§2 · Agent 2 · Integral models · 2026-10-18T03:23:14Z

Summary 2.10 · Verdict on (Q2): (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 for \(\mathcal M_\theta\) over every such \(\mathcal O\)

proved audit: correct ★ key step

Second audit: correct. Follows from Theorem 2.8: a complete DVR of odd residue characteristic has 2 invertible, so (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 for M_θ over it in mixed and equal characteristic, p=3 included, with Sakellaridis's scheme structure and the complement clause for all 270 pairs. 'Settles item (I) of S4-13' is accurate: S4-13 (I) is exactly Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over O for this model, and the argument does not distinguish ramified from unramified finite F/Q_p. The scope disclaimer (nothing about (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7)., (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7)., (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits., (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5., (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). or Theorem 8.0.2 in characteristic p) is accurate; it inherits only the one-line gaps of Proposition 2.5 and Lemma 2.7.

(Q2) is answered positively: for every complete DVR of odd residue characteristic, mixed or equal characteristic, \(p=3\) included, Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. holds for \(E_6/F_4\) as \(\mathcal O\)-schemes, complement clause included; no pair fails.

Summary 2.10.

For \(\mathcal M_\theta\) and every DVR \(\mathcal O\ni\frac12\): the \(B_F\)-orbits are the \(45\) orbits \(B\dot wx_0\) (\(ww_Lw^{-1}\in\mathcal C\)), all of maximal rank; Sakellaridis's \(Y\) of each is the smooth \(B/B_x\); \(YP_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2\backslash\mathrm{PGL}_2\) for the \(90\) pairs with \(\alpha\in w(\Phi_L)\) and \(\simeq B_2\backslash\mathrm{PGL}_2=\mathbb P^1_{\mathcal O}\) for the other \(180\); the complement clause holds for all \(270\) pairs.

  1. Proof: Theorem 2.8 from Lemma 2.2–Lemma 2.7; audit Audit 2.9.
  2. This settles item (I) of S4-13 for every finite \(F/\mathbb Q_p\), \(p\) odd, and supplies (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. over \(\mathbb F_q((t))\) for (Q1), (Q3).
  3. Not claimed: (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7)., (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7)., (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits., (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5., (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). or Theorem 8.0.2 in characteristic \(p\).

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