You are agent A5, an independent auditor in a bounded team attacking one mathematical question. You did not write the claims below and you receive no opinion about them.

SETTING (Proposition VI of the satake-two-levels pages: Sakellaridis's theorem, restated for a local field of characteristic p).

Fields. p is an odd prime, q a power of p. F is a non-archimedean local field with ring of integers O, uniformizer ϖ, residue field F_q, |ϖ| = q^{-1}. Two cases that must never be confused:
  char F = 0 : F/Q_p finite ("p-adic"). Sakellaridis assumes F/Q_p unramified "to avoid complicated formulas", saying "the modifications needed to remove this assumption are trivial" (§1.1, p. 2).
  char F = p : F = F_q((t)), O = F_q[[t]], ϖ = t.
"p odd" is the residue characteristic in both cases; the characteristic of F is 0 or p.

Groups. G is a split reductive group scheme over O with connected fibres and simply connected derived group; B ⊇ A a Borel subgroup and a split maximal torus over O; U the unipotent radical of B; W the Weyl group; K = G(O); J the Iwahori subgroup (preimage of B(F_q)). H_K = C_c(K\G(F)/K), vol K = 1; the Satake isomorphism S : H_K ≅ C[A^*]^W, A^* the complex dual torus (unramified characters of A(F)).

Variety. Sakellaridis uses right actions and writes X = H\G; our pages use left actions, X = G/H (read x·g as g^{-1}x). X is a G-scheme over O with an open B_F-orbit X̊_F; X̊ := X \ closure(X \ X̊_F); x0 ∈ X̊(O), H its stabilizer.
Invariants (Sakellaridis §2.1–2.2, §6.1, §7.1, §8): P(X) = {p : X̊p = X̊}; L(X) a Levi of P(X) with [L(X),L(X)] ⊆ H; A_X = L(X)/(L(X)∩H); 𝒳(X) the B-eigencharacters of F(X)^(B); Λ_X = Hom(𝒳(X), Z); Λ_X^+ the monoid of G-invariant valuations, V its cone; A_X^+ ⊆ A_X(F) the points with label in Λ_X^+ (via Λ_X ≅ A_X(F)/A_X(O), λ ↦ e^λ(ϖ)); W_X the little Weyl group (Knop's action of W on the B_F̄-orbits of maximal rank is transitive, the stabilizer of X̊ is W(X) = W_X ⋉ W_{P(X)}); δ(X) = |d(X)|, d(X) the modular character of L(X)∩B; δ(X)^{1/2}A_X^* ⊆ A^* the translate of the image of A_X^* → A^*; Δ_X the normalized spherical roots; colours, virtual weighted colours, Θ, Θ^+.
Functions. C_K = C_c^∞(X(F))^K, a module over H_K by (f∗φ)(x) = Σ_{gK∈G(F)/K} f(g)φ(g^{-1}x); φ0 = 1_{X(O)}; (SG): H_K∗φ0 = C_K. The restriction map r : C[A^*]^W → S := C[δ(X)^{1/2}A_X^*]^{W_X} (Sakellaridis (8.2), p. 65).

HYPOTHESES, relabelled (page numbers of arXiv:0905.4244v3 = printed pages).
  (A1₀) char F = 0 (§1.1, p. 2).
  (A1ₚ) char F = p, i.e. F = F_q((t)). Not in the paper.
  (A1)  G split reductive over O, connected fibres, simply connected derived group (§1.7, p. 6).
  (A2)  Integral model (§1.7, p. 7): X is a quasi-affine G-scheme over O; X̊ is smooth and surjective over O and its special fibre is homogeneous under B_{F_q}; X has a G-eigenmeasure and X̊ a B-invariant measure.
  (A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1, p. 9): for every B_F-orbit Y_F ⊆ X_F of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is isomorphic to one of PGL2\PGL2, SU\PGL2 (S ⊆ N(U)), T\PGL2 (T a non-trivial torus over O) or N(T)\PGL2; and the complement of YP_α is the closure of the complement of (YP_α)_F. Here the O-scheme Y is "the complement of the closure of the complement of Y_F in its closure" (p. 9).
  (A3b) Cartan decomposition (Axiom 2.4.1, p. 10): A_X^+ contains a complete set of representatives of the K-orbits on X(F); elements of A_X^+ with distinct images in Λ_X^+ lie in distinct K-orbits.
  (A3c) Iwahori orbits (Axiom 2.4.2, p. 11): xJ ⊆ xB(O) for x ∈ A_X^+.
  (A4)  Spherical roots and colours, over F̄: (i) no pair (Y, α) of maximal rank is of type N, and δ(X)^{-1/2}χ̃ is unramified (opening of §6, p. 41); (ii) L_γ is a classical group for every γ ∈ Δ_X (§7, p. 58); (iii) Statement 6.3.1 (pp. 44–45), which follows from (i) and (ii) by Theorem 6.3.2 (p. 45); (iv) Statement 7.1.5 (p. 60).
  (A5)  X is homogeneous and affine, and X̊(F) is a single B(F)-orbit (§8, p. 63).

PUBLISHED, characteristic 0. Theorem 8.0.2 (p. 64): under (A1₀), (A1)–(A5) there is a canonical isomorphism C_K ≅ S, compatible with the H_K-structure and the Satake isomorphism (f acts on S as multiplication by r(S f)), with 1_{X(O)} ↦ 1. Corollary 8.0.4 (p. 65): if r is onto, C_K is a principal H_K-module generated by φ0. Hence, in characteristic 0, (SG) ⟺ r onto.

THE CASE E6/F4. G_Z the split simply connected Chevalley group of type E6, pinning (T, B, {x_{α_i}}), Bourbaki labels (chain 1–3–4–5–6, node 2 on node 4). θ0 = Int(σ_L)∘C, with C the Chevalley involution and σ_L the Tits lift of w_L, the longest element of W(D4) on nodes 2,3,4,5; θ0|_T = −w_L; type E IV; θ0 is an involution defined over Z (S4-11). The model M_θ: X_O := (G_Z/G_Z^{θ0}) ⊗ O with p odd; H_O = G_O^{θ0}, smooth with fibres of type F4; x0 = eH. Λ_X = Λ_S = Ze1 ⊕ Ze6 with e_i = (1−θ0)α_i^∨; W_X = S3; restricted roots A2, all multiplicities 8; P(X) the standard parabolic of nodes {2,3,4,5}, L = Z_G(S), S = (T^{−θ0})°; 45 Borel orbits of maximal rank (|W(E6)|/(|W_X|·|W(D4)|) = 51840/(6·192)).
Known for p-adic F (char 0), p odd (salvage returns S4, S4b): (A1), (A2), (A3b), (A3c), (A4), (A5) hold for M_θ, and the types in (A3a) over F̄ are only G and U; (A3a) as O-schemes is open. Proposition VII (proved, every q): r is not onto; so wherever Theorem 8.0.2 holds for M_θ, (SG) fails.

TEAM QUESTION.
 (Q1) Formulate and prove Sakellaridis's Theorem 8.0.2 over a local field of characteristic p: F = F_q((t)), p odd, for a symmetric X = G/G^θ (θ an involution of G over O) satisfying (A1)–(A5), every statement over F̄ read over an algebraic closure of F_q((t)). State exactly which further hypotheses, if any, the characteristic-p proof needs. At minimum: X = M_θ (E6/F4).
 (Q2) Prove (A3a) for M_θ over every complete discrete valuation ring O with odd residue characteristic, mixed (p-adic) and equal (F_q[[t]]) characteristic, or exhibit the pairs (Y, α) where it fails.
 (Q3) Verify (A1), (A2), (A3b), (A3c), (A4), (A5) for M_θ over F_q((t)), p odd, replacing every characteristic-0 citation.
A complete answer gives the characteristic-p theorem with its exact hypotheses and a proof in which every step is proved or cited with its number and checked to hold in characteristic p; (A3a) for M_θ over every O as above; and the hypotheses verified for M_θ over F_q((t)). Then, for E6/F4 over F_q((t)) and over p-adic F (p odd), (SG) ⟺ r onto, so (SG) fails.

DOES NOT COUNT. "The proof carries over verbatim" without checking each characteristic-0 input: Knop [Kn91, Kn94, Kn95, Kn96], Brion [Br01], Brion–Luna–Vust [BLV86], Luna [Lu01], Sakellaridis [Sa06, Sa08, Sa12], Braverman–Kazhdan [BK98], Casselman and Casselman–Shalika [Ca80, CS80], Wasserman [Was96], Akhiezer [Ak83], Tate's thesis in §5.3. A p-adic result presented as a characteristic-p result. (A3a) checked over F̄ only (the types), or only on one fibre, or "at almost every place", instead of as an isomorphism of O-schemes. A close-fields transfer that assumes Theorem 8.0.2 for ramified p-adic fields without checking Sakellaridis's normalizations (§3.8, §5.3). Numerical checks for small q only. A statement about E6/F4 over F_q((t)) obtained by citing Theorem 8.0.2 in characteristic 0.

CHECKLIST (the ways a wrong answer typically fails; audits check every one).
 1. Hypothesis transfer: a characteristic-0 theorem used in characteristic p.
 2. Residue characteristic confused with the characteristic of F.
 3. Generic fibre, special fibre and O-scheme confused; in particular Sakellaridis's scheme structure on Y.
 4. A type over F̄ taken for an isomorphism of O-schemes.
 5. Non-reduced stabilizers and inseparable orbit maps in characteristic p.
 6. Finiteness of B(F)-orbits and of Galois cohomology in characteristic p (H^1(F, μ_p) and Artin–Schreier classes are infinite).
 7. Normalizations: measures, the conductor of ψ, "unramified over Q_p".
 8. Left versus right actions; Λ_X^+ versus Λ_S^+; anti-dominant versus dominant labels.
 9. A statement "at almost every place" used at a given place.
 10. Circularity: the conclusion of Theorem 8.0.2 in characteristic p used to verify its hypotheses.
 11. Bad primes: p = 3 is bad for E6 and F4.

TERMINOLOGY TRAPS. The type of (Y, α) (G, U, T, N) is a statement over F̄; (A3a) is a statement about O-schemes. X(O) and Kx0 are different sets in general (equal for M_θ by Hensel and Lang). K-orbits, B(F)-orbits and B_F̄-orbits are different things. Λ_X^+ is the valuation monoid (for symmetric X the image of the anti-dominant chamber); our pages label K-orbits by dominant μ ∈ Λ_S^+ through t^μ x0. W_X for E6/F4 is S3. "Unramified over Q_p" is a simplification of the paper, not part of the characteristic-p question. "Symmetric" means H = G^θ (or an open subgroup of it) for an involution θ.

FILES (read HTML with  sed -e 's/<[^>]*>/ /g' FILE | tr -s ' \n'  and PDFs with  pdftotext -layout FILE - ).
 Sakellaridis, Spherical functions on spherical varieties, arXiv:0905.4244v3: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sources/sakellaridis-v3.pdf ; as text: /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve-vi/sak.txt (§2.2.1 near line 503, §3.3 near 720, Theorem 4.2.2 near 1410, §5.4 near 1823, §6.19 near 3388, Theorem 7.2.1 near 3631, §8 near 3830).
 Proposition VI page (hypotheses with their status, the E6/F4 verification): /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sakellaridis.html
 Salvage returns S4, S4b (the p-adic verification for E6/F4; nodes S4-1 … S4-14): /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/audit/S4.json and S4b.json
 E6/F4 root data and the symmetric pair: /Users/miltonlin/Projects/Trees/subpages/tools/proof_graphs/spherical-generation/families/e6-f4.html ; Proposition VII: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/e6-f4-module.html
 Other local sources: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sources/{conrad-reductive-group-schemes,milne-iag-v2,chen-yi-v4,zhu-satake-v2,bf-satake-v2,stacks-algebra,stacks-more-algebra}.pdf

RULES.
- Return concrete lemmas, constructions, computations or counterexamples, each with a proof or an explicit gap. No status reports, no optimism. Label every claim: proved (here, or cited with exact result number) · heuristic · gap. An analogue in one characteristic does not certify an inference in the other.
- Check the exact meaning of every term you use (see the traps above).
- Do not search the web for a solution of this question; ordinary mathematical background (statements and standing hypotheses of standard theorems and papers) may be looked up. Prefer the local sources. When you cite a paper you could not open, say so.
- You may run short Python checks with `python3 -I` (standard library only; no sympy or numpy) for finite computations: single process, a few seconds each, no parallelism. Keep scripts in /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve-vi/A5/ (create it). Do not modify anything under /Users/miltonlin/Projects/Trees.
- Timestamps: at the start of every node run `date -u +%Y-%m-%dT%H:%M:%SZ` and record it in the node. Work in nodes: an idea, a lemma with proof, a computation, a check, an objection, a dead end (status blocked, with the exact missing lemma), or a verdict. Produce 5–14 nodes.
- Write every node as a stage of a mathematics paper: a plain-words name, a one-sentence summary, the precise statement, and the argument as numbered steps, readable by a mathematician outside the subfield. Cite other nodes as [[A5-3]], never by bare code in prose; state computations as mathematics (what was computed, the result, why it suffices), not as code or program output.
- Concept keys for nodes: setup · transfer · integral · hypotheses · close · synthesis.
- FINAL ANSWER: only one JSON object inside a ```json fence, with the schema below; also write the identical JSON to /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve-vi/A5/return.json. Escape JSON strings properly (double every backslash, escape quotes and newlines). Nothing outside the fence.

RETURN SCHEMA
{"agent":"A5",
 "family":"approach family, at most twelve words",
 "nodes":[
  {"id":"A5-1","t":"2026-10-18T02:10:12Z",
   "kind":"idea|lemma|computation|check|objection|blocked|verdict",
   "status":"proved|refuted|blocked|open|heuristic|superseded",
   "label":"Setup|Definition|Strategy|Lemma|Proposition|Theorem|Corollary|Computation|Example|Counterexample|Remark|Objection|Audit|Conjecture|Dead end|Summary",
   "name":"tree label: at most 44 characters, plain text (Unicode, no TeX), says what the step establishes",
   "title":"page title, at most 90 characters, inline TeX allowed",
   "summary":"one or two sentences: what this step establishes and why it matters",
   "statement":"the precise claim (Markdown + TeX); empty for Strategy and Remark",
   "parents":["A5-0"],
   "body":"the argument: Markdown; math as \\( \\) inline and $$ $$ display; proofs as numbered steps; other nodes as [[A5-1]]; results cited with exact numbers and the file they come from",
   "concept":"one of: setup, transfer, integral, hypotheses, close, synthesis",
   "key":false,
   "sources":["sakellaridis-v3.pdf, Axiom 2.2.1, p. 9"]}],
 "result":{"status":"proved|partial|refuted|blocked",
           "statement":"precise statement of what was established",
           "gap":"exact remaining obligation, or none"},
 "blocked_routes":["route and the theorem-strength lemma it would need"],
 "scripts":["paths of scripts run"]}

CLAIMS TO AUDIT (copied verbatim from the returns of the agents who wrote them; the full returns are /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/sakellaridis-char-p/returns/A4.json and A3.json).

