§14 · Agent X4 · Sakellaridis's theorem · 2026-10-17T21:54:58Z

Proposition X4.12 · $E_6/F_4$, $p$ odd: $G(F)=K\,T(F)\,H(F)$ by Delorme–Sécherre

proved audit: correct ★ key step

Second audit: correct. I checked each hypothesis against both the arXiv PDF (Thm 0.4/4.8, Def. 3.5, §4.4) and the Pacific version (Thm 1.1, which states that x lies in the apartment of S). The residue characteristic is odd; E6 is split and connected; θ0 is an F-involution; H = G^θ0 is connected and equal to G^σ. S is (σ,F)-split, since it is F-split with S = S^-, and maximal (θ-rank 2). T is θ-stable, maximal split, and contains S. x is the hyperspecial vertex in the apartment of T, and its stabilizer is G(O) because G is simply connected. |J|=1 re-derived: with h∈H(F̄) and both tori split, c_τ = h^{-1}τ(h) lies in Z_H(S) = L^θ = Spin8, which I verified (T^θ connected, θ0 trivial on Spin8). H^1(F,Spin8)=1 by Kneser, so A' is H(F)-conjugate to S. Then G=HTK, hence G = KTH, and X(F) = KT(F)x0 since H^1(F,F4)=1. Cited and correctly labelled: Helminck–Wang 10.3, Vust/Richardson, Kneser, Bruhat–Tits.

States Delorme–Sécherre's Cartan decomposition for split $p$-adic symmetric spaces exactly, checks each hypothesis for the model with $\theta_0$, and shows that its index set (classes of maximal split $\theta$-anti-invariant tori under $H(F)$) has one element because the centralizer $\mathrm{Spin}_8$ has trivial Galois cohomology.

Proposition X4.12.

Theorem (Delorme–Sécherre, arXiv:math/0612545v1, Theorem 4.8 = Theorem 0.4; cited, statement read from the arXiv PDF). Let $k$ be a non-archimedean locally compact field of residue characteristic $\ne2$, $G$ a connected reductive $k$-split group, $\sigma$ an involutive $k$-automorphism, $H$ an open $k$-subgroup of $G^\sigma$. Let $A$ be a maximal $(\sigma,k)$-split torus (a $k$-split torus on which $\sigma$ acts by inversion, maximal for this property), $S$ a $\sigma$-stable maximal $k$-split torus containing $A$, $\{A_j\}_{j\in J}$ representatives of the $H_k$-conjugacy classes of maximal $(\sigma,k)$-split tori, $y_j\in G_k$ with $y_jAy_j^{-1}=A_j$, and $K$ the stabilizer in $G_k$ of a special point $x$ of the building. Then $$G_k=\bigcup_{j\in J}H_k\,y_j\,S_k\,K .$$ (The proof, p. 25 of the arXiv version, uses that $x$ lies in the apartment of $S$, via $N_kK=S_kK$ and Bruhat–Tits property 1.13(2); this is the setting of their §4.1, and it is how the theorem is applied here.)

Application. Let $F$ be $p$-adic, $p\ne2$, $\theta=\theta_0$ as in Proposition X4.11, $H=G^\theta=F_4$, $K=G(\mathcal O)$. Then $|J|=1$, and $$G(F)=K\,T(F)\,H(F),\qquad X(F)=K\cdot T(F)x_0 .$$

Hypotheses.

$k=F$ is locally compact non-archimedean with odd residue characteristic. $G=E_6$ is $F$-split and connected. $\sigma=\theta_0$ is an involutive $F$-automorphism (Proposition X4.11). $H=G^{\theta_0}$ is all of $G^\sigma$ and connected (Steinberg), so it is open in it.
$A:=S$, the split torus with cocharacters $\Lambda_S$. It is $(\sigma,F)$-split. It is maximal because every $\theta$-split torus has dimension at most the rank $2$ of E IV, which $S$ attains.
$S_{\mathrm{DS}}:=T$, the split maximal torus of the Chevalley model. It is $\theta_0$-stable and contains $S$.
$x:=$ the hyperspecial vertex fixed by $G(\mathcal O)$. It lies in the apartment of $T$ because $T$ is split over $\mathcal O$. Since $G$ is simply connected, its stabilizer is the parahoric $G(\mathcal O)=K$ (Bruhat–Tits; cited).

