoint union of double
cosets

$G_{\mathbb{R}}=\prod_{[n]\in W_{s}}B_{\mathbb{R}}nB_{\mathbb{R}}.$

The group $B_{\mathbb{R}}$ is semialgebraic and the Bruhat decomposition can
be extended to $G_{\mathbb{F}}$.

######  Theorem 6.10 ($G=BWB$).

Every element $g\in G_{\mathbb{F}}$ can be written as $g=b_{1}nb_{2}$ with
$b_{1},b_{2}\in B_{\mathbb{F}}$ and $n\in N_{\mathbb{F}}$. In this
decomposition $n$ is unique up to multiplying by an element in
$M_{\mathbb{F}}$. We have a disjoint union of double cosets

$G_{\mathbb{F}}=\prod_{[n]\in W_{s}}B_{\mathbb{F}}nB_{\mathbb{F}}.$

###### Proof.

The existence of the decomposition follows directly from the transfer
principle and Theorem 6.9. For uniqueness we utilize the first-order formula

$\displaystyle\varphi\colon\quad$ $\displaystyle\forall
b_{1},b_{2},b_{1}^{\prime},b_{2}^{\prime}\in B,n,n^{\prime}\in N$
$\displaystyle b_{1}nb_{2}=b_{1}^{\prime}n^{\prime}b_{2}^{\prime}\to
n^{-1}n^{\prime}\in M,$

which holds over $\mathbb{R}$ by Theorem 6.9. Since
$W_{s}=N_{\mathbb{F}}/M_{\mathbb{F}}=N_{\mathbb{R}}/M_{\mathbb{R}}$ by
Proposition 6.4, $G_{\mathbb{F}}$ is a disjoint union as described. ∎

Let

$A_{\mathbb{F}}^{+}=\left\\{a\in A_{\mathbb{F}}\colon\chi_{\alpha}(a)\geq
1\text{ for all }\alpha\in\Delta\right\\},$

where $\chi_{\alpha}$ is the algebraic character associated to
$\alpha\in\Delta$. Then we may choose $a\in A_{\mathbb{F}}^{+}$ in the Cartan
decomposition $G_{\mathbb{F}}=K_{\mathbb{F}}A_{\mathbb{F}}^{+}K_{\mathbb{F}}$
which is then unique, since the Weyl group acts simply transitively on the set
of Weyl chambers.

###  6.6 Baker-Campbell-Hausdorff formula

Given $X,Y\in\mathfrak{g}_{\mathbb{R}}$, the Baker-Campbell-Hausdorff formula
gives a formal power series description of $Z=\log(\exp(X)\exp(Y))$ in terms
of $X,Y$ and iterated commutators of $X$ and $Y$, see for instance [Jac79,
Proposition V.1] or [Mag54]. The formal power series converges in a
neighborhood of the identity [Hoc65, Theorem 3.1 in X.3], but may not converge
everywhere in general. If $X,Y\in\mathfrak{n}_{\mathbb{R}}$, then the power
series is given by a polynomial and thus converges everywhere [Wei63]. Since
only finitely many terms are involved, one can see directly or invoke the
transfer principle to obtain the Baker-Campbell-Hausdorff formula for elements
in
$\mathfrak{n}_{\mathbb{F}}:=\bigoplus_{\alpha>0}(\mathfrak{g}_{\alpha})_{\mathbb{F}}$.

######  Proposition 6.11.

For every $X,Y\in\mathfrak{n}_{\mathbb{F}}$, there is
$Z\in\mathfrak{n}_{\mathbb{F}}$ such that $\exp(X)\exp(Y)=\exp(Z)$. The
element $Z$ is given by a finite sum of iterated commutators, the first terms
of which are given by

$Z=X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}\left(\left[X,\left[X,Y\right]\right]-\left[Y,\left[Y,X\right]\right]\right)-\frac{1}{24}\left[Y,\left[X,\left[X,Y\right]\right]\right]+\ldots$

There are various variations in the literature. We will make use of the
Zassenhaus formula

$\exp(X+Y)=\exp(X)\exp(Y)\exp\left(-\frac{1}{2}\left[X,Y\right]\right)\exp\left(\frac{1}{3}[Y,[X,Y]]+\frac{1}{6}[X,[X,Y]]\right)\cdots$

for $X,Y\in\mathfrak{n}_{\mathbb{F}}$, which can be obtained from the above by
calculating $\exp(-X)\exp(X+Y)=\exp(Z)$ iteratively [Mag54].

###  6.7 The unipotent group $U$

The root spaces $\mathfrak{g}_{\alpha}$ in the root decomposition

$\mathfrak{g}=\mathfrak{g}_{0}\oplus\bigoplus_{\alpha\in\Sigma}\mathfrak{g}_{\alpha}.$

corresponding to positive roots, consist of simultaneously nilpotent elements
by Lemma 5.6. Therefore the group

$U=\exp(\mathfrak{n})=\exp\left(\bigoplus_{\alpha\in\Sigma_{>0}}\mathfrak{g}_{\alpha}\right)$

is unipotent.

Let $\alpha\in\Sigma$. When $2\alpha\notin\Sigma$, the root space
$\mathfrak{g}_{\alpha}$ is an ideal. In any case,
$\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha}$ is an ideal, where
$\mathfrak{g}_{2\alpha}=0$ if $2\alpha\notin\Sigma$. Thus, for every
$\alpha\in\Sigma$ there is a unipotent subgroup
$U_{\alpha}=\exp(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})<U$, called
the _root group_ which is also an algebraic group, since the exponential
function is a polynomial on nilpotent elements. Note that
$U_{2\alpha}<U_{\alpha}$ if both exist.

The following Lemma shows that $A_{\mathbb{F}}$ normalizes the root groups
$(U_{\alpha})_{\mathbb{F}}$.

######  Lemma 6.12.

Let
$\alpha\in\Sigma,X\in\mathfrak{g}_{\alpha},X^{\prime}\in\mathfrak{g}_{2\alpha}$
and $a\in A_{\mathbb{F}}$. Then

$a\exp\left(X+X^{\prime}\right)a^{-1}=\exp\left(\chi_{\alpha}(a)X+\chi_{\alpha}(a)^{2}X^{\prime}\right),$

where $\chi_{\alpha}\colon A_{\mathbb{F}}\to\mathbb{F}_{>0}$ is the algebraic
character from Lemma 6.1.

###### Proof.

We know that $\operatorname{Ad}_{a}(X)=\chi_{\alpha}(a)X$ and
$\operatorname{Ad}_{a}(X^{\prime})=\chi_{2\alpha}(a)X^{\prime}=\chi_{\alpha}(a)^{2}X^{\prime}$.
We then use distributivity of matrix multiplication to obtain

$\displaystyle a\exp(X+X^{\prime})a^{-1}$
$\displaystyle=a\sum_{n=0}^{\infty}\frac{(X+X^{\prime})^{n}}{n!}a^{-1}=\sum_{n=0}^{\infty}\frac{1}{n!}a(X+X^{\prime})^{n}a^{-1}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\left(a(X+X^{\prime})a^{-1}\right)^{n}=\exp\left(aXa^{-1}+aX^{\prime}a^{-1}\right)$
$\displaystyle=\exp\left(\operatorname{Ad}_{a}(X)+\operatorname{Ad}_{a}(X^{\prime})\right)=\exp\left(\chi_{\alpha}(a)X+\chi_{\alpha}(a)^{2}X^{\prime}\right)$

∎

We consider $\Theta\subseteq\Sigma_{>0}$ _closed under addition_ , meaning
that for any $\alpha,\beta\in\Theta$, if the sum $\alpha+\beta\in\Sigma$, then
$\alpha+\beta\in\Theta$. For any $\Theta\subseteq\Sigma_{>0}$ closed under
addition,

$\mathfrak{g}_{\Theta}:=\bigoplus_{\alpha\in\Theta}\mathfrak{g}_{\alpha}$

is an ideal, since
$\left[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}\right]\subseteq\mathfrak{g}_{\alpha+\beta}$
for any $\alpha,\beta\in\Sigma$, see Proposition 5.4. Hence

$U_{\Theta}:=\exp(\mathfrak{g}_{\Theta})$

is a real algebraic group, in fact for $\Theta=\Sigma_{>0}$ we recover
$U=U_{\Sigma_{>0}}$. As a consequence of the Baker-Campbell-Hausdorff-formula
we obtain the following description of $(U_{\Theta})_{\mathbb{F}}$.

######  Lemma 6.13.

Let $\Theta=\left\\{\alpha_{1},\ldots,\alpha_{k}\right\\}\subseteq\Sigma_{>0}$
be a subset closed under addition with $\alpha_{1}>\ldots>\alpha_{k}$. Then

$(U_{\Theta})_{\mathbb{F}}:=\exp\left(\bigoplus_{\alpha\in\Theta}(\mathfrak{g}_{\alpha})_{\mathbb{F}}\right)=\prod_{i=1}^{k}\exp\left(\left(\mathfrak{g}_{\alpha_{i}}\right)_{\mathbb{F}}\right)=\langle
u\in U_{\alpha}\colon\alpha\in\Theta\rangle.$

###### Proof.

We start by proving

$\exp\left(\bigoplus_{\alpha\in\Theta}(\mathfrak{g}_{\alpha})_{\mathbb{F}}\right)\subseteq\prod_{i=1}^{k}\exp\left(\left(\mathfrak{g}_{\alpha_{i}}\right)_{\mathbb{F}}\right)$

using induction over $|\Theta|$. If $|\Theta|=1$, then the statement is
immediate. Now assume that $\Theta=\\{\alpha_{1}\\}\cup\Theta^{\prime}$ with
$\alpha_{1}>\beta$ for all $\beta\in\Theta^{\prime}$ and such that

$\exp\left(\bigoplus_{\alpha\in\Theta^{\prime}}(\mathfrak{g}_{\alpha})_{\mathbb{F}}\right)\subseteq\prod_{i=2}^{k}\exp\left(\left(\mathfrak{g}_{\alpha_{i}}\right)_{\mathbb{F}}\right).$

Let $X_{i}\in(\mathfrak{g}_{\alpha_{i}})_{\mathbb{F}}$. Recall that
$[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]\subseteq\mathfrak{g}_{\alpha+\beta}$
for all $\alpha,\beta\in\Sigma$, see Proposition 5.4. Thus we can use the
Zassenhaus-formula in Section 6.6 about the Baker-Campbell-Hausdorff formula
to obtain the finite product

$\displaystyle\exp\left(\sum_{i=1}^{k}X_{i}\right)$
$\displaystyle=\exp(X_{1})\exp\left(\sum_{i=2}^{k}X_{i}\right)\exp\left(-\frac{1}{2}\left[X_{1},\sum_{i=2}^{k}X_{i}\right]\right)\ldots$

and then repeatedly apply the original version in Proposition 6.11 to simplify
the expression back to

$\displaystyle\exp(X_{1})\exp\left(\sum_{i=2}^{k}\tilde{X}_{i}\right)$

for some new $\tilde{X}_{i}\in(\mathfrak{g}_{\alpha_{i}})_{\mathbb{F}}$.
Applying the induction hypothesis, we conclude

$\displaystyle\exp\left(\sum_{i=1}^{k}X_{i}\right)$
$\displaystyle\in\exp(X_{1})\prod_{i=2}^{k}\exp\left(\left(\mathfrak{g}_{\alpha_{i}}\right)_{\mathbb{F}}\right)\subseteq\prod_{i=1}^{k}\exp\left(\left(\mathfrak{g}_{\alpha_{i}}\right)_{\mathbb{F}}\right).$

Next, we notice that
$\prod_{i=1}^{k}\exp((\mathfrak{g}_{\alpha_{i}})_{\mathbb{F}})\subseteq\langle
u\in(U_{\alpha})_{\mathbb{F}}\colon\alpha\in\Theta\rangle$. Finally, we prove
the inclusion

$\langle u\in
U_{\alpha}\colon\alpha\in\Theta\rangle\subseteq\exp\left(\bigoplus_{\alpha\in\Theta}(\mathfrak{g}_{\alpha})_{\mathbb{F}}\right)$

using induction over the word length of an element in $\langle u\in
U_{\alpha}\colon\alpha\in\Theta\rangle$. If the word length is $1$, then the
statement holds. Now assume that

$v=\exp\left(\sum_{i=1}^{k}X_{i}\right)$

for some $X_{i}\in(\mathfrak{g}_{\alpha_{i}})_{\mathbb{F}}$ and consider
$u=\exp(X)$ for some $X\in(\mathfrak{g}_{\alpha})_{\mathbb{F}}$ and some
$\alpha\in\Theta$. We apply Proposition 6.11 to obtain

$\displaystyle uv$ $\displaystyle=\exp(X)\exp\left(\sum_{i=1}^{k}X_{i}\right)$
$\displaystyle=\exp\left(X+\sum_{i=1}^{k}X_{i}+\frac{1}{2}\left[X,\sum_{i=1}^{k}X_{i}\right]+\ldots\right)\in\exp\left(\bigoplus_{\alpha\in\Theta}(\mathfrak{g}_{\alpha})_{\mathbb{F}}\right)$

concluding the proof. ∎

We point out that the order of the product expression in Lemma 6.13 starts
with $\exp((\mathfrak{g}_{\alpha})_{\mathbb{F}})$ corresponding to the largest
$\alpha=\alpha_{1}$ followed in decreasing order. Writing elements of
$U_{\Theta}$ as the inverses of elements in $U_{\Theta}$, also gives an
expression starting with the smallest root, followed by an increasing order.
The following technical Lemma will be used in the proof of the mixed Iwasawa
decomposition, Theorem 8.42.

######  Lemma 6.14.

Let $\Theta\subseteq\Sigma_{>0}$ be a subset closed under addition and
$\alpha\in\Sigma_{>0}$ such that $\alpha>\beta$ for all $\beta\in\Theta$. Then
$uU_{\Theta}u^{-1}=U_{\Theta}$ for all
$u\in\exp((\mathfrak{g}_{\alpha})_{\mathbb{F}})$.

