Audit 6.3 · Audit of the Borel-orbit classification in every characteristic \(\ne2\) (claim A2-3)

1. Torus conjugacy (item 1). For a \(\theta\)-stable maximal torus \(T'\), \(\theta\) acts on \(\mathfrak t'\) with eigenspaces of dimensions \(a=\dim T'_+\) and \(6-a\) (this uses \(\frac12\) and smoothness of \(T'^\theta\)). Each pair \(\{\gamma,\theta\gamma\}\) of real or complex roots contributes one line to \(\mathfrak g^{\theta}\) and one to \(\mathfrak g^{-\theta}\); an imaginary root contributes to exactly one of them. With \(\dim\mathfrak g^{\pm\theta}=52,26\) (Audit 6.2 step 7):

The imaginary roots are \(\Phi(T')\) intersected with the \(+1\)-eigenspace of \(\theta\) on \(X^*(T')_{\mathbb Q}\), a simply laced root subsystem of rank \(\le a\); for \(a\le3\) it has at most 12 roots (\(A_3\); \(A_2{+}A_1\) has 8, \(A_1^3\) has 6). Then \(i_c\le12<26\le i_c-i_n\), absurd. So \(a=4\). Since \(Z_H(T^\theta)=T\cap H=T^\theta\simeq\mathbb G_m^4\) (Audit 6.2 steps 4–5), \(\operatorname{rank}H=4\), so \(T'_+\) is a maximal torus of the connected group \(H\), \(hT'_+h^{-1}=T^\theta\) for some \(h\in H(k)\), and \(hT'h^{-1}\subseteq Z_G(T^\theta)=T\). Verified; nothing depends on \(p\) beyond \(p\ne2\).

2. Every orbit meets \(N_G(T)x_0\) (item 2), with the repair. Let \(B_1=g^{-1}Bg\), \(B_2=\theta(B_1)\). Choose a maximal torus \(T_1\subseteq B_1\) and \(g'\) with \(B_2=g'B_1g'^{-1}\). By the Bruhat decomposition (Conrad, Thm. 1.4.12, over the field \(k\)) \(g'=bnb'\) with \(b,b'\in B_1(k)\), \(n\in N(T_1)(k)\); then \(bT_1b^{-1}=bnT_1n^{-1}b^{-1}\subseteq bnB_1n^{-1}b^{-1}=B_2\) and \(bT_1b^{-1}\subseteq B_1\). So \((B_1\cap B_2)^\circ\) is a \(\theta\)-stable connected solvable group containing a maximal torus of \(G\); Steinberg's Thm. 7.5 (an involution is semisimple when \(\operatorname{char}k\ne2\)) gives a \(\theta\)-stable maximal torus \(T'\subseteq B_1\). By step 1, \(T'=h^{-1}Th\); \(hB_1h^{-1}\supseteq T\), so \(hB_1h^{-1}=n'^{-1}Bn'\) with \(n'\in N_G(T)(k)\), whence \(gh^{-1}n'^{-1}\in N_G(B)=B\) and \(gx_0\in Bn'x_0\).
3. Distinct orbits (item 2). \(\theta\) acts on \(W=N_G(T)/T\) by \(w\mapsto w_Lww_L\) (\(\theta|_T=-w_L\) and \(-1\) is central in \(\mathrm{GL}(X^*(T))\)); hence \(\dot w\theta(\dot w)^{-1}\sigma_L\sigma_0\) represents \(ww_Lw^{-1}w_L\cdot w_Lw_0=xw_0\). \(C(B)=B^-=\sigma_0B\sigma_0^{-1}\) gives \(\theta(B)=\sigma_L\sigma_0B(\sigma_L\sigma_0)^{-1}\), so \(\tilde\tau(bz)=b\,\tilde\tau(z)\,\beta(b)^{-1}\) with \(\beta(b)=(\sigma_L\sigma_0)^{-1}\theta(b)\sigma_L\sigma_0\in B\), and \(\tilde\tau(By_x)\subseteq B\,\dot{(xw_0)}\,B\). Bruhat cells are disjoint, so distinct \(x\) give distinct orbits. Conversely lifts of a coset \(vC_W(w_L)\) give points of one \(T\)-orbit, because \(N_H(T)\to C_W(w_L)\) is onto (Audit 6.2 step 6). Verified.
4. Dimension and rank (item 3). \(B_x=(B\cap\theta_xB)^{\theta_x}=T^{\theta_x}\ltimes(U_{\Phi^+\cap\theta_x\Phi^+})^{\theta_x}\) (fixed points of a semidirect product with both factors stable), Lie algebra of dimension \(4+12+c_x\); \(\ell(x)=12+2c_x\) (Audit 6.1); the \(B\)-eigencharacters of \(k(B/B_x)\) are the characters of \(T\) trivial on \(T^{\theta_x}\), a group of rank \(6-4=2\). Verified.
5. Item 4 and the closing remark (Knop's action is \(x\mapsto s_\alpha xs_\alpha\) in both types, i.e. left multiplication on \(W/C_W(w_L)\)) are correct; the remark is not used.
6. Not verified locally. Steinberg's Thms. 7.5 and 8.1, CGP A.8.10 and Edixhoven 3.4 are not among the local sources; the statements recalled above are the ones used, and they hold over algebraically closed fields of every characteristic (8.1 needs \(G\) simply connected, true here).

Outcome: holds with the repair of step 2.