== Part 1, main line ==
--- A4-2 · Lemma · Lemma: truncated rings agree and valuation conditions are fixed mod $\varpi^D$
STATUS CLAIMED: proved
PARENTS: A4-1
STATEMENT:
Let \(V_i\) (\(i=1,\dots,6\)) be the Weyl module of highest weight \(\omega_i\) over \(\Lambda\) (free, with a basis of \(T\)-weight vectors, \(G_\Lambda\)-stable), \(\rho_i\) the action, \(\ell_i\) the coordinate of the lowest weight vector (weight \(w_0\omega_i\)). For a matrix \(A\) over \(E\) in such a basis put \(\operatorname{val}(A)=\min\) of the valuations of its entries. For \(c,d\in\Lambda_T\otimes\mathbb Q\) write \(c\preceq d\) if \(\langle d-c,\omega_j\rangle\ge0\) for all \(j\).

(a) If \(F'\supseteq\mathbb Q_q\) has residue field \(\mathbb F_q\) and ramification index \(e\ge m\), there is a \(\Lambda\)-algebra isomorphism \(\mathcal O'/\varpi'^m\cong\mathbb F_q[t]/(t^m)\) with \(\bar\varpi'\mapsto\bar t\). So \(F'_e\) and \(F\) are \(e\)-close.

(b) For a smooth affine \(\Lambda\)-scheme \(Y\) (here \(G\), \(H\), \(X\)), \(Y(\mathcal O_E)\to Y(\mathcal O_E/\varpi^m)\) is onto, and \(Y(\mathcal O_E/\varpi^m)\) depends only on the \(\Lambda\)-algebra \(\mathcal O_E/\varpi^m\); \(K/K(m)=G(\mathcal O_E/\varpi^m)\).

(c) (Cartan detection.) For \(g\in G(E)\) and \(\nu\in\Lambda_T^+\): \(g\in Kt^\nu K\iff\operatorname{val}\rho_i(g)=\langle\nu,w_0\omega_i\rangle\) for \(i=1,\dots,6\). In particular \(g\in K\iff\operatorname{val}\rho_i(g)=0\) for all \(i\). If \(g\in Kt^aK\), \(g'\in Kt^bK\), \(gg'\in Kt^cK\) (\(a,b,c\) dominant) then \(c\preceq a+b\); the set \(\{c\in\Lambda_T^+:c\preceq d\}\) is finite.

(d) (Iwasawa detection.) For \(g\in G(E)\) and \(\nu\in\Lambda_T\): \(g\in U(E)t^\nu K\iff\operatorname{val}(\ell_i\rho_i(g))=\langle\nu,w_0\omega_i\rangle\) for all \(i\).

(e) (Finite determinacy.) Let \(\mathcal C\) be a finite Boolean combination of conditions \(\operatorname{val}M(z)\ge c\), \(\operatorname{val}M(z)=c\) on \(z\in\prod_jY_j(\mathcal O_E)\), \(Y_j\in\{G,H\}\), where the entries of \(M(z)\) are polynomials over \(\Lambda\) in \(\varpi^{\pm1}\) and in the matrix coefficients of \(\rho_i(z_j^{\pm1})\), \(\rho_i(\theta(z_j)^{\pm1})\). There is \(D=D(\mathcal C)\), independent of \(E\), such that \(\mathcal C(z)\) depends only on \(\bar z\in\prod_jY_j(\mathcal O_E/\varpi^D)\), and for \(D\)-close \(E,E'\) the subsets of \(\prod_jY_j(\mathcal O/\varpi^D)\cong\prod_jY_j(\mathcal O'/\varpi'^D)\) cut out by \(\mathcal C\) correspond. Hence counts of solutions, existence statements over \(Y_j(\mathcal O)\), and equivalence relations defined in this way agree over \(D\)-close fields.
PROOF / BODY:
**Proof.**
1. *(a)* \(p\in\varpi'^e\mathcal O'\subseteq\varpi'^m\mathcal O'\), so \(R=\mathcal O'/\varpi'^m\) is a local Artinian \(\mathbb F_p\)-algebra with residue field \(\mathbb F_q\). The map \(\Lambda\to R\) factors through \(\Lambda/p=\mathbb F_q\); its image is a coefficient field (the Teichmüller representatives). The \(\mathbb F_q\)-algebra map \(\mathbb F_q[T]\to R\), \(T\mapsto\bar\varpi'\), is onto (every element of \(\mathcal O'\) is \(\sum_i[a_i]\varpi'^i\)) and kills \(T^m\); both sides have length \(m\), so it is an isomorphism, compatible with \(\Lambda\to\mathbb F_q\). This is the close-rings step of Lemma 1.7 (A1-7) as audited in A4-17.
2. *(b)* Hensel's lemma for smooth schemes over a complete discrete valuation ring (EGA IV\(_4\) 18.5.17); \(Y(\mathcal O_E/\varpi^m)\) is the functor \(Y\) evaluated on that \(\Lambda\)-algebra.
3. *(c)* \(V_i\otimes\mathcal O\) is \(G(\mathcal O)\)-stable, so \(\rho_i(K)\subseteq\mathrm{GL}_n(\mathcal O)\). For \(B,C\in\mathrm{GL}_n(\mathcal O)\), \(\operatorname{val}(BAC)\ge\operatorname{val}(A)\) and \(A=B^{-1}(BAC)C^{-1}\), so equality holds. For dominant \(\nu\), \(\rho_i(t^\nu)\) is diagonal with entries \(\varpi^{\langle\nu,\chi\rangle}\), \(\chi\) running over the weights of \(V_i\); every \(\chi-w_0\omega_i\) is a sum of positive roots, so the minimum is \(\langle\nu,w_0\omega_i\rangle\). By the Cartan decomposition \(G(E)=\bigsqcup_{\nu'\in\Lambda_T^+}Kt^{\nu'}K\) (Bruhat–Tits, valid over every non-archimedean local field; for \(\mathbb F_q(\!(t)\!)\) see also Zhu, (2.1.2)), \(\operatorname{val}\rho_i(g)=\langle\nu',w_0\omega_i\rangle\) for the \(\nu'\) of \(g\). The \(w_0\omega_i=-\omega_{i^*}\) span \(X^*(T)_{\mathbb Q}\), so the six values determine \(\nu'\). Superadditivity \(\operatorname{val}(AB)\ge\operatorname{val}A+\operatorname{val}B\) gives \(\langle c,w_0\omega_i\rangle\ge\langle a+b,w_0\omega_i\rangle\), i.e. \(\langle a+b-c,\omega_{i^*}\rangle\ge0\) for all \(i\). A dominant \(c\) with \(d-c\) in the cone of positive coroots lies in a bounded region, hence in a finite set.
4. *(d)* Positive root groups send each weight space of \(V_i\) into the sum of strictly higher ones, so \(\ell_i\rho_i(u)=\ell_i\) for \(u\in U(E)\); \(\ell_i\rho_i(t^\nu)=\varpi^{\langle\nu,w_0\omega_i\rangle}\ell_i\) (the lowest weight space has rank one); \(\ell_i\rho_i(k)\) is a row of a matrix in \(\mathrm{GL}_n(\mathcal O)\), of valuation \(0\). The Iwasawa decomposition \(G(E)=\bigsqcup_{\nu\in\Lambda_T}U(E)t^\nu K\) and the injectivity of \(\nu\mapsto(\langle\nu,w_0\omega_i\rangle)_i\) on \(\Lambda_T\) finish.
5. *(e)* Let \(N\) be the largest negative exponent of \(\varpi\) in \(M\); then \(\varpi^NM(z)\) has entries in \(\Lambda[\text{coordinates}][\varpi]\). "\(\operatorname{val}M(z)\ge c\)" means \(\varpi^NM(z)\equiv0\bmod\varpi^{N+c}\); "\(=c\)" adds \(\not\equiv0\bmod\varpi^{N+c+1}\). The coordinates of \(\rho_i(z^{-1})=\rho_i(z)^{-1}\) and of \(\rho_i(\theta(z))\) are polynomials over \(\Lambda\) in those of \(z\) (unit determinant; \(\theta\) is defined over \(\Lambda\)). So replacing \(z_j\) by \(z_j'\equiv z_j\bmod\varpi^D\) changes \(\varpi^NM(z)\) by a matrix in \(\varpi^DM_n(\mathcal O)\); \(D:=\max(N+c+1)\) over the finitely many conditions works. A \(\Lambda\)-algebra isomorphism \(\mathcal O/\varpi^D\cong\mathcal O'/\varpi'^D\) with \(\bar\varpi\mapsto\bar\varpi'\) carries these reduced polynomials to each other and, by (b), \(Y_j(\mathcal O/\varpi^D)\) to \(Y_j(\mathcal O'/\varpi'^D)\); quantifiers over \(Y_j(\mathcal O)\) become quantifiers over \(Y_j(\mathcal O/\varpi^D)\) by (b). \(\square\)

**Why it suffices.** Every later count (orbits in \(Kt^\lambda X(\mathcal O)\), structure constants, Satake coefficients) will be written as a count of solutions of conditions of type (e) in \(G(\mathcal O/\varpi^D)\) and \(H(\mathcal O/\varpi^D)\); membership in \(K\), in a Cartan double coset or in an Iwasawa cell is of that type by (c), (d). No closed embedding of \(X\) is needed for this, unlike Lemma 1.7.
SOURCES: satake-converse/nodes/A1-7.html (close rings, cosets) and A4-17.html; zhu-satake-v2.pdf, (2.1.2) Cartan decomposition and §5.6; EGA IV_4 18.5.17 (Hensel for smooth schemes; standard, not opened); Weyl modules over Z (Jantzen, Representations of Algebraic Groups, II.8; standard, not opened); Bruhat–Tits I (Cartan and Iwasawa decompositions over any non-archimedean local field; not opened)
--- A4-4 · Computation · Computation: $\Lambda_S=(1-\theta)X_*(T)$ and $(T^\theta)^\circ$ is maximal in $F_4$
STATUS CLAIMED: proved
PARENTS: A4-1
STATEMENT:
In \(X_*(T)=Q^\vee\) (simply connected \(E_6\), Bourbaki labels), with \(\theta=-w_L\):
1. \(e_1:=(1-\theta)\alpha_1^\vee=2\alpha_1^\vee+\alpha_2^\vee+2\alpha_3^\vee+2\alpha_4^\vee+\alpha_5^\vee=2\omega_1^\vee-\omega_6^\vee\), \(e_6:=(1-\theta)\alpha_6^\vee=\alpha_2^\vee+\alpha_3^\vee+2\alpha_4^\vee+2\alpha_5^\vee+2\alpha_6^\vee=-\omega_1^\vee+2\omega_6^\vee\), and \((1-\theta)\alpha_j^\vee=0\) for \(j=2,3,4,5\).
2. \(X_*(S)=\{\mu:\theta\mu=-\mu\}=\mathbb Ze_1\oplus\mathbb Ze_6=(1-\theta)X_*(T)=\Lambda_S\).
3. \(\operatorname{rank}X_*(T)^\theta=4\); so \(T_H:=(T^\theta)^\circ\), the split subtorus with \(X_*(T_H)=X_*(T)^\theta\), is a maximal torus of every fibre of \(H\).
4. \(3\omega_1^\vee=2e_1+e_6\).
Consequently, for \(\kappa\in\Lambda_S\) and any \(\mu\) with \((1-\theta)\mu=\kappa\), the point \(x_\kappa:=t^\mu x_0\in X(E)\) does not depend on \(\mu\) and \(\tau(x_\kappa)=t^\kappa\).
PROOF / BODY:
**What was computed.** \(w_L\) was found as the unique element of the group generated by \(s_2,\dots,s_5\) (order \(192=|W(D_4)|\), enumerated exactly) that negates \(\alpha_2^\vee,\dots,\alpha_5^\vee\); it has length \(12\). Then \(\theta=-w_L\) was applied to the simple coroots, the fundamental coweights were obtained by solving \(\langle\alpha_j,\omega_i^\vee\rangle=\delta_{ij}\) over \(\mathbb Q\), and items 1, 3, 4 were checked by exact rational arithmetic (script a4_lattice.py). As a consistency check, all \(19\) vectors \(\mu\in Q^\vee\) with coordinates in \([-4,4]\) and \(\theta\mu=-\mu\) lie in \(\mathbb Ze_1\oplus\mathbb Ze_6\).

**Proof of 2 (all of \(X_*(S)\), not only the box).**
1. \(\theta\mu=-\mu\iff w_L\mu=\mu\iff\langle\alpha_j,\mu\rangle=0\) for \(j=2,\dots,5\) (\(w_L=-1\) on the \(D_4\) span and \(+1\) on its orthogonal). So \(X_*(S)=Q^\vee\cap(\mathbb Z\omega_1^\vee\oplus\mathbb Z\omega_6^\vee)\).
2. \(P^\vee/Q^\vee\cong\mathbb Z/3\) and \(\omega_1^\vee\notin Q^\vee\) (its coordinates are \((\tfrac43,1,\tfrac53,2,\tfrac43,\tfrac23)\)), so \(X_*(S)\) has index \(3\) in \(\mathbb Z\omega_1^\vee\oplus\mathbb Z\omega_6^\vee\).
3. \(\mathbb Ze_1\oplus\mathbb Ze_6\subseteq X_*(S)\) has index \(\det\begin{pmatrix}2&-1\\-1&2\end{pmatrix}=3\) there too; hence equality. Since \((1-\theta)\) kills \(\alpha_2^\vee,\dots,\alpha_5^\vee\), \((1-\theta)X_*(T)=\mathbb Ze_1+\mathbb Ze_6\).

**Proof of the consequence.** \(\tau(t^\mu x_0)=t^\mu\theta(t^\mu)^{-1}=t^{\mu-\theta\mu}\). If \((1-\theta)\mu=(1-\theta)\mu'\), then \(\mu-\mu'\in X_*(T_H)\), \(t^{\mu-\mu'}\in T_H(E)\subseteq H(E)\), and \(t^\mu x_0=t^{\mu'}x_0\). \(T_H\) is split over \(\Lambda\) because \(X_*(T)^\theta=\ker(\theta-1)\) is saturated; it lies in \(H\) because \(\theta\) fixes it pointwise; it is maximal because \(\operatorname{rank}F_4=4\).