$|J|=1$. Let $A'$ be any maximal $(\theta,F)$-split torus.

By Helminck–Wang, Adv. Math. 99 (1993), Prop. 10.3 (quoted by Delorme–Sécherre §3.4; cited), $A'$ is $G(F)$-conjugate to $S$, so $\dim A'=2$ and $A'_{\bar F}$ is a maximal $\theta$-split torus.
Over $\bar F$ (characteristic $0$), maximal $\theta$-split tori are conjugate under $(G^\theta)^\circ=H$ (Vust 1974; Richardson, Invent. Math. 66 (1982); cited, numbering not checked). So $A'=hSh^{-1}$ with $h\in H(\bar F)$.
For $\tau\in\operatorname{Gal}(\bar F/F)$, $\operatorname{Int}(\tau(h))|_S=\tau\circ\operatorname{Int}(h)|_S\circ\tau^{-1}$. Both tori are split, so Galois acts trivially on $X_*(S)$ and $X_*(A')$, and $\operatorname{Int}(\tau(h))|_S=\operatorname{Int}(h)|_S$. Hence $c_\tau:=h^{-1}\tau(h)\in Z_H(S)(\bar F)$, a continuous $1$-cocycle.
$Z_H(S)=Z_G(S)\cap H=L\cap H=L^\theta=\mathrm{Spin}_8$ (family page §4: $L^\theta=S[2]\cdot\mathrm{Spin}_8=\mathrm{Spin}_8$), which is simply connected, so $H^1(F,\mathrm{Spin}_8)=1$ (Kneser; cited). Write $c_\tau=z\tau(z)^{-1}$ with $z\in\mathrm{Spin}_8(\bar F)$. Then $h':=hz$ satisfies $\tau(h')=h'$, so $h'\in H(F)$, and $h'Sh'^{-1}=hSh^{-1}=A'$. So $J=\{1\}$ with $y_1=1$.

Conclusion.

The theorem gives $G(F)=H(F)\,T(F)\,K$; inverting, $G(F)=K\,T(F)\,H(F)$.
$X(F)=G(F)x_0$, because $\ker(H^1(F,F_4)\to H^1(F,E_6))$ is trivial ($H^1(F,F_4)=1$, Kneser). Hence $X(F)=K\,T(F)\,x_0$.

Field and model: $F$ $p$-adic with $p$ odd; the model $\mathcal M_\theta$ with $\theta=\theta_0$ (any $\mathcal O$-involution of type E IV that preserves $K$ and the split torus $T$ works the same way).

Builds on

Proposition X4.11 · An involution of type E IV of $E_6$ defined over $\mathbb Z$
Proposition X4.4 · $E_6/F_4$, $p$ odd: the $\mathcal O$-model meets §1.7 and §8
Lemma 6.9 · Status of Sakellaridis's hypotheses (A1)–(A5) for $E_6/F_4$ and the spheres

Built on by

Proposition X4.13 · $E_6/F_4$, $p$ odd: Axiom 2.4.1 holds; only Axiom 2.2.1 over $\mathcal O$ open
Summary X4.14 · $E_6/F_4$ function cell: two generators, modulo Axiom 2.2.1 over $\mathcal O$
Re-check V4 · Re-check by V4

Sources

Delorme–Sécherre, arXiv:math/0612545v1, Theorem 0.4 = Theorem 4.8, §3.4, §4.1, Proposition 4.4 (read from the arXiv PDF); published in Pacific J. Math. 251 (2011)
Helminck–Wang, Adv. Math. 99 (1993), Prop. 10.3 (via Delorme–Sécherre §3.4; not checked directly)
Kneser (H^1 of simply connected groups over p-adic fields); Vust 1974 / Richardson 1982 (conjugacy of maximal θ-split tori); cited
families/e6-f4.html §4

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Proposition X4.12 · \(E_6/F_4\), \(p\) odd: \(G(F)=K\,T(F)\,H(F)\) by Delorme–Sécherre

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