For every subset $\Psi\subseteq\Sigma_{>0}$, elements $u\in U_{\Theta}$ can be
expressed as $u=u^{\prime}u^{\prime\prime}$ with

$u^{\prime}\in\prod_{\alpha\in\Theta\cap\Psi}\exp((\mathfrak{g}_{\alpha})_{\mathbb{F}})\quad\text{
and }\quad
u^{\prime\prime}\in\prod_{\alpha\in\Theta\setminus\Psi}\exp((\mathfrak{g}_{\alpha})_{\mathbb{F}}).$

###### Proof.

Let $X\in(\mathfrak{g}_{\alpha})_{\mathbb{F}}$ and $\exp(Y)\in U_{\Theta}$.
Then we can apply the Baker-Campbell-Hausdorff formula, Proposition 6.11,
twice to obtain

$\displaystyle\exp(X)\exp(Y)\exp(X)^{-1}$
$\displaystyle=\exp\left(X+Y+\frac{1}{2}[X,Y]+\ldots\right)\exp(-X)$
$\displaystyle=\exp\left(Y+\frac{1}{2}[X,Y]+\ldots\right)\in U_{\Theta}.$

We show the second statement using induction over the size of $\Theta$. If
$|\Theta|=1$, the statement is clear. Now consider the subset closed under
addition $\Theta^{\prime}:=\Theta\setminus\\{\alpha\\}$ where $\alpha$ is the
largest element of $\Theta$. For $u\in U_{\Theta}$, we use Lemma 6.13 to
obtain $u=u_{\alpha}\bar{u}$ with $\bar{u}\in U_{\Theta^{\prime}}$. If
$\alpha\notin\Psi$, then the first part of this Lemma can be used to instead
write $u=\bar{u}u_{\alpha}$. Either way, the induction assumption gives

$\bar{u}^{\prime}\in\prod_{\beta\in\Theta^{\prime}\cap\Psi}\exp((\mathfrak{g}_{\beta})_{\mathbb{F}})\quad\text{
and
}\quad\bar{u}^{\prime\prime}\in\prod_{\beta\in\Theta^{\prime}\setminus\Psi}\exp((\mathfrak{g}_{\beta})_{\mathbb{F}})$

such that $\bar{u}=\bar{u}^{\prime}\bar{u}^{\prime\prime}$. Then
$u=u_{\alpha}\bar{u}^{\prime}\bar{u}^{\prime\prime}$ or
$u=\bar{u}^{\prime}\bar{u}^{\prime\prime}u_{\alpha}$ as required. ∎

Lemmas 6.12 and 6.13 can be used to prove that $A_{\mathbb{F}}$ normalizes all
of $U_{\mathbb{F}}$.

######  Proposition 6.15.

Let $\Theta\subseteq\Sigma_{>0}$ closed under addition. Then $A_{\mathbb{F}}$
normalizes $(U_{\Theta})_{\mathbb{F}}$: for all $a\in
A_{\mathbb{F}},u\in(U_{\Theta})_{\mathbb{F}}\colon
aua^{-1}\in(U_{\Theta})_{\mathbb{F}}$. In particular
$aU_{\mathbb{F}}a^{-1}=U_{\mathbb{F}}$ for all $a\in A_{\mathbb{F}}$.

###### Proof.

Let $a\in A_{\mathbb{F}}$. By Lemma 6.13, any $u\in(U_{\Theta})_{\mathbb{F}}$
can be written as $u=u_{\alpha_{1}}\cdot\ldots\cdot u_{\alpha_{k}}$ with
$u_{\alpha_{i}}\in(U_{\alpha_{i}})_{\mathbb{F}}$ where
$\Theta=\\{\alpha_{1},\ldots,\alpha_{k}\\}$. By Lemma 6.12,
$au_{\alpha_{i}}a^{-1}\in(U_{\alpha_{i}})_{\mathbb{F}}$, so

$aua^{-1}=au_{\alpha_{1}}a^{1}\cdot\ldots\cdot
au_{\alpha_{k}}a^{-1}\in\prod_{i=1}^{k}(U_{\alpha_{i}})_{\mathbb{F}}=(U_{\Theta})_{\mathbb{F}},$

where we used Lemma 6.13 again. ∎

###  6.8 Jacobson-Morozov Lemma for algebraic groups

The Jacobson-Morozov Lemma on the level of algebraic groups seems to be
folklore. Since we could not find a detailed treatment in the literature we
will give its statement and proof here. The classical Jacobson-Morozov Lemma
applies to semisimple Lie algebras over fields of characteristic $0$. The
following formulation is more general than what we proved in Lemma 5.9.

######  Proposition 6.16.

[Bou08b, VIII §11.2 Prop. 2] Let $\mathfrak{g}$ be a semisimple Lie algebra
and $x\in\mathfrak{g}$ a non-zero nilpotent element. Then there exist
$h,y\in\mathfrak{g}$ such that $(x,y,h)$ is an $\mathfrak{sl}_{2}$-triplet,
meaning that

$[h,x]=2x,\quad[h,y]=-2y,\quad[x,y]=h$

and hence the Lie algebra generated by $x,h,y$ is isomorphic to
$\mathfrak{sl}_{2}$.

We will use [Bor91, Chapter 7] and [PR93] to show the following version of the
Jacobson-Morozov Lemma for algebraic groups over an algebraically closed field
$\mathbb{D}$ of characteristic $0$. Note that any semisimple linear algebraic
group with Lie algebra $\mathfrak{sl}_{2}$ is isomorphic to either
$\operatorname{SL}_{2}$ or $\operatorname{PGL}_{2}$ [Hum75, Corollary 32.2].

######  Proposition 6.17.

Let $g\in G$ be any unipotent element in a semisimple linear algebraic group
$G$ over an algebraically closed field $\mathbb{D}$ of characteristic $0$.
Then there is an algebraic subgroup $\operatorname{SL}_{g}<G$ with Lie algebra
$\operatorname{Lie}(\operatorname{SL}_{g})\cong\mathfrak{sl}_{2}$ and
$g\in\operatorname{SL}_{g}$. The element $\log(g)\in\operatorname{Lie}(G)$
corresponds to

$\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\in\mathfrak{sl}_{2}.$

Moreover, if $g\in G_{\mathbb{F}}$ for a field
$\mathbb{F}\subseteq\mathbb{D}$, then $\operatorname{SL}_{g}$ can be assumed
to be defined over $\mathbb{F}$.

###### Proof.

Since $g\in G$ is unipotent, the nilpotent element $x=\log(g)\in\mathfrak{g}$
exists. By the Jacobson-Morozov Lemma, Proposition 6.16, there are
$y,h\in\mathfrak{g}$ such that $(x,y,h)$ is an $\mathfrak{sl}_{2}$-triplet.
Let $\mathfrak{sl}_{g}$ denote the subalgebra of $\mathfrak{g}$ generated by
$x,y$ and $h$. Thus, $x\in\operatorname{sl}_{g}$ corresponds to

$\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\in\mathfrak{sl}_{2}$

under the isomorphism $\mathfrak{sl}_{g}\cong\mathfrak{sl}_{2}$. We follow
[Bor91, 7.1] and define

$\mathcal{A}(\mathfrak{sl}_{g})=\bigcap\left\\{H\colon H\text{ algebraic
subgroup of $G$ with
}\mathfrak{sl}_{g}\subseteq\operatorname{Lie}(H)\right\\},$

which is a connected, normal algebraic subgroup. Let

$\operatorname{SL}_{g}=[\mathcal{A}(\mathfrak{sl}_{g}),\mathcal{A}(\mathfrak{sl}_{g})]$

be its commutator group, which is an algebraic group by [Bor91, 2.3] since
$\mathcal{A}(\mathfrak{sl}_{g})$ is connected. We then use [Bor91] to obtain

$\displaystyle\operatorname{Lie}(\operatorname{SL}_{g})$
$\displaystyle\stackrel{{\scriptstyle
7.8}}{{=}}[\operatorname{Lie}(\mathcal{A}(\mathfrak{sl}_{g})),\operatorname{Lie}(\mathcal{A}(\mathfrak{sl}_{g}))]\stackrel{{\scriptstyle
7.9}}{{=}}[\mathfrak{sl}_{g},\mathfrak{sl}_{g}]=\mathfrak{sl}_{g}$

where the last inequality follows from
$\mathfrak{sl}_{g}\cong\mathfrak{sl}_{2}$. The map

$\displaystyle\alpha\colon\mathbb{G}_{a}$ $\displaystyle\to G$ $\displaystyle
t$ $\displaystyle\mapsto\exp(tx)=\sum_{n=0}^{\infty}\frac{(tx)^{n}}{n!}$

is a polynomial and hence a morphism of algebraic groups. We have
$\operatorname{Lie}(\alpha(\mathbb{G}_{a}))=\langle
x\rangle\subseteq\mathfrak{sl}_{g}$ and hence
$g\in\alpha(\mathbb{G}_{a})\subseteq\operatorname{SL}_{g}$ by [Bor91, 7.1(2)].

If $g\in G_{\mathbb{F}}$, $\log(g)\in\mathfrak{g}_{\mathbb{F}}$, and we may
assume $y,h\in\mathfrak{g}_{\mathbb{F}}$ as well. Thus
$\mathfrak{sl}_{g}\subseteq\mathfrak{g}_{\mathbb{F}}$. Now
$\mathcal{A}(\mathfrak{sl}_{g})$ is defined over $\mathbb{F}$ as in [Bor91,
2.1(b)]. Then by [Bor91, 2.3], $\operatorname{SL}_{g}$ is defined over
$\mathbb{F}$. ∎

Let us return to the case of real closed fields
$\mathbb{K}\subseteq\mathbb{F}\cap\mathbb{R}$. Recall from Section 6.7, that
$U_{\mathbb{F}}$ has subgroups $(U_{\alpha})_{\mathbb{F}}$ consisting of
unipotent elements for $\alpha\in\Sigma_{>0}$ defined as the semialgebraic
extensions of
$U_{\alpha}=\exp(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})$. The
following is another variation of the Jacobson-Morozov Lemma that only works
when $\Sigma$ is reduced.

######  Proposition 6.18.

Let $\alpha\in\Sigma$ and assume $\mathfrak{g}_{2\alpha}=0$. Let
$u\in(U_{\alpha})_{\mathbb{F}}$. Then there are
$X\in(\mathfrak{g}_{\alpha})_{\mathbb{F}}$ and $t\in\mathbb{F}$ such that
$u=\exp(tX)$ and $(-v)(X)=0$. Let $(X,Y,H)$ be the $\mathfrak{sl}_{2}$-triplet
of Lemma 5.9. Then there is a morphism of algebraic groups
$\varphi\colon\operatorname{SL}(2,\mathbb{D})\to G$ defined over $\mathbb{F}$
such that $\varphi$ has finite kernel and

$\displaystyle\varphi\begin{pmatrix}1&t\\\
0&1\end{pmatrix}=u=\exp(tX)\quad\text{and}\quad\varphi\begin{pmatrix}1&0\\\
t&1\end{pmatrix}=\exp(tY).$

If $\varphi$ is not injective, then $\ker(\varphi)\cong\mathbb{Z}/2\mathbb{Z}$
and $\varphi$ factors through the isomorphism

$\operatorname{PGL}(2,\mathbb{D}):=\operatorname{SL(2,\mathbb{D})}/\ker(\varphi)\xrightarrow{\sim}\varphi(\operatorname{SL}(2,\mathbb{D}))$

which is also defined over $\mathbb{F}$. Moreover
$\varphi(g{{}^{\mathsf{T}}\\!})=\varphi(g){{}^{\mathsf{T}}\\!}$, for any
$g\in\operatorname{SL}(2,\mathbb{F})$.

###### Proof.

Let $u,X,t$ as in the statement. We apply Proposition 6.17 to
$\exp(X)\in(U_{\alpha})_{\mathbb{F}}$ to obtain an algebraic group
$\operatorname{SL}_{\exp(X)}(\mathbb{D})<G_{\mathbb{D}}$ defined over
$\mathbb{F}$ with Lie algebra $\operatorname{sl}_{2}(\mathbb{D})$. By [Hum75,
Corollary 32.3], the only algebraic groups with Lie algebra
$\mathfrak{sl}_{2}(\mathbb{D})$ are $\operatorname{SL}(2,\mathbb{D})$ and
$\operatorname{PGL}(2,\mathbb{D}):=\operatorname{SL}(2,\mathbb{D})/Z(\operatorname{SL(2,\mathbb{D})})$.
In both cases we obtain an algebraic homomorphism
$\varphi\colon\operatorname{SL}_{2}(\mathbb{D})\to\operatorname{SL}_{\exp(X)}$
with finite kernel.

We note that since $X\in(\mathfrak{g}_{\alpha})_{\mathbb{F}}$, the
$\mathfrak{sl}_{2}$-triplet is described by $(X,Y,H)$ in Lemma 5.9 and we have
a Lie algebra isomorphism
$\mathfrak{sl}_{2}(\mathbb{D})\cong\operatorname{Lie}(\operatorname{SL}_{\exp(X)}(\mathbb{D}))$.
We note that the two Lie algebra isomorphisms

$\displaystyle\varphi\colon\operatorname{SL}_{2}(\mathbb{D})\xrightarrow{\sim}$
$\displaystyle\ \operatorname{SL}_{\exp(X)}(\mathbb{D})$
$\displaystyle\mathfrak{sl}_{2}(\mathbb{D})\cong$ $\displaystyle\
\mathfrak{sl}_{\exp(X)}(\mathbb{D})$
$\displaystyle\cong\mathfrak{sl}_{2}(\mathbb{D})$ $\displaystyle\ X$
$\displaystyle\mapsfrom\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$

may not coincide. Since $\operatorname{sl}_{2}(\mathbb{D})$ does not have any
outer automorphisms ([Hum75, Theorem 14.1] and [FH04, Proposition D.40]), the
two isomorphisms only differ by $\operatorname{Ad}(g)$ for some
$g\in\operatorname{SL}(2,\mathbb{D})$. Up to conjugation we may therefore
assume that
$\operatorname{d}\\!\varphi\colon\mathfrak{sl}_{2}(\mathbb{D})\to\operatorname{Lie}(\operatorname{SL}_{\exp(X)}(\mathbb{D}))$
maps

$\displaystyle\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\mapsto
X,\quad\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\mapsto
H\quad\text{and}\quad\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}\mapsto Y.$

Even though the conjugation may be by an element in
$\operatorname{SL}_{2}(\mathbb{D})$, the explicit description of
$\operatorname{d}\\!\varphi$ shows that $\varphi$ is still defined over
$\mathbb{F}$. The description of $\exp(tX)$ and $\exp(tY)$ in the statement of
the Proposition also follows. Finally, note that transposition is
$\operatorname{d}\\!\varphi$-equivariant and hence also $\varphi$-equivariant.
∎

Since in the $\mathfrak{sl}_{2}$-triplet, $H\in\mathfrak{a}_{\mathbb{F}}$ we
obtain multiplicative one-parameter groups

$\displaystyle\mathbb{F}_{>0}$ $\displaystyle\to A_{\mathbb{F}}$
$\displaystyle\lambda$ $\displaystyle\mapsto\varphi\begin{pmatrix}\lambda&0\\\
0&\lambda^{-1}\end{pmatrix}$

which satisfy the following property that will be useful later.

