§2 · Agent 2 · Integral models · 2026-10-18T03:19:13Z

Proposition 2.5 · Rank-one quotients: a point (type $G$) or $\mathbb P^1_{\mathcal O}$ (type $U$)

proved audit: minor defect ★ key step

Minor defect (second audit). The construction is right: π(t)=diag(α(t),1) on L_α/ker α=SL2/μ₂; for α∈Φ_x, φ_α(SL2)⊆G^{θ_x}; for xα>0, Ψ=(Φ⁺∩θ_xΦ⁺)⊔{−α,δ} and dim Lie(P_α)y=17+c_x=dim B′_y (recomputed for all 90 pairs), u(c)=x{−α}(c)x_δ(±c) is θ_x-fixed, the image of (P_α)y in PGL2 is B₂⁻ (PGL2 would force xα=±α), the ideal argument over O is right, and the proper-monomorphism argument (EGA IV 18.12.6) gives the immersion. Two defects that change no claim: (i) in step 5 the bound 'Z′=closure of B′y has fibres of dimension ≤ dim Y_x−1' is asserted without proof (it holds because Z′ is the closure of the flat irreducible (B′y)_F, or by Z′=ṡ_αZ{x′} and Lemma 2.7 item 3, non-circularly since Lemma 2.7 uses only the geometric-fibre part of Proposition 2.5); (ii) Sakellaridis writes V/Γ for a geometric quotient (p. 7 of arXiv:0905.4244v3), while Proposition 2.5 proves only that the fppf quotient of 𝒫 (resp. Y_x) by R(P_α) is P_α/B′≅ℙ¹_O (resp. Spec O); since 𝒫→P_α/B′ is fppf and 𝒫×_{ℙ¹}𝒫 is the R(P_α)-orbit relation, it is also a geometric quotient. Both lines are supplied in Audit 6.5, which I checked (its 'categorical' argument covers the geometric-quotient properties).

If $\alpha\in\Phi_x$, $P_\alpha Y_x=Y_x$ and the rank-one quotient is a point. Otherwise the $P_\alpha$-orbit is an immersed smooth $\mathcal O$-scheme $Y_x\sqcup Y_{s_\alpha xs_\alpha}$ with quotient $\mathbb P^1_{\mathcal O}$ by $R(P_\alpha)$; types $T,N$ never occur.

Proposition 2.5.

$\mathcal O$ a DVR, $2\in\mathcal O^\times$; $x\in\mathcal C$, $\alpha\in\Delta$, $y=y_x$, $\pi:P_\alpha\to P_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2$ ($\pi(t)=\operatorname{diag}(\alpha(t),1)$).

(G) If $\alpha\in\Phi_x$: $\varphi_\alpha(\mathrm{SL}_2)\subseteq G^{\theta_x}$, $P_\alpha Y_x=Y_x$, $Y_x/R(P_\alpha)=\operatorname{Spec}\mathcal O$.
(U, $Y_x$ open) If $\alpha\notin\Phi_x$, $x\alpha>0$: with $x'=s_\alpha xs_\alpha$, $B'=\dot s_\alpha B\dot s_\alpha^{-1}=\pi^{-1}(B_2^-)$: (a) $(P_\alpha)_y=B'_y$, smooth, $\pi((P_\alpha)_y)=B_2^-$; (b) $P_\alpha/(P_\alpha)_y\to X_{\mathcal O}$ is an immersion onto a smooth $\mathcal P=P_\alpha Y_x=P_\alpha Y_{x'}$, $|\mathcal P|=|Y_x|\sqcup|Y_{x'}|$, $\dim Y_{x'}=\dim Y_x-1$; (c) $py\mapsto\pi(p)[0:1]$ gives $P_\alpha$-equivariantly

$$\mathcal P/R(P_\alpha)\simeq\mathrm{PGL}_2/B_2^-\simeq\mathrm{PGL}_2/B_2=SU\backslash\mathrm{PGL}_2\ (S=T_2),$$

with $Y_x$ over $\{[z:1]\}$ and $Y_{x'}$ over $[1:0]$.

(U, $Y_x$ small) If $\alpha\notin\Phi_x$, $x\alpha<0$: $(s_\alpha xs_\alpha,\alpha)$ is as in 2.

