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

1. \(\theta\)-stable tori in Borels. For a Borel \(B'\), \(R=B'\cap\theta(B')\) is connected solvable, \(\theta\)-stable and contains a maximal torus of \(G\); \(\theta\) has order 2, prime to \(\mathrm{char}\,k\), so it is a semisimple automorphism and fixes a maximal torus of \(R\) (Steinberg, Mem. AMS 80 (1968), Thm 7.5; characteristic-free, cited).
2. (1). \(b\in B_g\) fixes \(x_0\) iff \(\theta(b)=b\), so \(B_g\cap H=R^\theta\) scheme-theoretically; \(R=T'\ltimes R_u\) with \(\theta\) preserving both factors gives \(R^\theta=T'^\theta\ltimes R_u^\theta\), smooth (Proposition 3.2 step 2). Rational \(B_g\)-eigenfunctions on \(B_g/R^\theta\) have as weights the characters of \(T'\) trivial on \(T'^\theta=\ker(\theta-1)\), i.e. \((1-\theta^*)X^*(T')\), of rank \(\dim T'^{-\theta}\le2\) (Proposition 3.4(a)).
3. (2), orbits. If \(\mathrm{rk}\,Y=2\), \((T'^{-\theta})^\circ\) is a maximal \(\theta\)-split torus, so it equals \(hSh^{-1}\), \(h\in H\) (Proposition 3.4(c)); \(h^{-1}T'h\subseteq Z_G(S)=L\) is a maximal torus \(S\cdot T_8''\) of \(L\), and \(T_8''=mT_8m^{-1}\) with \(m\in\mathrm{Spin}_8\subseteq H\). So \(T'=h'Th'^{-1}\), \(h'\in H\), \(h'^{-1}B_gh'=n_w^{-1}Bn_w\) for some \(w\), and \(Y=gB_gx_0=(gh'n_w^{-1})Bn_wx_0\) with \(gh'n_w^{-1}\in N_G(B)=B\): \(Y=Y_w\). Conversely \(Y_w=n_w(n_w^{-1}Bn_w)x_0\) with \(T\) \(\theta\)-stable in \(n_w^{-1}Bn_w\), so \(\mathrm{rk}\,Y_w=\dim T^{-\theta}=2\).
4. (2), the group. \(\mathrm{Stab}_T(n_wx_0)=w(T^\theta)\), while every \(y=uax_0\in\mathring X\simeq U_P\times A_X\) has \(\mathrm{Stab}_T(y)\subseteq T^\theta\) (Proposition 3.2(4)). So \(n_wx_0\in\mathring X\) forces \(w(T_8)=T_8\), i.e. \(w\in N_W(\Phi_L)\); this group has order \(51840/45=1152\) (the \(W\)-orbit of the subsystem \(\Phi_L\) has 45 elements; computed). Conversely \(W_L\subseteq W(X)\) (Tits lifts of \(s_2,\dots,s_5\) lie in \(\mathrm{Spin}_8\subseteq H\)) and \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H(\mathbb Z)\) lifts \(s_{\alpha_i}s_{\beta_i}\) (S4-13, an identity in \(G(\mathbb Z)\)). Computed: \(\langle s_{\alpha_1}s_{\beta_1},s_{\alpha_6}s_{\beta_6}\rangle\) has order 6 and acts on \(\mathcal X(X)\) by \(\omega_1\mapsto\omega_6-\omega_1,\ \omega_6\mapsto\omega_6\) and \(\omega_1\mapsto\omega_1,\ \omega_6\mapsto\omega_1-\omega_6\), i.e. by \(s_{\gamma_1},s_{\gamma_6}\); \(W_L\) acts trivially, so the intersection is trivial and this complement times \(W_L\) (order 1152) is all of \(N_W(\Phi_L)\). As \(W_L\) is simply transitive on the positive systems of \(\Phi_L\), each coset of \(W_L\) in \(N_W(\Phi_L)\) contains exactly one element preserving \(\Phi_L^+\) (its element of minimal length); these form the canonical complement \(W_X\). Computed: \(w_{\gamma_1}=s_1s_3s_4s_2s_5s_4s_3s_1\) and \(w_{\gamma_6}\) preserve \(\Phi_L^+\), are involutions of length 8, act as \(s_{\gamma_1},s_{\gamma_6}\), and generate a group of order 6 (while \(s_{\alpha_1}s_{\beta_1}\), of length 14, does not preserve \(\Phi_L^+\)).
5. (3). Conjugating by \(n_w^{-1}\), \((Y_w,\alpha)\) becomes \((B^wx_0,\beta)\), \(B^w=n_w^{-1}Bn_w\), \(\beta=w^{-1}\alpha\) simple for \(B^w\). Let \(P_\beta\) be the minimal parabolic of \(B^w\) for \(\beta\) and \(R=P_\beta\cap\theta(P_\beta)\), smooth connected and generated by \(T\) and the \(U_\gamma\), \(\gamma\in\Psi=(\Phi_w^+\cup\{-\beta\})\cap\theta(\Phi_w^+\cup\{-\beta\})\). Then \(\mathrm{Stab}_{P_\beta}(x_0)=R^\theta\) is smooth and \(\hat S\) is its image in \(P_\beta/R(P_\beta)\simeq\mathrm{PGL}_2\), smooth. (a) \(\beta\in\Phi_L\): \(U_{\pm\beta}\subseteq\mathrm{Spin}_8\subseteq H\), so \(\hat S=\mathrm{PGL}_2\). (b) \(\beta\notin\Phi_L\): then \(\theta\beta\ne\pm\beta\) (computed: the \(\theta\)-fixed roots are \(\Phi_L\), and no root has \(\theta\beta=-\beta\)); exactly one of \(\pm\beta\), say \(\varepsilon\beta\), lies in \(\Psi\) (\(\beta\in\Psi\iff\theta\beta\in\Phi_w^+\)), so the image of \(R\) is the Borel \(\bar T\ltimes\bar U_{\varepsilon\beta}\) and \(\hat S\) lies in it; \(\beta|_{T^\theta}\ne1\) (otherwise \(\theta\beta=-\beta\)), so \(\hat S\supseteq\bar T\); \(\varepsilon\theta\beta\in\Psi\) and \(\beta+\theta\beta\) is never a root (computed), so \(\{u_{\varepsilon\beta}(c)\theta(u_{\varepsilon\beta}(c))\}\subseteq R^\theta\) maps onto \(\bar U_{\varepsilon\beta}\). Thus \(\hat S=\bar T\ltimes\bar U_{\varepsilon\beta}\). No classification of subgroups of \(\mathrm{PGL}_2\) in characteristic \(p\) is needed.
6. (4). In type \(G\), \(Y_wP_\alpha=Y_w\ni\dot s_\alpha n_wx_0\); in type \(U\), \(Y_wP_\alpha\) is the union of the two distinct orbits \(Y_w\) and \(Y_{s_\alpha w}\) (\(P_\alpha=B\cup Bs_\alpha B\); they map to the two \(\bar B\)-orbits of \(\mathbb P^1\)), both of maximal rank. So \(s_\alpha\cdot Y_w=Y_{s_\alpha w}\) is the reflection Sakellaridis describes, depends only on the orbit, and extends to the action \(w'\cdot Y_w=Y_{w'w}\); the stabilizer of \(Y_1\) is \(\{w:n_wx_0\in\mathring X\}=W(X)\) (step 4).

The types agree with S4-5(3); all stabilizers are smooth, so no rank-one quotient is inseparable (checklist item 5).