§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:05:22Z

Audit 5.10 · Audit of A3-6: the 45 maximal-rank Borel orbits and their types in char ≠ 2

proved audit: correct

Second audit: correct. My Weyl-group enumeration confirms |W.Phi_L| = 45, |N_W(Phi_L)| = 1152, W_X = <w_g1, w_g6> of order 6 (involutions of length 8 preserving Phi_L^+, acting by omega1 -> omega6-omega1, omega6 -> omega6), W_X.W_L = N_W(Phi_L), and that b + theta b is never a root. Audit 5.10's repair (equal orbits imply w(T_8) = w'(T_8), by conjugacy of maximal tori in the connected stabilizer and B -> B/U) is valid. The gap is milder than stated: Proposition 3.6 step 6 already shows s_a.Y_w = Y_{s_a w} with the left side depending only on the orbit, which gives well-definedness by induction and then the general converse from step 4. This changes no conclusion. Correct.

The description of the maximal-rank \(B\)-orbits as \(Bn_wx_0\), the group \(W(X)=N_W(\Phi_L)\) of order 1152, \(W_X\simeq S_3\), and the types \(G\)/\(U\) are correct in characteristic \(\ne2\). One implication, needed for the action \(w'\cdot Y_w=Y_{w'w}\) to be well defined, is proved only for \(w=1\); a short argument supplies it.

Audit 5.10.

Outcome: holds with one repair in (2). The body of Proposition 3.6 proves "\(Y_1=Y_w\Rightarrow w\in N_W(\Phi_L)\)" and "\(w^{-1}w'\in W(X)\Rightarrow Y_w=Y_{w'}\)". Part (4) also uses the general converse "\(Y_w=Y_{w'}\Rightarrow w^{-1}w'\in W(X)\)", without which \(w'\cdot Y_w:=Y_{w'w}\) need not be well defined. Repair (Lemma). If \(Bn_wx_0=Bn_{w'}x_0\), then \(w(T_8)=w'(T_8)\), so \(w^{-1}w'\in N_W(T_8)=N_W(\Phi_L)=W(X)\). Computations confirmed. The \(W\)-orbit of \(\Phi_L\) has \(45\) elements, so \(|N_W(\Phi_L)|=1152\). \(w_{\gamma_1}=s_1s_3s_4s_2s_5s_4s_3s_1\) and \(w_{\gamma_6}\) preserve \(\Phi_L^+\), are involutions of length \(8\), act on \(\mathcal X(X)\) by \(\omega_1\mapsto\omega_6-\omega_1,\ \omega_6\mapsto\omega_6\) (resp. symmetrically), and generate a group of order \(6\). For every root \(\beta\notin\Phi_L\), \(\beta+\theta\beta\) is not a root.

Proof of the repair.

  1. \(\operatorname{Stab}_B(n_wx_0)=n_w(B^w\cap H)n_w^{-1}\), where \(B^w\cap H=R^\theta\) and \(R=B^w\cap\theta(B^w)=T\ltimes R_u\). Then \(R^\theta=T^\theta\ltimes R_u^\theta=T_8\ltimes R_u^\theta\) is smooth (Proposition 3.6 step 2), and \(T_8\) is a maximal torus of it, because \(R_u^\theta\) is unipotent.
  2. Suppose \(n_{w'}x_0=bn_wx_0\) with \(b\in B\). Then \(\operatorname{Stab}_B(n_{w'}x_0)=b\operatorname{Stab}_B(n_wx_0)b^{-1}\). Both \(w'(T_8)=n_{w'}T_8n_{w'}^{-1}\) and \(bw(T_8)b^{-1}\) are maximal tori of this smooth group, so they are conjugate by some \(c\) in its identity component \(\subseteq B\).
  3. If \(\beta S\beta^{-1}\subseteq T\) with \(\beta\in B\) and \(S\subseteq T\), then \(\beta S\beta^{-1}=S\): project \(B\to B/U=T\), which is the identity on \(T\) and kills conjugation. Hence \(w'(T_8)=w(T_8)\), so \(w^{-1}w'\) normalizes \(X_*(T_8)_{\mathbb Q}=\operatorname{span}\Phi_L^\vee\), i.e. preserves \(\Phi_L\). \(\square\)

Re-derivation of the rest.

  1. (1) Rank. \(B'\cap\theta B'\) is connected solvable and \(\theta\)-stable, and \(\theta\) (order \(2\), prime to \(p\)) fixes a maximal torus of it (Steinberg 1968, Thm 7.5; characteristic-free). The characters trivial on the scheme \(T'^\theta=\ker(\theta-1)\) are \((1-\theta^*)X^*(T')\), of rank \(\dim(T'^{-\theta})^\circ\le2\) by Proposition 3.4(a).
  2. (2) Orbits. For rank \(2\), \((T'^{-\theta})^\circ=hSh^{-1}\) (Proposition 3.4(c)). Inside \(L=S\cdot\mathrm{Spin}_8\) the maximal tori are \(\mathrm{Spin}_8\)-conjugate, and \(gh'n_w^{-1}\in N_G(B)=B\). So the rank-\(2\) orbits are exactly the \(Y_w\).
  3. (3) Types. The stabilizer of \([x_0]\) in \(P_\beta/R(P_\beta)\) is the image of \(\operatorname{Stab}_{P_\beta}(x_0)=R^\theta\), because \(R(P_\beta)\) is normal. For \(\beta\in\Phi_L\), \(U_{\pm\beta}\subseteq\mathrm{Spin}_8\subseteq H\), giving type \(G\). For \(\beta\notin\Phi_L\), \(\theta\beta\ne\pm\beta\), and exactly one of \(\pm\beta\) lies in \(\Psi\) (since \(\Psi\) is \(\theta\)-stable). \(T_8\) maps onto \(\bar T\) because \(\beta|_{T_8}\ne1\). The elements \(u\theta(u)\), \(u\in U_{\varepsilon\beta}\), are \(\theta\)-fixed (\(\varepsilon(\beta+\theta\beta)\) is not a root, recomputed) and map onto \(\bar U_{\varepsilon\beta}\), because \(U_{\varepsilon\theta\beta}\subseteq U_{P_\beta}\). So the stabilizer is a Borel: type \(U\) with \(S=T\). The stabilizers are smooth, so no rank-one quotient is inseparable (checklist item 5). No classification of subgroups of \(\mathrm{PGL}_2\) in characteristic \(p\) is used.
  4. (4) Action. With the repair, \(w'\cdot Y_w:=Y_{w'w}\) is well defined. It agrees with the geometric reflections of p. 9: in type \(G\), \(Y_wP_\alpha=Y_w\ni\dot s_\alpha n_wx_0\); in type \(U\), \(Y_wP_\alpha=Y_w\sqcup Y_{s_\alpha w}\). It is transitive, and the stabilizer of \(Y_1\) is \(W(X)\). This construction replaces Knop's characteristic-\(0\) theorem for this \(X\) (checklist item 1).

Builds on

  • Proposition 3.6 · Char \(\ne2\): 45 maximal-rank Borel orbits \(Bn_wx_0\), all pairs of type \(G\) or \(U\)

Built on by

  • Summary 5.13 · Verdict of audit A5: transfer route A4-2…A4-9 and char-p hypotheses A3-1…A3-10

Sources