######  Lemma 6.19.

Let $\alpha\in\Sigma$ such that $(\mathfrak{g}_{2\alpha})_{\mathbb{F}}=0$. Let
$u\in(U_{\alpha})_{\mathbb{F}}$ and
$\varphi\colon\operatorname{SL}(2,\mathbb{D})\to G$ as in Proposition 6.18.
For every $\lambda\in\mathbb{F}_{>0}$

$\chi_{\alpha}\left(\varphi\begin{pmatrix}\lambda&0\\\
0&\lambda^{-1}\end{pmatrix}\right)=\lambda^{2}.$

###### Proof.

We note that for all $g\in\operatorname{SL}(2,\mathbb{F})$, the diagrams

simplicial buildingsspherical buildings$\operatorname{rank}/\dim$discrete
affine buildings Euclidean buildings [KL97]$\mathbb{R}$-buildings [BT72,
Par00]$\Lambda$-buildings [Ben94, HIL24]
[KT02]trees$\mathbb{R}$-trees$\Lambda$-trees$1$$0$affine
buildings$1$$2$$2$$3$$3$$4$$\vdots$$\vdots$${{\mathfrak{sl}(2,\mathbb{F})}}$${{\mathfrak{sl}(2,\mathbb{F})}}$${{\operatorname{SL}(2,\mathbb{F})}}$${{\operatorname{SL}(2,\mathbb{F})}}$${\mathfrak{g}_{\mathbb{F}}}$${\mathfrak{g}_{\mathbb{F}}}$${G_{\mathbb{F}}}$${G_{\mathbb{F}}}$$\scriptstyle{\operatorname{Ad}(g)}$$\scriptstyle{\operatorname{d}\\!\varphi}$$\scriptstyle{\operatorname{d}\\!\varphi}$$\scriptstyle{\varphi}$$\scriptstyle{c_{g}}$$\scriptstyle{\varphi}$$\scriptstyle{\operatorname{Ad}(\varphi(g))}$$\scriptstyle{c_{\varphi(g)}}$

commute. For

$a=\varphi\begin{pmatrix}\lambda&0\\\ 0&\lambda^{-1}\end{pmatrix}\quad\text{
and }\quad X=\operatorname{d}\\!\varphi\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\in(\mathfrak{g}_{\alpha})_{\mathbb{F}}$

we then have

$\displaystyle\operatorname{Ad}\left(a\right)X$
$\displaystyle=\operatorname{d}\\!\varphi\left(\operatorname{Ad}\begin{pmatrix}\lambda&0\\\
0&\lambda^{-1}\end{pmatrix}\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\right)=\operatorname{d}\\!\varphi\begin{pmatrix}0&\lambda^{2}\\\
0&0\end{pmatrix}=\lambda^{2}X$

and since $\chi_{\alpha}(a)$ is defined by
$\operatorname{Ad}(a)X=\chi_{\alpha}(a)X$ we have
$\chi_{\alpha}(a)=\lambda^{2}$. ∎

######  Lemma 6.20.

Let $\alpha\in\Sigma$ such that $(\mathfrak{g}_{2\alpha})_{\mathbb{F}}=0$. Let
$u\in(U_{\alpha})_{\mathbb{F}}$ and
$X\in(\mathfrak{g}_{\alpha})_{\mathbb{F}}$, $t\in\mathbb{F}$ and
$\varphi\colon\operatorname{SL}(2,\mathbb{D})\to G$ as in Proposition 6.18, so
that $u=\exp(tX)$. The element

$m(u):=\varphi\begin{pmatrix}0&t\\\ -1/t&0\end{pmatrix}\in G_{\mathbb{F}}$

is contained in $\operatorname{Nor}_{G_{\mathbb{F}}}(A_{\mathbb{F}})$ and

$\chi_{\alpha}(m(u)\cdot a\cdot m(u)^{-1})=\chi_{\alpha}(a)^{-1}$

for any $a\in A_{\mathbb{F}}$.

When $t=1$, even $m(u)\in
N_{\mathbb{F}}=\operatorname{Nor}_{K_{\mathbb{F}}}(A_{\mathbb{F}})$ and

$m(u)\cdot(U_{\alpha})_{\mathbb{F}}\cdot
m(u)^{-1}=(U_{-\alpha})_{\mathbb{F}}.$

###### Proof.

Let $a\in A_{\mathbb{F}}$ and

$a_{\perp}:=\varphi\begin{pmatrix}\sqrt{\chi_{\alpha}(a)}&0\\\
0&\sqrt{\chi_{\alpha}(a)}^{-1}\end{pmatrix},$

then $a_{0}:=aa_{\perp}^{-1}$ satisfies

$\chi_{\alpha}(a_{0})=\chi_{\alpha}(a)\chi_{\alpha}(a_{\perp})^{-1}=\chi_{\alpha}(a)\frac{1}{\chi_{\alpha}(a)}=1$

by Lemma 6.19. For any
$u^{\prime}\in\exp((\mathfrak{g}_{\alpha})_{\mathbb{F}})$ or
$u^{\prime}\in\exp((\mathfrak{g}_{-\alpha})_{\mathbb{F}})$,
$a_{0}u^{\prime}=u^{\prime}a_{0}$ by Lemma 6.12. We note that

$\displaystyle m(u)$ $\displaystyle=\varphi\begin{pmatrix}0&t\\\
-1/t&0\end{pmatrix}=\varphi\left(\begin{pmatrix}1&0\\\
-1/t&1\end{pmatrix}\begin{pmatrix}1&t\\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\\
-1/t&1\end{pmatrix}\right)$
$\displaystyle\in\exp((\mathfrak{g}_{-\alpha})_{\mathbb{F}})\cdot
u\cdot\exp((\mathfrak{g}_{-\alpha})_{\mathbb{F}})$

which implies $a_{0}m(u)=m(u)a_{0}$. Now

$\displaystyle m(u)\cdot a\cdot m(u)^{-1}$ $\displaystyle=m(u)\cdot a_{0}\cdot
a_{\perp}\cdot m(u)^{-1}=a_{0}\cdot m(u)\cdot a_{\perp}\cdot m(u)^{-1}$
$\displaystyle=a_{0}\cdot\varphi\left(\begin{pmatrix}0&t\\\
-1/t&0\end{pmatrix}\begin{pmatrix}\sqrt{\chi_{\alpha}(a)}&0\\\
0&\sqrt{\chi_{\alpha}(a)}^{-1}\end{pmatrix}\begin{pmatrix}0&-t\\\
1/t&0\end{pmatrix}\right)$
$\displaystyle=a_{0}\cdot\varphi\begin{pmatrix}\sqrt{\chi_{\alpha}(a)}^{-1}&0\\\
0&\sqrt{\chi_{\alpha}(a)}\end{pmatrix}=a_{0}a_{\perp}^{-1}\in A_{\mathbb{F}}$

and thus $m(u)=\operatorname{Nor}_{G_{\mathbb{F}}}(A_{\mathbb{F}})$. We see
directly that

$\chi_{\alpha}(m(u)\cdot a\cdot m(u)^{-1})=\chi_{\alpha}(a_{0}\cdot
a_{\perp}^{-1})=\chi_{\alpha}(a_{0})\cdot\chi_{\alpha}(a_{\perp})^{-1}=\chi_{\alpha}(a)^{-1}.$

Now if $t=1$ we can use that $\varphi$ preserves transposition by Proposition
6.18, to show

$\displaystyle m(u)\cdot m(u){{}^{\mathsf{T}}\\!}$
$\displaystyle=\varphi\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\left(\varphi\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\right){{}^{\mathsf{T}}\\!}$
$\displaystyle=\varphi\left(\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\right)=\varphi\begin{pmatrix}1&0\\\
0&1\end{pmatrix}=\operatorname{Id}$

and hence $m(u)\in K_{\mathbb{F}}$, so $m(u)\in
N_{\mathbb{F}}=\operatorname{Nor}_{K_{\mathbb{F}}}(A_{\mathbb{F}})$. Thus
$m(u)$ is a representative of an element $w=[m(u)]$ of the spherical Weyl
group $W_{s}=N_{\mathbb{F}}/M_{\mathbb{F}}$. While
$u\in(U_{\alpha})_{\mathbb{F}}$,

$\displaystyle m(u)\cdot u\cdot m(u)^{-1}$
$\displaystyle=\varphi_{\mathbb{F}}\left(\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}1&t\\\ 0&1\end{pmatrix}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\right)$
$\displaystyle=\varphi_{\mathbb{F}}\begin{pmatrix}1&0\\\
-t&1\end{pmatrix}\in(U_{-\alpha})_{\mathbb{F}},$

and thus we have $w(\alpha)=-\alpha$. Then

$m(u)\cdot(U_{\alpha})_{\mathbb{F}}\cdot
m(u)^{-1}=(U_{w(\alpha)})_{\mathbb{F}}=(U_{-\alpha})_{\mathbb{F}}.$

∎

###  6.9 Rank 1 subgroups

Let $\alpha\in\Sigma$. In this section we investigate the group generated by
$U_{\alpha}$ and $U_{-\alpha}$. When $\dim(U_{\alpha})=1$, then this group is
given by the image of the Jacobson-Morozov-morphism
$\varphi\colon\operatorname{SL}(2,\mathbb{D})\to G$ from Proposition 6.18. In
general, when $\dim(U_{\alpha})$ is not $1$, the group generated is larger
than the image of $\varphi$, but is still rank $1$.

######  Theorem 6.21.

Let $\alpha\in\Sigma$. Then there is a connected semisimple self-adjoint
linear algebraic group $L_{\pm\alpha}$ defined over $\mathbb{K}$ such that

  1. (i)

$\operatorname{Lie}(L_{\pm\alpha})=(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})\oplus(\mathfrak{g}_{-\alpha}\oplus\mathfrak{g}_{-2\alpha})\oplus([\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]+[\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}])$.

  2. (ii)

$\operatorname{Rank}_{\mathbb{R}}(L_{\pm\alpha})=\operatorname{Rank}_{\mathbb{F}}(L_{\pm\alpha})=1$.

###### Proof.

We consider the semisimple $\mathbb{D}$-Lie algebra

$\mathfrak{l}:=(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})\oplus(\mathfrak{g}_{-\alpha}\oplus\mathfrak{g}_{-2\alpha})\oplus([\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]+[\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}]).$

Similar to the proof of Proposition 6.17 we follow Borel [Bor91, 7.1] by
defining the connected normal algebraic subgroup

$\mathcal{A}(\mathfrak{l})=\bigcap\left\\{H\colon H<G\text{ is an algebraic
subgroup }\mathfrak{l}\subseteq\operatorname{Lie}(H)\right\\}$

of $G$. We set
$L_{\pm\alpha}:=[\mathcal{A}(\mathfrak{l}),\mathcal{A}(\mathfrak{l})]$. Then
(i) follows by

$\operatorname{Lie}(L_{\pm\alpha})=[\mathcal{A}(\mathfrak{l}),\mathcal{A}(\mathfrak{l})]=[\mathfrak{l},\mathfrak{l}]=\mathfrak{l},$

where we used that $\mathfrak{l}$ is semisimple in the step
$[\mathfrak{l},\mathfrak{l}]=\mathfrak{l}$. The algebraic group
$\mathcal{A}(\mathfrak{l})$ is defined over $\mathbb{K}$ by [Bor91, 2.1(b)].
Thus $L_{\pm\alpha}$ is defined over $\mathbb{K}$ and connected by [Bor91,
2.3]. Since $\mathfrak{l}$ is semisimple, so is $L_{\pm\alpha}$. Since
$\theta(X)=-X{{}^{\mathsf{T}}\\!}$ and
$\theta(\mathfrak{g}_{\alpha})=\mathfrak{g}_{-\alpha}$ and
$\theta(\mathfrak{g}_{2\alpha})=\mathfrak{g}_{-2\alpha}$ by Proposition 5.4,
$\theta([\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}])=[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]$
and
$\theta([\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}])=[\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}]$,
and hence $\mathfrak{l}$ and $L_{\pm\alpha}$ are self-adjoint.

For (ii), Lemma 5.8 tells us that the real rank of $\mathfrak{l}_{\mathbb{R}}$
is $1$ and hence $\operatorname{Rank}_{\mathbb{R}}(L_{\pm\alpha})=1$. By the
theorem on split tori, Theorem 5.17, we then have
$\operatorname{Rank}_{\mathbb{R}}(L_{\pm\alpha})=\operatorname{Rank}_{\mathbb{F}}(L_{\pm\alpha})=1$.
∎

######  Lemma 6.22.

$(L_{\pm\alpha})_{\mathbb{F}}\subseteq\operatorname{Cen}_{G_{\mathbb{F}}}(\left\\{a\in
A_{\mathbb{F}}\colon\chi_{\alpha}(a)=1\right\\})$.

###### Proof.

This is a semialgebraic statement, hence it suffices to prove it over the real
numbers. Let $a\in A_{\mathbb{R}}$ with $\chi_{\alpha}(a)=1$. We have

$\operatorname{Lie}(L_{\pm\alpha})=(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})\oplus(\mathfrak{g}_{-\alpha}\oplus\mathfrak{g}_{-2\alpha})\oplus([\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]\oplus[\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}]).$

For $X\in(\mathfrak{g}_{\alpha})_{\mathbb{R}}$ and
$X^{\prime}\in(\mathfrak{g}_{2\alpha})_{\mathbb{R}}$, we have by Lemma 6.12

$\displaystyle a\exp(X)a^{-1}$ $\displaystyle=\exp(\chi_{\alpha}(a)X)=\exp(X)$
$\displaystyle a\exp(X^{\prime})a^{-1}$
$\displaystyle=\exp(\chi_{\alpha}(a)^{2}X^{\prime})=\exp(X^{\prime})$

and the same argument shows $a\exp(Y)a^{-1}=\exp(Y)$ for
$Y\in(\mathfrak{g}_{-\alpha})_{\mathbb{R}}\oplus(\mathfrak{g}_{-2\alpha})_{\mathbb{R}}$.
Since
$\mathfrak{g}_{0}\cap\mathfrak{l}=\mathfrak{z}_{\mathfrak{k}\cap\mathfrak{l}}(\mathfrak{a}\cap\mathfrak{l})\oplus(\mathfrak{a}\cap\mathfrak{l})$,
also $a\exp(H)a^{-1}=\exp(H)$ for $H\in\mathfrak{g}_{0}\cap\mathfrak{l}$ is
clear. We have shown that

$\operatorname{Ad}(a)\colon\operatorname{Lie}((L_{\pm\alpha})_{\mathbb{R}})\to\operatorname{Lie}((L_{\pm\alpha})_{\mathbb{R}})$

is the identity, and conjugation by $a$ is constant on connected components of
$(L_{\pm})_{\mathbb{R}}$. Since $L_{\pm\alpha}$ is connected as an algebraic
group and $c_{a}^{-1}(\operatorname{Id})$ is a closed algebraic set,
conjugation by $a$ is the identity. ∎

We can treat $L_{\pm\alpha}$ from Theorem 6.21 as an example of the theory we
have developed so far, in particular we can apply group decompositions as
outlined in the beginning of Section 6. Since $S<G$ is a self-adjoint maximal
split torus, whose Lie algebra contains
$[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]\oplus[\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}]$,
$S_{\pm\alpha}:=S\cap L_{\pm\alpha}$ is a self-adjoint maximal split torus of
$L_{\pm\alpha}$. Then
$K_{\pm\alpha}=L_{\pm\alpha}\cap\operatorname{SO}_{n}=L_{\pm\alpha}\cap K$.
Considering the $\mathbb{F}$-points, the semialgebraic connected component
$(A_{\pm\alpha})_{\mathbb{K}}$ of $(S_{\pm\alpha})_{\mathbb{K}}$ containing
the identity can be semialgebraically extended to
$(A_{\pm\alpha})_{\mathbb{F}}$. The following is a version on Lemma 6.19, but
now for the rank 1 group $(L_{\pm\alpha})_{\mathbb{F}}$.