**Why it matters.** The labels need no \(\sqrt\varpi\) (unlike Lemma 1.7's \(x_\mu=\mu(\sqrt\varpi)x_0\)), so "\(x_\kappa\in Kt^\lambda X(\mathcal O)\)" can be tested through \(H(E)\) alone, and \(H(E)=\bigsqcup_\eta H(\mathcal O)t^\eta H(\mathcal O)\) with \(\eta\in X_*(T_H)\subseteq X_*(T)\). Item 4 identifies the extra generator \(\mathbf 1_{Kx_{3\omega_1^\vee}}\) of Proposition VII as a label in \(\Lambda_S^+\).
SOURCES: script a4_lattice.py (exact arithmetic, standard library); spherical-generation/families/e6-f4.html (E6/F4 data); satake-two-levels/audit/S4b.json S4-13 (e_i, W_X), S4.json S4-5 (uncovered labels 3kω1∨)
--- A4-5 · Proposition · Proposition: $K\backslash X(\mathbb F_q((t)))\simeq\Lambda_S^+$ by counting orbits
STATUS CLAIMED: proved
PARENTS: A4-2, A4-3, A4-4
STATEMENT:
Let \(p\) be odd, \(F=\mathbb F_q(\!(t)\!)\), \(\mathcal O=\mathbb F_q[\![t]\!]\), \(K=G(\mathcal O)\). Then
$$X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa,\qquad x_\kappa=t^\mu x_0\ \ ((1-\theta)\mu=\kappa),$$
equivalently \(G(F)=K\,T(F)\,H(F)\); so Axiom 2.4.1 holds for \(\mathcal M_\theta\) over \(\mathbb F_q[\![t]\!]\) and \(K\backslash X(F)\simeq\Lambda_S^+\). More precisely, for every test field \(E\) and \(\lambda\in\Lambda_T^+\), with \(Z_\lambda(E):=Kt^\lambda X(\mathcal O_E)\):
(a) \(Kx_\kappa=Kx_{\kappa'}\Rightarrow\kappa=\kappa'\), and \(X(E)=\bigcup_\lambda Z_\lambda(E)\);
(b) \(N_\lambda(E):=\#K\backslash Z_\lambda(E)\) and \(M_\lambda(E):=\{\kappa\in\Lambda_S^+:x_\kappa\in Z_\lambda(E)\}\) are finite and depend only on \((\mathcal O_E/\varpi^D,\bar\varpi)\) for \(D\ge D(\lambda)\), \(D(\lambda)\) independent of \(E\);
(c) if \(E,E'\) are \(D(\lambda)\)-close and every \(K\)-orbit in \(Z_\lambda(E')\) contains some \(x_\kappa\), then every \(K\)-orbit in \(Z_\lambda(E)\) contains some \(x_\kappa\).
PROOF / BODY:
**Inputs, valid over every test field \(E\) (characteristic \(0\) or \(p\)).**
(E1) Cartan decompositions \(G(E)=\bigsqcup_{\nu\in\Lambda_T^+}Kt^\nu K\) and \(H(E)=\bigsqcup_\eta H(\mathcal O)t^\eta H(\mathcal O)\), \(\eta\in X_*(T_H)\) dominant for a Borel subgroup of \(H\): \(H_\Lambda\) is split reductive with split maximal torus \(T_H\subseteq T\) ([[A4-4]]); Bruhat–Tits (any non-archimedean local field).
(E2) \(X(\mathcal O_E)=Kx_0\): S4-4 step 6 (Hensel for the smooth \(H\), Lang for its connected special fibre); smoothness (Edixhoven) and connectedness (Steinberg, Thm 8.1) need only \(p\ne2\), so this holds over \(\mathbb F_q[\![t]\!]\).
(E3) \(X(E)=G(E)x_0\): by S4-4 step 3 (scheme-theoretic, any \(\mathcal O\) with \(2\in\mathcal O^\times\)), \(\mathring X=B/(B\cap H)\) with \(B\cap H\) a Borel subgroup scheme of \(\mathrm{Spin}_8\), smooth, connected, split solvable, so \(H^1(E,B\cap H)=1\) and \(\mathring X(E)=B(E)x_0\). \(G(E)\)-orbits in \(X(E)\) are open (the orbit maps are smooth) and \(\mathring X(E)\) is dense in the analytic manifold \(X(E)\), so every \(G(E)\)-orbit meets \(B(E)x_0\). (No \(H^1(E,F_4)\) is needed.)
(E4) \(\theta(K)=K\), \(\tau(kx)=k\tau(x)\theta(k)^{-1}\), \(\tau(x_\kappa)=t^\kappa\), \(\tau(X(\mathcal O))\subseteq K\).

**Proof.**
1. *Separation.* By (E4), \(\tau(Kx_\kappa)\subseteq Kt^\kappa K\); distinct dominant \(\kappa\) give disjoint double cosets (E1).
2. *Covering.* \(X(E)=G(E)x_0=\bigcup_\lambda Kt^\lambda Kx_0=\bigcup_\lambda Z_\lambda(E)\) by (E3), (E1), (E2). This proves (a).
3. *Orbits in \(Z_\lambda\) as classes in \(K\).* Every \(K\)-orbit in \(Z_\lambda\) contains a point \(t^\lambda kx_0\), \(k\in K\), and \(t^\lambda kx_0\in Kt^\lambda k'x_0\) iff there is \(h\in H(E)\) with \(t^\lambda khk'^{-1}t^{-\lambda}\in K\) (put \(a=t^\lambda khk'^{-1}t^{-\lambda}\) in \(at^\lambda k'x_0=t^\lambda kx_0\)).
4. *Bounding \(h\).* Then \(h\in Kt^{-\lambda}Kt^\lambda K\), so by [[A4-2]](c) \(h\in Kt^cK\) with \(c\preceq\lambda+\lambda^*\), \(\lambda^*=-w_0\lambda\). Writing \(h\in H(\mathcal O)t^\eta H(\mathcal O)\) (E1) gives \(h\in Kt^{\eta^+}K\), \(\eta^+\) the dominant \(W\)-conjugate. So \(\eta\in B(\lambda):=\{\eta\in X_*(T_H)\ H\text{-dominant}:\eta^+\preceq\lambda+\lambda^*\}\), a finite set. Hence \(k\sim k'\) iff for some \(\eta\in B(\lambda)\) and \(h_1,h_2\in H(\mathcal O)\): \(\operatorname{val}\rho_i(t^\lambda kh_1t^\eta h_2k'^{-1}t^{-\lambda})=0\) for all \(i\) ([[A4-2]](c)).
5. *Determinacy of \(N_\lambda\).* For each of the finitely many \(\eta\) this is a condition of type [[A4-2]](e) on \((k,k',h_1,h_2)\in G(\mathcal O)^2\times H(\mathcal O)^2\). With \(D_1(\lambda)\) the largest of the resulting \(D\), \(\sim\) is the pullback of the equivalence relation \(\approx\) on \(G(\mathcal O/\varpi^{D_1})\): \(\bar k\approx\bar k'\) iff some \(\eta\in B(\lambda)\) and \(\bar h_1,\bar h_2\in H(\mathcal O/\varpi^{D_1})\) satisfy the reduced congruences. So \(N_\lambda(E)=\#\big(G(\mathcal O/\varpi^{D_1})/\!\approx\big)\), a function of \((\mathcal O/\varpi^{D_1},\bar\varpi)\).
6. *Determinacy of \(M_\lambda\).* Fix lifts \(\mu_\kappa\) with \((1-\theta)\mu_\kappa=\kappa\). By (E2), \(x_\kappa\in Z_\lambda\) iff \(t^{\mu_\kappa}h\in Kt^\lambda K\) for some \(h\in H(E)\); such \(h\) lies in \(Kt^{-\mu_\kappa}Kt^\lambda K\), hence in \(H(\mathcal O)t^\eta H(\mathcal O)\) with \(\eta^+\preceq(-\mu_\kappa)^++\lambda\); so membership is "\(\exists\eta,h_1,h_2:\operatorname{val}\rho_i(t^{\mu_\kappa}h_1t^\eta h_2)=\langle\lambda,w_0\omega_i\rangle\ \forall i\)", of type [[A4-2]](e). Moreover \(x_\kappa\in Z_\lambda\) gives \(t^\kappa=\tau(x_\kappa)\in Kt^\lambda Kt^{-\theta\lambda}K\) (E4), so \(\kappa\preceq\lambda+(-\theta\lambda)^+\): \(M_\lambda(E)\) lies in a finite set independent of \(E\), and is determined modulo \(\varpi^{D_2(\lambda)}\). This proves (b) with \(D(\lambda)=\max(D_1,D_2)\).
7. *Counting.* Let \(E,E'\) be \(D(\lambda)\)-close; then \(N_\lambda(E)=N_\lambda(E')\) and \(M_\lambda(E)=M_\lambda(E')\). If every orbit in \(Z_\lambda(E')\) contains some \(x_\kappa\), then \(\kappa\mapsto Kx_\kappa\) is a bijection \(M_\lambda(E')\to K\backslash Z_\lambda(E')\) (onto by hypothesis, injective by step 1), so \(N_\lambda(E')=|M_\lambda(E')|\). Over \(E\), \(\kappa\mapsto Kx_\kappa\) injects \(M_\lambda(E)\) into \(K\backslash Z_\lambda(E)\), and \(|M_\lambda(E)|=|M_\lambda(E')|=N_\lambda(E')=N_\lambda(E)\); an injection between finite sets of equal size is onto. This proves (c).
8. *Over \(\mathbb F_q(\!(t)\!)\).* For each \(\lambda\) take \(E'=F'_e\), \(e\ge D(\lambda)\): it is \(D(\lambda)\)-close to \(F\) ([[A4-2]](a)), and (H-Cartan) holds over it (S4-13, any ramification; [[A4-3]] item 3). By (c) every \(K\)-orbit in \(Z_\lambda(F)\) is some \(Kx_\kappa\); by step 2 and step 1, \(X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa\). As \(X(F)=G(F)x_0\) and \(x_\kappa\in T(F)x_0\), \(G(F)=KT(F)H(F)\). The same holds over \(\mathbb F_{q^n}(\!(t)\!)\) for every \(n\). \(\square\)

