§14 · Agent X4 · Sakellaridis's theorem · 2026-10-17T21:53:56Z

Proposition X4.11 · An involution of type E IV of $E_6$ defined over $\mathbb Z$

proved correction ★ key step

Correction. Corrects Proposition X4.4 · $E_6/F_4$, $p$ odd: the $\mathcal O$-model meets §1.7 and §8.

Settles the existence of the integral model: the Chevalley involution composed with the Tits lift of the longest element of the black $D_4$ is an involution of the split simply connected $E_6$ over $\mathbb Z$, acting by $-w_L$ on the torus, trivially on $\mathrm{Spin}_8$, with 52-dimensional fixed group $F_4$ on every fibre of characteristic $\ne2$, and with the standard $T\subseteq B\subseteq P$ adapted to it.

Proposition X4.11.

Let $G_{\mathbb Z}$ be the split simply connected Chevalley group of type $E_6$ with pinning $(T,B,\{x_{\alpha_i}\})$ (Bourbaki labels). Let $C$ be the Chevalley involution: $C(t)=t^{-1}$ on $T$ and $C(x_{\pm\alpha_i}(c))=x_{\mp\alpha_i}(-c)$. Put $\sigma_\alpha=x_\alpha(1)x_{-\alpha}(-1)x_\alpha(1)$, let $\sigma_L$ be the Tits lift of $w_L$ (the longest element of $W(D_4)$, black nodes $2,\dots,5$), and set $$\theta_0:=\operatorname{Int}(\sigma_L)\circ C .$$ Then:

$\theta_0$ is an automorphism of $G_{\mathbb Z}$ with $\theta_0^2=\mathrm{id}$ and $\theta_0|_T=-w_L$.
$\theta_0$ is the identity on the derived group $\mathrm{Spin}_8$ of the black Levi $L$.
On every geometric fibre of characteristic $\ne2$, $\theta_0$ is an outer involution with $\dim\mathfrak g^{\theta_0}=52$; hence it has type E IV and $G^{\theta_0}$ is connected of type $F_4$.
$\theta_0(U_P)=U_P^-$ for the standard parabolic $P\supseteq B$ of $\{2,3,4,5\}$, so $P$ is minimal $\theta_0$-split with $\theta_0$-stable Levi $L=Z_G(S)$, $S=(T^{-\theta_0})^\circ$ of rank $2$.

So the model $\mathcal M_\theta$ of Proposition X4.4 exists, with $\theta=\theta_0$ defined over $\mathbb Z$; its existence is no longer a cited input.

