Proof. Let S be a maximal K-split torus in $G_K$ . Let $\Gamma $ be the Galois group $\operatorname {\mathrm {Gal}}(K^{\operatorname {\mathrm {nr}}}/K)$ . By [Reference LourençoLou21, Lemme 4.8], there exists a maximal $K^{\operatorname {\mathrm {nr}}}$ -split torus $S^{\operatorname {\mathrm {nr}}}$ in $G_{K^{\operatorname {\mathrm {nr}}}}$ that contains $S_{K^{\operatorname {\mathrm {nr}}}}$ and is defined over K. Denote by $\Phi = \Phi (G_K, S)$ and $\Phi ^{\operatorname {\mathrm {nr}}} = \Phi (G_{K^{\operatorname {\mathrm {nr}}}}, S^{\operatorname {\mathrm {nr}}})$ the corresponding root systems. The apartment $A(G_{K^{\operatorname {\mathrm {nr}}}}, S^{\operatorname {\mathrm {nr}}})$ is then stable under the action of $\Gamma $ and admits a fixed point y corresponding to a $\Gamma $ -invariant valuation $\varphi ^{\operatorname {\mathrm {nr}}}$ of the root group datum $(Z_{G_{K^{\operatorname {\mathrm {nr}}}}}(S^{\operatorname {\mathrm {nr}}})(K^{\operatorname {\mathrm {nr}}}), (U_{a^{\operatorname {\mathrm {nr}}}}(K^{\operatorname {\mathrm {nr}}}))_{a^{\operatorname {\mathrm {nr}}} \in \Phi ^{\operatorname {\mathrm {nr}}}})$ . It then follows from [Reference LourençoLou21, Théorème 4.9] that the valuation $\varphi ^{\operatorname {\mathrm {nr}}}$ descends to a valuation $\varphi $ of the root group datum $(Z_{G_K}(S)(K), (U_a(K))_{a\in \Phi })$ , corresponding to a point $x \in A(G_K, S)$ . By [Reference Bruhat and TitsBT72, 9.1.17], there is then a $G(K)$ -equivariant isometry

$$ \begin{align*}p_{K^{\operatorname{\mathrm{nr}}}/K}: \mathcal B(G, K) \to \mathcal B(G, K^{\operatorname{\mathrm{nr}}})\end{align*} $$

such that $p_{K^{\operatorname {\mathrm {nr}}}/K}(x) = y$ . We then have $p_{K^{\operatorname {\mathrm {nr}}}/K}(A(G_K, S)) = A(G_{K^{\operatorname {\mathrm {nr}}}}, S^{\operatorname {\mathrm {nr}}})^{\Gamma }$ and, by $G(K)$ -equivariance, the inclusion $p_{K^{\operatorname {\mathrm {nr}}}/K}(\mathcal B(G, K)) \subset \mathcal B(G, K^{\operatorname {\mathrm {nr}}})^{\Gamma }$ , which completes the proof of the existence statement. By [Reference Bruhat and TitsBT72, 9.2.14 and 9.2.15], the map $p_{K^{\operatorname {\mathrm {nr}}}/K}$ is in fact the unique $G(K)$ -equivariant isometry from $\mathcal B(G, K)$ to $\mathcal B(G, K^{\operatorname {\mathrm {nr}}})$ that maps into $\mathcal B(G, K^{\operatorname {\mathrm {nr}}})^{\Gamma }$ , which establishes the uniqueness statement.

The additional statements follow directly from [Reference LourençoLou21, Théorème 4.9].

Proof. First of all, note that Conditions 1 and 3 of Proposition 6 imply that

$$ \begin{align*}p_{K^{\operatorname{\mathrm{nr}}}/K}(\mathcal B(Z_{G_K}(S), G, K)) \subset \mathcal B(Z_{G_K}(S), G, K^{\operatorname{\mathrm{nr}}})^{\operatorname{\mathrm{Gal}}(K^{\operatorname{\mathrm{nr}}}/K)}.\end{align*} $$

For the reverse inclusion, we apply Proposition 6 to the group $Z_{G_K}(S)$ and use the fact that the building $\mathcal B(Z_{G_K}(S), K^{\operatorname {\mathrm {nr}}})$ is a quotient of the façade $\mathcal B(Z_{G_K}(S), G, K^{\operatorname {\mathrm {nr}}})$ . Let T be a maximal K-split torus of $G_K$ containing S. By [Reference LourençoLou21, Lemme 4.8], we have at our disposal a maximal $K^{\operatorname {\mathrm {nr}}}$ -split torus $T^{\operatorname {\mathrm {nr}}}$ of $G_{K^{\operatorname {\mathrm {nr}}}}$ containing $T_{K^{\operatorname {\mathrm {nr}}}}$ that is defined over K, and we have

$$ \begin{align*}p_{K^{\operatorname{\mathrm{nr}}}/K}(A(G_K, T)) = A(G_{K^{\operatorname{\mathrm{nr}}}}, T^{\operatorname{\mathrm{nr}}})^{\operatorname{\mathrm{Gal}}(K^{\operatorname{\mathrm{nr}}}/K)}.\end{align*} $$

Consider the vector spaces

$$ \begin{align*}\begin{array}{ccc} V^{\operatorname{\mathrm{nr}}} & = & \ker(V(G_{K^{\operatorname{\mathrm{nr}}}}, T^{\operatorname{\mathrm{nr}}}) \to V((Z_{G_K}(S))_{K^{\operatorname{\mathrm{nr}}}}, T^{\operatorname{\mathrm{nr}}})) \\ V & = & \ker(V(G_K, T) \to V(Z_{G_K}(S), T)). \end{array}\end{align*} $$

The map $X_*(T) \to X_*(T^{\operatorname {\mathrm {nr}}})$ given by extension of scalars induces a linear injection $\begin {array}{ccc}V & \hookrightarrow & V^{\operatorname {\mathrm {nr}}}\\v & \mapsto & v_{K^{\operatorname {\mathrm {nr}}}}\end {array}$ that identifies V with the set of $\operatorname {\mathrm {Gal}}(K^{\operatorname {\mathrm {nr}}}/K)$ -fixed points of $V^{\operatorname {\mathrm {nr}}}$ . Indeed, by Theorem 1 (ii), the action of $\operatorname {\mathrm {Gal}}(K^{\operatorname {\mathrm {nr}}}/K)$ on $V(G_{K^{\operatorname {\mathrm {nr}}}}, T^{\operatorname {\mathrm {nr}}})$ is induced by the pullback action on cocharacters.

Then, we claim that the restriction of $p_{K^{\operatorname {\mathrm {nr}}}/K}$ to $\mathcal B(Z_{G_K}(S), G, K)$ is V-equivariant, i.e. that for each $x \in \mathcal B(Z_{G_K}(S), G, K)$ and each $v \in V$ , we have

$$ \begin{align*}p_{K^{\operatorname{\mathrm{nr}}}/K}(x+v) = p_{K^{\operatorname{\mathrm{nr}}}/K}(x) + v_{K^{\operatorname{\mathrm{nr}}}}.\end{align*} $$

Because the action of V commutes with that of $Z_{G_K}(S)(K)$ (Property 2 of the Appendix), it is sufficient to prove the above for $x \in A(G_K, T)$ .

Recall that the centraliser $Z_{G_K}(T)(K)$ acts on $A(G_K, T)$ by translation via the map

$$ \begin{align*}\nu: Z_{G_K}(T)(K) \longrightarrow V(G_K, T)\end{align*} $$

such that, for each $z \in Z_{G_K}(T)(K)$ and $\alpha \in \Phi (G_K, T)$ , we have

$$ \begin{align*}\langle \alpha, \nu(z) \rangle = - \omega(\alpha(z)).\end{align*} $$

Likewise, we have an analogously defined map

$$ \begin{align*}\nu^{\operatorname{\mathrm{nr}}}: Z_{G_{K^{\operatorname{\mathrm{nr}}}}}(T^{\operatorname{\mathrm{nr}}})(K^{\operatorname{\mathrm{nr}}})\longrightarrow V(G_{K^{\operatorname{\mathrm{nr}}}}, T^{\operatorname{\mathrm{nr}}}).\end{align*} $$

In particular, the centre $Z(Z_{G_K}(S))(K)$ acts on $A(G_K, T)$ by translations by elements of V and on $A(G_{K^{\operatorname {\mathrm {nr}}}}, T^{\operatorname {\mathrm {nr}}})$ by elements of $V^{\operatorname {\mathrm {nr}}}$ and we have, for each $z \in Z(Z_{G_K}(S))(K)$

$$ \begin{align*}\nu^{\operatorname{\mathrm{nr}}}(z) = \nu(z)_{K^{\operatorname{\mathrm{nr}}}}.\end{align*} $$

Because $p_{K^{\operatorname {\mathrm {nr}}}/K}$ is $G(K)$ -equivariant, we have

$$ \begin{align*}p_{K^{\operatorname{\mathrm{nr}}}/K}(x + \nu(z)) = p_{K^{\operatorname{\mathrm{nr}}}/K}(x) + \nu(z)_{K^{\operatorname{\mathrm{nr}}}}\end{align*} $$

for each $x \in A(G_K, T)$ and $z \in Z(Z_{G_K}(S))(K)$ . Because both sides of the equation are affine and because the convex hull of $\{\nu (z), z \in Z(Z_{G_K}(S))(K)\}$ is V, we get the result.

Now let

$$ \begin{align*}\widetilde \pi^{\operatorname{\mathrm{nr}}}: \mathcal B(Z_{G_K}(S), G, K^{\operatorname{\mathrm{nr}}}) \to \mathcal B(Z_{G_K}(S), K^{\operatorname{\mathrm{nr}}})\end{align*} $$

be the essentialisation map described in the Appendix. Because it is invariant under the action of $V^{\operatorname {\mathrm {nr}}}$ , the map $\widetilde \pi ^{\operatorname {\mathrm {nr}}} \circ p_{K^{\operatorname {\mathrm {nr}}}/K}$ is invariant under the action of V. Therefore, there exists a unique map j that makes the following diagram commute

Moreover:

1. 1. The restriction of j to $A(Z_{G_K}(S), T)$ fits into the following commutative diagram

   

   In particular, the restriction of j to $A(Z_{G_K}(S), T)$ is an affine injection.

2. 2. The map j is $G(K)$ -equivariant. In particular, the restriction of j to each apartment $A(G_K, T')$ is an affine injection into an apartment $A(G_{K^{\operatorname {\mathrm {nr}}}}, (T')^{\operatorname {\mathrm {nr}}})$ for some maximal $K^{\operatorname {\mathrm {nr}}}$ -split torus $(T')^{\operatorname {\mathrm {nr}}}$ containing $(T')_{K^{\operatorname {\mathrm {nr}}}}$ .

