A2 was asked to prove (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary., Sakellaridis's Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O., for \(E_6/F_4\) over every complete discrete valuation ring of odd residue characteristic: the one hypothesis the earlier \(p\)-adic check had left open. It worked once over \(\mathbb Z[1/2]\), with \(\theta\)-stable tori and Bruhat cells and no input from Knop's characteristic-0 theory.
First it showed, by counting eigenspaces of \(\theta\) on the Lie algebra, that every \(\theta\)-stable maximal torus is conjugate under \(F_4\) to the standard one. So in every characteristic \(\ne2\) the Borel orbits are exactly the 45 orbits \(B\dot wx_0\), all of maximal rank ( (Proposition 2.3)). Each orbit is a smooth locally closed subscheme \(B/B_x\) over \(\mathbb Z[1/2]\); its stabilizer is the fixed group of an involution on the intersection of two Borel subgroups ( (Lemma 2.4)). The rank-one quotient is a point when \(\alpha\) is one of the 24 compact imaginary roots, and \(\mathbb P^1\) otherwise ( (Proposition 2.5)); a script sorted the 270 pairs into 90 and 180. Finally an induction on closures, from the closed orbit up, identified Sakellaridis's \(\mathcal O\)-scheme, the complement of the closure of the complement, with the orbit scheme, and gave the complement clause ( (Lemma 2.7), (Theorem 2.8)).
The audit A6 recomputed the table and found every step correct; three needed a line more. The result closes the open item of the \(p\)-adic theorem and supplies the one input the close-field transfer needed.
A fresh review re-derived every step and recomputed the orbit data by other methods; it found no error.
Result as the agent reported it proved
For \(\mathcal M_\theta\) (\(E_6/F_4\), \(X_{\mathcal O}=(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0})\otimes\mathcal O\)) and every DVR \(\mathcal O\) with \(2\in\mathcal O^\times\) (every complete DVR of odd residue characteristic, \(\operatorname{char}F=0\) or \(p\), \(p=3\) included), Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. holds as a statement about \(\mathcal O\)-schemes. The \(B_F\)-orbits are exactly the \(45\) orbits \(B\dot wx_0\), \(ww_Lw^{-1}\in\mathcal C\) (\(\simeq W/W_H(T)\), \(W_H(T)=C_W(w_L)\simeq W(F_4)\)), all of maximal rank and defined over \(\mathbb Z[1/2]\). Sakellaridis's \(\mathcal O\)-scheme of each is the smooth orbit scheme \(B/B_x\), \(B_x=(B\cap\theta_xB)^{\theta_x}\) smooth of fibre dimension \(16+c_x\). \(YP_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2\backslash\mathrm{PGL}_2\) for the \(90\) pairs with \(\alpha\in w(\Phi_L)\) and \(\simeq B_2\backslash\mathrm{PGL}_2\simeq\mathbb P^1_{\mathcal O}\) (type \(U\), \(S=T_2\)) for the other \(180\), \(P_\alpha\)-equivariantly; the complement clause holds for all \(270\) pairs; types \(T,N\) never occur.
Gap
None for (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. beyond cited standard results: Steinberg, Mem. AMS 80, Thms. 7.5, 8.1 (numbers not checked locally); CGP Prop. A.8.10 / Edixhoven Prop. 3.4; EGA IV 11.3.10, 11.10.10, 17.9.1, 18.12.6; Conrad, Reductive group schemes, Thm. 1.4.12, Thm. 5.1.13, Props. 5.1.14, 5.1.16; and the proved nodes S4-11 (step 3 re-checked) and S4-13 step 2. (Q1), (Q3) are not addressed.
Blocked routes
- Knop's W-action plus Sakellaridis's inductive remark (p. 9) for the 45-count and types: Knop's weight-lattice transport is a characteristic-0 input and the remark presupposes the axiom; replaced by the theta-stable-torus count of Proposition 2.3.
- Identifying Y^can via flat closures of Bruhat cells under tau~: would need flatness over Z of Schubert schemes with Schubert-variety fibres (Seshadri-Ramanathan type); avoided by the induction of Lemma 2.7.
- Y^can = orbit by Zariski's Main Theorem: would need normality of Y^can along the special fibre; replaced by the tau-immersion of Lemma 2.4 and reducedness.
Brief as sent
Target (Q2): (A3a), Sakellaridis's Axiom 2.2.1 (rank-one models over \(\mathcal O\)), for the \(E_6/F_4\) model over every complete discrete valuation ring of odd residue characteristic. Approach: Tits lifts and \(\theta\)-twisted Borel orbits over \(\mathbb Z[1/2]\): the 45 orbits \(B\dot wx_0\), smooth stabilizers \((B\cap\theta_wB)^{\theta_w}\), orbit immersions, the rank-one quotients, Sakellaridis's scheme structure and the complement clause, the \(45\times6\) table by script.