**Characteristic-\(p\) inputs.** Only (E1)–(E4). Not used: Knop, Luna–Vust, Gaitsgory–Nadler, Chen–Yi, Delorme–Sécherre or Bruhat–Tits III in characteristic \(p\), any statement over \(\overline{\mathbb F_q(\!(t)\!)}\), any part of Theorem 8.0.2. Delorme–Sécherre enters only over the \(p\)-adic fields \(F'_e\) (characteristic \(0\)). Since \(D(\lambda)\) grows with \(\lambda\), infinitely many \(F'_e\) are used, one for each \(\lambda\).
SOURCES: satake-two-levels/audit/S4.json S4-4 (steps 3, 6), S4-5 (distinct labels); S4b.json S4-12, S4-13 (Delorme–Sécherre Thm 4.8, Kneser, W_X in K); zhu-satake-v2.pdf (2.1.2) (Cartan decomposition over k((t))); Bruhat–Tits I (Cartan decomposition for split groups over any local field; not opened); Steinberg, Mem. AMS 80 (1968) Thm 8.1; Edixhoven, Compositio 81 (1992) Prop. 3.4 (as cited in S4-4); sakellaridis-v3.pdf, Axiom 2.4.1 p. 10 and the Remark after Axiom 2.4.2, p. 11
--- A4-6 · Proposition · Proposition: all structure constants and Satake coefficients agree over close fields
STATUS CLAIMED: proved
PARENTS: A4-2, A4-5
STATEMENT:
Let \(E,E'\) be test fields satisfying (H-Cartan) (by [[A4-5]] and S4-13: \(\mathbb F_q(\!(t)\!)\) and every \(p\)-adic \(F'\) with residue field \(\mathbb F_q\)). For \(\lambda\in\Lambda_T^+\), \(\mu,\nu\in\Lambda_S^+\) put
$$c^\nu_{\lambda\mu}(E):=(T_\lambda*\mathbf 1_{Kx_\mu})(x_\nu)=\#\{gK\subseteq Kt^\lambda K:\ g^{-1}x_\nu\in Kx_\mu\}.$$
(a) \(T_\lambda*\mathbf 1_{Kx_\mu}=\sum_{\nu\in N(\lambda,\mu)}c^\nu_{\lambda\mu}(E)\,\mathbf 1_{Kx_\nu}\), where \(N(\lambda,\mu):=\{\nu\in\Lambda_S^+:\nu\preceq\lambda+\mu+(-\theta\lambda)^+\}\) is finite and independent of \(E\).
(b) There is \(D(\lambda,\mu)\), independent of \(E\), with \(c^\nu_{\lambda\mu}(E)=c^\nu_{\lambda\mu}(E')\) for all \(\nu\) whenever \(E,E'\) are \(D(\lambda,\mu)\)-close.
(c) \(\mathcal S(T_\lambda)=\sum_{\nu\in\Lambda_T}q^{-\langle\rho,\nu\rangle}n_{\lambda\nu}(E)\,e^\nu\) with \(n_{\lambda\nu}(E)=\#\big((U(E)t^\nu K\cap Kt^\lambda K)/K\big)\), and there is \(D'(\lambda)\) with \(n_{\lambda\nu}(E)=n_{\lambda\nu}(E')\) for all \(\nu\) over \(D'(\lambda)\)-close fields (no Cartan decomposition of \(X\) is needed here). So \(\mathcal S(T_\lambda)\in\mathbb C[\Lambda_T]^W\) is the same element for \(E\) and \(E'\).
(d) For \(\mu=0\): \(c^\nu_{\lambda0}=(T_\lambda*\varphi_0)(x_\nu)\), which is Lemma 1.7(a).
PROOF / BODY:
**Proof.**
1. *Orbit test.* Under (H-Cartan), \(y\in Kx_\nu\iff\tau(y)\in Kt^\nu K\): if \(y\in Kx_{\nu'}\) then \(\tau(y)\in Kt^{\nu'}K\) by (E4) of [[A4-5]], and the double cosets of distinct dominant \(\nu'\) are disjoint.
2. *Cosets.* \(Kt^\lambda K/K\cong K/K_\lambda\), \(K_\lambda=K\cap t^\lambda Kt^{-\lambda}\supseteq K(c(\lambda))\), \(c(\lambda)=\max_\alpha|\langle\alpha,\lambda\rangle|\) (Iwahori factorization of \(K(c)\) into \(T(1+\varpi^c\mathcal O)\) and root groups \(U_\alpha(\varpi^c\mathcal O)\), as in Lemma 1.7). \(k\in K_\lambda\iff\operatorname{val}\rho_i(t^{-\lambda}kt^\lambda)=0\) for all \(i\) ([[A4-2]](c)), a condition of type [[A4-2]](e); so \(K_\lambda/K(D)\subseteq G(\mathcal O/\varpi^D)\) and its order are determined by \((\mathcal O/\varpi^D,\bar\varpi)\) for \(D\ge c(\lambda)\) large.
3. *Incidence.* For \(g=kt^\lambda\): \(\tau(g^{-1}x_\nu)=t^{-\lambda}k^{-1}\tau(x_\nu)\theta(k)\theta(t^\lambda)=t^{-\lambda}k^{-1}t^\nu\theta(k)t^{\theta\lambda}\). By step 1 and [[A4-2]](c),
$$c^\nu_{\lambda\mu}(E)=\#\{kK_\lambda\in K/K_\lambda:\ \operatorname{val}\rho_i\big(t^{-\lambda}k^{-1}t^\nu\theta(k)t^{\theta\lambda}\big)=\langle\mu,w_0\omega_i\rangle\ \forall i\}.$$
The matrix coefficients are \(\varpi^{-\langle\lambda,\chi\rangle+\langle\theta\lambda,\chi'\rangle}\sum_{\chi''}\rho_i(k^{-1})_{\chi\chi''}\varpi^{\langle\nu,\chi''\rangle}\rho_i(\theta(k))_{\chi''\chi'}\), polynomials over \(\Lambda\) in the coordinates of \(k\) and \(\varpi^{\pm1}\) (\(\theta\) is defined over \(\Lambda\)); the condition is right \(K_\lambda\)-invariant and of type [[A4-2]](e).
4. *Counting.* \(c^\nu_{\lambda\mu}(E)=\#\{\bar k\in G(\mathcal O/\varpi^D):\text{condition}\}\big/\#(K_\lambda/K(D))\) with \(D=D(\lambda,\mu,\nu)\); numerator and denominator are determined by \((\mathcal O/\varpi^D,\bar\varpi)\). Taking \(D(\lambda,\mu)=\max_{\nu\in N(\lambda,\mu)}D(\lambda,\mu,\nu)\) gives (b), once (a) shows that only \(\nu\in N(\lambda,\mu)\) occur.
5. *Support, (a).* \(T_\lambda*\mathbf 1_{Kx_\mu}\) is \(K\)-invariant, supported in \(Kt^\lambda Kx_\mu\), and \(\tau(Kt^\lambda Kx_\mu)\subseteq Kt^\lambda Kt^\mu Kt^{-\theta\lambda}K\); applying [[A4-2]](c) twice gives \(\nu\preceq\lambda+\mu+(-\theta\lambda)^+\) for every \(Kx_\nu\) in the support. Under (H-Cartan) the \(\mathbf 1_{Kx_\nu}\) form a basis of \(C_K(E)\).
6. *Satake, (c).* With \((\mathcal Sf)(t^\nu)=\delta_B(t^\nu)^{1/2}\int_{U(E)}f(t^\nu u)\,du\), \(\operatorname{vol}U(\mathcal O)=1\), \(\delta_B(t^\nu)^{1/2}=q^{-\langle\rho,\nu\rangle}\): \(\int_UT_\lambda(t^\nu u)\,du\) counts the cosets \(t^\nu uK\subseteq Kt^\lambda K\), i.e. \(n_{\lambda\nu}\) (two \(u\) give the same coset iff they differ by \(U(E)\cap K=U(\mathcal O)\)). Parametrizing \(Kt^\lambda K/K\) by \(K/K_\lambda\) as in step 2, \(kt^\lambda K\subseteq U(E)t^\nu K\iff\operatorname{val}(\ell_i\rho_i(kt^\lambda))=\langle\nu,w_0\omega_i\rangle\ \forall i\) ([[A4-2]](d)), a condition of type [[A4-2]](e); only \(\nu\) with \(\nu^+\preceq\lambda\) occur. So \(n_{\lambda\nu}\) agrees over \(D'(\lambda)\)-close fields. The normalization of \(\mathcal S\) is the same formula for both fields, so \(\mathcal S(T_\lambda)\) is the same element of \(\mathbb C[\Lambda_T]^W\). (For \(\mathbb F_q(\!(t)\!)\) this is consistent with Zhu, Lemma 5.6.3, which expresses \(\mathcal S^{-1}(\chi_\mu)\) through Kazhdan–Lusztig polynomials evaluated at \(q\).)
7. *(d)* is the case \(\mu=0\), since \(\mathbf 1_{Kx_0}=\varphi_0\) by (E2). Lemma 1.7 used a closed embedding \(X\subseteq V\) to test \(g^{-1}x_\mu\in X(\mathcal O)\); here the test is \(\tau(g^{-1}x_\mu)\in K\), valid under (H-Cartan) (or by [[A4-3]] item 1 in general). \(\square\)
SOURCES: satake-converse/nodes/A1-7.html (cosets, K(c) ⊆ K_λ, close rings); zhu-satake-v2.pdf, §5.6, Lemma 5.6.3 (classical Satake over F_q((t))); sakellaridis-v3.pdf §1.7 (Satake isomorphism, A^*), (8.2) p. 65; Bruhat–Tits I (Iwasawa decomposition; not opened)
--- A4-7 · Audit · Check: Theorem 8.0.2 over ramified $F'$ needs only Axiom 2.2.1 over $W(\mathbb F_q)$
STATUS CLAIMED: proved
PARENTS: A4-1
STATEMENT:
Let \(F'\) be a finite extension of \(\mathbb Q_q\) with residue field \(\mathbb F_q\) (any ramification), \(p\) odd.
(a) Fix \(\psi'\) of conductor \(\mathcal O'\) (exists: \(\psi'(x)=\psi_{\mathbb Q_p}(\operatorname{Tr}(cx))\) with \(c\mathcal O'=\mathfrak d_{F'/\mathbb Q_p}^{-1}\)) and \(dx(\mathcal O')=1\). Then (3.10), (4.1), Lemma 3.9.1, Lemma 4.2.3, (4.5), Tate's formulas of §5.3, (5.2) (case \(U\)) and (5.3) (case \((U,\psi)\)) hold over \(F'\) exactly as printed.
(b) (A1), (A2), (A3b), (A3c), (A4), (A5) hold for \(\mathcal M_\theta\) over \(\mathcal O'\). If Axiom 2.2.1 (A3a) holds for \(\mathcal M_\theta\) over \(\Lambda=W(\mathbb F_q)\) as \(\Lambda\)-schemes — call this \((I_\Lambda)\) — it holds over \(\mathcal O'\).
(c) Under \((I_\Lambda)\), Theorem 8.0.2 holds for \(\mathcal M_\theta\) over \(F'\), and the isomorphism \(\Phi_{F'}\) sends \(\mathbf 1_{K'x'_\kappa}\) to
$$P_\kappa=c^{-1}\,\delta_{P(X)}^{-1/2}(x_\kappa)\prod_{\check\theta\in\Theta^+}\big(1-q^{-4}T_{\check\theta}\big)s_{\check\lambda(\kappa)},\qquad c=\beta(\delta_{P(X)}^{1/2})^{-1},\quad\beta=\prod_{\check\gamma\in\check\Phi_X^+}\frac{1-e^{\check\gamma}}{1-q^{-4}e^{\check\gamma}},$$
\(\Theta^+=\check\Phi_X^+\), \(s\) the Schur polynomial of lowest weight \(\check\lambda(\kappa)\) (Sakellaridis's label of \(Kx_\kappa\)). \(P_\kappa\) depends only on \(q\), not on \(F'\).
(d) The statement of Theorem 8.0.2 involves no choice of \(\psi\) or of measures beyond \(\operatorname{vol}K=1\).
PROOF / BODY:
**(a) Normalizations.**
1. *Where the paper invokes the assumption.* §1.1 (p. 2): "we will assume that \(k\) is unramified over \(\mathbb Q_p\); the modifications needed to remove this assumption are trivial". §3.8 (p. 23): "\(o^2\) has measure \(1\) (this is the correct choice only under our simplifying assumption...)". §5.3 (p. 32): "here we are using our simplifying assumption...; these equations would have to be modified otherwise". In each place the computation uses only (i) \(\mathcal O\) is the largest fractional ideal on which \(\psi\) is trivial (conductor \(\mathcal O\)), (ii) \(dx(\mathcal O)=1\); by (i) the dual lattice of \(\mathcal O\) is \(\mathcal O\), so (ii) is the self-dual measure. §1.7 (p. 7), and §3.8 in the same sentence as the parenthetical, fix \(\psi\) "whose conductor is equal to the ring of integers"; with that \(\psi\) the parenthetical is not needed. The assumption would matter only for the standard character \(\psi_{\mathbb Q_p}\circ\operatorname{Tr}\), whose conductor is the inverse different, equal to \(\mathcal O\) exactly when \(k/\mathbb Q_p\) is unramified (Serre, *Local Fields*, III §3; not opened).
2. *(3.10).* For \(\psi'\) of conductor \(\mathcal O'\) and \(\omega=xw-yv\) on \(\mathcal O'^2\), the dual lattice of \(\mathcal O'^2\) is \(\mathcal O'^2\); so \(dx\,dy(\mathcal O'^2)=1\) is the self-dual measure, \(F\mathbf 1_{\mathcal O'^2}=\mathbf 1_{\mathcal O'^2}\), and the basic vector \(c_0\) is Fourier-invariant, as §3.8 requires.
3. *(4.1).* \(F\mathbf 1_{\mathfrak p\times\mathcal O}(v,w)=\int_{\mathfrak p}\psi(-xw)dx\int_{\mathcal O}\psi(yv)dy=q^{-1}\mathbf 1_{\mathfrak p^{-1}}(w)\mathbf 1_{\mathcal O}(v)\): uses (i), (ii).
4. *(5.2), case \(U\).* \(F(|x|^{s}dx\,dy)=\frac{1-q^{s}}{1-q^{-s-1}}\,\delta_0(v)|w|^{-s-1}dw\) is Tate's unramified formula of §5.3 with \(\chi=|\cdot|^{s+1}\); it uses (i), (ii). *(5.3), case \((U,\psi)\):* the \(y\)-integral gives \(\delta(v-1/x)\) for the self-dual \(dy\), then \(x=1/v\). Tate's ramified formula (Gauss sums, §5.3) uses (i), (ii) as well, and is needed only for ramified \(\tilde\chi\) in cases \(T\), \(N\), which do not occur for \(\mathcal M_\theta\) (S4-3, S4-5(3), S4-10).
5. *Lemma 3.9.1, Lemma 4.2.3, (4.5).* These use \(\mu(x_0J)=1\), \(\operatorname{vol}K=1\), \(\operatorname{vol}A_0=1\), \(Q=(U_0:U_1)\operatorname{Vol}(J)\) and Casselman–Shalika [CS80, Thm 5.4] (any \(p\)-adic field, \(\psi\) of conductor \(\mathcal O\)); all are expressed through \(q\).
6. The rest of the proof (§3.2–3.7, §6, §7, §8) is geometry over \(\bar F'\) (characteristic \(0\)) and combinatorics. So the paper's remark on p. 2 holds in the precise sense: no modification is needed once \(\psi'\) has conductor \(\mathcal O'\).

**(b) Hypotheses over \(\mathcal O'\).**
7. S4-4 (\(\mathcal O'\)-model, open orbit, \(X(\mathcal O')=K'x_0\), §8 conditions), S4-5 (Axiom 2.4.2 via Iwahori factorization, open cells and a square root of \(\zeta(\varpi')\); distinct labels; types \(G\), \(U\) only), S4-3 and S4-10 (A4, statements over \(\bar F'\)) and S4-12, S4-13 (Axiom 2.4.1 via Delorme–Sécherre for any non-archimedean locally compact field of odd residue characteristic, \(|J|=1\) by Kneser for any finite extension of \(\mathbb Q_p\)) use \(p\ne2\), Hensel and Lang, never \(e(F')=1\).
8. *Base change of (A3a).* \(\mathcal O'\) is finite free over \(\Lambda\), with \(\mathcal O'\otimes_\Lambda\mathbb Q_q=F'\). The maximal-rank \(B\)-orbits are defined over the base field (Sa06, Prop. 3.6.1, quoted on p. 9), so \(Y_{F'}=Y_{\mathbb Q_q}\otimes F'\). Scheme-theoretic closures commute with flat base change (Stacks, Tag 081I), and for the flat, open, generizing morphism \(X_{\mathcal O'}\to X_\Lambda\) closures of subsets pull back to closures. Hence Sakellaridis's \(\mathcal O'\)-scheme \(Y\) ("complement of the closure of the complement of \(Y_{F'}\) in its closure") is \(Y_\Lambda\otimes\mathcal O'\), likewise \(YP_\alpha\) and its complement; and quotients by \(R(P_\alpha)\) commute with flat base change. An isomorphism \(YP_\alpha/R(P_\alpha)\cong\) (one of the four models) over \(\Lambda\) therefore base-changes to \(\mathcal O'\), and so does the complement clause.

