A4 was asked to obtain Sakellaridis's theorem over \(\mathbb F_q(\!(t)\!)\) by transfer from characteristic 0. Its idea: \(\mathbb F_q(\!(t)\!)\) and the totally ramified fields \(\mathbb Q_q(p^{1/e})\) have the same truncated rings \(\mathcal O/\varpi^m\) for \(m\le e\), and every quantity in the Hecke module \(C_K\) is a count that can be read modulo a fixed power of \(\varpi\).
It proved that membership in Cartan and Iwasawa cells is decided by valuations of matrix coefficients in the fundamental representations, with a bound independent of the field ( (Lemma 4.2)). Counting \(K\)-orbits then carried the Cartan decomposition to \(\mathbb F_q(\!(t)\!)\) ( (Proposition 4.5)), and every structure constant of \(C_K\) and every Satake coefficient agreed with its value over \(\mathbb Q_q(p^{1/e})\) for \(e\) large ( (Proposition 4.6)). Sakellaridis's "unramified" simplification only fixes the additive character and the measure, so his theorem holds over the ramified fields once (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 there ( (Audit 4.7)). This gave the theorem for \(E_6/F_4\) over \(\mathbb F_q(\!(t)\!)\), with a unique isomorphism, conditional only on (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 \(W(\mathbb F_q)\) ( (Theorem 4.8)), and a general version for symmetric varieties under (T1)–(T5)Hypotheses of the characteristic-p transfer for symmetric X over W(F_q): (T1) H split with connected fibres; (T2) X(E) = G(E)x0; (T3) τ detects integral points; (T4) the Cartan decomposition over the ramified fields F'_e; (T5) Sakellaridis's theorem over F'_e = Q_q(p^{1/e}), e ≫ 0. ( (Proposition 4.9)).
A2 then proved (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., and the root checked that [Sa08] covers every local field of characteristic 0. The audit A5 found the main line sound; it added a bound on the Satake support that A4 had used without proof.
A fresh review found no error. It noted one more line needed for a general symmetric variety: choosing the fields along which the colour data stay constant ( (Proposition 4.9)).
Result as the agent reported it partial
For p odd, F = F_q((t)), X = M_θ (E6/F4), by Deligne–Kazhdan close fields: (1) [proved, modulo standard cited inputs] X(F) = ⊔{κ∈Λ_S^+} K t^{μ_κ} x_0 with (1−θ)μ_κ = κ, i.e. G(F) = K T(F) H(F) and Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. over F_q[[t]] in the strong form K\X(F) ≅ Λ_S^+: the number of K-orbits in K t^λ X(O) and the labels they contain are computed in G(O/ϖ^D), H(O/ϖ^D) (Cartan decompositions of G and of H = F4) and agree with those over Q_q(p^{1/e}), e ≥ D(λ), where Delorme–Sécherre's decomposition holds (Proposition 4.5). (2) [proved] every structure constant (T_λ*1{Kx_μ})(x_ν) of C_K over F equals its value over any p-adic F' with residue field F_q and e(F') ≥ D(λ,μ), and the Satake transforms S(T_λ) agree (Proposition 4.6); this extends Lemma 1.7 (Computation 1.7) to the whole module without its embedding hypothesis, which in any case holds for M_θ because τ: X → G is a closed immersion over W(F_q) (Audit 4.3). (3) [proved] Sakellaridis's normalizations (§1.1, §3.8, (4.1), §5.3, (5.2), (5.3), (4.5)) hold verbatim over every finite F'/Q_q with residue field F_q once ψ' has conductor O' (as §1.7 prescribes), the hypotheses (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). for M_θ over O' are ramification-free (S4/S4b), and Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over O' follows from (I_Λ) = Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over W(F_q) by flat base change; the image P_κ of 1_{Kx_κ} depends only on q (Audit 4.7). (4) [proved under (I_Λ)] Theorem 8.0.2 holds for M_θ over F_q((t)): Ψ(1_{Kx_κ}) = P_κ is an isomorphism C_K ≅ C[δ(X)^{1/2}A_X^*]^{W_X} with Ψ(f*φ) = r(Sf)Ψ(φ), Ψ(1_{X(O)}) = 1, and it is the unique such isomorphism; hence (SG) ⟺ r onto, and (SG) fails over F_q((t)) (and over every p-adic F with residue field F_q, any ramification), with C_K = H_Kφ_0 ⊕ C·1_{Kx_{3ω1∨}} and cokernel the character q^{ρ−ω4} (Theorem 4.8). No hypothesis of Sakellaridis over F_q[[t]] and no statement over an algebraic closure of F_q((t)) is used; p = 3 is not excluded by the transfer. (5) [proved] For a general symmetric X = G/G^θ the characteristic-p theorem follows from (T1) G^θ split with connected fibres, (T2) X(E) = G(E)x_0, (T3) τ detects O-points, over W(F_q) and F, together with (T4) the Cartan decomposition and (T5) Theorem 8.0.2 over Q_q(p^{1/e}) for e ≫ 0 (Proposition 4.9).