3. 3. Because $\widetilde \pi ^{\operatorname {\mathrm {nr}}}$ is $\operatorname {\mathrm {Gal}}(K^{\operatorname {\mathrm {nr}}}/K)$ -equivariant (Proposition A.2), we have

   $$ \begin{align*}\widetilde \pi^{\operatorname{\mathrm{nr}}}(p_{K^{\operatorname{\mathrm{nr}}}/K}(\mathcal B(Z_{G_K}(S), G, K))) \subset \mathcal B(Z_{G_K}(S), K^{\operatorname{\mathrm{nr}}})^{\operatorname{\mathrm{Gal}}(K^{\operatorname{\mathrm{nr}}}/K)}\end{align*} $$

   and therefore

   $$ \begin{align*}j(\mathcal B(Z_{G_K}(S), K)) \subset \mathcal B(Z_{G_K}(S), K^{\operatorname{\mathrm{nr}}})^{\operatorname{\mathrm{Gal}}(K^{\operatorname{\mathrm{nr}}}/K)}.\end{align*} $$

In other words, the map j is none other than the map

$$ \begin{align*}p^1_{K^{\operatorname{\mathrm{nr}}}/K}: \mathcal B(Z_{G_K}(S), K) \to \mathcal B(Z_{G_K}(S), K^{\operatorname{\mathrm{nr}}})\end{align*} $$

given by Proposition 6.

Now, let $y^{\operatorname {\mathrm {nr}}} \in \mathcal B(Z_{G_K}(S), G, K^{\operatorname {\mathrm {nr}}})^{\operatorname {\mathrm {Gal}}(K^{\operatorname {\mathrm {nr}}}/K)}$ . Because $\widetilde \pi ^{\operatorname {\mathrm {nr}}}$ is $\operatorname {\mathrm {Gal}}(K^{\operatorname {\mathrm {nr}}}/K)$ -equivariant, we have

$$ \begin{align*}\widetilde \pi^{\operatorname{\mathrm{nr}}}(y^{\operatorname{\mathrm{nr}}}) \in \mathcal B(Z_{G_K}(S), K^{\operatorname{\mathrm{nr}}})^{\operatorname{\mathrm{Gal}}(K^{\operatorname{\mathrm{nr}}}/K)}.\end{align*} $$

By Proposition 6, there exists $x \in \mathcal B(Z_{G_K}(S), K)$ such that

$$ \begin{align*}\widetilde \pi^{\operatorname{\mathrm{nr}}}(y^{\operatorname{\mathrm{nr}}}) = p^1_{K^{\operatorname{\mathrm{nr}}}/K}(x).\end{align*} $$

Because $\widetilde \pi $ is onto, there exists $z\in \mathcal B(Z_{G_K}(S), G, K)$ such that $\widetilde \pi (z) = x$ . Then, we have

$$ \begin{align*}\widetilde \pi^{\operatorname{\mathrm{nr}}}(p_{K^{\operatorname{\mathrm{nr}}}/K}(z)) = p^1_{K^{\operatorname{\mathrm{nr}}}/K}(\widetilde \pi(z)) = \widetilde \pi^{\operatorname{\mathrm{nr}}}(y^{\operatorname{\mathrm{nr}}}).\end{align*} $$

Therefore, there exists $v^{\operatorname {\mathrm {nr}}} \in V^{\operatorname {\mathrm {nr}}}$ such that $y^{\operatorname {\mathrm {nr}}} = p_{K^{\operatorname {\mathrm {nr}}}/K}(z) + v^{\operatorname {\mathrm {nr}}}$ . Because both $y^{\operatorname {\mathrm {nr}}}$ and $p^1_{K^{\operatorname {\mathrm {nr}}}/K}(z)$ are fixed under the action of the Galois group, so must $v^{\operatorname {\mathrm {nr}}}$ be (Proposition A.2, 3). Consequently, $v^{\operatorname {\mathrm {nr}}}$ can be written $v^{\operatorname {\mathrm {nr}}} = v_{K^{\operatorname {\mathrm {nr}}}}$ for some $v \in V$ and we thus have

$$ \begin{align*}y^{\operatorname{\mathrm{nr}}} = p_{K^{\operatorname{\mathrm{nr}}}/K}(z + v) \in p_{K^{\operatorname{\mathrm{nr}}}/K}(\mathcal B(Z_{G_K}(S), G, K)),\end{align*} $$

which completes the proof.

Proof. Because $G_K$ is quasi-split, the centraliser $Z_{G_K}(S)$ is nilpotent and therefore contains a unique maximal torus T that lies in the centre $Z(Z_{G_K}(S))$ [Reference MilneMil17, Theorem 16.47]. It immediately follows that $Z_{G_K}(S) = Z_{G_K}(T)$ . Moreover, for each K-algebra R, if $n \in N_{G_K}(S)(R)$ , then n normalises the subgroup $Z_{G_R}(S_R)$ and, because $T_R$ is characteristic in $Z_{G_R}(S_R)$ , n must also normalise $T_R$ . Hence $N_{G_K}(S) \subset N_{G_K}(T)$ .

Applying the same theorem to $Z_{G_L}(S_L) = (Z_{G_K}(S))_L$ , we note that $T_L$ is the only maximal torus of $G_L$ that contains $S_L$ . But $S_L$ is split and therefore contained in a maximal L-split torus of $G_L$ which, because $G_L$ is pseudo-split, must be a maximal torus and can thus only be $T_L$ . Hence, the torus $T_L$ must be split.

Because $T_L$ is defined over K, the action of $\operatorname {\mathrm {Gal}}(L/K)$ on $\mathcal B(G, L)$ stabilises its associated apartment $A(G_L, T_L)$ and induces an action by affine automorphisms. In particular, because $\operatorname {\mathrm {Gal}}(L/K)$ is compact, the action of $\operatorname {\mathrm {Gal}}(L/K)$ admits a fixed point. Setting this point as origin defines an isomorphism of affine spaces between $A(G_L, T_L)$ and $V(G_L, T_L)$ that identifies the action of $\operatorname {\mathrm {Gal}}(L/K)$ on $A(G_L, T_L)$ with the action on $V(G_L, T_L)$ induced by the pullback action on cocharacters of $T_L$ . Because the fixed locus of the $\operatorname {\mathrm {Gal}}(L/K)$ -action on $V(G_L, T_L) = X_*(T_L)_{\mathbb R}/X_*(Z(G_L))_{\mathbb R}$ is precisely $X_*(S_L)_{\mathbb R}/X_*(Z(G_L))_{\mathbb R}$ , we get the expected result.

Proof of Proposition 8

The spirit of the proof is very similar to that of Proposition 6. As in the previous case, we prove that, letting $\Phi = \Phi (G, S)$ be the relative root system of G and $\widetilde \Phi = \Phi (G_L, T_L)$ be its absolute root system, the valuation $(\varphi _{a}^y)_{a\in \widetilde \Phi }$ associated to y descends to the valuation $(\varphi _a^x)_{a\in \Phi }$ associated to x in the sense of [Reference Bruhat and TitsBT72, 9.1.11]. The strategy goes as follows: Letting x be a special vertex in $A(G, S)$ , we give an explicit construction of a valuation of the root data $(Z(L), (U_a(L))_{a\in \widetilde \Phi })$ that descends to $(\varphi _a^x)_{a\in \Phi }$ and prove that it is fixed under the Galois action. Proposition [Reference Bruhat and TitsBT72, 9.1.17] then directly yields a map satisfying 1–3 and Propositions [Reference Bruhat and TitsBT72, 9.2.14 and 9.2.15] then ensure the uniqueness.

Let x be a special vertex in $A(G, S)$ . We may assume without loss of generality that there exists a Chevalley quasi-system in $G^{\operatorname {\mathrm {pred}}}$ along S [Reference LourençoLou21, Définition 2.4], i.e. a family of isomorphisms $(\zeta _a)_{a\in \Phi }$ of Weil restrictions of groups of the form $\operatorname {\mathrm {SL}}_2, \operatorname {\mathrm {SU}}_3$ or $\operatorname {\mathrm {BC}}_1$ onto the universal covers $\widetilde G_a$ of the groups of K-rank 1 $G_a = \langle U_{a}, U_{-a}\rangle $ associated to each non-divisible root $a \in \Phi _{\operatorname {\mathrm {nd}}}$ , such that the valuation $(\varphi _a^x)_{a\in \Phi }$ associated to x is given by the formulas in [Reference LourençoLou21, Définition 3.1].

We construct an associated Chevalley quasi-system in $G_L$ along $T_L$ . Note that, under the assumption that $G_L$ is pseudo-split, each $(G_a)_L$ is pseudo-split with $(T \cap G_a)_L$ as a split maximal torus [Reference Conrad, Gabber and PrasadCGP15, Corollary A.2.7]. Because G is pseudo-split over K, there exists a pseudo-parabolic subgroup $B \subset G$ such that $B_L$ is a minimal pseudo-parabolic L-subgroup of $G_L$ . Denote by $\Delta \subset \Phi (G, S)$ the system of relative simple roots associated to B and by $\widetilde \Delta \subset \Phi (G_L, T_L)$ the system of absolute simple roots associated to $B_L$ . Recall [Reference Conrad, Gabber and PrasadCGP15, Proof of Theorem C.2.15, Step 5] that the restriction map $\alpha \mapsto \alpha _{|S}$ establishes a bijection between the set of $\operatorname {\mathrm {Gal}}(L/K)$ -orbits in $\widetilde \Delta $ and the set $\Delta $ . Let $a \in \Delta $ be a relative simple root, let $S_a = (S \cap G_a)^0_{\operatorname {\mathrm {red}}}$ and let $\widetilde S_a$ be the unique maximal L-split torus of $\widetilde G_a$ that maps onto $S_a$ .