$C$ and the Tits group. $C$ exists over $\mathbb Z$ by the isomorphism theorem for pinned split reductive group schemes (SGA 3, Exp. XXIII; cited): it realizes $-1$ on $\mathcal X(T)$ with the pinning $\{-X_{-\alpha_i}\}$. In $\mathrm{SL}_2$, $\sigma=x(1)y(-1)x(1)=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ and $C(\sigma)=y(-1)x(1)y(-1)=\begin{pmatrix}0&1\\-1&0\end{pmatrix}=\sigma$. Hence $C$ fixes every $\sigma_{\alpha_i}$, hence $\sigma_L$; and $C^2=\mathrm{id}$, being trivial on $T$ and on the simple root groups.
Order. $C\circ\operatorname{Int}(\sigma_L)=\operatorname{Int}(C(\sigma_L))\circ C=\operatorname{Int}(\sigma_L)\circ C$, so $\theta_0^2=\operatorname{Int}(\sigma_L^2)$. On $T$, $\theta_0=w_L\circ(-1)=-w_L$.
$\theta_0$ on $\mathrm{Spin}_8$ (proves 2). Restricted to $\mathrm{Spin}_8$, $\theta_0=\operatorname{Int}(\sigma_{w_0})\circ C$ for the induced pinning of $D_4$. In the split model $\mathfrak{so}_8$ with $f_i=e_i^{\mathsf T}$ and $C(X)=-X^{\mathsf T}$, exact integer computation along the reduced word $s_1s_2s_1s_3s_2s_1s_4s_2s_1s_3s_2s_4$ ($D_4$ labels, centre $2$) gives $\operatorname{Ad}(\sigma_{w_0})(C(e_i))=e_i$ and $\operatorname{Ad}(\sigma_{w_0})(C(f_i))=f_i$ for all $i$ (c7_theta0.py). So $\theta_0(x_{\pm\alpha}(c))=x_{\pm\alpha}(c)$ for the black simple roots, over $\mathbb Z$, and $\theta_0$ is trivial on $\mathrm{Spin}_8$.
$\sigma_L^2=1$. By step 3, $\operatorname{Int}(\sigma_L^2)=\theta_0^2$ is trivial on $\mathrm{Spin}_8$, so $\sigma_L^2\in Z(\mathrm{Spin}_8)\simeq\mu_2\times\mu_2$. Its image in $\mathrm{SO}_8$ (the vector representation) is $I$ (same computation). Triality $\tau$ preserves the $D_4$ pinning and $w_0$, so it fixes the Tits lift $\sigma_{w_0}$ (Tits lifts do not depend on the reduced word). Hence $\sigma_L^2$ also acts trivially in the two half-spin representations (vector $\circ\,\tau^{\pm1}$). The three kernels meet trivially in $Z(\mathrm{Spin}_8)$, so $\sigma_L^2=1$ and $\theta_0^2=\mathrm{id}$. (This agrees with Tits's formula $\sigma_{w_0}^2=(2\rho^\vee)(-1)$ and $\rho_L^\vee=3\alpha_2^\vee+3\alpha_3^\vee+5\alpha_4^\vee+3\alpha_5^\vee\in\Lambda_T$.)
Type (proves 3). $\theta_0$ acts on $\mathcal X(T)$ by $-w_L\notin W(E_6)$ (as $-1\notin W(E_6)$), so it is outer. $\dim\mathfrak g^{\theta_0}$ is the sum of: $4$ from $\mathfrak t$ (the $+1$-eigenspace of $-w_L$ is the black span); $24$ from the $48$ roots outside $L$, which $-w_L$ pairs without fixed points ($-w_L\alpha=\alpha$ iff $\alpha$ lies in the black span), giving one fixed line per pair; and $24$ from the root spaces of $L$, fixed by step 3. Total $52$. The outer involutions of $E_6$ have fixed algebras of type $C_4$ (dimension $36$) or $F_4$ (dimension $52$), so $\theta_0$ has type E IV. On a fibre of characteristic $p\ne2$, $G^{\theta_0}$ is smooth, connected (Steinberg, Mem. AMS 80, Thm 8.1) and reductive of rank $4$ and dimension $52$, hence of type $F_4$.
Parabolic (proves 4). For $\alpha\in\Phi^+\smallsetminus\Phi_L$, $w_L$ does not change the $\alpha_1$- and $\alpha_6$-coefficients, so $\theta_0\alpha=-w_L\alpha$ has them negated, one of them non-zero, and is negative; hence $\theta_0(U_P)=U_P^-$ and $P\cap\theta_0(P)=L$. $T^{-\theta_0}$ has cocharacters the $+1$-eigenspace of $w_L$, of rank $2$, the rank of E IV.

Builds on

Proposition X4.4 · $E_6/F_4$, $p$ odd: the $\mathcal O$-model meets §1.7 and §8

Built on by

Proposition X4.12 · $E_6/F_4$, $p$ odd: $G(F)=K\,T(F)\,H(F)$ by Delorme–Sécherre
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
Summary 9.6 · Verdict after the repairs: every input held or repaired; $E_6/F_4$ still the one divergence

Sources

SGA 3 Exp. XXIII (isomorphism theorem for pinned groups; cited)
Steinberg, Mem. AMS 80 (1968), Thm 8.1 (cited)
families/e6-f4.html §§1–4

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 X4.11 · An involution of type E IV of \(E_6\) defined over \(\mathbb Z\)

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