Gap
(I_Λ): Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. for M_θ over W(F_q) as W(F_q)-schemes — for each of the 45 maximal-rank Borel orbits and each simple root, YP_α/R(P_α) ≅ SU\PGL2 (type U) or PGL2\PGL2 (type G) with Sakellaridis's scheme structure on Y, and the complement clause. This is exactly the open item (I) of the p-adic theorem over the unramified field Q_q; the transfer cannot remove it (Dead end 4.10). Cited, not re-proved or not opened: Delorme–Sécherre Thm 4.8 and Kneser (via S4-12), Bruhat–Tits (Cartan and 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 (only for the side checks Audit 4.3, Proposition 4.9), Serre Local Fields III §3, Stacks 081I, Casselman–Shalika (inside Sakellaridis's proof).
Blocked routes
- Removing (I_Λ) by the transfer: the module transfer needs Sakellaridis's Theorem 8.0.2 at infinitely many ramified places Q_q(p^{1/e}), and his proof there uses Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over O'_e (normalization of eigenmeasures, §3.3; reduction to SL2, §5.1), which by flat base change is the single statement (I_Λ); needed lemma: (I_Λ) itself, e.g. by S4-13's route (points y ∈ Y(Z[1/2]) with smooth B_y and Y = B·y on all fibres, for the 45×6 pairs).
- (SG)-failure over F_q((t)) without (I_Λ), via two Hecke eigenfunctionals with the same character q^{ρ−ω4} at y1, σy1 over Q_q(p^{1/e}) built from the open-orbit morphisms only, then Lemma 1.7(c): needs regularity and linear independence of the eigenfunctions Ω_{y1}, Ω_{σy1} at these special parameters, which requires the exact cocycles b_w, i.e. (I_Λ).
- (H-Cartan) over F_q((t)) via the pages' IIb step 1 (input D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers. = Chen–Yi Prop. 17(i)): Chen–Yi prove 17(i) by citing Gaitsgory–Nadler Thm 8.2.9 and Nadler 2004 Thm 4.2, which are over C — a characteristic-0 citation in characteristic p; replaced here by the counting transfer of Proposition 4.5.
- Applying Delorme–Sécherre directly over F_q((t)): its quoted statement covers locally compact fields of residue characteristic ≠ 2, but its internal inputs (Helminck–Wang Prop. 10.3, conjugacy of θ-split tori) were not checked in characteristic p here; not used — Delorme–Sécherre enters only over the p-adic fields F'_e.
Brief as sent
Target (Q1) for \(E_6/F_4\), by transfer from characteristic 0. Approach: Deligne's truncated discrete valuation rings and Kazhdan's close-field method: compare the Hecke module \(C_K\) over \(\mathbb F_q(\!(t)\!)\), structure constant by structure constant, with \(C_K\) over totally ramified \(p\)-adic fields, where Theorem 8.0.2 applies; check the paper's normalizations for ramified fields; transfer the Cartan decomposition by counting. The root sent the IId run's close-field lemma (satake-converse, Lemma 1.7) as a cited reference.