Audit 6.7 · Audit of Sakellaridis's scheme \(Y\) and the complement clause (claims A2-1, A2-8)

1. Text. Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. (p. 9): "For every \(B_k\)-orbit \(Y_k\subset X_k\) of maximal rank and every simple root \(\alpha\), the o-scheme \(YP_\alpha/R(P_\alpha)\) is isomorphic to one of: \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\), \(SU\backslash\mathrm{PGL}_2\) (where \(S\subset N(U)\)), \(T\backslash\mathrm{PGL}_2\) … or \(N(T)\backslash\mathrm{PGL}_2\). Moreover, the complement of \(YP_\alpha\) is the closure of the complement of \((YP_\alpha)_k\)." Then: "Y is the complement of the closure of the complement of \(Y_k\) in its closure."
2. Which complements. (a) If the inner complement were taken in the \(\mathcal O\)-closure \(\overline Y\), it would contain the special fibre of \(\overline Y\), and \(Y\) would be \(Y_F\) with empty special fibre, contradicting Sakellaridis's next sentence (Y smooth and surjective over o). (b) If the outer complement were taken in \(X\), \(Y\) would be open in \(X\). So \(Y^{\rm can}\) is as in Strategy 2.1. (c) For the clause: if \((YP_\alpha)_F\) is not open it does not contain \(\mathring X_F\) (a \(B\)-stable set meeting \(\mathring X_F\) contains it, and \(P_\alpha\mathring X_F\) is open), so \(X_F\smallsetminus(YP_\alpha)_F\supseteq\mathring X_F\) has closure \(X\), and the \(X\)-reading would force \(YP_\alpha=\emptyset\). The closure reading is the only satisfiable one. Strategy 2.1 reads it so.
3. \(Y^{\rm can}=Y_x\). \(Y_x\) is locally closed and \(\mathcal O\)-flat, so its topological closure is \(\overline{Y_{x,F}}=Z_x\) and it is open in \(Z_x\). \(\partial_F=\bigcup_{v\in\Sigma_x\smallsetminus\{x\}}Y_{v,F}\) has closure \(\bigcup Z_v\). By Lemma 2.7 item 3 (audited in Audit 6.6) \(Z_v\cap X_\kappa=\bigcup_{\Sigma_v}Y_{u,\kappa}\), and \(\Sigma_v\subseteq\Sigma_x\smallsetminus\{x\}\) (transitivity, Audit 6.1 item 6; \(x\notin\Sigma_v\) by dimension). So \(Y^{\rm can}\cap X_\kappa=\overline{Y_{x,\kappa}}\smallsetminus\bigcup_{\Sigma_x\smallsetminus\{x\}}Y_{v,\kappa}=Y_{x,\kappa}\), and \(Y^{\rm can}_F=Y_{x,F}\). Both are reduced open subschemes of the reduced \(Z_x\) with the same support, hence equal.
4. Complement clause. Type G: \(P_\alpha Y_x=Y_x\) (Audit 6.5 step 2), and the clause is item 2 for \(x\). Type U: with \(u\) the open member of \(\{x,s_\alpha xs_\alpha\}\) and \(u'=s_\alpha us_\alpha\), \(\mathcal P=P_\alpha Y_x\) is smooth, flat and locally closed, so open in its closure \(Z_u\); \(E=\Sigma_u\smallsetminus\{u,u'\}\) is stable under \(v\mapsto\Sigma_v\) (if \(v\in E\) and \(u'\in\Sigma_v\) then \(\dim Y_v\ge\dim Y_{u'}=\dim Y_u-1\), forcing \(v=u'\)); so, as in step 3, \((P_\alpha Y_{x,F})^{\rm can}\) has special fibre \(Y_{u,\kappa}\sqcup Y_{u',\kappa}=|\mathcal P_\kappa|\) and equals \(\mathcal P\). The quotients are those of Audit 6.5.
5. Open saturations, \(X\)-reading. \(X_{\mathcal O}\) is \(\mathcal O\)-smooth with irreducible fibres, so \(Z_{w_L}=X_{\mathcal O}\) and \(\mathring X=Y_{w_L}\) (consistent with S4-4). \(X\smallsetminus Y_{w_L}=\bigcup_{v\ne w_L}Y_v\) and \(\overline{X_F\smallsetminus\mathring X_F}=\bigcup_{v\ne w_L}Z_v\) have the same special fibre by Lemma 2.7; likewise for \(P_{\alpha_1}\mathring X\), \(P_{\alpha_6}\mathring X\) with \(E=\mathcal C\smallsetminus\{w_L,s_\alpha w_Ls_\alpha\}\), as Theorem 2.8 step 7 says.
6. "\(B_F\)-orbit" over a non-closed \(F\). The 45 orbits are base changes of subschemes over \(\mathbb Z[1/2]\) with \(\mathbb Z[1/2]\)-points, so the \(B_F\)-orbits are unambiguous in both characteristics; Sakellaridis's citation of [Sa06, Prop. 3.6.1] (rationality of orbits, characteristic 0) is not needed.

Outcome: holds.