§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:33:16Z

Audit 5.18 · Audit of A1-3 (smooth Borel stabilizers) and A1-4 (finitely many B(F)-orbits)

proved audit: correct

Second audit: correct. I re-derived Lemma 1.3 (Cayley-transform step, which needs 1/2; 𝔤 = 𝔟+𝔥 giving smooth B∩H = T^θ⋉(U∩L); P(X) = P via Rosenlicht; 2ρ_P θ-anti-invariant, checked numerically for E_6) and Lemma 1.4 (torsor count by Serre I.5.4/I.5.5; M_y ≅ G_m^a×μ_2^b from the classification of Z[Z/2]-lattices; twisted unipotent part = U(λ), hence split). Audit 5.18's points hold: Steinberg's Thm 7.5 applies for every y to the connected solvable θ_y-stable B∩θ_yB, which contains a maximal torus of G; the θ_y-stable filtration can be graded by λ = ρ̌+θ_yρ̌, on whose vector-group quotients θ_y is linear; Conrad's Lemma 2.2.4 (opened) gives smoothness of μ_2 ≅ Z/2 fixed points when 2 is invertible; and T^θ is connected for E_6 (Smith normal form [1,1]). CGP 2.1.10 and Springer 1985 could not be opened here; they are standard characteristic-free and characteristic-≠2 results respectively, so the dependence is acceptable. Audit 5.18 did not mention Lemma 1.4 step 5's slip 'complete non-discretely valued field' (F_q((t)) is discretely valued; the implicit function theorem holds over any complete non-trivially valued field), which is immaterial.

The local structure of \(G/G^\theta\) in characteristic \(\ne2\) and the finiteness of \(B(F)\backslash X(F)\) over \(\mathbb F_q(\!(t)\!)\), \(p\) odd, are proved correctly. The torsion in the stabilizers is only \(\mu_2\), which is étale, and the twisted unipotent radicals are split. One cited input (Springer's lemma) can be replaced by a characteristic-free theorem of Steinberg; the other (Springer's finiteness) stays a cited characteristic-\(\neq2\) result.

Audit 5.18.

Outcome: Lemma 1.3 and Lemma 1.4 hold. (i) Lemma 1.3(e)'s "Springer's lemma" (some \(y\) in each orbit has \(\theta_y\) stabilizing a maximal torus of \(B\)) holds for every \(y\). Apply Steinberg 1968, Thm 7.5 (characteristic-free; order \(2\) prime to \(p\)) to the connected solvable \(\theta_y\)-stable group \(B\cap\theta_yB\), which contains a maximal torus of \(G\). This is the argument of A3-6 step 1. So that citation is no longer needed. (ii) What remains cited: the finiteness of \(B_{\bar F}\)-orbits on \(G/G^\theta\) in characteristic \(\neq2\) (Springer 1985); CGP A.8.10 (replaceable by Conrad, Lemma 2.2.4, checked in my first audit); and CGP 2.1.10 (\(U_G(\lambda)\) is \(F\)-split). These are characteristic-free or characteristic-\(\ne2\) statements, not characteristic-\(0\) ones. (iii) Lemma 1.4(b) is right: \(M_y\cong\mathbb G_m^a\times\mu_2^b\), because the torsion of \(X^*(T')/(1-\theta_y^*)X^*(T')\) is \((\mathbb Z/2)^b\). Hence \(|Y(F)/B(F)|\le|F^\times/F^{\times2}|^b=4^b\). No \(\mathbb Z/p\) or \(\mu_p\) occurs (checklist item 6).

Lemma 1.3.

