§2 · Agent 2 · Integral models · 2026-10-18T03:17:50Z

Proposition 2.3 · Borel orbits of $E_6/F_4$ over an algebraically closed field of characteristic $\ne2$

proved audit: correct ★ key step

Second audit: correct. Re-derived: 52=a+(r+c)/2+i_c and 26=(6−a)+(r+c)/2+i_n give i_c−i_n=32−2a; imaginary roots lie in the a-dimensional +1-eigenspace of θ on X*(T′)_Q and a root subsystem of E6 in a subspace of dimension ≤3 has at most 12 roots (enumerated), so a=4 and every θ-stable maximal torus is H(k)-conjugate to T (conjugacy inside H° suffices, so Steinberg's Thm 8.1 is not even needed). Steinberg, Mem. AMS 80, Thm 7.5 is the invariant Borel/maximal-torus theorem for a semisimple automorphism over an algebraically closed field (number confirmed through a citing paper, Chatterjee, Math. Z. 2011, ref. [21]; the memoir is not available locally); the τ̃-argument (θ(B)=σ_Lσ₀B(σ_Lσ₀)⁻¹, τ̃(ẇx₀) represents x·w₀) gives distinctness, and dim B_x=16+c_x, ℓ(x)=12+2c_x, rank 2 were recomputed. Independent second route in characteristic p: Σ_x|Y_x(F_q)|=Σ_x(q−1)²q^{24−c_x}=q¹²(q⁵−1)(q⁹−1)=|E6(q)|/|F4(q)| identically, so the 45 orbit schemes of Lemma 2.4 exhaust X over every F̄_q, p odd, p=3 included. Nit only: 'B″ and θB″ share a maximal torus' is the standard one-line corollary of the Bruhat decomposition that the step cites (Conrad Thm 1.4.12 is the decomposition itself); the in-run grade 'minor' (Audit 6.3) differs from mine only in threshold.

Over any algebraically closed $k$, $\operatorname{char}k\ne2$ ($\overline F$ and $\overline\kappa$, mixed or equal characteristic, $p=3$ included), the $B$-orbits on $X_k$ are exactly the $45$ orbits $By_x$, all of maximal rank. The proof uses $\theta$-stable tori and Bruhat cells, not Knop.

Proposition 2.3.

Let $k$ be algebraically closed, $\operatorname{char}k\ne2$.

Every $\theta$-stable maximal torus of $G_k$ is $H(k)$-conjugate to $T$.
$X(k)=\bigsqcup_{x\in\mathcal C}B(k)y_x$ (45 orbits, pairwise distinct).
$By_x\simeq B/B_x$ has dimension $26-c_x=32-\tfrac12\ell(x)$, $c_x=\#\{\text{pairs }\{\gamma,\theta_x\gamma\}\subseteq\Phi^+,\gamma\ne\theta_x\gamma\}$, and rank $2=\operatorname{rank}X$.
$\dot s_\alpha y_x\in By_{s_\alpha xs_\alpha}$ for $\alpha\in\Delta$.

$H_k$ is smooth (CGP A.8.10; Edixhoven 3.4), connected (Steinberg, Mem. AMS 80, Thm. 8.1), of dimension $52$; $Z_H(T^\theta)=T\cap H=T^\theta$ (Lemma 2.2, 5), so $T^\theta$ is a maximal torus and $\operatorname{rank}H=4$.
Item 1. Let $T'$ be $\theta$-stable, $T'_\pm$ the identity components of $\{\theta t=t^{\pm1}\}$, $a=\dim T'_+\le4$. Let $r,c$ count real and complex $T'$-roots, $i_c,i_n$ imaginary roots with $\theta=\pm1$ on the root space. By Lemma 2.2, 7:

$$52=a+\tfrac{r+c}2+i_c,\qquad26=(6-a)+\tfrac{r+c}2+i_n,\qquad i_c-i_n=32-2a.$$

Imaginary roots form a simply laced system of rank $\le a$; if $a\le3$ it has $\le12$ roots ($A_3$), so $i_c\le12<26\le i_c-i_n$: impossible. So $a=4$, $hT'_+h^{-1}=T^\theta$ ($h\in H(k)$), and $hT'h^{-1}\subseteq Z_G(T^\theta)=T$.