$\pi$. $Z(L_\alpha)=\ker\alpha$ is a split torus ($\alpha$ primitive in $X^*(T)$), $L_\alpha=\ker\alpha\cdot\varphi_\alpha(\mathrm{SL}_2)$, intersection $\alpha^\vee(\mu_2)$; so $L_\alpha/\ker\alpha\simeq\mathrm{PGL}_2$.
(G). $\theta_x$ fixes $U_{\pm\alpha}$ pointwise (Lemma 2.2, 2), so $\varphi_\alpha$ and $\theta_x\varphi_\alpha$ agree on $U^\pm$, on $\alpha^\vee(s)=w(s)w(1)^{-1}$, on the open cell, hence everywhere (schematic density, EGA IV 11.10.10). As $P_\alpha=B\cdot\varphi_\alpha(\mathrm{SL}_2)$, $P_\alpha y=By$; and $\pi((P_\alpha)_y)=\mathrm{PGL}_2$.
$\pi((P_\alpha)_y)\supseteq B_2^-$. $\pi(T^{\theta_x})=T_2$ since $\alpha|_{T^{\theta_x}}\ne0$ (Lemma 2.2, 5). With $\delta=-\theta_x\alpha\in\Phi^+\smallsetminus\{\alpha\}$, $-\alpha+\delta\notin\Phi\cup\{0\}$ (Lemma 2.2, 4), so $U_{-\alpha},U_\delta$ commute (Conrad 5.1.14) and $u(c)=x_{-\alpha}(c)\theta_x(x_{-\alpha}(c))=x_{-\alpha}(c)x_\delta(\pm c)\in(P_\alpha)_y$, $\pi(u(c))=\pi(x_{-\alpha}(c))$.
(a). Geometrically, $\operatorname{Lie}(P_\alpha)_y=(\mathfrak t\oplus\bigoplus_\Psi\mathfrak g_\gamma)^{\theta_x}$, $\Psi=\Phi(P_\alpha)\cap\theta_x\Phi(P_\alpha)=(\Phi^+\cap\theta_x\Phi^+)\sqcup\{-\alpha,\delta\}=s_\alpha\Phi^+\cap\theta_xs_\alpha\Phi^+$, dimension $17+c_x$, equal to $\operatorname{Lie}B'_y$ with $B'_y$ smooth (Lemma 2.4 for $\dot s_\alpha^{-1}\dot w$). So $(P_\alpha)_{y,\bar k}$ is smooth and $\pi$ of it is $B_2^-$ or $\mathrm{PGL}_2$. If $\mathrm{PGL}_2$, $P_\alpha y=R(P_\alpha)y\subseteq By$, so $By_{x'}=By_x$, $x'=x$, $x\alpha=\pm\alpha$: excluded. Hence $(P_\alpha)_{y,\bar k}=B'_{y,\bar k}$ on both fibres. Over $\mathcal O$ the ideal $I$ of $B'_y\subseteq(P_\alpha)_y$ has $I/\varpi I=0$ ($B'_y$ flat) and $\varpi^NI=0$ ($I\otimes F=0$), so $I=0$.
(b). On fibres, $py\mapsto\pi(p)[0:1]$ has $R(P_\alpha)$-orbit fibres, $R(P_\alpha)\subseteq B$: $\{[z:1]\}\leftrightarrow By$, $[1:0]\leftrightarrow B\dot s_\alpha y=Y_{x'}$. $c_{x'}=c_x+1$ ($s_\alpha$ maps the $\theta_x$-pairs in $(\Phi^+\smallsetminus\{\alpha\})\cup\{-\alpha\}$ onto the $\theta_{x'}$-pairs in $\Phi^+$). $B'y=\dot s_\alpha Y_{x'}$ is locally closed (Lemma 2.4). $\mathcal W=\{(gB',z):g^{-1}z\in B'y\}\subseteq\mathbb P^1\times X$ is $\simeq P_\alpha/(P_\alpha)_y$ and $\psi=\mathrm{pr}_2|_{\mathcal W}$ is the orbit map, a monomorphism. With $Z'=\overline{B'y}$ (fibres of dimension $\le\dim Y_x-1$) and $\partial'=Z'\smallsetminus B'y$: $Z'_k$ misses the larger $B'$-orbit $\dot s_\alpha Y_x$ of $P_\alpha y_k$, so $V=X\smallsetminus P_\alpha\partial'$ is an open neighbourhood of $|\mathcal P|$ over which $\mathcal W$ is closed in $\mathbb P^1\times V$. So $\psi$ is a proper monomorphism into $V$, a closed immersion (EGA IV 18.12.6).
(c). $(P_\alpha)_yR(P_\alpha)=B'$, so the fppf quotient is $P_\alpha/B'$; right translation by $\bar s$ gives $\mathrm{PGL}_2/B_2$.
3. $x'\alpha=s_\alpha(-x\alpha)>0$ as $-x\alpha\in\Phi^+\smallsetminus\{\alpha\}$, and $\alpha\notin s_\alpha\Phi_x=\Phi_{x'}$.

Builds on

Lemma 2.2 · The involutions $\theta_x$: compact imaginary roots, no real roots
Proposition 2.3 · Borel orbits of $E_6/F_4$ over an algebraically closed field of characteristic $\ne2$
Lemma 2.4 · Smooth stabilizers; orbit immersions $B/B_x\hookrightarrow X$ over $\mathbb Z[1/2]$

Built on by

Lemma 2.7 · Orbit closures over $\mathcal O$ have the expected special fibres
Theorem 2.8 · Theorem: Axiom 2.2.1 holds for $\mathcal M_\theta$ over every DVR with $2$ invertible
Audit 6.5 · Audit of the rank-one quotients $YP_\alpha/R(P_\alpha)$ over $\mathcal O$ (claim A2-5)

Sources

sakellaridis-v3.pdf, §2.2 and Axiom 2.2.1, p. 9
conrad-reductive-group-schemes.pdf, Prop. 5.1.14
EGA IV 11.10.10, 18.12.6 (cited)

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This conversation: the audit pass (eight reviewers, one salvage agent, re-checks) and the simplified VIb page05:21–07:33 UTC	267 M	$106
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Proposition 2.5 · Rank-one quotients: a point (type \(G\)) or \(\mathbb P^1_{\mathcal O}\) (type \(U\))

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