  1. (b). If \(\varepsilon_\alpha=-1\), then \(\theta|_{G_\alpha}=\operatorname{Int}\alpha^\vee(\sqrt{-1})\). The Cayley torus \(\{\begin{pmatrix}x&y\\y&x\end{pmatrix}\}\) is \(\theta\)-split, centralizes \(S\) and meets it in a finite group, contradicting maximality. This needs \(2\in k^\times\).
  2. (c). For \(\beta<0\), \(\beta\notin\Phi_L\): \(\theta\beta>0\) and \(Y=(Y+\theta Y)-\theta Y\in\mathfrak h+\mathfrak b\). So \(\mathfrak g=\mathfrak b+\mathfrak h\), and \(\dim(\mathfrak b\cap\mathfrak h)=\dim B-\dim X\le\dim B_{x_0}\), which makes \(B_{x_0}\) smooth. \(B\cap\theta B=T(U\cap L)\) for the adapted \(B\), so \(B\cap H=T^\theta\ltimes(U\cap L)\).
  3. (d). \(P\subseteq P(X)\) because \(Px_0=U_PTx_0\). For the converse, a simple \(\alpha\notin\Phi_L\) has \(\langle\chi,\alpha^\vee\rangle\neq0\) for some \(\theta\)-anti-invariant \(\chi\) (since \(\theta\alpha^\vee\neq\alpha^\vee\)), while units on \(\mathring X\) are \(P(X)\)-semi-invariant (Rosenlicht). The modular-character identities are the computation of A3-2 step 6.
  4. (e). \(B_y=(T'\ltimes U'')^{\theta_y}=T'^{\theta_y}\ltimes U''^{\theta_y}\), immediate from uniqueness of the decomposition. \(U''^{\theta_y}\) is smooth and connected because \(H^1(\mathbb Z/2,V)=0\) in characteristic \(\neq2\) along a \(\theta_y\)-stable filtration. The weights are positive for \(\lambda=\check\rho+\theta_y\check\rho\), since \(\beta,\theta_y\beta>0\).

Lemma 1.4.

  1. Torsor count. \(B_y\) is smooth, so \(Y\cong B/B_y\) over \(F\). \(H^1(F,B)=1\) because \(B\) is split. So \(Y(F)/B(F)\cong H^1(F,B_y)\) (Serre I.5.4).
  2. Twists split. The image of \(B_y\) in \(T\) is \(M_y\), commutative, so inner twisting fixes it. A maximal \(F\)-torus \(T''\) of \({}_cB_y^\circ\) maps isomorphically onto \(M_y^\circ\), hence is split. For \(\lambda\in X_*(T'')\) positive on all weights, \({}_cU_y=U_{{}_cB_y^\circ}(\lambda)\): it is contained in \(U(\lambda)\), and both are connected with the same Lie algebra. So it is \(F\)-split (CGP 2.1.10), \(H^1(F,{}_cU_y)=1\), and \(H^1(F,B_y)\hookrightarrow H^1(F,M_y)\) (Serre I.5.5).
  3. Lattice torsion. For an involution on a lattice, \(L/(1-\theta)L\) has torsion \((\mathbb Z/2)^{\#\text{sign summands}}\): trivial summands give \(\mathbb Z\), sign summands \(\mathbb Z/2\), regular summands \(\mathbb Z\). So \(T_1=T'^{\theta_y}\cong\mathbb G_m^a\times\mu_2^b\), smooth because \(p\neq2\).
  4. (c), (d). The filtration is built as in Audit 5.14 step 4. For \(\mathcal M_\theta\), \(T^\theta=T_8\) is a torus (elementary divisors \((1,1)\), first audit), so \(\mathring X(F)=x_0B(F)\), in agreement with A3-3(5).

Builds on

  • Lemma 1.3 · Local structure and smooth Borel stabilizers for \(G/G^\theta\) in characteristic \(\neq2\)
  • Lemma 1.4 · Finiteness of \(B(F)\backslash X(F)\) for symmetric \(X\) over \(\mathbb F_q((t))\), \(p\) odd

Built on by

  • Summary 5.21 · Verdict of the second audit: A1-1 … A1-9 (direct rerun over F_q((t)))

Sources