1. 1. Assume that the isomorphism $\zeta _a$ has as domain a Weil restriction $R_{K'/K}(\operatorname {\mathrm {SL}}_{2, K'})$ . The base change $R_{K'/K}(\operatorname {\mathrm {SL}}_{2, K'})_L$ decomposes into a direct product

   $$ \begin{align*}R_{K'/K}(\operatorname{\mathrm{SL}}_{2, K'})_L = R_{K'\otimes_KL/L}(\operatorname{\mathrm{SL}}_{2,K'\otimes L}) \simeq \prod_{i=1}^r R_{L^{\prime}_i/L}(\operatorname{\mathrm{SL}}_{2,L^{\prime}_i}),\end{align*} $$

   where $K' \otimes _K L = \prod _{i=1}^r L^{\prime }_i$ is the canonical decomposition of the finite reduced L-algebra $K'\otimes _KL$ into a product of extensions of L and we have a commutative diagram

   

   Here, the notation $\operatorname {\mathrm {diag}}\left (\begin {pmatrix}* & \\ & *\end {pmatrix}_L\right )$ refers to the diagonal embedding of the torus $\begin {pmatrix}* & \\ & *\end {pmatrix}_L$ of $\operatorname {\mathrm {SL}}_{2,L}$ in the product $\prod _i R_{L^{\prime }_i/L}(\operatorname {\mathrm {SL}}_{2,L^{\prime }_i})$ . Because $(G_a)_L$ is pseudo-split, so is $\prod _i R_{L^{\prime }_i/L}(\operatorname {\mathrm {SL}}_{2,L^{\prime }_i})$ , which implies that each $R_{L^{\prime }_i/L}(\operatorname {\mathrm {SL}}_{2,L^{\prime }_i})$ is pseudo-split and therefore [Reference Conrad, Gabber and PrasadCGP15, Proposition A.5.15 (2)] that each extension $L^{\prime }_i/L$ is purely inseparable. It follows that the torus $\prod _i \begin {pmatrix}* & \\ & *\end {pmatrix}_{L}$ is the unique maximal torus of $\prod _i R_{L^{\prime }_i/L}(\operatorname {\mathrm {SL}}_{2,L^{\prime }_i})$ containing the split torus $\operatorname {\mathrm {diag}}\left (\begin {pmatrix}* & \\ & *\end {pmatrix}_L\right )$ . The assumption that $\zeta _a$ maps the subgroup $R_{K' \otimes _K L/L}\begin {pmatrix}1 & \mathbb G_{a,K'\otimes _K L} \\ & 1 \end {pmatrix}$ to $U_a$ implies that a pulls back along $\zeta _a$ to the character $\begin {array}{ccc} \mathbb G_{m,L} & \to & \mathbb G_{m,L} \\ t & \mapsto & t^2 \end {array}$ . Consequently, pulling back along $(\zeta _a)_L$ establishes a Galois-equivariant one-to-one correspondence between the roots $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} = a_L$ and the roots of $\prod _{i=1}^r R_{L^{\prime }_i/L} (\operatorname {\mathrm {SL}}_{2, L^{\prime }_i})$ whose restriction to $\operatorname {\mathrm {diag}} \begin {pmatrix}* & \\ & * \end {pmatrix}_L$ is the character $\begin {pmatrix} t & \\ & t^{-1} \end {pmatrix} \mapsto t^2$ , namely the maps $\begin {array}{ccc} \mathbb G_{m,L}^r & \to & \mathbb G_{m,L}\\(t_1, \dots , t_r)& \mapsto & t_i^2\end {array}$ for $i \in [\![1, r]\!]$ . We relabel the factor fields of $K' \otimes _KL$ accordingly: $K' \otimes _K L = \prod _{\substack {\alpha \in \widetilde \Delta \\\alpha _{|S_L} = a_L}} L^{\prime }_{\alpha }$ .

   It follows that the factors $R_{L^{\prime }_\alpha /L}(\operatorname {\mathrm {SL}}_{2,L^{\prime }_\alpha })$ are the subgroups of L-rank 1 of $R_{K'\otimes _K L/L} (\operatorname {\mathrm {SL}}_{2,K' \otimes _K L})$ associated to the roots relative to $\prod _i \begin {pmatrix}* & \\ & * \end {pmatrix}_L$ whose restriction to $\operatorname {\mathrm {diag}} \begin {pmatrix}* & \\ & * \end {pmatrix}_L$ is the character $\begin {pmatrix} t & \\ & t^{-1} \end {pmatrix} \mapsto t^2$ . Accordingly, we have a direct product decomposition

   $$ \begin{align*}(\widetilde {G_a})_L = \prod_{\substack{\alpha \in \widetilde \Delta\\\alpha_{|S_L} = a_L}} \widetilde{(G_L)_{\alpha}}\end{align*} $$

   and the map $(\zeta _a)_L$ decomposes into a product

   $$ \begin{align*}(\zeta_a)_L = \prod_{\alpha_{|S_L} = a_L} \zeta_{\alpha},\end{align*} $$

   where each $\zeta _{\alpha }: R_{L^{\prime }_\alpha /L}(\operatorname {\mathrm {SL}}_{2, L^{\prime }_\alpha }) \to \widetilde {(G_L)_{\alpha }}$ is an isomorphism that maps the upper triangular matrices to the root group $U_{\alpha }$ and whose restriction to the torus $\begin {pmatrix} * & \\ & * \end {pmatrix}$ is the coroot $\alpha ^{\vee }$ .

   Observe also that, for each $\sigma \in \operatorname {\mathrm {Gal}}(L/K)$ , the automorphism induced by $\sigma $ on $K' \otimes _K L$ restricts to an isomorphism of valued fields $L^{\prime }_{\alpha } \overset {\sim }{\to } L^{\prime }_{\sigma (\alpha )}$ for each $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} = a_L$ . Because $(\zeta _a)_L$ is defined over K, we have

   $$ \begin{align*}\sigma \circ \zeta_{\alpha} \circ \sigma^{-1} = \zeta_{\sigma( \alpha)}\end{align*} $$

   for each $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} = a_L$ .

   Consequently, if for each $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} = a_L$ we define a function $\varphi _{\alpha }: U_{\alpha }(L) \to \mathbb R$ by setting, for each $x \in L^{\prime }_{\alpha }$ ,

   $$ \begin{align*}\varphi_{\alpha}\left(\zeta_{\alpha}\begin{pmatrix}1 & x \\ & 1\end{pmatrix}\right) = \omega(x),\end{align*} $$

   then we have

   $$ \begin{align*}\varphi_{\alpha} \circ \sigma^{-1} = \varphi_{\sigma(\alpha)}\end{align*} $$

   for each $\sigma \in \operatorname {\mathrm {Gal}}(L/K)$ .

   Moreover, under the identification $\zeta _a$ , the inclusion $U_a(K)\hspace{-1pt} \subset\hspace{-1pt} U_a(L)\hspace{-1pt} =\hspace{-1pt} \prod _{\substack {\alpha \in \widetilde \Delta \\\alpha _{|S_L} = a_L}}\hspace{-1pt} U_{\alpha }(L)$ pulls back to the diagonal map $K' \to K' \otimes _K L = \prod _{\substack {\alpha \in \widetilde \Delta \\\alpha _{|S_L} = a_L}} L^{\prime }_{\alpha }$ . Therefore, in the decomposition of an element $u \in U_{a}(K)\setminus \{1\}$ as $u = \prod _{\substack {\alpha \in \widetilde \Delta \\\alpha _{|S_L} = a_L}} u_{\alpha }$ , with $u_{\alpha } \in U_{\alpha }(L)$ , we have $\varphi ^x_a(u) = \varphi _{\alpha }(u_{\alpha })$ for each $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} = a_L$ .

2. 2. Next, assume that the isomorphism $\zeta _a$ has as domain a Weil restriction $R_{K'/K}(\operatorname {\mathrm {BC}}_{1, K'})$ . The exact same reductions as in step 1 apply and we may write $K' \otimes _K L = \prod _{\substack {\alpha \in \widetilde \Delta \\\alpha _{|S_L} = a_L}} L^{\prime }_{\alpha }$ and

   $$ \begin{align*}R_{K' \otimes_K L/L} (\operatorname{\mathrm{BC}}_{1, K' \otimes_K L}) = \prod_{\substack{\alpha \in \widetilde \Delta\\\alpha_{|S_L} = a_L}} R_{L^{\prime}_{\alpha}/L}(\operatorname{\mathrm{BC}}_{1, L^{\prime}_{\alpha}}).\end{align*} $$

   Likewise, we have a direct product decomposition

   $$ \begin{align*}(\widetilde {G_a})_L = \prod_{\substack{\alpha \in \widetilde \Delta\\\alpha_{|S_L} = a_L}} \widetilde{(G_L)_{\alpha}}\end{align*} $$

   and the map $(\zeta _a)_L$ decomposes into a product

   $$ \begin{align*}(\zeta_a)_L = \prod_{\alpha_{|S_L} = a_L} \zeta_{\alpha},\end{align*} $$

   where each $\zeta _{\alpha }: R_{L^{\prime }_\alpha /L}(\operatorname {\mathrm {BC}}_{1, L^{\prime }_\alpha }) \to \widetilde {(G_L)_{\alpha }}$ is an isomorphism that maps the ‘upper triangular unipotent matrices’ to the root group $U_{\alpha }$ and whose restriction to the torus $\begin {pmatrix} * & \\ & * \end {pmatrix}_L$ is the coroot $\alpha ^{\vee }$ . In this context, we callupper triangular unipotent matrices the elements of the root groupof $R_{L^{\prime }_{\alpha }/L}(\operatorname {\mathrm {BC}}_{1, L^{\prime }_{\alpha }})$ that lifts the group of upper triangular unipotentmatrices in $R_{(L^{\prime }_{\alpha })^{1/2}/L}(\operatorname {\mathrm {Sp}}_{2, (L^{\prime }_{\alpha })^{1/2}})$ . Like before, the fact that $(\zeta _a)_L$ is defined over K implies that

   $$ \begin{align*}\sigma \circ \zeta_{\alpha} \circ \sigma^{-1} = \zeta_{\sigma(\alpha)}\end{align*} $$

   for each $\sigma \in \operatorname {\mathrm {Gal}}(L/K)$ and $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} = a_L$ .

   Let $\eta \in K^{\prime 1/2} \setminus K'$ such that the function $\varphi _a^x: U_a(K) \to \mathbb R$ is given by

   $$ \begin{align*}\varphi_a^x(x_a(u, v)) = \frac{1}{2} \omega(\eta u^2 + v),\end{align*} $$

   where $x_a: R_{K^{\prime 1/2}/K}(\mathbb G_{a,K^{\prime 1/2}}) \times R_{K'/K}(\mathbb G_{a, K'}) \to U_a$ is the restriction of $\zeta _a$ to the subgroup of unipotent upper triangular matrices in $R_{K'/K}(\operatorname {\mathrm {BC}}_{1, K'})$ . Then, if for each $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} = a_L$ we define a function $\varphi _{\alpha }: U_{\alpha }(L) \to \mathbb R$ by setting for each $(u, v) \in (L^{\prime }_{\alpha })^{1/2}\times L^{\prime }_{\alpha }$

   $$ \begin{align*}\varphi_{\alpha}(x_{\alpha}(u, v)) = \frac{1}{2}\omega(\eta u^2 + v),\end{align*} $$

   we have for each such $\alpha $ and each $\sigma \in \operatorname {\mathrm {Gal}}(L/K)$

   $$ \begin{align*}\varphi_{\alpha} \circ \sigma^{-1} = \varphi_{\sigma(\alpha)}.\end{align*} $$

   Moreover, under the identification $\zeta _a$ , the inclusion $U_a(K) \subset U_a(L) = \prod _{\substack {\alpha \in \widetilde \Delta \\\alpha _{|S_L} = a_L}} U_{\alpha }(L)$ pulls back to the diagonal map $(K')^{1/2} \times K' \to \prod _{\substack {\alpha \in \widetilde \Delta \\\alpha _{|S_L} = a_L}} (L^{\prime }_{\alpha })^{1/2} \times L^{\prime }_{\alpha }$ . Therefore, in the decomposition of an element $u \in U_{a}(K)\setminus \{1\}$ as $u = \prod _{\substack {\alpha \in \widetilde \Delta \\\alpha _{|S_L} = a_L}} u_{\alpha }$ , with $u_{\alpha } \in U_{\alpha }(L)$ , we have $\varphi ^x_a(u) = \varphi _{\alpha }(u_{\alpha })$ for each $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} = a_L$ .

