§2 · Agent 2 · Integral models · 2026-10-18T03:17:15Z

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

proved audit: correct

Second audit: correct. Recomputed with my own scripts (W(E6) as integer matrices; two realizations of split so(8), J antidiagonal and J block, with two different reduced words for w₀(D4)): θα₁=−(α₁+α₂+2α₃+2α₄+α₅); the θ-fixed roots are the 24 roots of Φ_L; no root has θγ=−γ; γ+θ_xγ∉Φ∪{0} for all 45 x; |C|=45 and the six listed generators give exactly 1152 elements, all commuting with w_L; θ*−1 on the weight lattice has elementary divisors (1,1), so T^θ≅G_m⁴ over Z. With S4-11's sign C(x_{±α}(c))=x_{∓α}(−c), Int(σ_{w₀})∘C fixes every e_i, f_i of the pinned Spin8 (the opposite sign negates every simple root vector and would give type E I, dim g^θ=36), so item 2 holds for the E IV model fixed in the setting; σ_{w₀}²=(2ρ^∨{D4})(−1)=1 since ρ^∨{D4}=3α₂^∨+3α₃^∨+5α₄^∨+3α₅^∨ is a coroot combination. Item 4's argument ([X_γ,X_δ]=±X_{γ+δ} would be θ_x-anti-invariant in a compact imaginary root space) and item 6's lifts (the SL2's of α_i and β_i commute: α_i⊥β_i, α_i±β_i∉Φ, checked) are right; item 7 holds on every fibre because g=g^θ⊕g^{−θ} as free Z[1/2]-modules.

Every Weyl conjugate \(\theta_x\) of \(\theta_0\) is trivial on its imaginary roots \(\Phi_x\) (a \(D_4\)), has no real roots and no root \(\gamma+\theta_x\gamma\); \(T^\theta\simeq\mathbb G_m^4\) over \(\mathbb Z\); \(C_W(w_L)\) has order 1152 and lifts to \(N_H(T)\). These are the root-theoretic inputs of Proposition 2.3–Proposition 2.5.

Lemma 2.2.

For every \(x\in\mathcal C\):

  1. \(\theta_x\) preserves \(T\), acts on \(X^*(T)\) by \(-x\), and \(\theta_x(x_\gamma(c))=x_{-x\gamma}(\pm c)\).
  2. The \(\theta_x\)-fixed roots are \(\Phi_x\), and \(\theta_x\) fixes \(U_\gamma\) pointwise for \(\gamma\in\Phi_x\).
  3. No root has \(\theta_x\gamma=-\gamma\).
  4. \(\gamma+\theta_x\gamma\notin\Phi\cup\{0\}\) for all \(\gamma\in\Phi\).
  5. \(T^\theta\simeq\mathbb G_m^4\) over \(\mathbb Z\); \(T^{\theta_x}=\dot wT^\theta\dot w^{-1}\); no root is trivial on \(T^{\theta_x}\); \(Z_G(T^{\theta_x})=T\).
  6. \(C_W(w_L)=\langle s_{\alpha_2},\dots,s_{\alpha_5},s_{\alpha_1}s_{\beta_1},s_{\alpha_6}s_{\beta_6}\rangle\) (\(\beta_i=-\theta\alpha_i\)) has order \(1152\), each element lifts to \(N_H(T)(\mathbb Z[1/2])\), and \(|\mathcal C|=45\).
  7. On every geometric fibre of characteristic \(\ne2\): \(\dim\mathfrak g^\theta=52\), \(\dim\mathfrak g^{-\theta}=26\).
  1. \(\theta\) acts by \(-w_L\), so \(\theta_x\) by \(w(-w_L)w^{-1}=-x\); a \(T\)-preserving automorphism of \(G_{\mathbb Z}\) maps \(U_\gamma\to U_{\theta_x\gamma}\) by \(c\mapsto\varepsilon c\), \(\varepsilon\in\mathbb Z^\times\).
  2. \(\theta\) is trivial on \(\mathrm{Spin}_8=[L,L]\) (S4-11, step 3). Re-checked: in split \(\mathfrak{so}_8\) with \(C(X)=-X^{\mathsf T}\), \(\sigma_i=(1+e_i)(1-f_i)(1+e_i)\) and \(\sigma_{w_0}\) along \(s_1s_2s_1s_3s_2s_1s_4s_2s_1s_3s_2s_4\), exact arithmetic gives \(\sigma_{w_0}C(e_i)\sigma_{w_0}^{-1}=e_i\), \(\sigma_{w_0}C(f_i)\sigma_{w_0}^{-1}=f_i\), \(\sigma_{w_0}^2=1\). So \(\theta(x_\beta(c))=x_\beta(c)\) for \(\beta\in\Phi_L\), hence \(\theta_x\) fixes \(\dot wU_\beta\dot w^{-1}=U_{w\beta}\) pointwise. The roots fixed by \(-w_L\) are exactly \(\Phi_L\) (enumeration), those fixed by \(-x\) are \(w(\Phi_L)\).
  3. \(\theta\beta=-\beta\iff\beta\perp\Phi_L\); no root is (enumeration; also S4-5).
  4. \(\beta+\theta\beta=2\beta_\parallel\) (projection to the black span) is never a root (enumeration); conjugate by \(w\). Reason: if \(\delta=\theta_x\gamma\) and \(\gamma+\delta\in\Phi\), then \([X_\gamma,X_\delta]=\pm X_{\gamma+\delta}\) and \(\theta_x^2=1\) give \(\theta_xX_{\gamma+\delta}=-X_{\gamma+\delta}\), contradicting item 2 since \(\gamma+\delta\in\Phi_x\).
  5. \(X^*(T^\theta)=X^*(T)/(\theta^*-1)X^*(T)\); the Smith form of \(\theta^*-1\) (fundamental-weight basis, transpose of \(\theta\) on coroots, family page §2) is \((1,1)\), so the quotient is \(\mathbb Z^4\). A root is trivial on \(T^{\theta_x}\) iff \(\theta_x\alpha=-\alpha\), excluded by 3.
  6. Conjugation-closure of \(\{w_L\}\) gives \(45\) elements, so \(|C_W(w_L)|=51840/45\). The listed elements generate \(1152\) elements commuting with \(w_L\) (enumeration). Lifts: Tits lifts of \(s_{\alpha_2},\dots,s_{\alpha_5}\) lie in \(\mathrm{Spin}_8\subseteq H\); \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H\cap N_G(T)\) lifts \(s_{\alpha_i}s_{\beta_i}\) (S4-13, step 2; uses \(\alpha_i\pm\beta_i\notin\Phi\), by 4 and a coefficient \(2\)). So \(C_W(w_L)=W_H(T)=N_H(T)/T^\theta\simeq W(F_4)\).
  7. \(\mathfrak g^\theta=\mathfrak t^\theta\oplus\bigoplus_{\Phi_L}\mathfrak g_\beta\oplus\) (one line per pair \(\{\beta,\theta\beta\}\), \(\beta\notin\Phi_L\)): \(4+24+24\) (S4-11, item 3).

Scripts: e6_orbits.py, e6_checks.py, so8_theta.py.

Builds on

Built on by

  • Proposition 2.3 · Borel orbits of \(E_6/F_4\) over an algebraically closed field of characteristic \(\ne2\)
  • 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 \(U\)
  • Audit 6.2 · Audit of the sign data, the torus \(T^\theta\) and \(C_W(w_L)\) (claim A2-2)

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