Audit 6.5 · Audit of the rank-one quotients \(YP_\alpha/R(P_\alpha)\) over \(\mathcal O\) (claim A2-5)

1. \(P_\alpha/R(P_\alpha)\). \(\alpha\) is primitive in \(X^*(T)\) (its \(\omega\)-coordinates are a row of the Cartan matrix, containing \(-1\)), so \(\ker\alpha=Z(L_\alpha)\) is a split torus; \(T=\ker\alpha\cdot\alpha^\vee(\mathbb G_m)\) fppf (as \(\alpha\circ\alpha^\vee\) is squaring), so \(L_\alpha=\ker\alpha\cdot\varphi_\alpha(\mathrm{SL}_2)\) with intersection \(\alpha^\vee(\mu_2)\), and \(L_\alpha/\ker\alpha\simeq\mathrm{SL}_2/\mu_2=\mathrm{PGL}_2\) over \(\mathbb Z\); \(R(P_\alpha)=\ker\alpha\ltimes U_{P_\alpha}=\ker\pi\).
2. Type G. \(\theta_x\) fixes \(U_{\pm\alpha}\) pointwise (Audit 6.2 step 1, transported by \(\dot w\)). \(\theta_x\circ\varphi_\alpha\) and \(\varphi_\alpha\) agree on \(U^\pm\) and on \(\alpha^\vee(s)=w(s)w(1)^{-1}\), hence on the open cell of \(\mathrm{SL}_2\), which is schematically dense over any base because it is fibrewise dense in the flat \(\mathrm{SL}_2\) (EGA IV 11.10.10); the target is separated, so they agree. Then \(P_\alpha=B\cdot\varphi_\alpha(\mathrm{SL}_2)\) gives \(P_\alpha y=By\) and \((P_\alpha)_yR(P_\alpha)=P_\alpha\): the quotient is \(\operatorname{Spec}\mathcal O\). Verified.
3. Type U, the stabilizer. \(\delta:=-\theta_x\alpha=x\alpha\in\Phi^+\smallsetminus\{\alpha\}\) and \(\delta-\alpha\notin\Phi\cup\{0\}\) (Audit 6.1 item 5), so \(U_{-\alpha}\) and \(U_\delta\) commute (Conrad, Prop. 5.1.14, valid over any base) and \(u(c)=x_{-\alpha}(c)\theta_x(x_{-\alpha}(c))\in(P_\alpha)_y\). \(\operatorname{Lie}(P_\alpha)_y=\mathfrak p_\alpha\cap\mathfrak g^{\theta_x}\) has dimension \(17+c_x\) (recomputed for all 90 pairs, Audit 6.1 item 5), equal to \(\dim B'_y\) (\(B'_y\) is conjugate to the stabilizer of \(\dot s_\alpha^{-1}y\in Y_{x'}\), smooth of dimension \(16+c_{x'}=17+c_x\) by Lemma 2.4). So \((P_\alpha)_{y,\bar k}\) is smooth with \(B'_{y,\bar k}\) open in it; its image under \(\pi\) is a closed subgroup containing \(B_2^-\), hence \(B_2^-\) or \(\mathrm{PGL}_2\) (a subgroup containing a Borel subgroup is parabolic, hence connected); \(\mathrm{PGL}_2\) would give \(P_\alpha y=R(P_\alpha)y\subseteq By\), so \(x'=x\), \(x\alpha=\pm\alpha\), excluded by \(\alpha\notin\Phi_x\) and the absence of real roots. Hence \((P_\alpha)_{y,\bar k}=B'_{y,\bar k}\), and over \(\kappa\), \(F\) by descent. Over \(\mathcal O\), locally \(0\to I\to A\to A'\to0\) with \(A'\) flat: \(\operatorname{Tor}_1^{\mathcal O}(A',\kappa)=0\) gives \(I/\varpi I\hookrightarrow A/\varpi A\simeq A'/\varpi A'\), so \(I=\varpi I\); \(I\) is finitely generated over the Noetherian \(A\) with \(I\otimes F=0\), so \(\varpi^NI=0\) and \(I=\varpi^NI=0\). Verified.
4. Type U, the immersion, with the repair. \(\mathcal W\simeq P_\alpha\times^{B'}B'y\simeq P_\alpha/(P_\alpha)_y\) and \(\psi=\operatorname{pr}_2\) is the orbit map, a monomorphism. Now \(\dot s_\alpha^{-1}y=nx_0\) with \(n\) lifting \(s_\alpha w\), so \(B'y=\dot s_\alpha Y_{x'}\) (Lemma 2.4 item 4) and \(Z':=\overline{B'y}=\dot s_\alpha Z_{x'}\). By Lemma 2.7 item 3, \(Z'\cap X_\kappa=\dot s_\alpha\bigcup_{v\in\Sigma_{x'}}Y_{v,\kappa}\), and similarly over \(F\); since \(\dim Y_x>\dim Y_{x'}\), \(x\notin\Sigma_{x'}\), so \(Z'\) misses \(\dot s_\alpha Y_x\) on both fibres. (The proof of Lemma 2.7 uses only the orbit decomposition of \(P_\alpha y_{\bar k}\), i.e. step 3 over \(\bar k\), and the closedness of \(P_\alpha Z\); so there is no circularity. Alternatively one may cite that the special fibre of an \(\mathcal O\)-flat integral scheme of finite type has the dimension of its generic fibre; A2 states the bound without citation.) Hence \(P_\alpha\partial'\) (closed: image under the proper \(P_\alpha\times^{B'}\partial'\to X\)) misses \(|\mathcal P|\); over \(V=X\smallsetminus P_\alpha\partial'\) the closed set \(\overline{\mathcal W}=P_\alpha\times^{B'}Z'\subseteq\mathbb P^1\times X\) equals \(\mathcal W\); \(\psi\) is a proper monomorphism into \(V\), hence a closed immersion (EGA IV 18.12.6). \(\mathcal P\) is smooth because \((P_\alpha)_y\) is.
5. Type U, the quotient. \(\pi((P_\alpha)_y)=B_2^-\) as fppf sheaves: \(\subseteq\) because \((P_\alpha)_y=B'_y\subseteq B'=\pi^{-1}(B_2^-)\); \(\supseteq\) because \(\pi|_{T^{\theta_x}}=\alpha|_{T^{\theta_x}}\) is a split surjection onto \(T_2\) (Audit 6.2 step 5; so A2's caveat about divisibility is moot) and \(\pi(u(c))=\pi(x_{-\alpha}(c))\). So \((P_\alpha)_yR(P_\alpha)=B'\) and the fppf quotient is \(P_\alpha/B'\simeq\mathrm{PGL}_2/B_2^-\simeq\mathbb P^1_{\mathcal O}\). It is categorical: \(\mathcal P\to P_\alpha/B'\) is fppf, and if \(p'\in pB'=pR(P_\alpha)(P_\alpha)_y\), say \(p'=prh\), then \(p'y=(prp^{-1})\,py\) with \(prp^{-1}\in R(P_\alpha)\); so \(\mathcal P\times_{\mathbb P^1}\mathcal P\) is the \(R(P_\alpha)\)-orbit relation and every \(R(P_\alpha)\)-invariant morphism to a scheme descends uniquely.
6. Item 3. \(x'\alpha=-s_\alpha(x\alpha)\in\Phi^+\) because \(-x\alpha\in\Phi^+\smallsetminus\{\alpha\}\), and \(\alpha\notin s_\alpha\Phi_x=\Phi_{x'}\). Verified.

Outcome: holds with the repair in step 4.