**(c) The images \(P_\kappa\).**
9. The proof of Theorem 8.0.2 (p. 64) defines the isomorphism by \(\mathbf 1_{x_{\check\lambda}K}\mapsto P_{\check\lambda}=\Omega_{\chi\delta(X)^{1/2}}(x_{\check\lambda})/L_X^{1/2}(\chi)\) (8.1). By (7.4) (p. 60) and Definition 7.2.3 (p. 62), \(P_{\check\lambda}=c^{-1}\delta_{P(X)}^{-1/2}(x_{\check\lambda})\prod_{\Theta^+}(1-\sigma_{\check\theta}q^{-r_{\check\theta}}T_{\check\theta})s_{\check\lambda}\), \(c\) as in (7.3). For \(\mathcal M_\theta\), \(\Theta^+=\check\Phi_X^+\times\{(+,4)\}\) (S4-3). \(\delta_{P(X)}(x_{\check\lambda})\) is a power of \(q\) fixed by \(\check\lambda\); the torus \(A_X^*\), the point \(\delta(X)^{1/2}\), \(W_X\) and the dictionary \(\check\lambda\leftrightarrow\kappa\) (S4-5: Sakellaridis's \(x_0a\), \(a=t^{-\mu}\), is our \(t^\mu x_0\)) are the same for every \(F'\) with residue field \(\mathbb F_q\). So \(P_\kappa\) depends only on \(q\).
10. *(d).* \(\mathcal H_K\) (with \(\operatorname{vol}K=1\)), \(\mathcal S\), \(r\) and \(\mathbf 1_{X(\mathcal O)}\) are canonical; \(\psi\), Fourier transforms and eigenmeasures occur only inside the proof. For \(\mathcal M_\theta\) the isomorphism with \(\varphi_0\mapsto1\) is moreover unique ([[A4-8]], step 4). \(\square\)
SOURCES: sakellaridis-v3.pdf: §1.1 p. 2; §1.7 p. 7; §3.8 pp. 22–23 and (3.10); §3.9 Lemma 3.9.1 p. 24; §4.1 (4.1) p. 25; Lemma 4.2.3 pp. 26–27; §4.3 (4.5) p. 27; §5.3 p. 32; (5.2) and (5.3) p. 33; (7.3), (7.4) p. 60; Definition 7.2.3 p. 62; Theorem 8.0.2, (8.1) and proof p. 64; satake-two-levels/audit S4.json S4-3, S4-4, S4-5; S4b.json S4-10, S4-12, S4-13; Serre, Local Fields III §3 (different; not opened); Stacks Project Tag 081I; Casselman–Shalika 1980 Thm 5.4 (cited by the paper; not opened)
--- A4-8 · Theorem · Theorem: Theorem 8.0.2 for $\mathcal M_\theta$ over $\mathbb F_q((t))$; (SG) fails
STATUS CLAIMED: proved
PARENTS: A4-5, A4-6, A4-7
STATEMENT:
Let \(p\) be odd (\(p=3\) included), \(F=\mathbb F_q(\!(t)\!)\), \(\mathcal O=\mathbb F_q[\![t]\!]\), \(X=\mathcal M_\theta\), and assume
\((I_\Lambda)\): Axiom 2.2.1 holds for \(\mathcal M_\theta\) over \(\Lambda=W(\mathbb F_q)\) as \(\Lambda\)-schemes (for the 45 maximal-rank Borel orbits and the 6 simple roots).
Then the linear map \(\Psi:C_c^\infty(X(F))^K\to S=\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), \(\Psi(\mathbf 1_{Kx_\kappa})=P_\kappa\) (\(\kappa\in\Lambda_S^+\), \(P_\kappa\) as in [[A4-7]](c)), is an isomorphism with
$$\Psi(f*\varphi)=r(\mathcal Sf)\cdot\Psi(\varphi)\quad(f\in\mathcal H_K),\qquad\Psi(\mathbf 1_{X(\mathcal O)})=1,$$
and it is the unique \(\mathcal H_K\)-module isomorphism \(C_K\to S\) with \(\mathbf 1_{X(\mathcal O)}\mapsto1\). Consequently, over \(F\): \(\mathcal H_K*\varphi_0=\Psi^{-1}(\operatorname{im}r)\ne C_K\), so (SG) fails; \(C_K=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{Kx_{3\omega_1^\vee}}\), \(C_K\) is not cyclic and needs exactly two generators; \(C_K/\mathcal H_K\varphi_0\) is the character with Satake parameter \(q^{\rho-\omega_4}\). Without \((I_\Lambda)\): the Cartan decomposition [[A4-5]] and the equality of all structure constants with those over \(F'_e\), \(e\gg0\) ([[A4-6]]), still hold.
PROOF / BODY:
**Proof.**
1. *Bases.* By [[A4-5]] the indicators \(b_\kappa=\mathbf 1_{Kx_\kappa}\), \(\kappa\in\Lambda_S^+\), form a basis of \(C_K(F)\); the same holds over every \(F'_e\) (S4-13). By [[A4-7]](b),(c), \((I_\Lambda)\) gives Theorem 8.0.2 over every \(F'_e\), so \(\{P_\kappa\}=\Phi_{F'_e}(\{b'_\kappa\})\) is a basis of \(S\), the same set for all \(e\). Hence \(\Psi\) is a linear bijection.
2. *Basic function.* \(X(\mathcal O)=Kx_0\) ((E2) of [[A4-5]]), \(x_0=x_{\kappa=0}\), and \(P_0=1\) (\(P_0=1\) is stated in the proof on p. 64).
3. *Equivariance.* Fix \(\lambda\in\Lambda_T^+\), \(\mu\in\Lambda_S^+\) and choose \(e\ge\max(D(\lambda,\mu),D'(\lambda))\) ([[A4-6]]); \(F'_e\) is \(e\)-close to \(F\) ([[A4-2]](a)). Then
$$\Psi(T_\lambda*b_\mu)=\sum_{\nu\in N(\lambda,\mu)}c^\nu_{\lambda\mu}(F)P_\nu=\sum_{\nu\in N(\lambda,\mu)}c^\nu_{\lambda\mu}(F'_e)P_\nu=\Phi_{F'_e}(T'_\lambda*b'_\mu)=r(\mathcal ST'_\lambda)P_\mu=r(\mathcal ST_\lambda)\Psi(b_\mu),$$
using [[A4-6]](a),(b) for the first two equalities, Theorem 8.0.2 over \(F'_e\) for the fourth, and [[A4-6]](c) for the last. The \(T_\lambda\) span \(\mathcal H_K\) and the \(b_\mu\) span \(C_K\).
4. *Uniqueness.* Let \(\Psi'\) be another such isomorphism and \(\Delta=\Psi'\Psi^{-1}\): an automorphism of \(S\), linear over \(R=\operatorname{im}r\), with \(\Delta(1)=1\), hence \(\Delta|_R=\mathrm{id}\). By S4-7 (unconditional, every \(q\)), \(R=\{g\in S:g(y_1)=g(\sigma y_1)\}\); so the ideal \(\mathfrak f=\{g\in S:g(y_1)=g(\sigma y_1)=0\}\) of \(S\) is non-zero and \(\mathfrak f\subseteq R\). For \(s\in S\) and \(0\ne a\in\mathfrak f\): \(as\in\mathfrak f\subseteq R\), so \(a\Delta(s)=\Delta(as)=as\); \(S\) is a domain, so \(\Delta(s)=s\).
5. *Consequences.* \(\Psi(\mathcal H_K*\varphi_0)=r(\mathbb C[A^*]^W)\cdot1=R\), and \(R\ne S\) with \(\dim S/R=1\) (S4-7; Proposition VII). So (SG) fails over \(F\). S4-8 derives the remaining statements from an isomorphism \(C_K\simeq S\) with \(\varphi_0\mapsto1\) together with \(P_{3\omega_1^\vee}\notin R\); \(\Psi\) is such an isomorphism and \(\Psi(\mathbf 1_{Kx_{3\omega_1^\vee}})\) is the same \(P_{3\omega_1^\vee}\) (it depends only on \(q\)), so S4-8 applies verbatim over \(F\).
6. *Inputs.* Characteristic \(p\): only (E1)–(E4) of [[A4-5]] and the Iwasawa decomposition. Characteristic \(0\): Sakellaridis's Theorem 8.0.2 over \(F'_e\) for all \(e\), whose hypotheses are (b) of [[A4-7]] plus \((I_\Lambda)\); Delorme–Sécherre over \(F'_e\). No hypothesis (A2)–(A5) over \(\mathbb F_q[\![t]\!]\), and no statement over an algebraic closure of \(\mathbb F_q(\!(t)\!)\), is used. In particular Axiom 2.2.1 over \(\mathbb F_q[\![t]\!]\) (part of (Q2)) is not needed for (Q1) by this route. \(\square\)

**Comparison with Lemma 1.7(c).** Lemma 1.7(c) with S4-8 over \(F'_e\) already gives \(\mathbf 1_{Kx_{3\omega_1^\vee}}\notin\mathcal H_K\varphi_0\) over \(F\) under \((I_\Lambda)\) (its (H-Cartan) on the \(p\)-adic side is S4-13). The theorem above adds the whole module structure, uniqueness, and the Cartan decomposition over \(F\).
SOURCES: sakellaridis-v3.pdf, Theorem 8.0.2 and proof p. 64, (8.1), Corollary 8.0.4 p. 65; satake-two-levels/audit/S4.json S4-7 (im r), S4-8 (two generators); S4b.json S4-13; satake-two-levels/props/e6-f4-module.html (Proposition VII); satake-converse/nodes/A1-7.html (Lemma 1.7(c))

== Part 1, side claims ==
--- A4-3 · Audit · Check: $\tau$ is a closed immersion over $W(\mathbb F_q)$, giving (H-int)
STATUS CLAIMED: proved
PARENTS: A4-2
STATEMENT:
Let \(p\) be odd and \(\Lambda=W(\mathbb F_q)\).
1. \(\tau:X_\Lambda\to G_\Lambda\), \(gH\mapsto g\theta(g)^{-1}\), is a closed immersion. Hence \(X_\Lambda\) is affine, and for every discrete valuation ring \(\mathcal O\) over \(\Lambda\) (e.g. \(\mathbb F_q[\![t]\!]\) or \(\mathcal O_{F'}\)): \(X(\mathcal O)=\{y\in X(\operatorname{Frac}\mathcal O):\tau(y)\in G(\mathcal O)\}\).
2. (H-int) of Lemma 1.7 holds: if \(G_\Lambda\hookrightarrow\mathrm{GL}(W)\) is a closed immersion with \(W\) free over \(\Lambda\) with a \(T\)-weight basis, then \(y\mapsto(\tau(y),\tau(y)^{-1})\) is a \(G\)-equivariant closed immersion \(X\hookrightarrow V:=\operatorname{End}W\oplus\operatorname{End}W\) for the action \(g\cdot(A,B)=(gA\theta(g)^{-1},\theta(g)Bg^{-1})\), on which \(T\) acts diagonally, and \(x_0\mapsto(1,1)\in V(\Lambda)\).
3. (H-Cartan) of Lemma 1.7 holds over every finite extension \(F'\) of \(\mathbb Q_q\) with residue field \(\mathbb F_q\), of any ramification (S4-12, S4-13), and over \(\mathbb F_q(\!(t)\!)\) ([[A4-5]]). The points \(x_\kappa\) are rational without \(\sqrt\varpi\) ([[A4-4]]).
PROOF / BODY:
**Proof of 1.**
1. *Monomorphism.* For a \(\Lambda\)-algebra \(R\) and \(g_1,g_2\in G(R)\): \(g_1\theta(g_1)^{-1}=g_2\theta(g_2)^{-1}\iff g_2^{-1}g_1=\theta(g_2^{-1}g_1)\iff g_2^{-1}g_1\in H(R)\). So the fppf sheaf map \(G/H\to G\) is injective and \(\tau\) is a monomorphism.
2. *Target.* \(Q:=\{g:\theta(g)=g^{-1}\}\) is the fixed scheme of the involution \(\iota(g)=\theta(g)^{-1}\) of the smooth \(G_\Lambda\); as \(2\in\Lambda^\times\) it is smooth (Edixhoven 1992, Prop. 3.4, as used in S4-4). \(\tau\) lands in \(Q\) and is equivariant for the twisted action \(g\cdot q=gq\theta(g)^{-1}\) of \(G\) on \(Q\).
3. *Fibres.* Over an algebraically closed field \(k\), \(\operatorname{char}k\ne2\), let \(q\in Q(k)\) and \(\theta_q:=\operatorname{Int}(q)\circ\theta\), an involution since \(\theta(q)=q^{-1}\). The stabilizer of \(q\) is \(G^{\theta_q}\) (smooth). Writing tangent vectors at \(q\) as \(\frac{d}{d\epsilon}\exp(\epsilon Y)q\), one computes \(\iota(\exp(\epsilon Y)q)=\exp(-\epsilon\operatorname{Ad}(q)d\theta(Y))\,q\), so \(T_qQ\) is the \((-1)\)-eigenspace of \(d\theta_q\). Hence \(\dim G\cdot q=\dim\mathfrak g-\dim\mathfrak g^{\theta_q}=\dim T_qQ=\dim_qQ\): every twisted orbit is open in \(Q_k\), so each orbit is open and closed. At \(x_0\), \(d\tau=2\cdot\mathrm{id}\) on \(\mathfrak p=\mathfrak g/\mathfrak h=T_1Q\), and by equivariance \(d\tau\) is bijective everywhere; so \(\tau_k\) is an étale monomorphism onto the open and closed orbit \(G\cdot1\), i.e. an isomorphism onto it (EGA IV\(_4\) 17.9.1).
4. *Open immersion over \(\Lambda\).* \(X\) and \(Q\) are flat of finite presentation and \(\tau\) is flat on fibres, hence flat (EGA IV\(_3\) 11.3.10); it is unramified (a monomorphism of finite type, EGA IV\(_4\) 17.2.6), hence étale, and an étale monomorphism is an open immersion (17.9.1).
5. *Closedness.* Let \(y\in\overline{\tau(X)}\smallsetminus\tau(X)\). Since \(\tau(X)_\eta\) is closed in the open \(Q_\eta\), \(y\) lies in the special fibre, in a geometric twisted orbit \(\Omega\ne\tau(X)_{\bar{\mathbb F}_q}\). Pass to \(\Lambda_r=W(\mathbb F_{q^r})\) (finite étale, compatible with closures), pick a closed point \(y'\in\Omega(\mathbb F_{q^r})\), and lift it to \(q'\in Q(\Lambda_r)\) (Hensel, \(Q\) smooth). If \(q'_\eta=\tau(x)\) with \(x\in X(E_r)\), \(E_r=\Lambda_r[1/p]\), then by (H-Cartan) over the \(p\)-adic field \(E_r\) (S4-13) \(x\in K_rx_\kappa\), so \(\tau(x)\in K_rt^\kappa K_r\cap G(\Lambda_r)\), forcing \(\kappa=0\), \(x=kx_0\) and \(y'=\bar k\theta(\bar k)^{-1}\in\tau(X)\): contradiction. Otherwise the twisted orbit map \(g\mapsto g\cdot q'\) is smooth (fibrewise smooth onto open orbits, plus flatness), so its image \(U\) is open; \(U_\eta=G\cdot q'_\eta\) misses the open and closed \(G\)-stable \(\tau(X)_\eta\), and \(U_s=\Omega\) misses \(\tau(X)_s\). So \(U\ni y\) is an open set missing \(\tau(X)\): contradiction. Thus \(\tau(X)\) is closed in \(Q\), hence in \(G\).
6. *Consequences.* A closed subscheme of the affine \(G_\Lambda\) is affine. For \(\mathcal O\) a domain over \(\Lambda\), an \(\operatorname{Frac}\mathcal O\)-point of the closed \(X\subseteq G\) whose image lies in \(G(\mathcal O)\) is an \(\mathcal O\)-point of \(X\). Closed immersions are stable under base change, so this holds over \(\mathbb F_q[\![t]\!]\).

**Proof of 2.** \(\mathrm{GL}(W)\to\operatorname{End}W\oplus\operatorname{End}W\), \(A\mapsto(A,A^{-1})\), is a closed immersion; compose with \(X\hookrightarrow G\hookrightarrow\mathrm{GL}(W)\). Equivariance: \(\tau(gx)=g\tau(x)\theta(g)^{-1}\). On the matrix unit \(e_\chi e_{\chi'}^*\), \(t\in T\) acts by \(\chi(t)\chi'(\theta(t))^{-1}\). A closed immersion \(G_\Lambda\hookrightarrow\mathrm{GL}(W)\) exists for the split reductive \(G_\Lambda\) (affine flat of finite type over a discrete valuation ring; e.g. SGA 3, Exp. VI\(_B\), 11.11, or a fibrewise faithful sum of Weyl modules and SGA 3, Exp. XVI, 1.5; not opened), and \(T\) acts diagonally on \(W\) because it is split.

**Proof of 3.** S4-12 applies Delorme–Sécherre's Theorem 4.8 (stated for any non-archimedean locally compact field of residue characteristic \(\ne2\)) with \(|J|=1\) from Kneser's \(H^1(F',\mathrm{Spin}_8)=1\), valid for every finite extension of \(\mathbb Q_p\); S4-13 moves labels into \(\Lambda_S^+\) by \(\dot n_i\in H(\mathbb Z)\cap N_G(T)\). Neither uses \(e(F')=1\).

**Remark (the pages' characteristic-\(p\) route).** The pages obtain (H-Cartan) over \(\mathbb F_q(\!(t)\!)\) in IIb step 1 from input D0 = Chen–Yi, Prop. 17(i). Chen–Yi work over \(\bar{\mathbb F}_p\), \(p\ne2\), but prove Prop. 17(i) by citing Gaitsgory–Nadler, Thm 8.2.9, and Nadler 2004, Thm 4.2, which are over \(\mathbb C\) (chen-yi-v4.pdf, p. 10). That is a characteristic-\(0\) citation used in characteristic \(p\) (checklist item 1); [[A4-5]] gives an independent proof for \(\mathcal M_\theta\).