3. 3. Finally, assume that the isomorphism $\zeta _a$ has as domain a Weil restriction $R_{K^{\prime }_2/K}(\operatorname {\mathrm {SU}}_{3, K'/K^{\prime }_2})$ . Again, we decompose $K^{\prime }_2 \otimes _KL = \prod _{i\in I_a} L^{\prime }_i$ , where the $L^{\prime }_i$ are fields and because $R_{K^{\prime }_2/K}(\operatorname {\mathrm {SU}}_{3, K'/K^{\prime }_2})_L$ is pseudo-split, we have $K' \subset L^{\prime }_i$ for each i, each $L^{\prime }_i/L$ is purely inseparable, and

   $$ \begin{align*}R_{K^{\prime}_2/K}(\operatorname{\mathrm{SU}}_{3, K'/K^{\prime}_2})_L = \prod_{i\in I_a} R_{L^{\prime}_i/L}(\operatorname{\mathrm{SL}}_{3, L^{\prime}_i}).\end{align*} $$

   Likewise, we have a commutative diagram

   

   The torus $\prod _{i \in I_a} \begin {pmatrix}* & & \\ & * & \\& & *\end {pmatrix}_L$ is then the unique maximal torus of $\prod _{i\in I_a} R_{L^{\prime }_i/L}(\operatorname {\mathrm {SL}}_{3,L^{\prime }_i})$ containing the split torus $\operatorname {\mathrm {diag}}\left (\begin {pmatrix}* & & \\ & 1 & \\& & *\end {pmatrix}_L\right )$ .

   For each $i \in I_a$ , denote by $\alpha _i$ (resp. $\beta _i$ ) the character of $T_L$ that pulls back to

   $$ \begin{align*}\left(\begin{pmatrix}s_j & & \\ & s_j^{-1}t_j & \\& & t_j^{-1}\end{pmatrix}\right)_{j \in I_a} \mapsto s_i^2 t_i^{-1} \left( \text{resp.} \left(\begin{pmatrix}s_j & & \\ & s_j^{-1}t_j & \\& & t_j^{-1}\end{pmatrix}\right)_{j \in I_a} \mapsto t_i^2 s_i^{-1} \right)\end{align*} $$

   along $(\zeta _a)_L$ . The latter characters are the only roots of $\prod _{i \in I_a} R_{L^{\prime }_i/L}(SL_{3, L^{\prime }_i})$ that restrict to the character $\begin {pmatrix}t & & \\ & 1 & \\ & & t^{-1}\end {pmatrix}\mapsto t$ on $\operatorname {\mathrm {diag}} \left (\begin {pmatrix}* & &\\ & 1 & \\ & & *\end {pmatrix}_L\right )$ . Therefore, we have

   $$ \begin{align*}\{\alpha \in \widetilde \Delta, \alpha_{|S_L} = a_L\} = \bigcup_{i \in I_a}\{\alpha_i, \beta_i\} \text{ and } \{\alpha \in \widetilde \Phi, \alpha_{|S_L} = 2a_L\} = \{\alpha_i+\beta_i, i \in I_a\}\end{align*} $$

   For each $i \in I_a$ , set

   $$ \begin{align*}\zeta_{\alpha_i}: \begin{array}{ccc} R_{L^{\prime}_i/L}(SL_{2, L^{\prime}_i}) & \longrightarrow & G_{\alpha_i} \\ \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \longmapsto & \zeta_a\left(\begin{pmatrix} a & b & \\ c & d & \\ & & 1\end{pmatrix}\right)\end{array}\!\!,\end{align*} $$
   $$ \begin{align*}\zeta_{\beta_i}: \begin{array}{ccc} R_{L^{\prime}_i/L}(SL_{2, L^{\prime}_i}) & \longrightarrow & G_{\beta_i} \\ \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \longmapsto & \zeta_a\left(\begin{pmatrix} 1 & & \\ & a & b \\ & c & d \end{pmatrix}\right)\end{array}\!\!,\end{align*} $$

   and

   $$ \begin{align*}\zeta_{\alpha_i+\beta_i}: \begin{array}{ccc} R_{L^{\prime}_i/L}(SL_{2, L^{\prime}_i}) & \longrightarrow & G_{\alpha_i +\beta_i} \\ \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \longmapsto & \zeta_a\left(\begin{pmatrix} a & & b \\ & 1 & \\ c & & d \end{pmatrix}\right)\end{array}\!\!.\end{align*} $$

   Here, we think of $R_{L^{\prime }_i/L}(SL_{3, L^{\prime }_i})$ as a direct factor of $R_{K^{\prime }_2 \otimes _K L}(SL_{3, K^{\prime }_2\otimes _KL})$ .

   Then, for each $i \in I_a$ and $\sigma \in \operatorname {\mathrm {Gal}}(L/K)$ , we have

   $$ \begin{align*}\sigma \circ \zeta_{\alpha_i} \circ \sigma^{-1} = \zeta_{\alpha_{\sigma(i)}} = \zeta_{\sigma(\alpha_i)},\end{align*} $$
   $$ \begin{align*}\sigma \circ \zeta_{\beta_i} \circ \sigma^{-1} = \zeta_{\beta_{\sigma(i)}} = \zeta_{\sigma(\beta_i)},\end{align*} $$

   and

   $$ \begin{align*}\sigma \circ \zeta_{\alpha_i+\beta_i} \circ \sigma^{-1} = \zeta_{\sigma(\alpha_i+\beta_i)}.\end{align*} $$

   Once again, if, for each $\alpha \in \widetilde \Delta $ such that $\alpha _{|S_L} \in \{a_L, 2a_L\}$ , we define $\varphi _{\alpha }: U_{\alpha } \to \mathbb R$ by

   $$ \begin{align*}\varphi_{\alpha}\left(\zeta_{\alpha}\begin{pmatrix}1 & x\\ & 1\end{pmatrix} \right) = \omega(x),\end{align*} $$

   then for each $\sigma \in \operatorname {\mathrm {Gal}}(L/K)$ and each such $\alpha $ , we have

   $$ \begin{align*}\varphi_{\alpha} \circ \sigma^{-1} = \varphi_{\sigma(\alpha)}.\end{align*} $$

   Recall also that, denoting by $H_0(K', K_2')$ the $K^{\prime }_2$ -variety

   $$ \begin{align*}H_0(K', K_2') = \{(x, y) \in R_{K'/K^{\prime}_2}(\mathbb A_{K'}^2), y+\tau(y) = x\tau(x)\},\end{align*} $$

   where $\tau $ is the nontrivial element of $\operatorname {\mathrm {Gal}}(K'/K^{\prime }_2)$ , the group of upper-triangular unipotent matrices in $R_{K_2'/K }SU_{3, K'/K^{\prime }_2}$ is identified with $R_{K^{\prime }_2/K}(H_0(K', K^{\prime }_2))$ via the map

   $$ \begin{align*}\mu: (x, y) \mapsto \begin{pmatrix} 1 & -\tau(x) & -y \\ & 1 & x \\ & & 1\end{pmatrix}\!.\end{align*} $$

   Then, the inclusion $U_a(K) \subset U_a(L) = \prod _{i \in I_a} U_{\alpha _i}(L) U_{\alpha _i+\beta _i}(L) U_{\beta _i}(L)$ spells out explicitly as the map

   $$ \begin{align*}\zeta_a(\mu(x,y)) \mapsto \prod_{i \in I_a} \zeta_{\alpha_i}(-\tau(x))\zeta_{\alpha_i+\beta_i}(\tau(y))\zeta_{\beta_i}(x).\end{align*} $$

   Then, if $u = \zeta _a(\mu (x,y)) \in U_a(K) \setminus \{1\}$ decomposes as $u = \prod _{i \in I_a} u_{\alpha _i} u_{\alpha _i+\beta _i}u_{\beta _i}$ , we have for each $i \in I_a$ :

   $$ \begin{align*}\varphi_{\alpha_i}(u_{\alpha_i}) = \varphi_{\beta_i}(u_{\beta_i}) = \omega(x) \ge \varphi_a^x(u)\end{align*} $$

   and

   $$ \begin{align*}\varphi_{\alpha_i+\beta_i}(u_{\alpha_i+\beta_i}) = \omega(y) = 2\varphi_a^x(u).\end{align*} $$

We have thus constructed a quasi-épinglage $(\zeta _{\alpha })_{\alpha \in \widetilde \Delta }$ [Reference LourençoLou21, Définition 2.4] in $G_L$ along $T_L$ . By [Reference LourençoLou21, Proof of Proposition 2.5], this quasi-épinglage extends to a Chevalley quasi-system $(\zeta _{\alpha })_{\alpha \in \widetilde \Phi }$ in $G^{\operatorname {\mathrm {pred}}}_L$ along $T_L$ . Let us now extend our family $(\varphi _{\alpha })_{\alpha \in \widetilde \Delta }$ to a valuation $(\varphi _{\alpha })_{\alpha \in \widetilde \Phi _{\operatorname {\mathrm {nd}}}}$ of the root data $(Z_G(T)(L), (U_{\alpha }(L))_{\alpha \in \widetilde \Phi })$ . Our main tool is the fact that, for all non-divisible roots $\alpha $ and $\beta $ in $\widetilde \Phi _{\operatorname {\mathrm {nd}}}$ , the parametrisation $\operatorname {\mathrm {int}}(m_{\alpha }) \circ \zeta _{\beta }$ is similar to $\zeta _{s_{\alpha }(\beta )}$ , in the sense that it differs from $\zeta _{s_{\alpha }(\beta )}$ by precomposition with a field L-automorphism of $\bar L$ that maps the domain of $\zeta _{\beta }$ onto that of $\zeta _{s_{\alpha }(\beta )}$ , and by a possible sign change.