######  Lemma 6.23.

For every $t\in\mathbb{F}_{>0}$, there is an
$a\in(A_{\pm\alpha})_{\mathbb{F}}$ such that $\chi_{\alpha}(a)=t$.

###### Proof.

This is clearly a first-order statement and it therefore suffices to show it
for $\mathbb{R}=\mathbb{F}$. Let
$X\in(\mathfrak{g}_{\alpha})_{\mathbb{R}}\setminus 0$. From Lemma 5.8, we know
that $\exp([X,\theta(X)])\in A_{\pm\alpha}$. Given $t\in\mathbb{R}_{>0}$, let

$a:=\exp\left(\frac{\log(t)}{\alpha([X,\theta(X)])}\cdot[X,\theta(X)]\right).$

Then $\chi_{\alpha}(a)=e^{\alpha(\log(a))}=e^{\log(t)}=t$ by Lemma 6.1. ∎

The root space decomposition then gives
$U_{\pm\alpha}=\exp(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})=U_{\alpha}$,
and the spherical Weyl group
$W_{\pm\alpha}=(N_{\pm\alpha})_{\mathbb{F}}/(M_{\pm\alpha})_{\mathbb{F}}$ can
be defined from

$\displaystyle(N_{\pm\alpha})_{\mathbb{F}}$
$\displaystyle:=\operatorname{Nor}_{(K_{\pm\alpha})_{\mathbb{F}}}((A_{\pm\alpha})_{\mathbb{F}})$
$\displaystyle(M_{\pm\alpha})_{\mathbb{F}}$
$\displaystyle:=\operatorname{Cen}_{(K_{\pm\alpha})_{\mathbb{F}}}((A_{\pm\alpha})_{\mathbb{F}}).$

We note that as a consequence of Lemma 6.22, $N_{\pm\alpha}\subseteq
N_{\mathbb{F}}$ and also
$\operatorname{Nor}_{L_{\pm\alpha}}(A_{\pm\alpha})\subseteq\operatorname{Nor}_{K_{\mathbb{F}}}(A_{\mathbb{F}})$.

In the case that $\Sigma$ is reduced, there is an interpretation in terms of
the Jacobson-Morozov Lemma. Given $u\in(U_{\pm\alpha})_{\mathbb{F}}$, we note
that the morphism
$\varphi_{\mathbb{F}}\colon\operatorname{SL}(2,\mathbb{F})\to G_{\mathbb{F}}$
from Proposition 6.18 takes values in $(L_{\pm\alpha})_{\mathbb{F}}$. In fact,

$(A_{\pm\alpha})_{\mathbb{F}}=\varphi_{\mathbb{F}}\left(\left\\{\begin{pmatrix}\lambda&0\\\
0&\lambda^{-1}\end{pmatrix}\colon\lambda>0\right\\}\right)$

since over $\mathbb{R}$, both of these groups are connected and one-
dimensional. By Lemma 6.20, the element

$m=\varphi_{\mathbb{F}}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\in(N_{\pm\alpha})_{\mathbb{F}}$

is a representative of the only non-trivial element in the spherical Weyl
group $W_{\pm\alpha}=\\{[\operatorname{Id}],[m]\\}$. We can now apply the
Bruhat decomposition Theorem 6.10 to $(L_{\pm\alpha})_{\mathbb{F}}$.

######  Corollary 6.24.

Let
$(B_{\alpha})_{\mathbb{F}}:=(M_{\pm\alpha})_{\mathbb{F}}(A_{\pm\alpha})_{\mathbb{F}}(U_{\alpha})_{\mathbb{F}}$.
Then

$(L_{\pm\alpha})_{\mathbb{F}}=(B_{\alpha})_{\mathbb{F}}(N_{\pm\alpha})_{\mathbb{F}}(B_{\alpha})_{\mathbb{F}},$

and the element $m\in(N_{\pm\alpha})_{\mathbb{F}}$ is a representative of a
unique element in $W_{\pm\alpha}$, so

$(L_{\pm\alpha})_{\mathbb{F}}=(B_{\alpha})_{\mathbb{F}}\ \amalg\
(B_{\alpha})_{\mathbb{F}}\cdot m\cdot(B_{\alpha})_{\mathbb{F}}.$

We will use the following variation in Lemma 8.33.

######  Corollary 6.25.

For

$\displaystyle(B_{\alpha})_{\mathbb{F}}$
$\displaystyle:=(M_{\pm\alpha})_{\mathbb{F}}(A_{\pm\alpha})_{\mathbb{F}}(U_{\alpha})_{\mathbb{F}},$
$\displaystyle(B_{-\alpha})_{\mathbb{F}}$
$\displaystyle:=(M_{\pm\alpha})_{\mathbb{F}}(A_{\pm\alpha})_{\mathbb{F}}(U_{-\alpha})_{\mathbb{F}},$

we obtain the decompositions

$\displaystyle(L_{\pm\alpha})_{\mathbb{F}}$
$\displaystyle=(B_{\alpha})_{\mathbb{F}}\cdot(B_{-\alpha})_{\mathbb{F}}\
\amalg\ m\cdot(B_{-\alpha})_{\mathbb{F}},$
$\displaystyle(L_{\pm\alpha})_{\mathbb{F}}$
$\displaystyle=(B_{\alpha})_{\mathbb{F}}\cdot(B_{-\alpha})_{\mathbb{F}}\
\amalg\ (B_{\alpha})_{\mathbb{F}}\cdot m\cdot(B_{-\alpha})_{\mathbb{F}},$
$\displaystyle(L_{\pm\alpha})_{\mathbb{F}}$
$\displaystyle=(B_{\alpha})_{\mathbb{F}}(N_{\pm\alpha})_{\mathbb{F}}(B_{-\alpha})_{\mathbb{F}}$

where in the last one, the element in $(N_{\pm\alpha})_{\mathbb{F}}$ is a
representative of a unique element in
$W_{\pm\alpha}=\left\\{\operatorname{Id},[m]\right\\}$.

###### Proof.

We use $m^{-1}\cdot(U_{\alpha})_{\mathbb{F}}\cdot
m=(U_{-\alpha})_{\mathbb{F}}$ from Lemma 6.20. By Corollary 6.24, we choose
$m^{-1}$ as the representative of $[m^{-1}]=[m]\in W_{\pm\alpha}$ and obtain

$(L_{\pm\alpha})_{\mathbb{F}}=(B_{\alpha})_{\mathbb{F}}\ \amalg\
(B_{\alpha})_{\mathbb{F}}\cdot m^{-1}\cdot(B_{\alpha})_{\mathbb{F}},$

so

$(L_{\pm\alpha})_{\mathbb{F}}=m\cdot(B_{-\alpha})_{\mathbb{F}}\cdot m^{-1}\
\amalg\ (B_{\alpha})_{\mathbb{F}}\cdot(B_{-\alpha})_{\mathbb{F}}\cdot m^{-1}.$

Multiplying this expression on the right by $m\in(L_{\pm\alpha})_{\mathbb{F}}$
results in

$(L_{\pm\alpha})_{\mathbb{F}}=(B_{\alpha})_{\mathbb{F}}\cdot(B_{-\alpha})_{\mathbb{F}}\
\amalg\ m\cdot(B_{-\alpha})_{\mathbb{F}},$

and further multiplying by $(B_{\alpha})_{\mathbb{F}}$ on the left results in
the remaining two decompositions. ∎

###  6.10 Kostant convexity

Using the Iwasawa decomposition
$G_{\mathbb{F}}=U_{\mathbb{F}}A_{\mathbb{F}}K_{\mathbb{F}}$, Theorem 6.5, we
associate to every $g=uak\in G_{\mathbb{R}}$ its $A$-component
$a_{\mathbb{R}}(g)=a\in A_{\mathbb{R}}$. The following is Kostant’s convexity
theorem, which we will generalize to $G_{\mathbb{F}}$ in this chapter.

######  Theorem 6.26.

[Kos73, Theorem 4.1] For every $b\in A_{\mathbb{R}}$,

$\left\\{a_{\mathbb{R}}(kb)\in A_{\mathbb{R}}\colon k\in
K_{\mathbb{R}}\right\\}=\exp\left(\operatorname{conv}(W_{s}\log(b))\right),$

where $\log\colon A_{\mathbb{R}}\to\mathfrak{a}$ is the inverse of $\exp$,
$W_{s}$ is the spherical Weyl group acting on $\mathfrak{a}$ and
$\operatorname{conv}(W_{s}\log(b))$ is the convex hull of the Weyl group orbit
of $\log(b)$.

The left hand side of the equation in Theorem 6.26 is already a semialgebraic
set and we will reformulate the right hand side as a semialgebraic set as
well. For this, we first analyze the root system
$\Sigma\subseteq\mathfrak{a}^{\star}$.

Recall from Section 5.1, that we have a scalar product $B_{\theta}$ on
$\mathfrak{a}$, which can be used to set up an isomorphism
$\mathfrak{a}^{\star}\to\mathfrak{a},\alpha\mapsto H_{\alpha}$ that satisfies
the defining property $\lambda(H)=B_{\theta}(H_{\lambda},H)$ for all
$H\in\mathfrak{a}$. In this section we will denote $B_{\theta}$ as well as the
corresponding scalar product on $\mathfrak{a}^{\star}$ with brackets
$\langle\cdot,\cdot\rangle$. For $\lambda\in\mathfrak{a}^{\star}$
define101010In the notation of Section 3.3, $x_{\alpha}=H_{\alpha}^{\vee}$.

$x_{\lambda}=\frac{2}{\langle
H_{\lambda},H_{\lambda}\rangle}H_{\lambda}=\frac{2}{\langle\lambda,\lambda\rangle}H_{\lambda}$

which satisfies the property that for all $\alpha,\beta\in\Sigma$,

$\langle H_{\alpha},x_{\beta}\rangle=\frac{2\langle
H_{\alpha},H_{\beta}\rangle}{\langle H_{\beta},H_{\beta}\rangle}\in\mathbb{Z}$

since $\Sigma$ is a crystallographic root system, Theorem 5.5. We note that in
the language of Section 3.4, we have $x_{\alpha}=H_{\alpha^{\vee}}$ for all
$\alpha\in\Sigma$. Let $\Delta=\left\\{\delta_{1},\ldots,\delta_{r}\right\\}$
be a basis of $\Sigma$ and abbreviate $x_{i}:=x_{\delta_{i}}$ and
$H_{i}:=H_{\delta_{i}}$. Let
$\overline{\mathfrak{a}}^{+}=\left\\{H\in\mathfrak{a}\colon\delta(H)\geq
0\text{ for all }\delta\in\Delta\right\\}$. On the way to prove Theorem 6.26,
Kostant describes $\operatorname{conv}(W_{s}x)$ using the closed convex cone

$\mathfrak{a}_{p}:=\left\\{x\in\mathfrak{a}\colon
x=\sum_{i=1}^{r}\mathbb{R}_{\geq 0}x_{i}\right\\}$

illustrated in Figure 9.

######  Lemma 6.27.

[Kos73, Lemma 3.3.(2)] Let $x,y\in\overline{\mathfrak{a}}^{+}$. Then

$\displaystyle y\in\operatorname{conv}(W_{s}x)\quad\iff\quad
x-y\in\mathfrak{a}_{p}.$

Now we want to describe $\mathfrak{a}_{p}$ by inequalities. The cone
$\mathfrak{a}_{p}$ is an intersection of open half-spaces defined by the half-
planes

$E_{j}=\sum_{k\neq j}\mathbb{R}x_{k}.$

Any vector $x\in\mathfrak{a}$ orthogonal to $E_{j}$ satisfies $\langle
x,x_{i}\rangle=0$ for all $i\neq j$. Writing $x=\sum_{k}\lambda_{jk}H_{k}$, we
have for all $i\neq j$

$\langle x,x_{i}\rangle=\sum_{k}\lambda_{jk}\langle H_{k},x_{i}\rangle=0.$

For every $i\neq j$, this is a homogeneous linear equation with variables
$\lambda_{j1},\ldots,\lambda_{jr}$ and coefficients $\langle
H_{k},x_{i}\rangle\in\mathbb{Z}$. Therefore there is a rational (and hence
integer) solution for the $\lambda_{jk}$. This shows that $x\in
E_{j}^{\perp}\setminus\\{0\\}$ may be chosen to lie in the lattice
$\Gamma=\sum_{l=1}^{r}\mathbb{Z}H_{k}$. There are two primitive vectors in
$\Gamma\cap E_{j}^{\perp}$. Let $e_{j}$ be the unique one that is on the same
side of $\partial E_{j}$ as $\mathfrak{a}_{p}$. Thus

$\mathfrak{a}_{p}=\left\\{x\in\mathfrak{a}\colon\langle e_{j},x\rangle\geq
0\text{ for all }j\right\\}.$

Figure 9: Root system of type $A_{2}$ associated to $\operatorname{SL}_{3}$.
The convex cone $\mathfrak{a}_{p}$ (orange stripes) can be viewed as spanned
by the $H_{\alpha_{i}}$ or as the intersection of the half-spaces defined by
the primitive vectors $e_{i}$.

Under the isomorphism $\mathfrak{a}^{\star}\cong\mathfrak{a}$, the lattice
$\Gamma$ corresponds to $L=\sum_{\alpha\in\Delta}\mathbb{Z}\alpha$ and we
define $\gamma_{j}\in L$ to be the element corresponding to $e_{j}\in\Gamma$.
By Lemma 6.1, $\gamma_{j}$ defines an algebraic character $\chi_{j}\colon
A_{\mathbb{R}}\to\mathbb{R}$ satisfying

$\chi_{j}(\exp(H))=e^{\gamma_{j}(H)}$

for all $H\in\mathfrak{a}$. The multiplicative closed Weyl chamber is

$A^{+}_{\mathbb{R}}:=\left\\{a\in A_{\mathbb{R}}\colon\chi_{\delta}(a)\geq
1\text{ for all }\delta\in\Delta\right\\}=\exp\overline{\mathfrak{a}}^{+}$

which is a semialgebraic set and thus has an $\mathbb{F}$-extension
$A^{+}_{\mathbb{F}}$. Using the the Iwasawa decomposition
$G_{\mathbb{F}}=U_{\mathbb{F}}A_{\mathbb{F}}K_{\mathbb{F}}$, Theorem 6.6, we
associate to every $g=uak\in G_{\mathbb{F}}$ its $A$-component
$a_{\mathbb{F}}(g)=a\in A_{\mathbb{F}}$ as before. We can now conclude the
following version of Kostant’s convexity theorem for $G_{\mathbb{F}}$,
illustrated in Figure 10.