**Status.** 1 and 3 are proved from the cited inputs; 2 uses the cited existence of a closed immersion \(G_\Lambda\hookrightarrow\mathrm{GL}(W)\). The main line ([[A4-5]], [[A4-6]], [[A4-8]]) does not need 1 or 2.
SOURCES: satake-converse/nodes/A1-7.html (hypotheses (H-int), (H-Cartan)); satake-two-levels/audit/S4.json S4-4 (Edixhoven 1992 Prop. 3.4); S4b.json S4-12, S4-13 (Delorme–Sécherre Thm 4.8, Kneser); EGA IV_3 11.3.10; EGA IV_4 17.2.6, 17.9.1 (standard, not opened); SGA 3 Exp. VI_B 11.11 and Exp. XVI 1.5 (not opened); chen-yi-v4.pdf, Proposition 17 and proof, p. 10
--- A4-9 · Proposition · Proposition: what the characteristic-$p$ theorem needs for symmetric $X$
STATUS CLAIMED: proved
PARENTS: A4-5, A4-6, A4-8
STATEMENT:
Let \(p\) be odd, \(\Lambda=W(\mathbb F_q)\), \(G\) split reductive over \(\Lambda\) with connected fibres and simply connected derived group, \(\theta\) an involution of \(G_\Lambda\) stabilizing a split maximal torus \(T\) that contains a maximal \(\theta\)-split torus \(S\) (split over \(\Lambda\)), \(H=G^\theta\), \(X=G/H\), \(x_0=eH\), \(\Lambda_S^+=X_*(S)\cap\Lambda_T^+\), \(x_\kappa:=\kappa(\sqrt\varpi)x_0\) (rational because \(S[2]\subseteq H\), and \(E(\sqrt\varpi)/E\) is separable for \(p\) odd; \(\tau(x_\kappa)=t^\kappa\)). Assume:
(T1) \(H\) has connected fibres and is split over \(\Lambda\);
(T2) \(X(E)=G(E)x_0\) for \(E=F\) and \(E=F'_e\), \(e\gg0\) (e.g. from \(\mathring X(E)=B(E)x_0\), part of (A5) over \(E\), by the density argument of [[A4-5]] (E3));
(T3) \(X(\mathcal O_E)=\{y\in X(E):\tau(y)\in G(\mathcal O_E)\}\) for these \(E\) (e.g. if \(\tau\) is a closed immersion over \(\Lambda\), as in [[A4-3]]);
(T4) \(X(F'_e)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa\) for \(e\gg0\) (e.g. Delorme–Sécherre with one \(H(F'_e)\)-class of maximal \((\theta,F'_e)\)-split tori and \(W_X\) realized in \(K\));
(T5) Theorem 8.0.2 holds over \(F'_e\) for \(e\gg0\), with Sakellaridis's labelling of \(Kx_\kappa\) independent of \(e\) (e.g. (A1)–(A5) over \(\mathcal O'_e\), with \(\psi'\) of conductor \(\mathcal O'_e\) as in [[A4-7]](a); (A2) and (A3a) over \(\mathcal O'_e\) follow from their versions over \(\Lambda\) by flat base change, [[A4-7]] step 8).
Then over \(F=\mathbb F_q(\!(t)\!)\): \(X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa\); \((T_\lambda*\mathbf 1_{Kx_\mu})(x_\nu)\) equals its value over \(F'_e\) for \(e\ge D(\lambda,\mu)\); and \(\mathbf 1_{Kx_\kappa}\mapsto P_\kappa\) (the images over \(F'_e\), given by (8.1) and (7.4), which depend only on \(q\) and the combinatorial data of \(X\)) is an isomorphism \(C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) compatible with \(\mathcal H_K\) and the Satake isomorphism, with \(\mathbf 1_{X(\mathcal O)}\mapsto1\).
PROOF / BODY:
**Proof.** The proofs of [[A4-5]], [[A4-6]] and [[A4-8]] (steps 1–3) go through with these changes.
1. *Cartan decomposition of \(H\).* By (T1), \(H(E)=\bigsqcup_\eta H(\mathcal O)\eta(\varpi)H(\mathcal O)\), \(\eta\in X_*(T_H)\) for a split maximal torus \(T_H\subseteq H\) over \(\Lambda\), not necessarily inside \(T\). \(T_H\) lies in a split maximal torus \(T'\) of \(G_\Lambda\) (inside the smooth reductive centralizer \(Z_G(T_H)\), lift a split maximal torus of the special fibre containing \(T_{H,\mathbb F_q}\); a torus over the henselian \(\Lambda\) with split special fibre is split), and \(T'=gTg^{-1}\) with \(g\in G(\Lambda)\) (conjugacy of split maximal tori over a local ring, SGA 3 Exp. XXVI 7.16; not opened). So \(\eta(\varpi)=g\,t^{\eta'}g^{-1}\in Kt^{\eta'}K\) with \(\eta'\in X_*(T)\): the matrix coefficients of \(\rho_i(\eta(\varpi))\) are polynomials in the fixed \(\Lambda\)-point \(g\) and in \(\varpi^{\pm1}\), so [[A4-2]](e) applies, and the bound of [[A4-5]] step 4 gives a finite set \(B(\lambda)\).
2. *\(X(\mathcal O)=Kx_0\).* (T1) and Hensel–Lang, as in (E2).
3. *Labels.* Membership \(x_\kappa\in Kt^\lambda X(\mathcal O)\) is tested, by (T3), as \(\exists k\in K:\operatorname{val}\rho_i(t^{-\lambda}k^{-1}t^\kappa\theta(k)t^{\theta\lambda})=0\ \forall i\), a condition of type [[A4-2]](e); it replaces [[A4-5]] step 6, which used \(x_\kappa\in T(F)x_0\). Separation (step 1 of [[A4-5]]) uses only \(\tau(x_\kappa)=t^\kappa\).
4. *Covering* uses (T2); *counting* uses (T4) on the \(p\)-adic side; [[A4-6]] uses only (H-Cartan) on both sides, \(\theta\) over \(\Lambda\) and \(\tau(x_\nu)=t^\nu\).
5. *Module transfer* is [[A4-8]] steps 1–3 with (T5); \(P_\kappa\) is independent of \(e\) because (8.1) and (7.4) express it through \(q\), \(\Theta^+\) with its signs and exponents, \(\check\Phi_X\) and \(\delta_{P(X)}\), which are the same for all \(F'_e\). The uniqueness step of [[A4-8]] used the \(E_6/F_4\) conductor and is not claimed here. \(\square\)

**Reading for (Q1).** The characteristic-\(p\) proof needs, in characteristic \(p\), only (T1)–(T3) (and the Cartan and Iwasawa decompositions of \(G(F)\)); no statement over an algebraic closure of \(\mathbb F_q(\!(t)\!)\), no Knop–Brion–Luna theory and no axiom of Sakellaridis over \(\mathbb F_q[\![t]\!]\). The price is (T4), (T5) at infinitely many ramified \(p\)-adic places with residue field \(\mathbb F_q\). For \(\mathcal M_\theta\): (T1) by S4-4 and [[A4-4]]; (T2) by [[A4-5]] (E3); (T3) by [[A4-3]]; (T4) by S4-12, S4-13; (T5) by [[A4-7]] under \((I_\Lambda)\). Since \(\Lambda_S=(1-\theta)X_*(T)\) for \(\mathcal M_\theta\), the proof of [[A4-5]] there avoids (T3).
SOURCES: satake-converse/nodes/A1-7.html (x_μ = μ(√ϖ)x_0, rationality); sakellaridis-v3.pdf, §1.7 p. 7 (assumptions), §8 p. 63, (7.4) p. 60, (8.1) p. 64; satake-two-levels/audit S4.json S4-4; S4b.json S4-12, S4-13; Delorme–Sécherre, arXiv:math/0612545, Thm 4.8 (as quoted in S4-12; not opened here)

== Part 2 ==
--- A3-1 · Audit · Where the $p$-adic verification (S4, S4b) used characteristic 0, and what replaces it
STATUS CLAIMED: proved
PARENTS: 
STATEMENT:
**Setting.** \(p\) an odd prime, \(q=p^f\), \(F=\mathbb F_q(\!(t)\!)\), \(\mathcal O=\mathbb F_q[[t]]\), \(\varpi=t\), \(\bar F\) an algebraic closure and \(F^{s}\) the separable closure. \(G=G_{\mathbb Z}\otimes\mathcal O\) is the split simply connected group of type \(E_6\) with pinning \((T,B,\{x_{\alpha_i}\})\), Bourbaki labels; \(\theta=\theta_0=\mathrm{Int}(\sigma_L)\circ C\), an involution defined over \(\mathbb Z\) with \(\theta|_T=-w_L\) (S4-11). \(H:=G^{\theta}\) (scheme-theoretic fixed points), \(X:=G/H\) with the left action, \(x_0=eH\), \(K=G(\mathcal O)\), \(J\) the Iwahori subgroup, \(\tau(gH)=g\theta(g)^{-1}\). \(P=U_P\rtimes L\) is the standard parabolic of the black nodes \(\{2,3,4,5\}\), \(L=Z_G(S)\), \(S=(T^{-\theta})^\circ\), \(\Phi_L\) the 24 roots of the black \(D_4\), \(T_8\) the maximal torus of \([L,L]=\mathrm{Spin}_8\) (cocharacters \(\alpha_2^\vee,\dots,\alpha_5^\vee\)), \(\beta_i:=w_L\alpha_i=-\theta\alpha_i\), \(\gamma_1=\alpha_1+\beta_1=2\omega_1-\omega_6\), \(\gamma_6=\alpha_6+\beta_6=-\omega_1+2\omega_6\). \(A_X=T/T^\theta\), \(\Lambda_X=X_*(A_X)\), \(e_i\in\Lambda_X\) the image of \(\alpha_i^\vee\) (\(i=1,6\)); \(\varphi=1-\theta\) on \(\Lambda_T\), \(\Lambda_S=\varphi(\Lambda_T)\). \(B\)-eigenfunctions are normalized by \(f(b^{-1}x)=\chi(b)f(x)\), which is Sakellaridis's right-action convention under \(x\cdot g\leftrightarrow g^{-1}x\).

**Claim (audit).** The table lists every characteristic-0 or unchecked input that S4 and S4b (nodes S4-3 to S4-13) used for the hypotheses of (Q3), and its replacement. After the replacements, the only obligation of (Q3) not proved in characteristic \(p\) is the analytic content of Statement 6.3.1 ([[A3-11]]).

| S4/S4b step | input used there | replacement over \(\mathbb F_q(\!(t)\!)\) | node |
|---|---|---|---|
| S4-4 (C1): \(H_{\mathcal O}\) smooth | Edixhoven, Prop. 3.4 | Conrad, Lemma 2.2.4 (any base) | [[A3-2]] |
| S4-4 (C2): fibres of \(H\) connected | Steinberg 1968, Thm 8.1 | direct proof: \(T^\theta=T_8\), \(Z_G(T_8)=T\), \(\mathrm{Aut}(F_4)\) inner | [[A3-2]] |
| S4-4: \(X_{\mathcal O}\) quasi-affine | Alper 2014 (number unchecked) | Conrad, Thm 2.3.1 | [[A3-2]] |
| S4-4 step 7: \(X_F\) affine | Matsushima (char. 0) | Richardson 1977 / Haboush | [[A3-3]] |
| S4-12: \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\) | Kneser (char. 0) | Bruhat–Tits III, Thm 4.7 | [[A3-3]] |
| S4-12 step 5: \(A'\) is \(G(F)\)-conjugate to \(S\) | Helminck–Wang, Prop. 10.3 | not needed; direct proof | [[A3-4]] |
| S4-12 step 6: maximal \(\theta\)-split tori conjugate over \(\bar F\) | Vust (char. 0), Richardson | direct proof for char. \(\ne2\) | [[A3-4]] |
| S4-12: Delorme–Sécherre, Thm 4.8 | proof not checked | proof checked for \(\mathbb F_q(\!(t)\!)\); one lemma replaced | [[A3-5]] |
| S4-5(3), S4-10: only types \(G,U\) | Knop 1995 (lattice transport) | maximal-rank orbits via \(\theta\)-stable tori | [[A3-6]] |
| S4-3 step 3: valuation cone | Knop 1991, §5 (char. 0) | arc valuations + \(D_5\)-localization | [[A3-7]] |
| S4-3 step 6: colours | Helgason's theorem (char. 0) | \(\mathrm{Pic}\,X=0\), units, divisor map | [[A3-7]] |
| S4-3 step 9: colour of \(\gamma_1\) | Lemma 7.1.2 (case analysis) | first edge of every Brion path is \(\alpha_1\) | [[A3-10]] |
| S4-3 step 10: Statement 6.3.1 | Thm 6.3.2 with Thm 6.11.1 (Wasserman, char. 0) | combinatorics checked; analytic part open | [[A3-10]], [[A3-11]] |
| S4-5(1): distinct labels | Cartan decomposition of \(G(F)\) | valid for every local field | [[A3-8]] |
| S4-5(2): Axiom 2.4.2 | \(\sqrt\varpi\) | \(F(\sqrt t)/F\) separable, \(p\) odd | [[A3-9]] |
PROOF / BODY:
1. **How the table was made.** Every node S4-3, S4-4, S4-5, S4-10, S4-11, S4-12, S4-13 of the salvage returns was read and each cited input classified as characteristic-free, characteristic-0, or unchecked. Characteristic-free inputs are kept: existence of Chevalley group schemes over \(\mathbb Z\) and the isomorphism theorem (SGA 3, used in S4-11); EGA IV 17.9.1 and 21.12.7 (any locally noetherian scheme); Hensel plus Lang for \(H^1(\mathcal O,H)\) (Conrad, discussion after Remark 7.2.13, for every non-archimedean local field); Bruhat–Tits theory of the building (every non-archimedean local field); additive and multiplicative Hilbert 90 (every field).
2. **What changes in characteristic \(p\) and how it is met.** (a) \(\sqrt\varpi\): \(F(\sqrt t)/F\) is separable because \(p\ne2\) ([[A3-9]]). (b) \(F^\times/F^{\times2}\) has order 4 because \(p\ne2\) (used only in Delorme–Sécherre, Lemma 3.6, which Theorem 4.8 does not need). (c) Inseparable orbit maps: every stabilizer used is computed scheme-theoretically and is smooth: \(H\), \(B\cap H\), \(P\cap H\), \(Q\cap H\), \(M^\theta\), the stabilizers of the rank-one and rank-two quotients, \(T^\theta\), \(N_H(S)\) ([[A3-2]], [[A3-6]]). (d) Infinite Galois cohomology: \(H^1(F,\mathbb Z/p)\) and \(H^1(F,\mu_p)\) are infinite, but the only finite group schemes that occur are \(S[2]\simeq\mu_2^2\) (étale) and \(N_H(S)/Z_H(S)\simeq S_3\) (constant); the descent cocycle of [[A3-4]] lands in \(Z_H(S)=\mathrm{Spin}_8\), never in \(N_H(S)\), so \(H^1(F,S_3)\), infinite for \(p=3\), does not enter. (e) The bad prime \(p=3\): the centre \(\mu_3\) of \(E_6\) is infinitesimal in characteristic 3 but is never used; \(X^*(T)/(1-\theta^*)X^*(T)\) is torsion-free, so \(T^\theta\) contains no \(\mu_3\).
3. **Scope.** Not covered here: (A3a) as \(\mathcal O\)-schemes (Q2), the proof of Theorem 8.0.2 in characteristic \(p\) (Q1), and the analytic part of Statement 6.3.1, which is part of (Q1) ([[A3-11]]).
SOURCES: S4.json and S4b.json (salvage returns, nodes S4-3 to S4-13); sakellaridis-v3.pdf, §1.1 p. 2; §1.7 pp. 6–7; Axioms 2.2.1, 2.4.1, 2.4.2 pp. 9–11; opening of §6 p. 41; §7 p. 58; §8 p. 63; conrad-reductive-group-schemes.pdf, Lemma 2.2.4, Thm 2.3.1, discussion after Remark 7.2.13
--- A3-2 · Proposition · (A1), (A2) for $\mathcal M_\theta$ over any complete DVR with 2 invertible
STATUS CLAIMED: proved
PARENTS: A3-1
STATEMENT:
Let \(\mathcal O\) be a complete discrete valuation ring with \(2\in\mathcal O^\times\) and finite residue field \(\mathbb F_q\) (for instance \(\mathbb F_q[[t]]\), \(p\) odd), \(F\) its fraction field. Then:
1. (A1) \(G\) is split reductive over \(\mathcal O\) with connected fibres and simply connected derived group.
2. \(H=G^\theta\) is a smooth closed subgroup scheme with \(\mathrm{Lie}\,H=\mathfrak g^\theta\) of rank 52; each geometric fibre of \(H\) is connected reductive of type \(F_4\), with maximal torus \(T^\theta=T_8\).
3. \(X=G/H\) is represented by a smooth quasi-affine \(\mathcal O\)-scheme of relative dimension 26, and \(\tau:X\to G\) is a monomorphism.
4. \(T^\theta=T_8\) is a split torus; \(B\cap H=B_L\cap\mathrm{Spin}_8=:B_8\) and \(P\cap H=\mathrm{Spin}_8\) scheme-theoretically; \((u,a)\mapsto uax_0\) is an open immersion \(U_P\times A_X\hookrightarrow X\) whose image is \(\mathring X:=X\smallsetminus\overline{(X\smallsetminus\mathring X_F)}\); \(\mathring X\) is smooth and surjective over \(\mathcal O\) and \(\mathring X_{\mathbb F_q}\) is one \(B_{\mathbb F_q}\)-orbit with smooth stabilizer.
5. \(X\) carries a \(G\)-invariant relative volume form and \(\mathring X\) a \(B\)-invariant one, over \(\mathcal O\); their absolute values give a \(G(F)\)-invariant measure on \(X(F)\) and a \(B(F)\)-invariant measure on \(\mathring X(F)\).
6. \(X(\mathcal O)=Kx_0\), so \(\varphi_0=\mathbf 1_{X(\mathcal O)}=\mathbf 1_{Kx_0}\).

