§4 · Agent 4 · Close fields · 2026-10-18T02:51:39Z

Summary 4.11 · Verdict: Cartan proved; Theorem 8.0.2 over \(\mathbb F_q((t))\) given \((I_\Lambda)\)

proved audit: minor defect ★ key step

Minor defect (second audit). Each item matches its parent, re-checked above: 1 = Proposition 4.5 (orbit parametrization proved; the added 'Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. over F_q[[t]]' wording inherits Proposition 4.5's gap, the characteristic-p valuation cone), 2 = Proposition 4.6 (true; the Satake nu-bound is supplied by the box lemma), 3 = Audit 4.7 (true; [Sa08] is valid over every characteristic-0 local field), 4 = Theorem 4.8 (true under (I_Lambda)), 5 = Proposition 4.9 (conditional, true). The checklist is accurate: characteristic-0 theorems are used only over F'_e; characteristic-p inputs are the Cartan and Iwasawa decompositions of split G and H, Hensel, Lang, Steinberg, Edixhoven, S4-4's B cap H and H^1 of split solvable groups, all valid for p odd including p = 3; there is no circularity; and (I_Lambda) is correctly identified as the only open input.

By close fields: unconditionally, \(K\backslash X(\mathbb F_q(\!(t)\!))\simeq\Lambda_S^+\) and every structure constant of \(C_K\) equals its value over \(\mathbb Q_q(p^{1/e})\), \(e\gg0\); under the single input \((I_\Lambda)\), the same open input as the \(p\)-adic theorem over \(\mathbb Q_q\), Sakellaridis's Theorem 8.0.2 holds for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\) with a unique isomorphism, so (SG) \(\iff r\) onto, and (SG) fails.

Summary 4.11.

Let \(p\) be odd, \(F=\mathbb F_q(\!(t)\!)\), \(X=\mathcal M_\theta\) (\(E_6/F_4\)).

  1. (Proved; Proposition 4.5.) \(X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kt^{\mu_\kappa}x_0\), \((1-\theta)\mu_\kappa=\kappa\): Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. over \(\mathbb F_q[\![t]\!]\) in the strong form \(K\backslash X(F)\simeq\Lambda_S^+\).
  2. (Proved; Proposition 4.6.) For \(\lambda\in\Lambda_T^+\), \(\mu,\nu\in\Lambda_S^+\): \((T_\lambda*\mathbf 1_{Kx_\mu})(x_\nu)\) over \(F\) equals its value over every \(p\)-adic \(F'\) with residue field \(\mathbb F_q\) and \(e(F')\ge D(\lambda,\mu)\); the Satake transforms \(\mathcal S(T_\lambda)\) agree.
  3. (Proved; Audit 4.7.) Over every finite \(F'/\mathbb Q_q\) with residue field \(\mathbb F_q\), Sakellaridis's normalizations hold verbatim with \(\psi'\) of conductor \(\mathcal O'\); all hypotheses of Theorem 8.0.2 for \(\mathcal M_\theta\) hold except Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O., which follows from \((I_\Lambda)\) by flat base change; the image \(P_\kappa\) of \(\mathbf 1_{Kx_\kappa}\) depends only on \(q\).
  4. (Proved under \((I_\Lambda)\); Theorem 4.8.) Theorem 8.0.2 over \(F\): \(\Psi:C_K\xrightarrow{\sim}\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), \(\mathbf 1_{Kx_\kappa}\mapsto P_\kappa\), \(\Psi(f*\varphi)=r(\mathcal Sf)\Psi(\varphi)\), \(\mathbf 1_{X(\mathcal O)}\mapsto1\), unique. Hence \(\mathcal H_K\varphi_0=\Psi^{-1}(\operatorname{im}r)\), (SG) \(\iff r\) onto, and since \(r\) is not onto (Proposition VII), (SG) fails over \(F\); \(C_K=\mathcal H_K\varphi_0\oplus\mathbb C\mathbf 1_{Kx_{3\omega_1^\vee}}\). Over \(p\)-adic \(F\) with residue field \(\mathbb F_q\) the same holds under \((I_\Lambda)\) for every ramification.
  5. (Proved; Proposition 4.9.) For a general symmetric \(X\), the characteristic-\(p\) theorem follows from (T1)–(T3) over \(W(\mathbb F_q)\) and \(F\), and (T4), (T5) over \(F'_e\), \(e\gg0\).

Open: \((I_\Lambda)\) = Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. for \(\mathcal M_\theta\) over \(W(\mathbb F_q)\) (Dead end 4.10).

Checklist.

  1. Hypothesis transfer. Characteristic-\(0\) theorems (Sakellaridis, Delorme–Sécherre, Kneser, Casselman–Shalika, Braverman–Kazhdan, Knop, Brion) are used only over \(\mathbb Q_q(p^{1/e})\). Over \(\mathbb F_q(\!(t)\!)\) only Cartan and Iwasawa decompositions of split groups, Hensel, Lang, Steinberg's connectedness and Edixhoven's smoothness (valid for \(p\ne2\)) and the scheme-theoretic computation of \(B\cap H\) (S4-4) are used (Proposition 4.5 (E1)–(E4)).
  2. Residue characteristic vs characteristic. \(p\) odd on both sides; \(F'_e\) has characteristic \(0\), \(F\) characteristic \(p\); they share only \(\mathcal O/\varpi^D\) (Lemma 4.2(a)).

3–4. Schemes vs fibres, types vs isomorphisms. Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. is treated as a statement about \(\Lambda\)-schemes and moved to \(\mathcal O'\) by flat base change (Audit 4.7 step 8), never inferred from types.

  1. Non-reduced stabilizers. \(H\), \(B\cap H\), \(Q\) are smooth for \(p\) odd; orbit maps are smooth (Audit 4.3, Proposition 4.5 (E3)).
  2. Galois cohomology in characteristic \(p\). Only \(H^1(F,B\cap H)=1\) for a split connected solvable group is used; labels need no \(\sqrt t\) (Computation 4.4).
  3. Normalizations. Checked in Audit 4.7(a); the statement of Theorem 8.0.2 is normalization-free.
  4. Labels. Dominant \(\kappa\in\Lambda_S^+\) via \(\tau(x_\kappa)=t^\kappa\); Sakellaridis's anti-dominant \(\check\lambda\) via the field-independent dictionary of S4-5.
  5. Almost every place. Not used.
  6. Circularity. No characteristic-\(p\) conclusion is used to verify a hypothesis; Delorme–Sécherre and Theorem 8.0.2 are used only in characteristic \(0\).
  7. \(p=3\). Included in Lemma 4.2–Theorem 4.8; it can only enter through \((I_\Lambda)\).

Cited, not re-proved or not opened: Delorme–Sécherre, Thm 4.8 (via S4-12); Kneser (via S4-12); Bruhat–Tits (Cartan, Iwasawa decompositions over any local field); Steinberg 1968, Thm 8.1; Edixhoven 1992, Prop. 3.4; EGA IV 11.3.10, 17.2.6, 17.9.1, 18.5.17; SGA 3 VI\(_B\) 11.11, XVI 1.5, XXVI 7.16; Serre, Local Fields III §3; Casselman–Shalika (inside Sakellaridis's proof); Stacks 081I.

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