Every orbit meets $N_G(T)x_0$. For $z=gx_0$, $B''=g^{-1}Bg$ and $\theta B''$ share a maximal torus (Bruhat, Conrad Thm. 1.4.12); $\theta$ (order 2, semisimple) fixes a maximal torus $T'$ of $(B''\cap\theta B'')^\circ$ (Steinberg, Thm. 7.5). By item 1, $T'=h^{-1}Th$, so $hB''h^{-1}=n^{-1}Bn$, $n\in N_G(T)$; then $gh^{-1}n^{-1}\in B$ and $z\in Bnx_0=B\dot vx_0$.
Same orbit iff same $x$. If $v'=vu$, $u\in C_W(w_L)$, lift $u$ to $h\in N_H(T)$ (Lemma 2.2, 6): $\dot v'x_0\in\dot vhTx_0=T\dot vx_0$. Conversely, with $\sigma_0$ lifting $w_0$ and $\tilde\tau(z)=\tau(z)\sigma_L\sigma_0$: $\theta(B)=\sigma_L\sigma_0B(\sigma_L\sigma_0)^{-1}$, so $\tilde\tau(bz)=b\tilde\tau(z)\beta(b)^{-1}$ with $\beta(b)\in B$, and $\tilde\tau(By_x)\subseteq B\dot w\theta(\dot w)^{-1}\sigma_L\sigma_0B$, the Bruhat cell of $w(w_Lww_L)^{-1}w_Lw_0=xw_0$. Cells are disjoint.
Item 3. $B_x=(B\cap\theta_xB)^{\theta_x}=T^{\theta_x}\ltimes(U_{\Phi^+\cap\theta_x\Phi^+})^{\theta_x}$ (over $\mathbb Z$: Lemma 2.4, step 2), of dimension $4+12+c_x$ by Lemma 2.2, 2–3. The $B$-eigencharacters of $k(By_x)$ are the characters of $T$ trivial on $T^{\theta_x}$: rank $2$. $\ell(x)$ counts the $12$ positive roots of $\Phi_x$ and the $2c_x$ positive $\gamma$ with $-x\gamma>0$.
Item 4. $\dot s_\alpha\dot w$ lifts $s_\alpha w$ and $(s_\alpha w)w_L(s_\alpha w)^{-1}=s_\alpha xs_\alpha$.

Remark: Knop's action is left multiplication on $W/C_W(w_L)$; the stabilizer of the open orbit is $C_W(w_L)=W_X\ltimes W_L$, so the count $45$ of S4-5(3) holds in every characteristic $\ne2$.

Builds on

Lemma 2.2 · The involutions $\theta_x$: compact imaginary roots, no real roots

Built on by

Lemma 2.4 · Smooth stabilizers; orbit immersions $B/B_x\hookrightarrow X$ over $\mathbb Z[1/2]$
Proposition 2.5 · Rank-one quotients: a point (type $G$) or $\mathbb P^1_{\mathcal O}$ (type $U$)
Computation 2.6 · The $45\times6$ pairs: 90 of type $G$, 180 of type $U$
Lemma 2.7 · Orbit closures over $\mathcal O$ have the expected special fibres
Theorem 2.8 · Theorem: Axiom 2.2.1 holds for $\mathcal M_\theta$ over every DVR with $2$ invertible
Audit 6.3 · Audit of the Borel-orbit classification in every characteristic $\ne2$ (claim A2-3)

Sources

Steinberg, Mem. AMS 80 (1968), Thms. 7.5, 8.1 (cited, not checked against a local source)
conrad-reductive-group-schemes.pdf, Thm. 1.4.12
CGP Prop. A.8.10; Edixhoven 1992 Prop. 3.4 (cited)
scripts e6_orbits.py

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Proposition 2.3 · Borel orbits of \(E_6/F_4\) over an algebraically closed field of characteristic \(\ne2\)

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