In particular (A1) and (A2) hold for \(\mathcal M_\theta\) over \(\mathbb F_q[[t]]\) for every odd \(p\), including \(p=3\).
PROOF / BODY:
1. **(A1).** Chevalley–Demazure group scheme over \(\mathbb Z\), base-changed to \(\mathcal O\).
2. **Smoothness of \(H\).** Let \(G':=G\rtimes(\mathbb Z/2)_{\mathcal O}\), the generator acting by \(\theta\) (an automorphism of order 2 over \(\mathbb Z\), S4-11). Since \(2\in\mathcal O^\times\), \((\mathbb Z/2)_{\mathcal O}\simeq\mu_{2,\mathcal O}\) is of multiplicative type, so by Conrad, Lemma 2.2.4, its centralizer \(Z_{G'}(\mu_2)\) is smooth with Lie algebra \(\mathrm{Lie}(G')^{\mu_2}\). An element \(g\varepsilon\) centralizes the generator iff \(\theta(g)=g\), so \(Z_{G'}(\mu_2)=H\times\mu_2\) and \(H\), the fibre over the identity section of the étale \(\mu_2\), is smooth with \(\mathrm{Lie}\,H=\mathfrak g^\theta\). Over \(\mathbb Z[1/2]\) the idempotent \((1+d\theta)/2\) splits \(\mathfrak g\), and the rank of \(\mathfrak g^\theta\) is \(4+24+24=52\): the \(+1\)-eigenspace of \(-w_L\) on \(\mathfrak t\) is the black coroot span; \(\theta\) is the identity on \(\mathrm{Spin}_8\) (S4-11); the 48 other roots form 24 orbits \(\{\beta,\theta\beta\}\) with \(\theta\beta\ne\beta\) (exact computation: the \(\theta\)-fixed roots are exactly \(\Phi_L\), no root has \(\theta\beta=-\beta\)).
3. **Fibres are \(F_4\) and connected.** Let \(k\) be an algebraically closed field of characteristic \(\ne2\). (a) The character group of \(T^\theta=\ker(\theta-1)\) is \(X^*(T)/(1-\theta^*)X^*(T)\); the elementary divisors of \(1-\theta^*\) are \((1,1)\) (computed), so the quotient is free of rank 4: \(T^\theta\) is a torus, equal to \(T_8\), in every characteristic. (b) No root of \(E_6\) is orthogonal to all of \(\alpha_2^\vee,\dots,\alpha_5^\vee\) (computed), so \(Z_G(T_8)=T\). (c) Hence \(T_8\) is a maximal torus of \(H_k^\circ\) (its centralizer there lies in \(T\cap H=T_8\)). (d) \(H_k^\circ\) is reductive: identity component of the fixed points of an automorphism of order prime to \(\mathrm{char}\,k\) of a reductive group (Prasad–Yu 2002, Thm 2.1; CGP, Prop. A.8.12, as quoted in Conrad, Remark 3.1.5; Steinberg 1968; all characteristic-free, not opened). Its \(T_8\)-roots on \(\mathfrak g^\theta\) are the 24 roots of \(\Phi_L\) and the 24 restrictions \(\beta|_{T_8}\) of the pairs, each with a one-dimensional space: the root system \(F_4\). So \(H_k^\circ\simeq F_4\). (e) Let \(g\in H(k)\). \(\mathrm{Int}(g)\) preserves \(H_k^\circ\), and every automorphism of \(F_4\) is inner (Conrad, Thm 7.1.9: automorphisms modulo \(G_{\mathrm{ad}}\) are automorphisms of the based root datum, trivial for \(F_4\)); so \(\mathrm{Int}(g)|_{H^\circ}=\mathrm{Int}(h)\) with \(h\in H^\circ(k)\), and \(z=h^{-1}g\) centralizes \(T_8\). By (b) \(z\in T\cap H=T^\theta=T_8\subseteq H^\circ\). Thus \(H(k)=H^\circ(k)\), and \(H_k\), being smooth, is connected. (This replaces Steinberg's Thm 8.1, which is characteristic-free in any case.)
4. **\(X\) quasi-affine.** Conrad, Thm 2.3.1, applied to the smooth affine \(G'\) and its multiplicative-type subgroup \(\mu_2\): \(G'/Z_{G'}(\mu_2)\) is a smooth quasi-affine \(\mathcal O\)-scheme (built from Artin's theorem, Knutson's representability for quasi-finite separated algebraic spaces and Zariski's Main Theorem). Since \(G'=G\cdot\mu_2\) and \(G\cap(H\times\mu_2)=H\), it is \(G/H\). Relative dimension \(78-52=26\). \(\tau\) is a monomorphism: \(g\theta(g)^{-1}=g'\theta(g')^{-1}\iff g^{-1}g'\in H\) on points with values in any \(\mathcal O\)-algebra.
5. **The open orbit.** For \(g=ul\in B\cap H\) (\(u\in U_P\), \(l\in B_L\)): \(\theta(u)=u\,l\theta(l)^{-1}\in\theta(U_P)\cap P=U_P^-\cap P=1\) (\(\theta\) maps \(\Phi^+\smallsetminus\Phi_L\) into \(\Phi^-\), computed; \(U_P^-\times P\to G\) is an open immersion over \(\mathbb Z\)), so \(u=1\) and \(l\in B_L\cap L^\theta\). \(L=S\cdot\mathrm{Spin}_8\) with \(S\cap\mathrm{Spin}_8=S[2]\simeq\mu_2^2\) étale, \(\theta(sm)=s^{-1}m\), so \(L^\theta=S[2]\cdot\mathrm{Spin}_8=\mathrm{Spin}_8\) on geometric fibres (\(S[2]\subseteq T_8\), family page §4); \(L^\theta\) is smooth (step 2) and contains the smooth \(\mathrm{Spin}_8\) with the same fibres, so they are equal (EGA IV 17.9.5). Hence \(B\cap H=B_8\) (smooth, connected fibres, relative dimension 16) and \(P\cap H=\mathrm{Spin}_8\). The orbit map \(B/B_8\to X\) is a monomorphism of smooth \(\mathcal O\)-schemes of relative dimension \(42-16=26\); on each fibre it is unramified between smooth varieties of equal dimension, hence étale and an open immersion; by the fibrewise flatness criterion it is flat, hence an open immersion (EGA IV 17.9.1). \(B/B_8\simeq U_P\times B_L/B_8=U_P\times T/T_8=U_P\times A_X\) is affine, and the purity argument of S4-4 step 4 (EGA IV 21.12.7 in the regular 27-dimensional \(X\)) gives \(\mathring X=U_P\times A_X\). The stabilizer \(B_8\) is smooth, so the orbit map of the special fibre is separable.
6. **Forms.** \(\Omega^{26}_{X/\mathcal O}=G\times^H\det(\mathfrak g/\mathfrak h)^\vee\) and \(H\) has no non-trivial characters (connected semisimple fibres), so the bundle is \(G\)-equivariantly trivial. On \(\mathring X=B/B_8\), \(B_8\) acts on \(\det(\mathfrak b/\mathfrak b_8)\) by \(2\rho-2\rho_L\) restricted to \(T_8\), which pairs to \(1\cdot2-1\cdot2=0\) with \(\alpha_j^\vee\), \(j=2,\dots,5\) (\(\langle\rho_L,\alpha_j^\vee\rangle=(-3,1,1,1,1,-3)\), computed). Weil's construction of measures from volume forms works over every non-archimedean local field.
7. **\(X(\mathcal O)=Kx_0\).** \(G\to X\) is an \(H\)-torsor; \(H^1(\mathcal O_{\text{ét}},H)=1\) for smooth affine \(H\) with connected fibres over the henselian \(\mathcal O\) with finite residue field (Hensel reduces to \(H^1(\mathbb F_q,H_{\mathbb F_q})\), which vanishes by Lang): the argument recorded in Conrad after Remark 7.2.13 for any non-archimedean local field. So every \(x\in X(\mathcal O)\) lifts to \(G(\mathcal O)\).