Figure 10: Root system of type $A_{2}$ associated to $\operatorname{SL}_{3}$.
The convex set in Kostant’s convexity Theorem 6.28 defined by inequalities is
illustrated in purple.

######  Theorem 6.28.

For all $b\in A_{\mathbb{F}}^{+}$, we have

$\left\\{a\in A_{\mathbb{F}}^{+}\colon\exists k\in
K_{\mathbb{F}},a_{\mathbb{F}}(kb)=a\right\\}=\left\\{a\in
A_{\mathbb{F}}^{+}\colon\chi_{i}(a)\leq\chi_{i}(b)\text{ for all }i\right\\}.$

###### Proof.

We first verify that the theorem holds for $\mathbb{F}=\mathbb{R}$. An element
$a\in A_{\mathbb{R}}^{+}$ satisfies $a_{\mathbb{R}}(kb)=a$, for some $k\in
K_{\mathbb{R}}$, if and only if $a\in
A_{\mathbb{R}}^{+}\cap\exp(\operatorname{conv}(W_{s}\log(b)))$ by Theorem
6.26. This means $a=\exp(H)$ for some $H\in\overline{\mathfrak{a}}^{+}$ and
$H\in\operatorname{conv}(W_{s}\log(b))$. By Lemma 6.27, this is equivalent to

$\log(b)-H\in\mathfrak{a}_{p}=\left\\{x\in\mathfrak{a}\colon\langle
e_{j},x\rangle\geq 0\text{ for all }j\right\\},$

and taking exponents this is equivalent to

$\chi_{i}(ba^{-1})\geq 1$

for all $i$, so

$\chi_{i}(a)\leq\chi_{i}(b),$

as in the Theorem. This concludes the case $\mathbb{F}=\mathbb{R}$. Now we
note that being part of either set in the statement can be formulated as a
first-order formula. Since over $\mathbb{R}$ the two formulae imply each
other, they also imply each other over $\mathbb{F}$ by the transfer principle.
This concludes the proof. ∎

The following Lemma will be useful later, when we prove that the Iwasawa-
retraction is distance-diminishing in Theorem 7.4.

######  Lemma 6.29.

For all $\eta\in L=\sum_{\delta\in\Delta}\mathbb{Z}\delta$, we have

$\eta=\sum_{\ell=1}^{r}\frac{\langle\eta,\delta_{\ell}\rangle}{\langle\gamma_{\ell},\delta_{\ell}\rangle}\gamma_{\ell}$

and for $\eta^{+}:=\sum_{\alpha\in\Sigma_{>0}}\alpha$, $\eta^{+}$ is a
positive linear combination of the $\gamma_{\ell}$,

$\eta^{+}=\sum_{\alpha\in\Sigma_{>0}}\alpha\in\sum_{\ell=1}^{r}\mathbb{Q}_{>0}\gamma_{\ell}.$

###### Proof.

By definition $\langle x_{j},e_{\ell}\rangle=0$ for all $\ell\neq j$, but
$\langle x_{\ell},e_{\ell}\rangle\neq 0$, since otherwise $e_{\ell}=0$. This
implies that $\left\\{e_{1},\ldots,e_{r}\right\\}$ and hence
$\left\\{\gamma_{1},\ldots,\gamma_{r}\right\\}$ are linearly independent: for
if $x=\sum_{\ell}\lambda_{\ell}e_{\ell}=0$, then $\langle
x_{j},x\rangle=\lambda_{j}\langle x_{j},e_{j}\rangle=0$, so $\lambda_{j}=0$
for all $j$.

Therefore we can find $n_{i\ell}\in\mathbb{Q}$ such that

$\delta_{i}=\sum_{\ell=1}^{r}n_{i\ell}\gamma_{\ell}.$

Since $\langle\gamma_{\ell},\delta_{k}\rangle=0$ when $\ell\neq k$, we have
$\langle\delta_{i},\delta_{k}\rangle=n_{ik}\langle\gamma_{k},\delta_{k}\rangle$.
Since $x_{\ell}\in\mathfrak{a}_{p}$, $\langle e_{\ell},x_{\ell}\rangle\geq 0$
and since $e_{j}\neq 0$ and $\langle e_{\ell},x_{j}\rangle=0$ when $\ell\neq
j$, actually $\langle e_{\ell},x_{\ell}\rangle>0$. Thus also
$\langle\gamma_{\ell},\delta_{\ell}\rangle>0$, so we can divide
$\langle\delta_{i},\delta_{\ell}\rangle=n_{i\ell}\langle\gamma_{\ell},\delta_{\ell}\rangle$
by $\langle\gamma_{\ell},\delta_{\ell}\rangle$ to get

$n_{i\ell}=\frac{\langle\delta_{i},\delta_{\ell}\rangle}{\langle\gamma_{\ell},\delta_{\ell}\rangle}.$

For $\eta\in L=\sum_{\delta\in\Delta}\delta$ we have

$\displaystyle\eta$
$\displaystyle=\sum_{i=1}^{r}\lambda_{i}\delta_{i}=\sum_{i=1}^{r}\sum_{\ell=1}^{r}\lambda_{i}\frac{\langle\delta_{i},\delta_{\ell}\rangle}{\langle\gamma_{\ell},\delta_{\ell}\rangle}\gamma_{\ell}$
$\displaystyle=\sum_{\ell=1}^{r}\frac{\langle\sum_{i=1}^{r}\lambda_{i}\delta_{i},\delta_{\ell}\rangle}{\langle\gamma_{\ell},\delta_{\ell}\rangle}\gamma_{\ell}=\sum_{\ell=1}^{r}\frac{\langle\eta,\delta_{\ell}\rangle}{\langle\gamma_{\ell},\delta_{\ell}\rangle}\gamma_{\ell}.$

For $\delta\in\Delta$, recall that the reflection $\sigma_{\delta}$ permutes
the elements of $\Sigma_{>0}\setminus\\{\delta,2\delta\\}$, [Bou08a, VI.1.6
Cor. 1]. Let

$\eta^{+}=\sum_{\alpha\in\Sigma^{+}}\alpha.$

If $\sigma_{\delta}(\eta-\delta)=\eta-\delta$, for instance when $\Sigma$ is
reduced, then we can use that the reflection $\sigma_{\delta}$ preserves the
scalar product to obtain

$\displaystyle\langle\eta,\delta\rangle$
$\displaystyle=\langle\sigma_{\delta}(\eta),-\delta\rangle=\langle\sigma_{\delta}\left(\eta-\delta\right)+\sigma_{\delta}(\delta),-\delta\rangle$
$\displaystyle=\langle(\eta-\delta)-\delta,-\delta\rangle=-\langle\eta,\delta\rangle+2\langle\delta,\delta\rangle.$

so $\langle\eta,\delta\rangle=\langle\delta,\delta\rangle>0$. If $\Sigma$ is
not reduced and $\delta,2\delta\in\Sigma$, then
$\sigma(\eta-3\delta)=\eta-3\delta$ and

$\displaystyle\langle\eta,\delta\rangle$
$\displaystyle=\langle\sigma_{\delta}(\eta),-\delta\rangle=\langle\sigma_{\delta}\left(\eta-3\delta\right)+\sigma_{\delta}(3\delta),-\delta\rangle$
$\displaystyle=\langle(\eta-3\delta)-3\delta,-\delta\rangle=-\langle\eta,\delta\rangle+6\langle\delta,\delta\rangle,$

so $\langle\eta,\delta\rangle=3\langle\delta,\delta\rangle>0$. This shows that
$\eta$ is a positive linear combination of elements $\gamma_{\ell}$. ∎

Note that for the slightly simpler $\eta=\sum_{\delta\in\Delta}\delta$, $\eta$
may not be a positive linear combination of elements in
$\sum\mathbb{Z}\gamma_{\ell}$, see Figure 11 for an example of type
$\operatorname{G}_{2}$.

Figure 11:  The root system of type $\operatorname{G}_{2}$ with basis
$\delta_{1},\delta_{2}$, the coroots $\delta_{1}^{\vee},\delta_{2}^{\vee}$
corresponding to the coroots $x_{1},x_{2}$ and the elements
$\gamma_{1},\gamma_{2}$ spanning the Weyl chamber
$\overline{(\mathfrak{a}^{\star})}^{+}$. The element
$\eta^{+}:=\sum_{\alpha>0}\alpha$ from Lemma 6.29 lies in
$\overline{(\mathfrak{a}^{\star})}^{+}$, while the other candidate
$\eta:=\delta_{1}+\delta_{2}$ does not.

##  7 Definition of the building $\mathcal{B}$

Let $\mathbb{K}$ and $\mathbb{F}$ be real closed fields such that
$\mathbb{K}\subseteq\mathbb{R}\cap\mathbb{F}$. Often we assume $\mathbb{F}$ to
be non-Archimedean with order compatible valuation
$v\colon\mathbb{F}\to\Lambda\cup\\{\infty\\}$. Let $G$ be a semisimple
connected self-adjoint algebraic $\mathbb{K}$-group and $S$ a maximal
$\mathbb{K}$-split torus that satisfies $s=s{{}^{\mathsf{T}}\\!}$ for all
$s\in S$. In addition to the setting of Section 6, we assume that
$G<\operatorname{SL}_{n}$ for some $n\in\mathbb{N}$. Let $A_{\mathbb{F}}$ be
the semialgebraic extension of the semialgebraic connected component of
$S_{\mathbb{K}}$ containing the identity and let
$K=G\cap\operatorname{SO}_{n}$.

For $\mathfrak{a}=\operatorname{Lie}(A_{\mathbb{R}})$, let
$\Sigma\subseteq\mathfrak{a}^{\star}$ be the root system whose elements
$\alpha\in\Sigma$ correspond to $\mathbb{K}$-roots
$\chi_{\alpha}\in\vphantom{\Phi}_{\mathbb{K}}\Phi\subseteq\hat{S}$, see
Section 6.2. Then $W_{s}=\vphantom{W}_{\mathbb{K}}W$ is its spherical Weyl
group. After choosing a basis $\Delta\subseteq\Sigma$ we let $U$ be the
unipotent group associated to the positive root spaces and

$A_{\mathbb{F}}^{+}=\left\\{a\in A_{\mathbb{F}}\colon\chi_{\alpha}(a)\geq
1\text{ for all }\alpha\in\Delta\right\\}.$

###  7.1 Non-standard symmetric spaces

In the theory of symmetric spaces,

$P_{\mathbb{R}}=\left\\{x\in\mathbb{R}^{n\times n}\colon
x=x{{}^{\mathsf{T}}\\!},\det(x)=1,x\text{ is positive definite }\right\\}$

is a model for the symmetric space of non-compact type associated to
$\operatorname{SL}(n,\mathbb{R})$. The group $\operatorname{SL}(n,\mathbb{R})$
acts transitively on $P_{\mathbb{R}}$ by

$g.x=gxg^{T}.$

for $g\in\operatorname{SL}(n,\mathbb{R})$, $x\in P_{\mathbb{R}}$. The orbit
$\mathcal{X}_{\mathbb{R}}=G_{\mathbb{R}}.\operatorname{Id}\subseteq
P_{\mathbb{R}}$ is a closed subset and a model for the symmetric space
associated to $G_{\mathbb{R}}$. We note that $P$ is a semialgebraic set
defined over $\mathbb{Q}$ and consider its $\mathbb{F}$-extension
$P_{\mathbb{F}}$. The action is algebraic, so the orbit can be
semialgebraically extended to

$\mathcal{X}_{\mathbb{F}}=G_{\mathbb{F}}.P_{\mathbb{F}}.$

When $\mathbb{F}$ is non-Archimedean, we call $\mathcal{X}_{\mathbb{F}}$ the
_non-standard symmetric space associated to $G_{\mathbb{F}}$_.

######  Proposition 7.1.

  * (a)

The group $G_{\mathbb{F}}$ acts transitively on $\mathcal{X}_{\mathbb{F}}$.

  * (b)

The stabilizer of $\operatorname{Id}\in\mathcal{X}_{\mathbb{F}}$ is
$K_{\mathbb{F}}$.

  * (c)

For any pair $x,y\in\mathcal{X}_{\mathbb{F}}$ there is a $g\in G_{\mathbb{F}}$
such that $g.x=\operatorname{Id}$ and $g.y$ lies in the closed Weyl chamber

$A_{\mathbb{F}}^{+}.\operatorname{Id}=\left\\{a.\operatorname{Id}\in\mathcal{X}_{\mathbb{F}}\colon\chi_{\alpha}(a)\geq
1\text{ for all }\alpha\in\Sigma\right\\}.$

###### Proof.

Transitivity and stabilizer of $\operatorname{Id}$ follow directly from the
definitions. Use transitivity obtain $h\in G_{\mathbb{F}}$ with
$h.x=\operatorname{Id}$. Use transitivity again to obtain $h^{\prime}\in
G_{\mathbb{F}}$ with $h^{\prime}.h.y=\operatorname{Id}$. Now decompose
$h^{\prime}=kak^{\prime}\in
G_{\mathbb{F}}=K_{\mathbb{F}}A_{\mathbb{F}}K_{\mathbb{F}}$ using the Cartan
decomposition Theorem 6.8, where we may assume $a^{-1}\in A_{\mathbb{F}}^{+}$
after applying an element of the spherical Weyl group $W_{s}$. Setting
$g=k^{\prime}h$ now results in the claimed

$\displaystyle g.x$
$\displaystyle=k^{\prime}h.x=k^{\prime}.\operatorname{Id}=\operatorname{Id}$
$\displaystyle g.y$
$\displaystyle=k^{\prime}h.y=k^{\prime}(h^{\prime})^{-1}.\operatorname{Id}=a^{-1}k^{-1}.\operatorname{Id}=a^{-1}.\operatorname{Id}\in
A_{\mathbb{F}}^{+}.\operatorname{Id}.$

∎

###  7.2 The pseudo-distance

The symmetric space $\mathcal{X}_{\mathbb{R}}$ admits an explicit distance
formula. Here we mimic this process to define a pseudo-distance on
$\mathcal{X}_{\mathbb{F}}$. Let $x,y\in\mathcal{X}_{\mathbb{F}}$ be two
points. We will first send $x$ and $y$ to a common flat, on which we define a
multiplicative norm $N_{\mathbb{F}}$. In the real case, the logarithm would
then be applied to obtain an additive distance. For the non-standard symmetric
space we instead use the valuation $v\colon\mathbb{F}_{>0}\to\Lambda$.

By Proposition 7.1(a), there is for every $x,y\in\mathcal{X}_{\mathbb{F}}$ a
$g\in G_{\mathbb{F}}$ with Cartan decomposition $g=kak^{\prime}\in
G_{\mathbb{F}}=K_{\mathbb{F}}A_{\mathbb{F}}K_{\mathbb{F}}$, Theorem 6.8, such
that $g.x=y$. The corresponding $a\in A_{\mathbb{F}}$ can be chosen to lie in
$A_{\mathbb{F}}^{+}$ and is then unique. We can view this process as a map,
called the _Cartan projection_.