Recall from [Reference LourençoLou21, Définition 2.4] that, if $(\zeta _a)_{a\in \Phi }$ is a Chevalley quasi-system, we let

$$ \begin{align*}m_a = \left\{\begin{array}{cl} \zeta_a\begin{pmatrix} & 1 \\ -1 & \end{pmatrix} & \text{if } \operatorname{\mathrm{dom}}(\zeta_a) \text{ is of the form } R_{K'/K}(SL_{2, K'}) \\ \zeta_a\begin{pmatrix}& & -1 \\ & - 1 & \\ -1 & &\end{pmatrix} & \text{if } \operatorname{\mathrm{dom}}(\zeta_a) \text{ is of the form } R_{K_2'/K}(SU_{3, K'/K^{\prime}_2}) \\ x_{-a}(0, -1)x_a(0, 1)x_{-a}(0, -1) & \text{if } \operatorname{\mathrm{dom}}(\zeta_a) \text{ is of the form } R_{K'/K}(BC_{1, K'}) \end{array}\right.\! .\end{align*} $$

The computations we made above allow us to relate the $(m_a)_{a\in \Phi _{\operatorname {\mathrm {nd}}}}$ and the $(m_{\alpha })_{\alpha \in \widetilde \Phi _{\operatorname {\mathrm {nd}}}}$ as follows:

1. 1. If $a \in \Delta $ is such that $\zeta _a$ has as domain a Weil restriction $R_{K'/K}(SL_{2, K'})$ or $R_{K'/K}(BC_{1, K'})$ , then we have $m_a = \prod _{\substack {\alpha \in \widetilde \Phi \\ \alpha _{|S} = a}} m_{\alpha }.$

2. 2. If $a \in \Delta $ is such that $\zeta _a$ has as domain a Weil restriction $R_{K^{\prime }_2/K}(SU_{3, K'/K^{\prime }_2})$ , then for some order on the roots, we have $m_a = \lambda (-1) \left (\prod _{\substack {\alpha \in \widetilde \Phi \\ \alpha _{|S} \in \{a, 2a\}}}m_{\alpha }\right )$ , for some cocharacter $\lambda \in X_*(T)$ .

This allows one to define valuation functions by precomposing those already constructed by products of $\operatorname {\mathrm {int}}(m_{\alpha })$ , and to transfer properties to the general case that were proved above for the $\varphi _{\alpha }$ , where $\widetilde \alpha \in \widetilde \Phi $ with $\alpha _{|S} \in \Delta \cup (2\Delta \cap \Phi )$ . The reason we expect this transfer to work is that, for valuations of root data, we do have such a compatibility with conjugacy by specific elements of $N_G(T)(L)$ , see [Reference Bruhat and TitsBT72, Proposition 6.2.7] – with the notations of this proposition, we have $m_{\alpha } \in M_{\alpha , 0}$ for each $\alpha $ .

Observe [Reference BourbakiBou81, Ch. VI, §1, Proposition 15] that each non-divisible root $a \in \Phi _{\operatorname {\mathrm {nd}}}$ can be mapped to $\Delta $ by some element $w\in W(G, S)(K)$ . For each $a\in \Phi _{\operatorname {\mathrm {nd}}} \setminus \Delta $ , fix once and for all a simple root $b_a \in \Delta $ and an element $w_a \in W(G, S)(K)$ such that $w_a(a) = b_a$ .

Finally, recall that, as observed in the proof of Lemma 9, we have an inclusion

$$ \begin{align*}W(G, S)(K)\hookrightarrow W(G_L, T_L)(L).\end{align*} $$

This inclusion has the property that, for any character $\chi $ and any $w \in W(G, S)(K)$ , we have $w(\chi )_{|S} = w(\chi _{|S})$ .

Now, let $\alpha \in \widetilde \Phi _{\operatorname {\mathrm {nd}}} \setminus \widetilde \Delta $ whose restriction to S is a root $ja$ , with $j \in \{1,2\}$ and $a \in \Phi _{\operatorname {\mathrm {nd}}} \setminus \Delta $ . Set

$$ \begin{align*}\varphi_{\alpha} = \varphi_{\beta_{\alpha}} \circ \operatorname{\mathrm{int}}(m_{\alpha_1}) \circ\dots \operatorname{\mathrm{int}}(m_{\alpha_r}),\end{align*} $$

where $\alpha _1, \dots , \alpha _r, \beta _{\alpha } \in \widetilde \Delta $ , with $\beta _{\alpha } = w_a(\alpha )$ and $w_a = s_{\alpha _1}\dots s_{\alpha _r}$ . This function does not depend on the decomposition of $w_a$ as a product of simple reflections. Indeed, if $\zeta _{\alpha }$ has as domain a Weil restriction $R_{L'/L}(SL_{2, L'})$ , then because $(\zeta _{\alpha })_{\alpha \in \widetilde \Phi _{\operatorname {\mathrm {nd}}}}$ is a Chevalley quasi-system, there exists a $\tau \in \operatorname {\mathrm {Aut}}_L(\overline L)$ such that, for each $x \in L'$ , we have

$$ \begin{align*}\operatorname{\mathrm{int}}(m_{\alpha_1}) \circ\dots \operatorname{\mathrm{int}}(m_{\alpha_r}) \circ \zeta_{\alpha}\begin{pmatrix} 1 & x \\ & 1\end{pmatrix} = \zeta_{\beta_{\alpha}}\begin{pmatrix} 1 & \pm \tau(x) \\ & 1 \end{pmatrix},\end{align*} $$

and therefore

$$ \begin{align*}\varphi_{\alpha}\left( \zeta_{\alpha}\begin{pmatrix} 1 & x \\ & 1\end{pmatrix}\right) = \varphi_{\beta_{\alpha}}\left(\zeta_{\beta_{\alpha}}\begin{pmatrix} 1 & \pm \tau(x) \\ & 1 \end{pmatrix}\right) = \omega(x).\end{align*} $$

On the other hand, if $\zeta _{\alpha }$ has as domain a Weil restriction $R_{L'/L}(BC_{1, L'})$ , then likewise, there exists $\tau \in \operatorname {\mathrm {Aut}}_L(\overline L)$ such that, for each $(u, v) \in L^{\prime 1/2}\times L'$ , we have

$$ \begin{align*}\operatorname{\mathrm{int}}(m_{\alpha_1}) \circ\dots \operatorname{\mathrm{int}}(m_{\alpha_r}) \circ \zeta_{\alpha}(x_{\alpha}(u,v)) = \zeta_{\beta_{\alpha}}(x_{\beta_{\alpha}}( \tau(u), \tau(v))).\end{align*} $$

Therefore, there exists $\eta \in L^{\prime 1/2} \setminus L'$ such that

$$ \begin{align*}\varphi_{\alpha}(x_{\alpha}(u,v)) = \varphi_{\beta_{\alpha}}(x_{\beta_{\alpha}}(\tau(u), \tau(v))) = \frac{1}{2}\omega(\eta u^2 + v)\end{align*} $$

for all $(u, v) \in L^{\prime 1/2} \times L'$ .

It now follows from [Reference LourençoLou21, Proposition 3.2] that the collection $(\varphi _{\alpha })_{\alpha \in \widetilde \Phi _{\operatorname {\mathrm {nd}}}}$ defined above constitutes a valuation of the root data $(Z_G(T)(L), (U_{\alpha }(L))_{\alpha \in \widetilde \Phi })$ corresponding to a point $z \in A(G_L, T)$ . Furthermore, we claim that, if $a \in \Phi _{\operatorname {\mathrm {nd}}}$ is a non-divisible root and $u \in U_a(K)\setminus \{1\}$ , we have the equivalence

$$ \begin{align*}u \in U_{a, r}^x \iff u \in \prod_{\substack{\alpha \in \widetilde \Phi_{\operatorname{\mathrm{nd}}}\\ \alpha_{|S_L} = a_L}} U_{\alpha, r}^z \times \prod_{\substack{\alpha \in \widetilde \Phi_{\operatorname{\mathrm{nd}}}\\ \alpha_{|S_L} = 2a_L}} U_{\alpha, 2r}^z,\end{align*} $$

that is

$$ \begin{align*}\varphi_a^x(u) = \sup\left\{r \in \mathbb R, u \in \prod_{\substack{\alpha \in \widetilde \Phi_{\operatorname{\mathrm{nd}}}\\ \alpha_{|S_L} = a_L}} U_{\alpha, r}^z \times \prod_{\substack{\alpha \in \widetilde \Phi_{\operatorname{\mathrm{nd}}}\\ \alpha_{|S_L} = 2a_L}} U_{\alpha, 2r}^z \right\}.\end{align*} $$

This follows from earlier computations in the case where $a \in \Delta $ . The general case will be reduced to that of simple roots. Let $a \in \Phi _{\operatorname {\mathrm {nd}}}\setminus \Delta $ . Let $a_1, \dots , a_p \in \Delta $ be such that $w_a$ admits the following decomposition as a product of simple reflections

$$ \begin{align*}w_a = s_{a_1} \dots s_{a_p}.\end{align*} $$

Let $u \in U_a(K) \setminus \{1\}$ . On the one hand, by [Reference Bruhat and TitsBT72, Proposition 6.2.7], we have

$$ \begin{align*}\varphi_{a}^x(u) = \varphi_{b_a}^x ( \operatorname{\mathrm{int}}(m_{a_1})\circ \cdots \operatorname{\mathrm{int}}(m_{a_p})(u)).\end{align*} $$

On the other hand, if $\alpha \in \widetilde \Phi $ is such that $\alpha _{|S} \in \{ a, 2a\}$ , then because

$$ \begin{align*}w_a = \left(\prod_{\substack{\alpha' \in \widetilde \Phi\\ \alpha^{\prime}_{|S} \{a_1, 2a_1\}}}s_{\alpha'}\right) \cdots \left(\prod_{\substack{\alpha' \in \widetilde \Phi\\ \alpha^{\prime}_{|S} \in \{a_p, 2a_p\}}}s_{\alpha'}\right)\end{align*} $$

is a decomposition of $w_a$ as a product of reflections in $W(G, T)(L)$ , the component $u_{\alpha }$ of u in $U_{\alpha }(L)$ satisfies

$$ \begin{align*}\begin{array}{ccl} \varphi_{\alpha}(u_{\alpha}) & = & \varphi_{\beta_{\alpha}}\left(\left(\prod_{\substack{\alpha' \in \widetilde \Phi\\ \alpha^{\prime}_{|S} \in \{a_1, 2a_1\}}}\operatorname{\mathrm{int}}(m_{\alpha'})\right) \circ \cdots \left(\prod_{\substack{\alpha' \in \widetilde \Phi\\ \alpha^{\prime}_{|S} \in\{ a_p, 2a_p\}}}\operatorname{\mathrm{int}}(m_{\alpha'})\right)(u_{\alpha})\right)\\ & = & \varphi_{\beta_{\alpha}}(\operatorname{\mathrm{int}}(\lambda(-1))\circ \operatorname{\mathrm{int}}(m_{a_1})\circ \cdots \operatorname{\mathrm{int}}(m_{a_p})(u_{\alpha}))\end{array}\end{align*} $$

for some $\lambda \in X_*(T)$ . We can then check explicitly using our formulas for the épinglage $(\varphi _{\alpha })_{\alpha \in \widetilde \Delta }$ that $\varphi _{\beta _{\alpha }}\circ \operatorname {\mathrm {int}}(\lambda (-1)) = \varphi _{\beta _{\alpha }}$ . Now, because $\beta _{\alpha } = w_a(\alpha )$ lifts $b_a = w_a(a)$ , and because $\operatorname {\mathrm {int}}(m_{a_1})\circ \cdots \operatorname {\mathrm {int}}(m_{a_p})(u_{\alpha })$ is the component of $\operatorname {\mathrm {int}}(m_{a_1})\circ \cdots \operatorname {\mathrm {int}}(m_{a_p})(u)$ in $U_{\beta _{\alpha }}(L)$ , we deduce from previous computations that

$$ \begin{align*}\begin{array}{cc} \varphi_{\alpha}(u_{\alpha}) \ge \varphi_a^x(u) & \text{if } \alpha_{|S} = a\\ \varphi_{\alpha}(u_{\alpha}) = 2\varphi_a^x(u) & \text{ if } \alpha_{|S} = 2a \end{array}\!\!.\end{align*} $$