######  Lemma 7.2.

The Cartan-projection

$\displaystyle\delta_{\mathbb{F}}\colon\mathcal{X}_{\mathbb{F}}\times\mathcal{X}_{\mathbb{F}}$
$\displaystyle\to A_{\mathbb{F}}^{+}$ $\displaystyle(x,y)$
$\displaystyle\mapsto a$

is well-defined and invariant under the action of $G_{\mathbb{F}}$. For all
$x,y\in\mathcal{X}_{\mathbb{F}}$, $\delta_{\mathbb{F}}(y,x)$ is in the Weyl-
group orbit of $\delta(x,y)^{-1}$.

###### Proof.

We first assume $x=\operatorname{Id}$. If both $g=ka\bar{k}$ and
$g^{\prime}=k^{\prime}a^{\prime}\bar{k}^{\prime}$ satisfy $y=g.x=g^{\prime}.x$
with $a,a^{\prime}\in A_{\mathbb{F}}^{+}$ and
$k,\bar{k},k^{\prime},\bar{k}^{\prime}\in K_{\mathbb{F}}$, then we can apply
the Cartan decomposition, Theorem 6.8, to
$kaa{{}^{\mathsf{T}}\\!}k{{}^{\mathsf{T}}\\!}=ka\bar{k}.\operatorname{Id}=k^{\prime}a^{\prime}\bar{k}^{\prime}.\operatorname{Id}=k^{\prime}a^{\prime}(a^{\prime}){{}^{\mathsf{T}}\\!}(k^{\prime}){{}^{\mathsf{T}}\\!}$
to get that
$aa{{}^{\mathsf{T}}\\!}=a^{\prime}(a^{\prime}){{}^{\mathsf{T}}\\!}$ as
elements in $A_{\mathbb{F}}^{+}$. Note that $aa{{}^{\mathsf{T}}\\!}\in
A_{\mathbb{F}}^{+}$, since $a=a{{}^{\mathsf{T}}\\!}$, by our assumption on the
torus $S$. Since $A_{\mathbb{F}}$ is an abelian group without torsion,
$aa{{}^{\mathsf{T}}\\!}=a^{\prime}(a^{\prime}){{}^{\mathsf{T}}\\!}$ implies
$a=a^{\prime}$, showing that
$\delta_{\mathbb{F}}(\operatorname{Id},g.\operatorname{Id})=a$ is well
defined.

For general $x,y$, we can always find an $h\in G_{\mathbb{F}}$ such that
$x=h.\operatorname{Id}$ and $y=g.h.\operatorname{Id}$. As the definition of
$\delta_{\mathbb{F}}$ only depends on $g$ and not on $h$ in this case,
$\delta_{\mathbb{F}}$ is invariant under $G_{\mathbb{F}}$, and thus well-
defined everywhere.

If $g=kak^{\prime}$ satisfies $g.x=y$, then
$g^{-1}=(k^{\prime})^{-1}a^{-1}k^{-1}$ satisfies $g^{-1}.y=x$. By definition,
the unique element of $A_{\mathbb{F}}^{+}$ in the Weyl orbit of $a^{-1}$ is
$\delta_{\mathbb{F}}(y,x)$. ∎

We use a basis $\Delta$ of the root system $\Sigma$ to define a notion of
positive roots $\Sigma_{>0}$. For $\alpha\in\Sigma$, let $\chi_{\alpha}\colon
A_{\mathbb{R}}\to\mathbb{R}^{\times}$ be the corresponding
($\mathbb{R}$-points of the) algebraic characters. We define a continuous,
semialgebraic $W_{s}$-invariant map

$\displaystyle N_{\mathbb{R}}\colon A_{\mathbb{R}}$
$\displaystyle\to\mathbb{R}^{\times}$ $\displaystyle a$
$\displaystyle\mapsto\prod_{\alpha\in\Sigma}\max\left\\{\chi_{\alpha}(a),\chi_{\alpha}(a)^{-1}\right\\}$

which is a multiplicative norm, meaning that for all $a,b\in A_{\mathbb{R}}$

  * (1)

$N_{\mathbb{R}}(a)\geq 1$ and $N_{\mathbb{R}}(a)=1$ if and only if
$a=\operatorname{Id}$,

  * (2)

$N_{\mathbb{R}}(ab)\leq N_{\mathbb{R}}(a)N_{\mathbb{R}}(b)$.

We call $N_{\mathbb{R}}$ the _semialgebraic norm_ 111111There are many
continuous semialgebraic multiplicative norms satisfying (1) and (2), but as
norms on finite dimensional vector spaces, they are equivalent and it suffices
for our purposes to fix $N_{\mathbb{R}}$.. Since $N_{\mathbb{R}}$ is
semialgebraic, we can extend it to a map $N_{\mathbb{F}}\colon
A_{\mathbb{F}}\to\mathbb{F}^{\times}$ which is still a $W_{s}$-invariant
multiplicative norm satisfying (1) and (2) by the transfer principle and
$N_{\mathbb{F}}$ is given by the same formula involving the characters.

For $\mathbb{F}$ non-Archimedean, we now use the Cartan projection
$\delta_{\mathbb{F}}$ together with the semialgebraic norm $N_{\mathbb{F}}$
and the valuation $v\colon\mathbb{F}_{>0}\to\Lambda$ to define

$\displaystyle d\colon X_{\mathbb{F}}\times X_{\mathbb{F}}$
$\displaystyle\to\Lambda$ $\displaystyle(x,y)$
$\displaystyle\mapsto(-v)(N_{\mathbb{F}}(\delta_{\mathbb{F}}(x,y))).$

We will show in Theorem 7.5 that $d$ is a pseudo-distance on
$\mathcal{X}_{\mathbb{F}}$. The pseudo-distance $d$ fails to be positive
definite essentially due to the fact that $v$ is not injective. The proof of
the triangle inequality uses Kostant’s convexity theorem and the Iwasawa
retraction

$\displaystyle\rho\colon\mathcal{X}_{\mathbb{F}}$ $\displaystyle\to
A_{\mathbb{F}}.\operatorname{Id}$ $\displaystyle
g.\operatorname{Id}=uak.\operatorname{Id}$ $\displaystyle\mapsto
a.\operatorname{Id}.$

using the Iwasawa-decomposition
$G_{\mathbb{F}}=U_{\mathbb{F}}A_{\mathbb{F}}K_{\mathbb{F}}$, Theorem 6.6.

######  Lemma 7.3.

For all $a\in A_{\mathbb{F}}$, $x\in\mathcal{X}_{\mathbb{F}}$,
$\rho(a.x)=a.\rho(x)$.

###### Proof.

Let $g=ua^{\prime}k\in
G_{\mathbb{F}}=U_{\mathbb{F}}A_{\mathbb{F}}K_{\mathbb{F}}$ such that
$g.\operatorname{Id}=x$. By Proposition 6.15, $aua^{-1}\in U_{\mathbb{F}}$, so

$\displaystyle\rho(a.x)$
$\displaystyle=\rho(aua^{\prime}k.\operatorname{Id})=\rho((aua^{-1}aa^{\prime}.\operatorname{Id})=aa^{\prime}.\operatorname{Id}=a.\rho(ua^{\prime}k.\operatorname{Id})=a.\rho(x).$

∎

We use Kostant’s convexity theorem to prove that $\rho$ is a $d$-diminishing
retraction.

######  Theorem 7.4.

The map $\rho\colon\mathcal{X}_{\mathbb{F}}\to
A_{\mathbb{F}}.\operatorname{Id}$ is a $d$-diminishing,

$\forall x,y\in\mathcal{X}_{\mathbb{F}}\colon d(\rho(x),\rho(y))\leq d(x,y),$

retraction to $A_{\mathbb{F}}.\operatorname{Id}$.

###### Proof.

It is clear that $\rho$ is a retraction, meaning that $\forall a\in
A_{\mathbb{F}},\rho(a.\operatorname{Id})=a.\operatorname{Id}$. Let
$a_{\mathbb{F}}(g)=a_{\mathbb{F}}(uak)=a$ denote the
$A_{\mathbb{F}}$-component of $g\in G_{\mathbb{F}}$ as in Section 6.10 on
Kostant convexity. To show that $\rho$ is $d$-diminishing, we first claim that
for all $b\in A_{\mathbb{F}}^{+}$ and for all $k\in K_{\mathbb{F}}$

$d(\operatorname{Id},\rho(kb.\operatorname{Id}))\leq
d(\operatorname{Id},b.\operatorname{Id}).$

For $b\in A_{\mathbb{F}}^{+}$ the set

$S_{\mathbb{F}}^{b}=\left\\{a\in A_{\mathbb{F}}\colon
a=a_{\mathbb{F}}(kb)\text{ for some }k\in K_{\mathbb{F}}\right\\}$

is semialgebraic and since over $\mathbb{R}$, $S_{\mathbb{R}}^{b}$ is closed
under the action of the spherical Weyl group (this is a consequence of the
real Kostant convexity Theorem 6.26). The statement
$W(S_{\mathbb{R}}^{b})\subseteq S_{\mathbb{R}}^{b}$ can be formulated as a
first-order formula, so $S_{\mathbb{F}}^{b}$ is also closed under the action
of $W_{s}$. Note that
$\rho(kb.\operatorname{Id})=a_{\mathbb{F}}(kb).\operatorname{Id}$. While
$a_{\mathbb{F}}(kb)$ may not lie in $A_{\mathbb{F}}^{+}$, there is a $w\in W$
such that $w(a_{\mathbb{F}}(kb))\in A_{\mathbb{F}}^{+}$. We apply Theorem 6.28
to $w(a_{\mathbb{F}}(kb))\in A_{\mathbb{F}}^{+}\cap S_{\mathbb{F}}^{b}$ get

$\chi_{\gamma_{i}}(w(a_{\mathbb{F}}(kb)))\leq\chi_{\gamma_{i}}(b)$

for all $\gamma_{i}$ described in Section 6.10. By Lemma 6.29 for every
$\alpha\in\Sigma$ there are positive rational numbers $n_{\alpha
i}\in\mathbb{Q}_{>0}$ such that

$\alpha=\sum_{i=1}^{r}n_{\alpha i}\gamma_{i}.$

We can now prove

$\displaystyle d(\operatorname{Id},\rho(kb.\operatorname{Id}))$
$\displaystyle=\prod_{\alpha\in\Sigma_{>0}}\max\left\\{\chi_{\alpha}(a_{\mathbb{F}}(kb)),\chi_{\alpha}(a_{\mathbb{F}}(kb))^{-1}\right\\}$
$\displaystyle=\prod_{\alpha\in\Sigma_{>0}}\chi_{\alpha}(w(a_{\mathbb{F}}(kb)))$
$\displaystyle=\prod_{\alpha\in\Sigma_{>0}}\prod_{i=1}^{r}\chi_{\gamma_{i}}(w(a_{\mathbb{F}}(kb)))^{n_{\alpha
i}}$
$\displaystyle\leq\prod_{\alpha\in\Sigma_{>0}}\prod_{i=1}^{r}\chi_{\gamma_{i}}(b)^{n_{\alpha
i}}$
$\displaystyle=\prod_{\alpha\in\Sigma_{>0}}\chi_{\alpha}(b)=d(\operatorname{Id},b.\operatorname{Id}),$

where we used that $n_{\alpha i}>0$, proving the claim.

Now let $x,y\in\mathcal{X}_{\mathbb{F}}$ arbitrary. By Proposition 7.1(c) and
Theorem 6.6, we can find $g=uak\in
G_{\mathbb{F}}=U_{\mathbb{F}}A_{\mathbb{F}}K_{\mathbb{F}}$ such that
$x=g.\operatorname{Id}$ and $y=g.b.\operatorname{Id}$ for some $b\in
A_{\mathbb{F}}$. Now we use Lemma 7.3 and the above to conclude

$\displaystyle d(\rho(x),\rho(y))$
$\displaystyle=d(\rho(uak.\operatorname{Id}),\rho(uakb.\operatorname{Id}))=d(a.\operatorname{Id},\rho(akb.\operatorname{Id}))$
$\displaystyle=d(\operatorname{Id},a^{-1}.\rho(akb.\operatorname{Id}))=d(\operatorname{Id},\rho(kb.\operatorname{Id}))$
$\displaystyle\leq
d(\operatorname{Id},b.\operatorname{Id})=d(g.\operatorname{Id},g.b.\operatorname{Id})=d(x,y)$

concluding the proof. ∎

######  Theorem 7.5.

The function
$d\colon\mathcal{X}_{\mathbb{F}}\times\mathcal{X}_{\mathbb{F}}\to\Lambda$ is a
pseudo-distance.

###### Proof.

We note that by definition, $N_{\mathbb{F}}(a)=N_{\mathbb{F}}(a^{-1})$ for all
$a\in A_{\mathbb{F}}$. By Weyl group invariance and the last part of Lemma 7.2
we then obtain
$N_{\mathbb{F}}(\delta_{\mathbb{F}}(x,y))=N_{\mathbb{F}}(\delta_{\mathbb{F}}(y,x))$
for all $x,y\in\mathcal{X}_{\mathbb{F}}$, whence $d$ is symmetric. Since
$N_{\mathbb{F}}(A_{\mathbb{F}})\subseteq\mathbb{F}_{\geq 1}$, $(-v)(1)=0$ and
$-v$ is order-preserving, $d$ is positive. It is also clear that $d(x,x)=0$
for all $x\in\mathcal{X}_{\mathbb{F}}$. It remains to show that the triangle
inequality holds. We start by analyzing the distance on the non-standard
maximal flats $A_{\mathbb{F}}.\operatorname{Id}$. Let $a,b,c\in
A_{\mathbb{F}}$, then we can use property (2) of the semialgebraic norm
$N_{\mathbb{F}}\colon\mathbb{A}_{\mathbb{F}}\to\mathbb{F}_{>0}$ to deduce

$\displaystyle d(a.\operatorname{Id},b.\operatorname{Id})$
$\displaystyle=d(\operatorname{Id},a^{-1}b.\operatorname{Id})=(-v)(N_{\mathbb{F}}(a^{-1}b))$
$\displaystyle=(-v)(N_{\mathbb{F}}(a^{-1}cc^{-1}b))\leq-(v)(N_{\mathbb{F}}(a^{-1}c)N_{\mathbb{F}}(c^{-1}b))$
$\displaystyle=-v(N_{\mathbb{F}}(a^{-1}c))+(-v)(N_{\mathbb{F}}(c^{-1}b))$
$\displaystyle=d(a.\operatorname{Id},c.\operatorname{Id})+d(c.\operatorname{Id},b.\operatorname{Id}),$

which settles the triangle inequality for points in
$A_{\mathbb{F}}.\operatorname{Id}$. For the general case we use the Iwasawa
retraction from Theorem 7.4, as suggested in [KT02, Lemma 1.2]. Let
$x,y,z\in\mathcal{X}_{\mathbb{F}}$. By Proposition 7.1(c) there is a $g\in
G_{\mathbb{F}}$ with $g.x,g.y\in A_{\mathbb{F}}.\operatorname{Id}$. Then

$\displaystyle d(x,y)$ $\displaystyle=d(g.x,g.y)=d(\rho(g.x),\rho(g.y))$
$\displaystyle\leq d(\rho(g.x),\rho(g.z))+d(\rho(g.z),\rho(g.y))$
$\displaystyle\leq d(g.x,g.z)+d(g.z,g.y)=d(x,z)+d(z,y)$

concludes the proof. ∎

###  7.3 The apartment

Over the reals, the orbit
$A_{\mathbb{R}}.\operatorname{Id}\subseteq\mathcal{X}_{\mathbb{R}}$ is a
maximal flat in the symmetric space $\mathcal{X}$. We take a closer look at
the group $A_{\mathbb{F}}$ and its orbit
$A_{\mathbb{F}}.\operatorname{Id}\subseteq\mathcal{X}_{\mathbb{F}}$ to define
a space $A_{\Lambda}$ which will play the role of the model apartment for an
affine $\Lambda$-building. Let $O$ be an order convex valuation ring of the
non-Archimedean real closed field $\mathbb{F}$ and
$(-v)\colon\mathbb{F}\to\Lambda\cup\\{\infty\\}$ the associated order
preserving valuation. We define the group

$A_{\Lambda}=A_{\mathbb{F}}/\\{a\in A_{\mathbb{F}}\colon N_{\mathbb{F}}(a)\in
O\\}.$

The goal of this section is to prove Theorem 7.8 which states that
$A_{\Lambda}$ can be given the structure of a model apartment
$\mathbb{A}=\mathbb{A}(\vphantom{\Phi}_{\mathbb{K}}\Phi,\Lambda,A_{\Lambda})$,
as defined in Section 3.4.