This completes the proof of the fact that the valuation $(\varphi _a^x)_{a \in \Phi }$ is induced by the valuation $(\varphi _{\alpha })_{\alpha \in \widetilde \Phi }$ in the sense of [Reference Bruhat and TitsBT72, Définition 9.1.11]. By [Reference Bruhat and TitsBT72, Proposition 9.1.17], there exists a $G(K)$ -equivariant map

$$ \begin{align*}j: \mathcal B(G, K) \to \mathcal B(G, L)\end{align*} $$

that maps x to z and is an isometric embedding if the metric on $V(G_K, S)$ is chosen to be induced by a choice of inner product on $V(G_L, T)$ . In particular, it is affine on apartments, which establishes condition 1.

Our last task is to check that $z \in \mathcal B(G, L)^{\operatorname {\mathrm {Gal}}(L/K)}$ , which amounts to proving that the valuation $(\varphi _{\alpha })_{\alpha \in \widetilde \Phi _{\operatorname {\mathrm {nd}}}}$ is $\operatorname {\mathrm {Gal}}(L/K)$ -invariant. We proved that, for all $\alpha \in \widetilde \Phi $ and $\sigma \in \operatorname {\mathrm {Gal}}(L/K)$ , we have

$$ \begin{align*}\varphi_{\alpha} \circ \sigma^{-1} = \varphi_{\sigma(\alpha)}.\end{align*} $$

Now, if $\alpha \in \widetilde \Phi _{\operatorname {\mathrm {nd}}}$ satisfies $\alpha _{|S} \in \{a, 2a\}$ with $a \in \Phi _{\operatorname {\mathrm {nd}}} \setminus \Delta $ , then writing $w_a = s_{\alpha _1} \circ \cdots s_{\alpha _r}$ as a product of simple reflections in the absolute Weyl group $W(G,T)(L)$ , we have for each $\sigma \in \operatorname {\mathrm {Gal}}(L/K)$ :

$$ \begin{align*}\begin{array}{ccl}\varphi_{\alpha}\circ \sigma^{-1} & = & \varphi_{\beta_{\alpha}}\circ \operatorname{\mathrm{int}}(m_{\alpha_1}) \circ \cdots \circ \operatorname{\mathrm{int}}(m_{\alpha_r})\circ \sigma^{-1}\\ &= & \varphi_{\beta_{\alpha}} \circ \sigma^{-1} \circ \operatorname{\mathrm{int}}(\sigma(m_{\alpha_1})) \circ \dots \operatorname{\mathrm{int}}(\sigma(m_{\alpha_r}))\\ & = & \varphi_{\sigma(\beta_{\alpha})} \circ \operatorname{\mathrm{int}}(\sigma(m_{\alpha_1})) \circ \dots \operatorname{\mathrm{int}}(\sigma(m_{\alpha_r})) \end{array}\!\!.\end{align*} $$

Because $w_a \in W(G, S)(K)$ is $\operatorname {\mathrm {Gal}}(L/K)$ -invariant, we have

$$ \begin{align*}\sigma(\beta_{\alpha}) = \sigma(w_a(\alpha)) = w_a(\sigma(\alpha)) = \beta_{\sigma(\alpha)}.\end{align*} $$

Moreover, given the compatibility between the parameterisations $\zeta _{\alpha }$ for $\alpha \in \widetilde \Delta $ and the action of $\operatorname {\mathrm {Gal}}(L/K)$ , we have $\sigma (m_{\alpha _i}) = m_{\sigma (\alpha _i)}$ for each $i \in [\![1,r]\!]$ . We deduce that

$$ \begin{align*}\begin{array}{ccl} \varphi_{\alpha} \circ \sigma^{-1} & = & \varphi_{\beta_{\sigma(\alpha)}}\circ \operatorname{\mathrm{int}}(m_{\sigma(\alpha_1)}) \circ \cdots \operatorname{\mathrm{int}}(m_{\sigma(\alpha_r)})\\ & = & \varphi_{\sigma(\alpha)}\end{array}\!\!,\end{align*} $$

as $w_a = \sigma (w_a) = s_{\sigma (\alpha _1)} \cdots s_{\sigma (\alpha _r)}$ . This completes the proof of the $\operatorname {\mathrm {Gal}}(L/K)$ -invariance of $(\varphi _{\alpha })_{\alpha \in \widetilde \Phi _{\operatorname {\mathrm {nd}}}}$ , and therefore z is fixed under the action of $\operatorname {\mathrm {Gal}}(L/K)$ on $\mathcal B(G, L)$ .

By $G(K)$ -equivariance of j, we deduce that $j(G(K) \cdot x) \subset \mathcal B(G, L)^{\operatorname {\mathrm {Gal}}(L/K)}$ . Finally, because $\mathcal B(G, K)$ is the convex hull of $G(K) \cdot x$ , we get $j(\mathcal B(G, K)) \subset \mathcal B(G, L)^{\operatorname {\mathrm {Gal}}(L/K)}$ .

The above map j is therefore a map from $\mathcal B(G, K)$ to $\mathcal B(G, L)$ that satisfies conditions 1–3, which completes the proof of Proposition 8.

Proof.

1. 1. Let $\gamma \in \operatorname {\mathrm {Gal}}(L/L_0)$ and $\gamma _K = \operatorname {\mathrm {res}}_{L/K}(\gamma )$ . Then, by Theorems 1 and 5, the map

   $$ \begin{align*}p_{\gamma} \circ p_{L/K} \circ p_{\gamma_K^{-1}} : \mathcal B(G, K) \to \mathcal B(G, L)\end{align*} $$

   satisfies conditions 1 and 2 of Theorem 1 with $\sigma :K \to L$ being the inclusion, hence we must have $p_{\gamma} \circ p_{L/K} \circ p_{\gamma_K^{-1}} = p_{L/K}$ .

2. 2. Let $\gamma \in \operatorname {\mathrm {Gal}}(L/K_0)$ and $\gamma _K = \operatorname {\mathrm {res}}_{L/K}(\gamma )$ . To prove that $p_{\gamma} \circ p_{L/K} \circ p_{\gamma_K^{-1}} = p_{L/K}$ , it will be sufficient, because of the uniqueness properties of Propositions 6 and 8, to prove that the map $p_{\gamma} \circ p_{L/K} \circ p_{\gamma_K^{-1}}$ satisfies conditions 1–3. As previously observed, conditions 1 and 2 follow directly from Theorem 5. Lastly, we need to check that

   $$ \begin{align*}p_{\gamma} \circ p_{L/K} \circ p_{\gamma_K^{-1}}(\mathcal B(G, K)) \subset \mathcal B(G, L)^{\operatorname{\mathrm{Gal}}(L/K)},\end{align*} $$

   that is

   $$ \begin{align*}p_{\gamma} \circ p_{L/K}(\mathcal B(G, K)) \subset \mathcal B(G, L)^{\operatorname{\mathrm{Gal}}(L/K)}.\end{align*} $$

   It is sufficient to check that $p_{\gamma }$ maps the $\operatorname {\mathrm {Gal}}(L/K)$ -fixed locus of $\mathcal B(G, L)$ to itself, which is immediate because $\operatorname {\mathrm {Gal}}(L/K)$ is normal in $\operatorname {\mathrm {Gal}}(L/K_0)$ .

Proof. We follow the proof of [Reference RousseauRou77, Théorème 5.2], with minimal adjustments. We first establish the uniqueness assertion.

(Uniqueness) Let $(p_{L/K})_{k \subset K \subset L}$ be a system of maps satisfying conditions 1–4. Observe that, if $K \subset L$ , with $L = K^{\operatorname {\mathrm {nr}}}$ , then $p_{L/K}$ must be the map constructed in Proposition 6. Likewise, if $K \subset L$ , with $G_K$ quasi-split and $G_L$ pseudo-split, then $p_{L/K}$ must be the map constructed in Proposition 8. Now, let K and L be arbitrary finite separable extensions of k with $K \subset L$ .

Consider the strict henselisation $K^{\operatorname {\mathrm {nr}}}$ (resp. $L^{\operatorname {\mathrm {nr}}}$ ) of K (resp. L). Then, the groups $G_{K^{\operatorname {\mathrm {nr}}}}$ and $G_{L^{\operatorname {\mathrm {nr}}}}$ are quasi-split by [Reference LourençoLou21, Corollaire 2.3]. There is thus a unique minimal finite Galois extension $\widetilde {K^{\operatorname {\mathrm {nr}}}}/K^{\operatorname {\mathrm {nr}}}$ (resp. $\widetilde {L^{\operatorname {\mathrm {nr}}}}/L^{\operatorname {\mathrm {nr}}}$ ) such that $G_{\widetilde {K^{\operatorname {\mathrm {nr}}}}}$ (resp. $G_{\widetilde {L^{\operatorname {\mathrm {nr}}}}}$ ) is pseudo-split. Then, the extensions $K^{\operatorname {\mathrm {nr}}}$ and $\widetilde {K^{\operatorname {\mathrm {nr}}}}$ (resp. $L^{\operatorname {\mathrm {nr}}}$ and $\widetilde {L^{\operatorname {\mathrm {nr}}}}$ ) are Galois over K (resp. L), the group $G_{K^{\operatorname {\mathrm {nr}}}}$ (resp. $G_{L^{\operatorname {\mathrm {nr}}}}$ ) is quasi-split and the group $G_{\widetilde {K^{\operatorname {\mathrm {nr}}}}}$ (resp. $G_{\widetilde {L^{\operatorname {\mathrm {nr}}}}}$ ) is pseudo-split.

It then follows from condition 4 and Propositions 6 and 8 that the following diagram commutes

Because all maps in the diagram are injective, the map $p_{L/K}$ is uniquely determined in terms of the maps $p_{K^{\operatorname {\mathrm {nr}}}/K}, p_{\widetilde {K^{\operatorname {\mathrm {nr}}}}/K^{\operatorname {\mathrm {nr}}}}, p_{\widetilde {L^{\operatorname {\mathrm {nr}}}}/\widetilde {K^{\operatorname {\mathrm {nr}}}}}, p_{\widetilde {L^{\operatorname {\mathrm {nr}}}}/L^{\operatorname {\mathrm {nr}}}}$ and $p_{L^{\operatorname {\mathrm {nr}}}/L}$ , each of which is uniquely determined in terms of G, K and L as a consequence of either Proposition 6 or Proposition 8.

(Existence) Let K and L be arbitrary henselian discretely valued extensions of k with perfect residue fields and excellent valuation rings with $K \subset L$ . We consider the extensions – with the same notations as above

$$ \begin{align*}K_1 = K^{\operatorname{\mathrm{nr}}}, K_2 = \widetilde{K^{\operatorname{\mathrm{nr}}}}, L_1 = L^{\operatorname{\mathrm{nr}}}, L_2 = \widetilde{L^{\operatorname{\mathrm{nr}}}}.\end{align*} $$

Consider

$$ \begin{align*}\begin{array}{ccc}\Gamma = \operatorname{\mathrm{Gal}}(K_2/K),& \Gamma_1 = \operatorname{\mathrm{Gal}}(K_1/K), & \Gamma_2 = \operatorname{\mathrm{Gal}}(K_2/K_1)\\ \Gamma' = \operatorname{\mathrm{Gal}}(L_2/L), & \Gamma^{\prime}_1 = \operatorname{\mathrm{Gal}}(L_1/L), & \Gamma^{\prime}_2 = \operatorname{\mathrm{Gal}}(L_2/L_1) \end{array}\!.\end{align*} $$

Restriction maps fit in a commutative diagram

By Theorem 5 and Propositions 6 and 8, we may then form the following diagram.

(2)

Because all maps in the diagram are injective, it is sufficient, in order to construct a map $\mathcal B(G, K) \to \mathcal B(G, L)$ that makes the rectangle (2) commute, to prove that

$$ \begin{align*}p_{L_2/K_2} \circ p_{K_2/K_1} \circ p_{K_1/K}(\mathcal B(G, K)) \subset p_{L_2/L_1} \circ p_{L_1/L}(\mathcal B(G, L)).\end{align*} $$

We proceed in two steps in order to exploit the features of the two special cases: first, we construct a map $p_{L_1/K_1}: \mathcal B(G, K_1) \to \mathcal B(G, L_1)$ making the right-hand square commute. Then, we exploit the uniqueness of this map to construct the map $p_{L/K}$ .

Step 1: By the same argument as before, it is sufficient to prove that

$$ \begin{align*}p_{L_2/K_2} \circ p_{K_2/K_1}(\mathcal B(G, K_1)) \subset p_{L_2/L_1}(\mathcal B(G, L_1)).\end{align*} $$

Furthermore, by $G(K_1)$ -equivariance, it is sufficient to check that

$$ \begin{align*}p_{L_2/K_2} \circ p_{K_2/K_1}(A(G_{K_1}, S_1)) \subset p_{L_2/L_1}(\mathcal B(G, L_1))\end{align*} $$

for some maximal $K_1$ -split torus $S_1 \subset G_{K_1}$ . Let $S_1$ be such a maximal $K_1$ -split torus in $G_{K_1}$ . By Proposition 8 and Lemma 9, the image $p_{K_2/K_1}(A(G_{K_1}, S_1))$ lies in the apartment $A(G_{K_2}, S_2)$ associated to the unique maximal $K_2$ -split torus $S_2 \subset G_{K_2}$ that contains $(S_1)_{K_2}$ . Now, by Theorem 1, the image $p_{L_2/K_2}(A(G_{K_2}, S_2))$ is precisely $A(G_{L_2}, T_2)$ , with $T_2 = (S_2)_{L_2}$ . As $T_2$ is defined over $K_1$ (Lemma 9), it is in particular defined over $L_1$ . Hence, the action of $\Gamma ^{\prime }_2$ on $B(G, L_2)$ leaves $A(G_{L_2}, T_2)$ invariant. Because $p_{L_2/K_2}$ is $\Gamma ^{\prime }_2$ -equivariant, by Proposition 10, we have the inclusion

$$ \begin{align*}p_{L_2/K_2} \circ p_{K_2/K_1}(A(G_{K_1}, S_1)) \subset A(G_{L_2}, T_2)^{\Gamma^{\prime}_2}.\end{align*} $$

To conclude, let $T_1$ be a maximal $L_1$ -split torus in $G_{L_1}$ containing $(S_1)_{L_1}$ . By the same reasoning as above, the image $p_{L_2/L_1}(A(G_{L_1}, T_1))$ lies in the apartment $A(G_{L_2}, T^{\prime }_2)$ associated to the unique maximal $L_2$ -split torus of $G_{L_2}$ containing $(T_1)_{L_2}$ . Furthermore, by construction of $p_{L_2/L_1}$ , we must then have

$$ \begin{align*}p_{L_2/L_1}(A(G_{L_1}, S_1)) = A(G_{L_2}, T^{\prime}_2)^{\Gamma^{\prime}_2}.\end{align*} $$

Observe that the torus $T^{\prime }_2$ must then contain $(S_1)_{L_2}$ . By uniqueness of the maximal torus in $(Z_{G_{K_1}}(S_1))_{L_2}$ , we must then have $T^{\prime }_2 = T_2$ . Hence, we have the inclusion

$$ \begin{align*}p_{L_2/K_2} \circ p_{K_2/K_1}(A(G_{K_1}, S_1)) \subset p_{L_2/L_1}(A(G_{L_1}, T_1)),\end{align*} $$

which completes the proof.

We have thus proved the existence of a map $p_{L_1/K_1}: \mathcal B(G, K_1) \to \mathcal B(G, L_1)$ such that the square

(3)

commutes. By $G(K_1)$ -equivariance and injectivity of $p_{L_2/K_2} \circ p_{K_2/K_1}$ and $p_{L_2/L_1}$ , the map $p_{L_1/K_1}$ must be $G(K_1)$ -equivariant. Moreover, it follows from the construction that it maps each apartment $A(G_{K_1}, S_1)$ to the apartment $A(G_{L_1}, T_1)$ associated to the unique maximal $L_1$ -split torus such that $(S_1)_{L_1} \subset T_1$ . The restriction map

$$ \begin{align*}p_{L_1/K_1}: A(G_{K_1}, S_1) \to A(G_{L_1}, T_1)\end{align*} $$

is then affine because, letting $S_2$ be the unique maximal $K_2$ -split torus of $G_{K_2}$ containing $S_1$ and $T_2 = (S_2)_{L_2}$ , we have the commutative diagram

where $p_{L_2/K_2} \circ p_{K_2/K_1}$ and $p_{L_2/L_1}$ are affine and injective.

We also prove that $p_{L_1/K_1}$ is $\Gamma ^{\prime }_1$ -equivariant, which will also follow formally from Galois-equivariance properties of the other maps in the square (3). Let $\gamma ^{\prime }_1 \in \Gamma ^{\prime }_1$ , a lift $\gamma '$ of $\gamma ^{\prime }_1$ to $L_2$ , the restriction $\gamma = \operatorname {\mathrm {res}}_{L_2/K_2}(\gamma ')$ , and the restriction $\gamma _1 = \operatorname {\mathrm {res}}_{K_2/K_1}(\gamma )$ . Then, Proposition 10 implies that

$$ \begin{align*}\begin{array}{ccll}p_{L_2/L_1} \circ p_{\gamma^{\prime}_1} \circ p_{L_1/K_1} \circ p_{\gamma_1}^{-1} & = & p_{\gamma'} \circ p_{L_2/L_1} \circ p_{L_1/K_1} \circ p_{\gamma_1}^{-1} & (\Gamma'\text{-equivariance of } p_{L_2/L_1}) \\& = & p_{\gamma'} \circ p_{L_2/K_2} \circ p_{K_2/K_1} \circ p_{\gamma_1}^{-1} & (\text{commutativity of the}\\& & & \text{diagram (3)})\\& = & p_{L_2/K_2} \circ p_{\gamma} \circ p_{K_2/K_1} \circ p_{\gamma_1}^{-1} & (\Gamma'\text{- equivariance of } p_{L_2/K_2})\\& = & p_{L_2/K_2} \circ p_{K_2/K_1} & (\Gamma\text{-equivariance of} p_{K_2/K_1})\\& = & p_{L_2/L_1} \circ p_{L_1/K_1}, \end{array}\!\end{align*} $$

and therefore

$$ \begin{align*}p_{\gamma^{\prime}_1} \circ p_{L_1/K_1} \circ p_{\gamma_1}^{-1} = p_{L_1/K_1}.\end{align*} $$

Step 2: Our second and final step is to deduce that, given an apartment $A(G_K, S)$ in $\mathcal B(G, K)$ , we have

$$ \begin{align*}p_{L_1/K_1} \circ p_{K_1/K}(A(G_K, S)) \subset p_{L_1/L}(A(G_L, T)),\end{align*} $$

for some maximal L-split torus T of $G_L$ such that $S_L \subset T$ . First of all, it follows from Proposition 6 and the construction of $p_{L_1/K_1}$ that the image $p_{L_1/K_1} \circ p_{K_1/K}(A(G_K, S))$ lies in the apartment $A(G_{L_1}, T_1)$ associated with a certain maximal $L_1$ -split torus $T_1$ containing S and is $\Gamma ^{\prime }_1$ -equivariant. Consequently, we have the inclusion

$$ \begin{align*}p_{L_1/K_1} \circ p_{K_1/K}(A(G_K, S)) \subset (\mathcal B(Z_{G_K}(S), G, L_1))^{\Gamma^{\prime}_1},\end{align*} $$

with the notations of the Appendix. By Proposition 7, we deduce

$$ \begin{align*}p_{L_1/K_1} \circ p_{K_1/K}(A(G_K, S)) \subset p_{L_1/L}(\mathcal B(Z_{G_K}(S), G, L)).\end{align*} $$

Let $x \in A(G_K, S)$ . The above inclusion implies that there exists a maximal L-split torus T of $G_L$ containing $S_L$ such that

$$ \begin{align*}p_{L_1/K_1} \circ p_{K_1/K}(x) \in p_{L_1/L}(A(G_L, T)).\end{align*} $$

Because the right-hand side is stable under the action of $S(K) \subset T(L)$ , we get that

$$ \begin{align*}p_{L_1/K_1} \circ p_{K_1/K}(S(K) \cdot x) \subset p_{L_1/L}(A(G_L, T))\end{align*} $$

and, passing to the convex hull,

$$ \begin{align*}p_{L_1/K_1} \circ p_{K_1/K}(A(G_K, S)) \subset p_{L_1/L}(A(G_L, T)).\end{align*} $$

We then deduce by the same arguments as before that there must exist a $G(K)$ -equivariant map

$$ \begin{align*}p_{L/K}: \mathcal B(G, K) \to \mathcal B(G, L)\end{align*} $$

such that $p_{L_1/K_1}\circ p_{K_1/K} = p_{L_1/L} \circ p_{L/K}$ . Moreover, we have $p_{L/K}(A(G_K, S)) \subset A(G_L, T)$ and, by $G(K)$ -equivariance, we find that for each S, there must exist a maximal L-split torus T containing $S_L$ such that $p_{L/K}(A(G_K, S)) \subset A(G_L, T)$ . The same argument as before then proves that the restriction of $p_{L/K}$ to $A(G_K, S)$ is affine, which establishes that $p_{L/K}$ satisfies condition 1. To complete the proof of Theorem 11, we now need to establish conditions 3 and 4.