The spherical Weyl group $W_{s}=N_{\mathbb{F}}/M_{\mathbb{F}}$ acts on
$A_{\mathbb{F}}$ by $[k].a=kak^{-1}$ for $a\in A_{\mathbb{F}},k\in
N_{\mathbb{F}}$. Since $N_{\mathbb{F}}$ is $W_{s}$-invariant, the action
descends to an action by automorphisms on $\mathbb{A}_{\Lambda}$.

Recall from Proposition 4.6 that there is a non-degenerate bilinear form

$\displaystyle\hat{S}\times X_{\star}(S)$ $\displaystyle\to\mathbb{Z}$
$\displaystyle(\chi,t)$ $\displaystyle\mapsto b(\chi,t)$

with the defining property that $\chi\circ t(x)=x^{b(\chi,t)}$ for all
$x\in\mathbb{G}_{m}$. Since $S$ is $\mathbb{K}$-split and hence
$\mathbb{F}$-split by Theorem 5.17, we can take the $\mathbb{F}$-points and
restrict to $A_{\mathbb{F}}$ and $\mathbb{F}_{>0}$ to obtain characters
$\chi_{\mathbb{F}}\colon A_{\mathbb{F}}\to\mathbb{F}_{>0}$ and one-parameter
subgroups $t_{\mathbb{F}}\colon\mathbb{F}_{>0}\to A_{\mathbb{F}}$ for
$\chi\in\hat{S}$ and $t\in X_{\star}(S)$. Note that since $\chi_{\mathbb{F}}$
is continuous and $A_{\mathbb{F}}$ is semialgebraically connected,
$\chi_{\mathbb{F}}(A_{\mathbb{F}})\subseteq\mathbb{F}_{>0}$ and similarly
since $t_{\mathbb{F}}$ is continuous and $\mathbb{F}_{>0}$ is connected,
$t_{\mathbb{F}}(\mathbb{F}_{>0})\subseteq A_{\mathbb{F}}$. We now use the
valuation $(-v)\colon\mathbb{F}_{>0}\to\Lambda$ to set up tools to prove that
$A_{\Lambda}$ is isomorphic to the model apartment $\mathbb{A}$.

######  Proposition 7.6.

Let
$\chi\in\operatorname{Span}_{\mathbb{Z}}(\vphantom{\Phi}_{\mathbb{K}}\Phi)$
and $t\in X_{\star}(S)$. The $\mathbb{F}$-valued characters
$\chi_{\mathbb{F}}$ and one-parameter subgroups $t_{\mathbb{F}}$ descend to
group homomorphisms $\chi_{\Lambda}$ and $t_{\Lambda}$ such that the diagram

simplicial buildingsspherical buildings$\operatorname{rank}/\dim$discrete
affine buildings Euclidean buildings [KL97]$\mathbb{R}$-buildings [BT72,
Par00]$\Lambda$-buildings [Ben94, HIL24]
[KT02]trees$\mathbb{R}$-trees$\Lambda$-trees$1$$0$affine
buildings$1$$2$$2$$3$$3$$4$$\vdots$$\vdots$${\mathbb{F}_{>0}}$${A_{\mathbb{F}}}$${\mathbb{F}_{>0}}$${\Lambda}$${A_{\Lambda}}$${\Lambda}$$\scriptstyle{-v}$$\scriptstyle{t_{\mathbb{F}}}$$\scriptstyle{\chi_{\mathbb{F}}}$$\scriptstyle{-v}$$\scriptstyle{t_{\Lambda}}$$\scriptstyle{\chi_{\Lambda}}$

commutes and such that $\chi_{\Lambda}\circ
t_{\Lambda}(\lambda)=b(\chi,t)\cdot\lambda.$

###### Proof.

We denote by $\pi\colon A_{\mathbb{F}}\to A_{\Lambda}=A_{\mathbb{F}}/\\{a\in
A_{\mathbb{F}}\colon N_{\mathbb{F}}(a)\in O\\}$ the projection. We first show
that $t_{\Lambda}\colon\Lambda\to
A_{\Lambda},\,(-v)(x)\mapsto\pi(t_{\mathbb{F}}(x))$ is well defined: we have
to show that if $(-v)(x)=0$, then $N(t_{\mathbb{F}}(x))\in O$. Indeed, let
$(-v)(x)=0$, so $x\in O^{\times}$. Then
$\alpha_{\mathbb{F}}(t_{\mathbb{F}}(x))=x^{b(\alpha,t)}\in O^{\times}$ for all
$\alpha\in\vphantom{\Phi}_{\mathbb{K}}\Phi$ and thus

$\displaystyle N_{\mathbb{F}}(t_{\mathbb{F}}(x))$
$\displaystyle=\prod_{\alpha\in\vphantom{\Phi}_{\mathbb{K}}\Phi}\max\left\\{\alpha_{\mathbb{F}}(t_{\mathbb{F}}(x)),\alpha_{\mathbb{F}}(t_{\mathbb{F}}(x))^{-1}\right\\}\in
O.$

To show that $\chi_{\Lambda}\colon A_{\Lambda}\to\Lambda,\,\pi(a)\mapsto
v(\chi_{\mathbb{F}}(a))$ is well-defined, we have to show that if
$N_{\mathbb{F}}(a)\in O$, then $(-v)(\chi_{\mathbb{F}}(a))=0$. Indeed, if
$N_{\mathbb{F}}(a)\in O$, then $\alpha_{\mathbb{F}}(a)\in O$ for all
$\alpha\in\vphantom{\Phi}_{\mathbb{K}}\Phi$ since

$\displaystyle N_{\mathbb{F}}(a)$
$\displaystyle=\prod_{\alpha\in\vphantom{\Phi}_{\mathbb{K}}\Phi}\max\left\\{\chi_{\alpha}(a),\chi_{\alpha}(a)^{-1}\right\\}\in
O$

is a product of elements that are larger than $1$. Since
$\chi\in\operatorname{Span}_{\mathbb{Z}}(\vphantom{\Phi}_{\mathbb{K}}\Phi)$,
$\chi_{\mathbb{F}}(a)$ is a product of $\alpha_{\mathbb{F}}(a)\in O$ and hence
in $O$.

The maps $t_{\Lambda}$ and $\chi_{\Lambda}$ were defined so that the diagrams
commute. For $\lambda\in\Lambda$, we can find $x\in\mathbb{F}_{>0}$ such that
$(-v)(x)=\lambda$. Then

$\displaystyle\chi_{\Lambda}\circ t_{\Lambda}(\lambda)$
$\displaystyle=\chi_{\Lambda}(\pi(t_{\mathbb{F}}(x)))=(-v)\left(\chi_{\mathbb{F}}\circ
t_{\mathbb{F}}(x)\right)$
$\displaystyle=v\left(x^{b(\chi,t)}\right)=b(\chi,t)\cdot(-v)(x).$

∎

Let $\Delta\subseteq\vphantom{\Phi}_{\mathbb{K}}\Phi$ be a basis and define
for $\delta\in\Delta$ a coroot $t_{\delta}\in X_{\star}(S)$ as in Lemma 6.2.
For general
$\chi\in\operatorname{Span}_{\mathbb{Z}}(\vphantom{\Phi}_{\mathbb{K}}\Phi)\subseteq\hat{S}$,
we use bilinearity to define $t_{\chi}\in X_{\star}(S)$ via

$\displaystyle
t_{\chi}(x)=\prod_{\delta\in\Delta}t_{\delta}(x)^{\lambda_{\delta}}\quad\text{
for
}\quad\chi=\prod_{\delta\in\Delta}\chi_{\delta}^{\lambda_{\delta}}\in\operatorname{Span}_{\mathbb{Z}}(\vphantom{\Phi}_{\mathbb{K}}\Phi),\quad\lambda_{\delta}\in\mathbb{Z},$

and we note that for all
$\chi,\nu\in\operatorname{Span}_{\mathbb{Z}}(\vphantom{\Phi}_{\mathbb{K}}\Phi)$,
$\chi(t_{\nu}(x))=x^{b(\chi,t_{\nu})}$ for $x\in G_{m}$.

Let $r$ denote the rank of the root system
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$.

######  Proposition 7.7.

There is a group isomorphism

$\displaystyle f_{\mathbb{F}}\colon(\mathbb{F}_{>0})^{r}$ $\displaystyle\to
A_{\mathbb{F}}$ $\displaystyle(x_{\delta})_{\delta\in\Delta}$
$\displaystyle\mapsto\prod_{\delta\in\Delta}(t_{\delta})_{\mathbb{F}}(x_{\delta})$

satisfying

$\delta(f_{\mathbb{F}}(x_{1},\ldots,x_{r}))=x_{\delta}$

for all $(x_{1},\ldots,x_{r})\in(\mathbb{F}_{>0})^{r}$ and $\delta\in\Delta$.

###### Proof.

Note that $f$ is a semialgebraic map defined over $\mathbb{K}$. We may write
$f_{\mathbb{R}}$ or $f_{\mathbb{F}}$ for the respective semialgebraic
extensions, when necessary, but forego the $\mathbb{F}$-subscript when there
is not danger of confusion. It is clear that $f$ is a homomorphism, we will
now find an inverse. We start by giving names to the elements of the basis
$\Delta=\\{\delta_{1},\ldots,\delta_{r}\\}$. We define the $r\times r$ integer
matrix $M$ by $M_{ij}=b(\delta_{i},t_{\delta_{j}})$. Since $M$ represents a
non-degenerate bilinear form, it is invertible and its inverse $M^{-1}$ has
rational entries. For $j=1,\ldots,r$, we define

$\chi_{j}=\prod_{k=1}^{r}\delta_{k}^{(M^{-1})_{jk}}\in\operatorname{Span}_{\mathbb{Q}}(\vphantom{\Phi}_{\mathbb{K}}\Phi)$

and use them to give an explicit right inverse

$\displaystyle f^{-1}\colon A_{\mathbb{F}}$
$\displaystyle\to(\mathbb{F}_{>0})^{r}$ $\displaystyle a$
$\displaystyle\mapsto(\chi_{1}(a),\ldots,\chi_{r}(a)).$

We calculate the $j$th entry

$\displaystyle f^{-1}(f(x_{1},\ldots,x_{r}))_{j}$
$\displaystyle=f^{-1}\left(\prod_{i=1}^{r}t_{\delta_{i}}(x_{i})\right)_{j}=\chi_{j}\left(\prod_{i=1}^{r}t_{\delta_{i}}(x_{i})\right)$
$\displaystyle=\prod_{i=1}^{r}\chi_{j}\left(t_{\delta_{i}}(x_{i})\right)=\prod_{i=1}^{r}\prod_{k=1}^{r}\delta_{k}^{(M^{-1})_{jk}}\left(t_{\delta_{i}}(x_{i})\right)$
$\displaystyle=\prod_{i=1}^{r}\prod_{k=1}^{r}(\delta_{k}\circ
t_{\delta_{i}})(x_{i})^{(M^{-1})_{jk}}=\prod_{i=1}^{r}\prod_{k=1}^{r}\left((x_{i})^{b(\delta_{k},t_{\delta_{i}})}\right)^{(M^{-1})_{jk}}$
$\displaystyle=\prod_{i=1}^{r}(x_{i})^{\sum_{k=1}^{r}(M^{-1})_{jk}M_{ki}}=\prod_{i=1}^{r}(x_{i})^{\operatorname{Id}_{ji}}=x_{j}.$

The existence of the right inverse proves that $f$ is surjective. We will now
argue why $f$ is also injective.

Note that $(\mathbb{R}_{>0})^{r}$ is a $r$-dimensional $\mathbb{R}$-vector
space, when equipped with addition defined by multiplication $(v,w)\mapsto vw$
and scalar multiplication defined by $(\lambda,v)\mapsto e^{\lambda}v$, where
$v,w\in\mathbb{R}_{>0},\lambda\in\mathbb{R}$. Similarly,
$A_{\mathbb{R}}=\exp(\mathfrak{a})$ is a $\mathbb{R}$-vector space since
$\mathfrak{a}$ is. By Proposition 6.3, $A_{\mathbb{R}}$ is also
$r$-dimensional. The map $f_{\mathbb{R}}$ is a surjective linear map between
vector spaces of the same finite dimension, and hence is bijective.

To get injectivity for $f_{\mathbb{F}}$, we write it as a first-order
statement

$\varphi\colon\quad\forall
x_{1},\ldots,x_{r}\colon\bigwedge_{i=1}^{r}x_{i}>0\wedge\left(f(x_{1},\ldots,x_{r})=\operatorname{Id}\to\bigwedge_{i=1}^{r}x_{i}=1\right),$

where we used the fact that the $t_{\delta}$ and thus $f$ are algebraic. By
the above, $\varphi$ holds over $\mathbb{R}$ and thus it holds over
$\mathbb{F}$. This concludes the proof that $f_{\mathbb{F}}$ is an
isomorphism. ∎

Finally we prove that $A_{\Lambda}$ is isomorphic to
$\mathbb{A}\cong\Lambda^{r}$.