Condition 4 is an easy consequence of the uniqueness conditions. Indeed, if $K \subset L \subset M$ is a tower of henselian discretely valued extensions of k with perfect residue fields and excellent valuation rings, then we have a commutative diagram

Furthermore, because $G_{K_2}$ is pseudo-split, we know by Theorem 1 that $p_{M_2/L_2} \circ p_{L_2/K_2} = p_{M_2/K_2}$ . Therefore, we have the commutative diagram

Because $p_{M/K}$ is, by construction, the only map $\mathcal B(G, K) \to \mathcal B(G, M)$ that makes the above rectangle commute, we must have

$$ \begin{align*}p_{M/K} = p_{M/L} \circ p_{L/K}.\end{align*} $$

Lastly, establishing condition 3 merely requires adapting the Galois-equivariance arguments we previously brought forth. Indeed, if $L/K$ is Galois, then $L_1$ and $L_2$ are Galois over K. Furthermore, setting

$$ \begin{align*}\widetilde \Gamma' = \operatorname{\mathrm{Gal}}(L_2/K), \widetilde \Gamma_1 = \operatorname{\mathrm{Gal}}(L_1/K), \Sigma = \operatorname{\mathrm{Gal}}(L/K),\end{align*} $$

we have a diagram

Recall that, by Theorem 5, we have the following actions:

Then, repeated applications of Proposition 10 yield

$$ \begin{align*}p_{L_2/K_2} \circ p_{K_2/K_1} \circ p_{K_1/K}(\mathcal B(G, K)) \subset \mathcal B(G, L_2)^{\widetilde \Gamma'}.\end{align*} $$

By commutativity of the diagram, we get

$$ \begin{align*}p_{L_2/L_1} \circ p_{L_1/L} \circ p_{L/K}(\mathcal B(G, K)) \subset \mathcal B(G, L_2)^{\widetilde \Gamma'}.\end{align*} $$

More precisely, we may write

$$ \begin{align*}p_{L_2/L_1} \circ p_{L_1/L} \circ p_{L/K}(\mathcal B(G, K)) \subset \mathcal B(G, L_2)^{\widetilde \Gamma'}\cap p_{L_2/L_1} \circ p_{L_1/L}(\mathcal B(G, L)).\end{align*} $$

Then, using Proposition 10 and the injectivity of $p_{L_2/L_1} \circ p_{L_1/L}$ , we get

$$ \begin{align*}p_{L_2/L_1} \circ p_{L_1/L} \circ p_{L/K}(\mathcal B(G, K)) \subset p_{L_2/L_1} \circ p_{L_1/L}(\mathcal B(G, L)^{\Sigma}),\end{align*} $$

and finally by injectivity

$$ \begin{align*}p_{L/K}(\mathcal B(G, K)) \subset \mathcal B(G, L)^{\Sigma},\end{align*} $$

which completes the proof.

Proof. Let $K^{\operatorname {\mathrm {nr}}}$ be the strict henselisation of K and $\widetilde K^{\operatorname {\mathrm {nr}}}$ be the minimal Galois extension that splits $G_{K^{\operatorname {\mathrm {nr}}}}$ . By [Reference LourençoLou21, Lemme 4.8], there exists a maximal $K^{\operatorname {\mathrm {nr}}}$ -split torus $S^{\operatorname {\mathrm {nr}}}$ that contains $S_{K^{\operatorname {\mathrm {nr}}}}$ and is defined over K. Because $G_{K^{\operatorname {\mathrm {nr}}}}$ is quasi-split, there exists a unique maximal torus T of $G_{K^{\operatorname {\mathrm {nr}}}}$ that contains $S^{\operatorname {\mathrm {nr}}}$ . The torus T is defined over K, because $S^{\operatorname {\mathrm {nr}}}$ is, and splits over $\widetilde K^{\operatorname {\mathrm {nr}}}$ . Moreover, Proposition 6 yields

$$ \begin{align*}p_{K^{\operatorname{\mathrm{nr}}}/K}(A(G, S)) \subset A(G_{K^{\operatorname{\mathrm{nr}}}}, S^{\operatorname{\mathrm{nr}}})\end{align*} $$

and Proposition 8 yields

$$ \begin{align*}p_{\widetilde K^{\operatorname{\mathrm{nr}}}/K^{\operatorname{\mathrm{nr}}}}(A(G_{K^{\operatorname{\mathrm{nr}}}}, S^{\operatorname{\mathrm{nr}}})) \subset A(G_{\widetilde K^{\operatorname{\mathrm{nr}}}}, T_{\widetilde K^{\operatorname{\mathrm{nr}}}}).\end{align*} $$

In particular,

$$ \begin{align*}p_{\widetilde K^{\operatorname{\mathrm{nr}}}/K}(A(G, S)) \subset A(G_{\widetilde K^{\operatorname{\mathrm{nr}}}}, T_{\widetilde K^{\operatorname{\mathrm{nr}}}}).\end{align*} $$

Now, let $L \subset \widetilde K^{\operatorname {\mathrm {nr}}}$ be a finite Galois extension of K such that $T_L$ is split. By Theorem 1, we have

$$ \begin{align*}p_{\widetilde K^{\operatorname{\mathrm{nr}}}/L}(A(G_L, T_L)) = A(G_{\widetilde K^{\operatorname{\mathrm{nr}}}}, T_{\widetilde K^{\operatorname{\mathrm{nr}}}}).\end{align*} $$

In other words,

$$ \begin{align*}p_{\widetilde K^{\operatorname{\mathrm{nr}}}/L} \circ p_{L/K}(A(G, S)) \subset p_{\widetilde K^{\operatorname{\mathrm{nr}}}/L}(A(G_L, T_L))\end{align*} $$

and, by injectivity, we have

$$ \begin{align*}p_{L/K}(A(G, S)) \subset A(G_L, T_L).\end{align*} $$

Because T is defined over K, the apartment $A(G_L, T_L)$ is stable under the action of $\operatorname {\mathrm {Gal}}(L/K)$ and the set of $\operatorname {\mathrm {Gal}}(L/K)$ -fixed points is a non-empty affine space under $V(G_L, T_L)^{\operatorname {\mathrm {Gal}}(L/K)} = V(G, S)$ . By considering dimensions, we thus see that

$$ \begin{align*}p_{L/K}(A(G, S)) = A(G_L, T_L)^{\operatorname{\mathrm{Gal}}(L/K)}.\\[-34pt]\end{align*} $$

Proof. The argument is a combination of Proposition 10 and some ideas of the proof of Theorem 11. Recall that, by construction, the map $p_{M/L}$ is the only map that makes the following diagram commute

where, as in the proof of Theorem 11, we let

$$ \begin{align*}L_1 = L^{\operatorname{\mathrm{nr}}}, L_2 = \widetilde{L^{\operatorname{\mathrm{nr}}}}, M_1 = M^{\operatorname{\mathrm{nr}}}, M_2 = \widetilde{M^{\operatorname{\mathrm{nr}}}}.\end{align*} $$

Note that all extensions in question are Galois over K under the assumption that L and M are Galois over K. Then, letting $\gamma ' \in \operatorname {\mathrm {Gal}}(M/K)$ and denoting by $\gamma ^{\prime }_1 \in \operatorname {\mathrm {Gal}}(M_1/K)$ a lift of $\gamma '$ , and by $\gamma ^{\prime }_2 \in \operatorname {\mathrm {Gal}}(M_2/K)$ a lift of $\gamma ^{\prime }_1$ , and letting $\gamma _2 = \operatorname {\mathrm {res}}^{M_2}_{L_2} (\gamma ^{\prime }_2) \in \operatorname {\mathrm {Gal}}(L_2/K)$ , as well as $\gamma _1 = \operatorname {\mathrm {res}}^{L_2}_{L_1}(\gamma _2) \in \operatorname {\mathrm {Gal}}(L_1/K)$ , and finally $\gamma = \operatorname {\mathrm {res}}^{L^{\operatorname {\mathrm {nr}}}}_L(\gamma ) \in \operatorname {\mathrm {Gal}}(L/K)$ , repeated applications of Proposition 10 yield:

$$ \begin{align*}\begin{array}{ccl} p_{M_2/M_1} \circ p_{M_1/M} \circ p_{\gamma'} \circ p_{M/L} & = & p_{M_2/M_1} \circ p_{\gamma^{\prime}_1} \circ p_{M_1/M} \circ p_{M/L} \\ & = & p_{\gamma^{\prime}_2} \circ p_{M_2/M_1} \circ p_{M_1/M} \circ p_{M/L} \\ & = & p_{\gamma^{\prime}_2} \circ p_{M_2/L_2} \circ p_{L_2/L_1} \circ p_{L_1/L} \\ & = & p_{M_2/L_2} \circ p_{\gamma_2} \circ p_{L_2/L_1} \circ p_{L_1/L}\\ & = & p_{M_2/L_2}\circ p_{L_2/L_1} \circ p_{\gamma_1} \circ p_{L_1/L} \\ & = & p_{M_2/L_2} \circ p_{L_2/L_1} \circ p_{L_1/L} \circ p_{\gamma}\end{array}.\end{align*} $$

It follows that $p_{\gamma '} \circ p_{M/L} \circ p_{\gamma }^{-1}$ makes the above rectangle commute, hence we must have

$$ \begin{align*}p_{\gamma'} \circ p_{M/L} \circ p_{\gamma}^{-1} = p_{M/L}.\\[-34pt]\end{align*} $$

Notation 16. For ease of notation, whenever K is a separable henselian discretely valued extension of k with perfect residue field and excellent valuation ring, and for each $x\in \mathcal B(G, k)$ , we will write $x_K$ instead of $p_{K/k}(x)$ in all of the following.