######  Theorem 7.8.

The group isomorphism $f_{\mathbb{F}}\colon(\mathbb{F}_{>0})^{r}\cong
A_{\mathbb{F}}$ descends to a group isomorphism

$\displaystyle f_{\Lambda}\colon\Lambda^{r}$ $\displaystyle\to A_{\Lambda}$
$\displaystyle(\lambda_{\delta})_{\delta\in\Delta}$
$\displaystyle\mapsto\sum_{\delta\in\Delta}(t_{\delta})_{\Lambda}(\lambda_{\delta}).$

###### Proof.

As a consequence of Proposition 7.6, the diagram

simplicial buildingsspherical buildings$\operatorname{rank}/\dim$discrete
affine buildings Euclidean buildings [KL97]$\mathbb{R}$-buildings [BT72,
Par00]$\Lambda$-buildings [Ben94, HIL24]
[KT02]trees$\mathbb{R}$-trees$\Lambda$-trees$1$$0$affine
buildings$1$$2$$2$$3$$3$$4$$\vdots$$\vdots$${(\mathbb{F}_{>0})^{r}}$${A_{\mathbb{F}}}$${\Lambda^{r}}$${A_{\Lambda}}$$\scriptstyle{f_{\mathbb{F}}}$$\scriptstyle{v^{r}}$$\scriptstyle{f_{\Lambda}}$

commutes. Since $v$ and the projection $\pi\colon A_{\mathbb{F}}\to
A_{\Lambda}$ are surjective, surjectivitiy of $f_{\Lambda}$ follows from the
surjectivity of $f_{\mathbb{F}}$, Proposition 7.7. It remains to show that
$f_{\Lambda}$ is injective. Let
$\Delta=\left\\{\delta_{1},\dots,\delta_{r}\right\\}$ and
$t_{i}=(t_{\delta_{i}})_{\mathbb{F}}$. For $a\in\mathbb{A}_{\mathbb{F}}$ with
$N_{\mathbb{F}}(a)\in O$, we showed in the proof of Proposition 7.6, that
$v(\delta_{j}(a))=0$ for all $j\in\\{1,\ldots,r\\}$. We use the inverse
function $f^{-1}_{\mathbb{F}}$ constructed in Proposition 7.7 to obtain

$\displaystyle v\left(\left(f^{-1}_{\mathbb{F}}(a)\right)_{j}\right)$
$\displaystyle=v\left(\prod_{k=1}^{r}\delta_{k}^{M_{jk}^{-1}}(a)\right)=\sum_{k=1}^{r}M_{jk}^{-1}v(\delta_{k}(a))=0$

where $M^{-1}_{jk}$ are some rational numbers. This shows
$\ker(f_{\Lambda})=0\in\Lambda^{r}$ and hence $f_{\Lambda}$ is injective. ∎

We now know that a choice of basis
$\Delta\subseteq\vphantom{\Phi}_{\mathbb{K}}\Phi$ leads to isomorphisms

simplicial buildingsspherical buildings$\operatorname{rank}/\dim$discrete
affine buildings Euclidean buildings [KL97]$\mathbb{R}$-buildings [BT72,
Par00]$\Lambda$-buildings [Ben94, HIL24]
[KT02]trees$\mathbb{R}$-trees$\Lambda$-trees$1$$0$affine
buildings$1$$2$$2$$3$$3$$4$$\vdots$$\vdots$${A_{\Lambda}}$${\Lambda^{r}}$${\mathbb{A}:=\operatorname{Span}_{\mathbb{Z}}(\vphantom{\Phi}_{\mathbb{K}}\Phi)\otimes_{\mathbb{Z}}\Lambda}$$\scriptstyle{f_{\Lambda}}$$\scriptstyle{g}$

where

$g((\lambda_{\delta})_{\delta\in\Delta})=\sum_{\delta\in\Delta}\delta\otimes\lambda_{\delta}.$

For $A_{\Lambda}$ to be a model apartment of type
$(\vphantom{\Phi}_{\mathbb{K}}\Phi,\Lambda,\Lambda^{r})$ as defined in Section
3.4, the isomorphism $A_{\Lambda}\cong\mathbb{A}$ has to be Weyl group
equivariant. The spherical Weyl group $W_{s}$ acts on
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ and hence on $\mathbb{A}$ as described in
Section 3.4. The group
$N_{\mathbb{F}}=\operatorname{Nor}_{K_{\mathbb{F}}}(A_{\mathbb{F}})$ acts on
$A_{\mathbb{F}}$ by isometries and hence on $A_{\Lambda}$. The spherical Weyl
group $W_{s}$ is isomorphic to $N_{\mathbb{F}}/M_{\mathbb{F}}$ by Section 6.2.
The translation group $T:=\Lambda^{r}$ acts on $A_{\Lambda}$ and $\mathbb{A}$
via $f_{\Lambda}$ and $g$ and this action coincides with the action on
$\mathbb{A}$ defined in Section 3.4.

######  Proposition 7.9.

The isomorphism $g\circ f_{\Lambda}^{-1}\colon A_{\Lambda}\to\mathbb{A}$ is
equivariant with respect to the actions of the affine Weyl group $W_{s}\ltimes
T$.

###### Proof.

Let $w\in W_{s}$ be an element of the spherical Weyl group of
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$. For $\delta,\eta\in\Delta$, there are
constants $k_{\delta,\eta}\in\mathbb{Z}$ such that
$w(\eta)=\sum_{\delta\in\Delta}k_{\delta,\eta}\delta$. Then

$\displaystyle w(g((\lambda_{\delta})_{\delta}))$
$\displaystyle=w\left(\sum_{\delta\in\Delta}\delta\otimes\lambda_{\delta}\right)=\sum_{\eta\in\Delta}w(\eta)\otimes\lambda_{\eta}=\sum_{\eta\in\Delta}\sum_{\delta\in\Delta}k_{\delta,\eta}\delta\otimes\lambda_{\eta}$
$\displaystyle=\sum_{\delta\in\Delta}\delta\otimes\left(\sum_{\eta\in\Delta}k_{\delta,\eta}\lambda_{\eta}\right)=g\left(\left(\sum_{\eta\in\Delta}k_{\delta,\eta}\lambda_{\eta}\right)_{\delta\in\Delta}\right).$

For $x\in\mathbb{F}_{>0}$ and $\eta\in\Delta$, we have
$w((t_{\eta})_{\mathbb{F}}(x))=(t_{w(\eta)})_{\mathbb{F}}(x)$ and hence

$\displaystyle w(f_{\Lambda}((\lambda_{\delta})_{\delta})))$
$\displaystyle=w\left(\prod_{\eta\in\Delta}(t_{\eta})_{\Lambda}(\lambda_{\eta})\right)=\prod_{\eta\in\Delta}(t_{w(\eta)})_{\Lambda}(\lambda_{\eta})=\prod_{\eta\in\Delta}(t_{\sum_{\delta}k_{\delta,\eta}\delta})_{\Lambda}(\lambda_{\eta})$
$\displaystyle=\prod_{\eta\in\Delta}\prod_{\delta\in\Delta}(t_{\delta})_{\Lambda}(\lambda_{\eta})^{k_{\delta,\eta}}=\prod_{\delta\in\Delta}(t_{\delta})_{\Lambda}(\lambda_{\eta})^{\sum_{\delta}k_{\delta,\eta}}=\prod_{\delta\in\Delta}(t_{\delta})_{\Lambda}\left(\sum_{\eta}k_{\delta,\eta}\lambda_{\eta}\right)$
$\displaystyle=f_{\Lambda}\left(\left(\sum_{\eta\in\Delta}k_{\delta,\eta}\lambda_{\eta}\right)_{\delta\in\Delta}\right)$

from which it follows that $g\circ f_{\Lambda}^{-1}$ is $W_{s}$-equivariant.
The way we defined $T=\Lambda^{r}$ to act, $g\circ f_{\Lambda}^{-1}$ is by
definition $W_{s}\ltimes T$-equivariant. ∎

###  7.4 The building

By Theorem 7.5, the non-standard symmetric space $\mathcal{X}_{\mathbb{F}}$
admits a $\Lambda$-pseudometric. We consider the quotient

$\mathcal{B}=\mathcal{X}_{\mathbb{F}}/\\!\sim$

where $x\sim y\in\mathcal{B}$ when $d(x,y)=0\in\Lambda$. We denote the induced
$\Lambda$-metric on $\mathcal{B}$ by the same letter $d$. We note that
$G_{\mathbb{F}}$ acts by isometries on $\mathcal{B}$. In Section 8 we will
show that the $\Lambda$-metric space $\mathcal{B}$ admits the structure of an
affine $\Lambda$-building in certain cases, see Theorem 8.1.

We denote the equivalence class of $\operatorname{Id}$ by $o\in\mathcal{B}$.
The stabilizer of $o$ has been calculated by [Tho02] when $\mathbb{F}$ is a
Robinson field and by [KT02] more generally. We first describe the special
case of the action of $A_{\mathbb{F}}$ on $\mathcal{B}$. For any semi-
algebraic subset $H_{\mathbb{F}}\subseteq G_{\mathbb{F}}$ we write
$H_{\mathbb{F}}(O):=H_{\mathbb{F}}\cap O^{n\times n}$, where $O$ is the
valuation ring.

######  Proposition 7.10.

The following are equivalent for $a\in A_{\mathbb{F}}$.

  * (i)

$a\in\operatorname{Stab}_{G_{\mathbb{F}}}(o)$.

  * (ii)

$N_{\mathbb{F}}(a)\in O$.

  * (iii)

$\chi_{\alpha}(a)\in O\ \forall\alpha\in\Sigma$.

  * (iv)

$\chi_{\alpha}(a)\in O^{\times}\ \forall\alpha\in\Delta$.

  * (v)

$a\in A_{\mathbb{F}}(O)$.

###### Proof.

If $a\in\operatorname{Stab}_{G_{\mathbb{F}}}(o)$, then
$d(\operatorname{Id},a.\operatorname{Id})=(-v)(N_{\mathbb{F}}(a))=0$, so
$N_{\mathbb{F}}(a)\in O$. This implies by the definition of $N_{\mathbb{F}}$
that $\chi_{\alpha}(a)\in O\cap O^{-1}=O^{\times}$ for all $\alpha\in\Sigma$,
in particular for all $\alpha\in\Delta$. Now use the algebraic map
$f_{\mathbb{F}}\colon(\mathbb{F}_{>0})^{r}\to A_{\mathbb{F}}$ from Proposition
7.7 to write

$a=f_{\mathbb{F}}((x_{\delta})_{\delta\in\Delta})=\prod_{\delta\in\Delta}t_{\delta}(x_{\delta})$

for some $(x_{\delta})_{\delta\in\Delta}\in(\mathbb{F}_{>0})^{r}$. Then

$\chi_{\delta}(a)=x_{\delta}\in O$

for all $\delta\in\Delta$. There is a constant $n_{0}\in\mathbb{N}$ such that
the first-order formula

$\varphi\colon\quad\forall x\ \left|t_{\alpha}(x)_{ij}\right|\leq nx^{n_{0}},$

holds over $\mathbb{R}$ (for
$t_{\alpha}^{\mathbb{R}}(e^{s})=\exp(sx_{\alpha})$, take a natural number
$n_{0}$ larger than $|(x_{\alpha})_{ij}|$ for all $i,j$.). By the transfer
principle, $\varphi$ holds over $\mathbb{F}$ as well. Since $x_{\delta}\in O$,
$t_{\delta}(x_{\delta})\in O^{n\times n}$ and hence $a\in A_{\mathbb{F}}(O)$.

On the other hand, if $a\in A_{\mathbb{F}}(O)$, then the linear map
$\operatorname{Ad}(a)\colon\mathfrak{g}\to\mathfrak{g},X\mapsto aXa^{-1}$
restricts to multiplication by $\chi_{\alpha}(a)$ on $\mathfrak{g}_{\alpha}$,
from which can be concluded that $\chi_{\alpha}(a)\in O$. Then
$N_{\mathbb{F}}(a)\in O$ and $d(o,a.o)=0$, so
$a\in\operatorname{Stab}_{G_{\mathbb{F}}}(o)$. ∎

######  Theorem 7.11.

The stabilizer of $o\in\mathcal{B}$ in $G_{\mathbb{F}}$ is
$G_{\mathbb{F}}(O)$.

###### Proof.

Let $g=kak^{\prime}\in
G_{\mathbb{F}}=K_{\mathbb{F}}A^{+}_{\mathbb{F}}K_{\mathbb{F}}$. If
$g\in\operatorname{Stab}_{G_{\mathbb{F}}}(o)$, then
$d(\operatorname{Id},g.\operatorname{Id})=(-v)(N_{\mathbb{F}}(a))=0$. By
Proposition 7.10, $a\in A_{\mathbb{F}}(O)$. Now since
$K_{\mathbb{F}}=K_{\mathbb{F}}(O)$, $g=kak^{\prime}\in G_{\mathbb{F}}(O)$.

If on the other hand we start with a $g\in G_{\mathbb{F}}(O)$, then
$a\in\mathbb{A}_{\mathbb{F}}(O)$, and by Proposition 7.10,
$N_{\mathbb{F}}(a)\in O$ and thus also
$d(\operatorname{Id},g.\operatorname{Id})=d(\operatorname{Id},a.\operatorname{Id})=0$,
hence $g\in\operatorname{Stab}_{G_{\mathbb{F}}}(o)$. ∎

As an application of the Iwasawa retraction, Theorem 7.4, we can give a group
decomposition for the stabilizer of $o$ in $\mathcal{B}$.

######  Corollary 7.12.

There is an Iwasawa decomposition
$G_{\mathbb{F}}(O)=U_{\mathbb{F}}(O)A_{\mathbb{F}}(O)K_{\mathbb{F}}$, meaning
that for every $g\in G_{\mathbb{F}}(O)$ there are unique $u\in
U_{\mathbb{F}}(O),a\in A_{\mathbb{F}}(O),k\in
K_{\mathbb{F}}=K_{\mathbb{F}}(O)$ with $g=uak$.

###### Proof.

Let $g=uak\in G_{\mathbb{F}}(O)\subseteq
U_{\mathbb{F}}A_{\mathbb{F}}K_{\mathbb{F}}$. We have
$\rho(g.\operatorname{Id})=a.\operatorname{Id}$. Since $g\in
G_{\mathbb{F}}(O)$, we have by Theorem 7.4

$\displaystyle d(\operatorname{Id},a.\operatorname{Id})$
$\displaystyle=d(\operatorname{Id},\rho(g.\operatorname{Id}))\leq
d(\operatorname{Id},g.\operatorname{Id})=0.$

This means that $a\in A_{\mathbb{F}}\cap G_{\mathbb{F}}(O)=A_{\mathbb{F}}(O)$.
Note that since $K_{\mathbb{F}}$ stabilizes
$\operatorname{Id}\in\mathcal{X}_{\mathbb{F}}$,