**Characteristic.** Only \(2\in\mathcal O^\times\) is used (steps 2, 5, 6); no step uses \(\mathrm{char}\,F=0\).
SOURCES: conrad-reductive-group-schemes.pdf, Lemma 2.2.4 (p. 64–66), Thm 2.3.1 (pp. 67–69), Remark 3.1.5, Thm 7.1.9, discussion after Remark 7.2.13 (p. 259); EGA IV 17.9.1, 17.9.5, 21.12.7 (not in local sources); Prasad–Yu, Invent. Math. 147 (2002), Thm 2.1; CGP Prop. A.8.12 (cited, not opened); families/e6-f4.html §4 (S[2] ⊆ T_8); S4-4, S4-11
--- A3-3 · Lemma · (A5) and Galois cohomology over $\mathbb F_q((t))$: $X(F)=G(F)x_0$, $X_F$ affine
STATUS CLAIMED: proved
PARENTS: A3-2
STATEMENT:
Let \(F=\mathbb F_q(\!(t)\!)\), \(p\) odd (the same holds for \(p\)-adic \(F\), \(p\) odd).
1. \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\) (Bruhat–Tits III, Thm 4.7: \(H^1(K,G)=1\) for \(G\) semisimple simply connected over a henselian discretely valued field \(K\) with perfect residue field of cohomological dimension \(\le1\)). \(H^1(F,T')=1\) for split tori and \(H^1(F,U')=1\) for split unipotent groups.
2. \(X(F)=G(F)\,x_0\).
3. \(A_X(F)x_0=L(F)x_0=T(F)x_0\), and \(A_X(\mathcal O)\) is the image of \(T(\mathcal O)\).
4. \(X_F\) is affine.
5. \(\mathring X(F)=B(F)\,x_0\).
Hence (A5) holds for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\). Only \(H^1\) of smooth connected groups is used, for which étale and fppf cohomology agree; \(H^1(F,\mu_2)=F^\times/F^{\times2}\) has order 4; no \(\mu_p\), \(\alpha_p\) or Artin–Schreier classes occur.
PROOF / BODY:
1. **(1).** Bruhat–Tits III (J. Fac. Sci. Univ. Tokyo 34 (1987)), Thm 4.7, cited (not in the local sources); its standing hypotheses cover every complete discretely valued field with perfect residue field of dimension \(\le1\), in particular \(\mathbb F_q(\!(t)\!)\) — it is the characteristic-free replacement of Kneser's theorem. \(\mathrm{Spin}_8\) and \(F_4\) are split simply connected. Split tori: Hilbert 90. Split unipotent: a central series with quotients \(\mathbb G_a\) and \(H^1(F,\mathbb G_a)=0\).
2. **(2).** Exact sequence of pointed sets \(G(F)\to X(F)\to H^1(F,H)\) for the étale \(H\)-torsor \(G\to X\) ([[A3-2]]) and \(H^1(F,H)=H^1(F,F_4)=1\).
3. **(3).** \(Lx_0\simeq L/L^\theta=L/\mathrm{Spin}_8=A_X\simeq T/T_8\) ([[A3-2]]); \(L(F)\to A_X(F)\) is onto since \(H^1(F,\mathrm{Spin}_8)=1\), \(T(F)\to A_X(F)\) is onto since \(H^1(F,T_8)=1\), and \(T(\mathcal O)\to A_X(\mathcal O)\) is onto since \(H^1(\mathcal O,T_8)=\mathrm{Pic}(\mathcal O)^4=1\).
4. **(4).** \(H_F\) is reductive, so \(G_F/H_F\) is affine: Richardson, Bull. LMS 9 (1977), Thm A (\(G/H\) affine iff \(H\) reductive, every characteristic, via Haboush's theorem); equivalently Haboush plus Mumford's GIT, Thm 1.1 and Amplification 1.3 (the free right action of \(H\) on the affine \(G\) has closed orbits, so \(\mathrm{Spec}\,F[G]^H\) is a geometric quotient). Cited, not opened; this replaces Matsushima's theorem used in S4-4.
5. **(5).** \(\mathring X=B/B_8\) ([[A3-2]]), \(B_8=T_8\ltimes U_8\) a split connected solvable group over \(\mathbb F_q\); \(H^1(F,B_8)\to H^1(F,T_8)=1\) has trivial fibre \(\mathrm{im}\,H^1(F,U_8)=1\), so \(H^1(F,B_8)=1\) and \(B(F)\to(B/B_8)(F)\) is onto.
SOURCES: Bruhat–Tits, Groupes algébriques sur un corps local III, J. Fac. Sci. Univ. Tokyo 34 (1987), Thm 4.7 (cited, not opened); Richardson, Bull. London Math. Soc. 9 (1977), Thm A; Haboush, Ann. Math. 102 (1975) (cited); sakellaridis-v3.pdf, §8 p. 63; S4-4 step 7, S4-12
--- A3-4 · Proposition · Maximal $\theta$-split tori: dimension 2, one $H(F)$-class (residue char. $\ne2$)
STATUS CLAIMED: proved
PARENTS: A3-2, A3-3
STATEMENT:
Let \(k\) be a field, \(\mathrm{char}\,k\ne2\). A torus \(A\subseteq G_k\) is \(\theta\)-split if \(\theta(a)=a^{-1}\) on \(A\), and \((\theta,k)\)-split if moreover it is \(k\)-split.
- (a) For every \(\theta\)-stable maximal torus \(T'\) of \(G_{\bar k}\), \(\dim(T'^{-\theta})^\circ\le2\).
- (b) Let \(A\) be a maximal \((\theta,k)\)-split torus contained in a \(\theta\)-stable maximal \(k\)-split torus \(S'\). Then \(\dim A=2\), \(M:=Z_G(A)\) is a Levi subgroup of type \(D_4\) with \(Z(M)^\circ=A\), and \([M,M]\subseteq H\).
- (c) Over \(\bar k\), every maximal \(\theta\)-split torus is \(H(\bar k)\)-conjugate to \(S\).
- (d) If \(k=F\) is a non-archimedean local field of residue characteristic \(p\ne2\) (characteristic \(0\) or \(p\), e.g. \(\mathbb F_q(\!(t)\!)\)), every maximal \((\theta,F)\)-split torus is \(H(F)\)-conjugate to \(S\): the index set \(J\) of Delorme–Sécherre has one element.
PROOF / BODY:
1. **A dimension identity.** Let \(T'\) be a \(\theta\)-stable maximal torus (split over the field considered), \(\theta'\) the induced involution of \(X^*(T')\), \(d=\dim(T'^{-\theta})^\circ\). Root spaces are lines and \(d\theta(\mathfrak g_a)=\mathfrak g_{\theta'a}\). On a \(\theta'\)-orbit \(\{a,\theta'a\}\) of size 2, \(d\theta\) swaps two lines and fixes a line; on a \(\theta'\)-fixed (imaginary) root space \(d\theta=\pm1\) (char. \(\ne2\)). With \(I\) the number of imaginary roots and \(c\le I\) the number on which \(d\theta=+1\):
$$52=\dim\mathfrak g^\theta=(6-d)+c+\tfrac{72-I}{2}\le42-d+\tfrac I2 .$$
2. **(a).** Imaginary roots lie in the \((+1)\)-eigenspace of \(\theta'\), of dimension \(6-d\), so they form a root subsystem of rank \(\le6-d\). If \(d\ge3\) this rank is \(\le3\) and \(I\le12\) (\(A_3\) is the largest rank-3 subsystem of \(E_6\)), contradicting \(I\ge20+2d\ge26\).
3. **The \(\mathrm{SL}_2\) trick.** Let \(A\), \(S'\) be as in (b), \(A\) maximal; then \(A=(S'^{-\theta})^\circ\). For \(a\in\Phi(M,S')\), \(a\) and \(a\circ\theta'\) agree on \((S'^{\theta})^\circ\) and are trivial on \(A\), so \(\theta'a=a\), \(\theta(U_a)=U_a\) and \(\theta(u_a(x))=u_a(\varepsilon x)\) with \(\varepsilon=\pm1\) (automorphisms of \(\mathbb G_a\) are scalars in every characteristic). If \(\varepsilon=-1\), then \(\theta\) fixes \(a^\vee(\mathbb G_m)\) pointwise and acts on \(G_a=\langle U_a,U_{-a}\rangle\) (\(\simeq\mathrm{SL}_2\) or \(\mathrm{PGL}_2\) over \(k\)) as \(\mathrm{Int}\,\mathrm{diag}(-1,1)\); the torus \(T_a=\{\begin{pmatrix}x&y\\y&x\end{pmatrix}:x^2-y^2=1\}\) is \(\theta\)-split and \(k\)-split (conjugate by \(\begin{pmatrix}1&1\\1&-1\end{pmatrix}\), invertible as \(2\in k^\times\)), commutes with \(A\), and meets \(A\) in a finite group, so \(A\cdot T_a\) is a larger \((\theta,k)\)-split torus. Hence \(\theta\) is trivial on every root group of \(M\), and \([M,M]\subseteq H\).
4. **(b).** Apply step 1 to \(S'\): \(\theta\) is trivial on \((S'^\theta)^\circ\) and on \([M,M]\), inversion on \(A\); roots of \(M\) are imaginary with \(d\theta=+1\); a root \(a\notin\Phi(M)\) has \(a|_A\ne1\), hence \((\theta'a)|_A=a|_A^{-1}\ne a|_A\). So, with \(d=\dim A\),
$$52=(6-d)+|\Phi(M)|+\tfrac{72-|\Phi(M)|}{2},\qquad |\Phi(M)|=20+2d .$$
\(\Phi(M)\) is a Levi subsystem of rank \(\le\mathrm{rank}\,H=4\) (\([M,M]\subseteq H\); [[A3-2]] step 3). Levi subsystems of \(E_6\) of rank \(\le4\) have \(0,2,4,6,8,10,12,14,20\) or \(24\) roots, and 24 only for the \(D_4\) on \(\{2,3,4,5\}\) (exact enumeration of the 57 subsets of at most four nodes). \(d=0\) would give \(M=G\) and \(\theta=\mathrm{id}\). So \(d=2\), \(|\Phi(M)|=24\), \(M\) is a \(D_4\)-Levi, and \(Z(M)^\circ\) (dimension \(6-4=2\)) contains \(A\), hence equals \(A\).
5. **(c).** Over \(\bar k\), let \(A\) be maximal \(\theta\)-split and \(M=Z_G(A)\) as in (b) (a \(\theta\)-stable maximal torus of \(M\) exists by Steinberg 1968, Thm 7.5, applied to \(\theta|_M\)). For generic \(\lambda\in X_*(A)\), \(Q:=P_G(\lambda)\) has Levi \(M\), \(\theta(Q)=P_G(\lambda^{-1})\) is opposite to \(Q\), and \(Q\cap\theta(Q)=M\). Since \(\{2,3,4,5\}\) is the only \(D_4\) subdiagram, \(Q\) is \(G\)-conjugate to \(P\), i.e. \(Q\in G/P\). \(H\cap Q\subseteq Q\cap\theta(Q)=M\), so \(H\cap Q=M^\theta=A[2]\cdot[M,M]\), of dimension 28, and the orbit \(H\cdot Q\subseteq G/P\) has dimension \(52-28=24=\dim G/P\): it is open. Likewise \(H\cap P=\mathrm{Spin}_8\) ([[A3-2]]) and \(H\cdot P\) is open. \(G/P\) is irreducible, so the two open orbits meet and coincide: \(Q=hPh^{-1}\) with \(h\in H(\bar k)\). Then \(M=Q\cap\theta(Q)=h(P\cap\theta(P))h^{-1}=hLh^{-1}\) (\(\theta(h)=h\); \(P\cap\theta(P)=L\), S4-11) and \(A=Z(M)^\circ=hSh^{-1}\).
6. **(d).** Let \(A'\) be a maximal \((\theta,F)\)-split torus. Delorme–Sécherre's Lemma 3.9 (a consequence of their Prop. 2.4; both checked for \(\mathbb F_q(\!(t)\!)\) in steps 2–3 of [[A3-5]], which do not use the present node) gives a \(\theta\)-stable maximal \(F\)-split torus \(S'\supseteq A'\). By (b) \(\dim A'=2\), so by (a) \(A'_{\bar F}\) is maximal \(\theta\)-split and by (c) \(A'=hSh^{-1}\) with \(h\in H(\bar F)\). The transporter \(\mathrm{Tr}=\{h\in H:hSh^{-1}=A'\}\) is a closed \(F\)-subscheme of \(H\), a torsor under \(N_H(S)\); \(N_H(S)^\circ=Z_H(S)=L\cap H=\mathrm{Spin}_8\) is smooth (Conrad, Lemma 2.2.4) and \(N_H(S)/Z_H(S)\) embeds in the constant group \(\mathrm{Aut}(S)\simeq\mathrm{GL}_2(\mathbb Z)\), so \(N_H(S)\) and \(\mathrm{Tr}\) are smooth and \(\mathrm{Tr}(F^s)\ne\emptyset\). For \(\sigma\in\mathrm{Gal}(F^s/F)\), \(\mathrm{Int}(h)|_S:S\to A'\) is a homomorphism of \(F\)-split tori, hence Galois-invariant, so \(c_\sigma:=h^{-1}\sigma(h)\) lies in \(Z_H(S)(F^s)=\mathrm{Spin}_8(F^s)\) and is a cocycle. \(H^1(F,\mathrm{Spin}_8)=1\) ([[A3-3]]), so \(c_\sigma=z\sigma(z)^{-1}\), and \(h'=hz\in H(F)\) satisfies \(h'Sh'^{-1}=A'\).

**Remarks.** The cocycle never lands in the disconnected \(N_H(S)\), whose cohomology involves \(H^1(F,S_3)\), infinite for \(p=3\); the bad prime causes no difficulty. Helminck–Wang's Prop. 10.3 (field hypotheses not checked) and the conjugacy theorems of Vust and Richardson are not used.
SOURCES: Delorme–Sécherre, arXiv:math/0612545v1, Lemma 3.9 and Prop. 2.4 (read from the arXiv PDF); conrad-reductive-group-schemes.pdf, Lemma 2.2.4; Steinberg, Mem. AMS 80 (1968), Thm 7.5 (cited); S4-11 item 4; S4-12 steps 5–8; scripts e6_charp_checks.py, e6_levi_sizes.py
--- A3-5 · Audit · Delorme–Sécherre over $\mathbb F_q((t))$, $p$ odd: $G(F)=H(F)T(F)K$
STATUS CLAIMED: proved
PARENTS: A3-4
STATEMENT:
**Theorem (Delorme–Sécherre, Thm 0.4 = Thm 4.8, arXiv v1).** Let \(k\) be a non-archimedean locally compact field of residue characteristic \(\ne2\), \(G\) a connected reductive \(k\)-split group, \(\sigma\) an involutive \(k\)-automorphism, \(H\) open in \(G^\sigma\), \(A\) a maximal \((\sigma,k)\)-split torus, \(S_{\mathrm{DS}}\supseteq A\) a \(\sigma\)-stable maximal \(k\)-split torus, \(\{A_j\}_{j\in J}\) representatives of the \(H_k\)-classes of maximal \((\sigma,k)\)-split tori, \(y_jAy_j^{-1}=A_j\), \(x\) a special point of the apartment of \(S_{\mathrm{DS}}\), \(K\) its stabilizer. Then \(G_k=\bigcup_{j}H_k\,y_j\,S_{\mathrm{DS},k}\,K\).

**Claim.** (i) The standing hypothesis of the paper (abstract, introduction, §§2–4) is exactly the one above, which includes \(\mathbb F_q(\!(t)\!)\), \(p\) odd. (ii) The proofs of Prop. 2.4, Lemma 3.9, Prop. 4.4 and Thm 4.8 use only: Bruhat–Tits theory and Landvogt's results on the building of a split group over a local field; that the groups \(U_\Omega^\pm\) are pro-\(p\) with \(p\) odd; \(2\in k^\times\) with \(|2|=1\); structure theory from Borel's and Humphreys' books; and Helminck–Helminck, Lemma 2.2. None of these uses \(\mathrm{char}\,k=0\). (iii) For \(G=E_6\), \(\sigma=\theta_0\), \(H=G^\theta\), \(S_{\mathrm{DS}}=T\), \(A=S\), \(x\) the hyperspecial vertex fixed by \(K=G(\mathcal O)\): \(J=\{1\}\) by [[A3-4]](d), and Helminck–Helminck's lemma is replaced by a direct argument. Hence, over \(F=\mathbb F_q(\!(t)\!)\),
$$G(F)=H(F)\,T(F)\,K=K\,T(F)\,H(F).$$
PROOF / BODY:
1. **The field.** The abstract and introduction say 'non Archimedean locally compact field of residue characteristic different from 2'; §1 'non Archimedean non discrete locally compact field'; §§2–4 'residue characteristic different from 2'. The characteristic of \(k\) is invoked once, in Lemma 3.6(iii) ('characteristic different from 2', for \([k^\times:k^{\times2}]<\infty\)), which holds for \(\mathbb F_q(\!(t)\!)\) with \(p\) odd and is not used in §4.
2. **Prop. 2.1 and 2.4 (every point of the building lies in a \(\sigma\)-stable apartment).** For \(\Omega=\{y,\sigma y\}\), \(y\) in a chamber, \(P_\Omega/N_\Omega\) is a projective limit of finite \(\sigma\)-sets \(P_\Omega/P_\Omega^i\) whose cardinalities are powers of \(p\) (Lemma 2.3, from the Iwahori decomposition \(P_\Omega=U_\Omega^-U_\Omega^+N_\Omega\) and pro-\(p\)-ness of \(U_\Omega^\pm\)), hence odd, and a \(\sigma\)-set of odd cardinality has a fixed point (Lemma 2.2). Inputs: Landvogt, Cor. 12.6, Cor. 13.7; Bruhat–Tits I, 7.4. For a split group over \(\mathbb F_q(\!(t)\!)\) these hold as for \(p\)-adic fields; the root-group filtrations are \(t^m\mathbb F_q[[t]]\), pro-\(p\).
3. **Lemma 3.9.** Prop. 2.4 applied to \(Z_G(A)^\circ\). No further input.
4. **Prop. 4.4.** Among \(\sigma\)-stable apartments through the special point \(x\), take one with \(\dim S_{\mathrm{DS}}^-\) maximal. If \(S_{\mathrm{DS}}^-\) is not maximal \((\sigma,k)\)-split, Lemma 4.6 (Humphreys 27.5(e); Lemma 4.3 from Borel 8.2, 14.2) gives a \(\sigma\)-invariant root \(a\) with \(U_a\not\subseteq H\); Lemma 4.7: \(\sigma=-1\) on \(U_a\simeq\mathbb G_a\) (automorphisms of \(\mathbb G_a\) are scalars in every characteristic); Lemma 4.1 writes \(w=v^2\) in \(U_{-a}(k)\simeq(k,+)\), i.e. \(v=w/2\), with the same filtration index because \(|2|=1\), and builds \(c\in U_x\) with \(c^{-1}\sigma(c)\) the affine reflection of \(s_a\) fixing \(x\) (Landvogt 7.4(ii), 7.7); then \(c\cdot S_{\mathrm{DS}}\) is a \(\sigma\)-stable torus through \(x\) with larger split part. Uses \(p\ne2\), never \(\mathrm{char}\,k=0\).
5. **Thm 4.8.** \(g\cdot x\) lies in a \(\sigma\)-stable apartment \(\mathcal A(S')\) with \(S'^-\) maximal (Prop. 4.4); \(S'^-=hy_jAy_j^{-1}h^{-1}\); Helminck–Helminck, Lemma 2.2 gives \(S'=hy_jS_{\mathrm{DS}}\); Bruhat–Tits 1.13(2) (two points of an apartment conjugate under \(G_k\) are conjugate under \(N_k\)) and \(N_kK=S_{\mathrm{DS},k}K\) (\(x\) special) give \(g\in Hy_jS_{\mathrm{DS}}K\).
6. **Replacing Helminck–Helminck for \(E_6/F_4\).** Here \(J=\{1\}\), \(y_1=1\). If \(S'^-=hSh^{-1}\) with \(h\in H(F)\) ([[A3-4]](d)), then \(h^{-1}S'h\) is a \(\theta\)-stable maximal \(F\)-split torus containing \(S\), hence contained in \(Z_G(S)=L=S\cdot\mathrm{Spin}_8\), so it equals \(S\cdot T_8'\) with \(T_8'\) a maximal \(F\)-split torus of the split group \(\mathrm{Spin}_8\). Maximal \(F\)-split tori of \(\mathrm{Spin}_8\) are \(\mathrm{Spin}_8(F)\)-conjugate (Borel–Tits, every field), and \(\mathrm{Spin}_8(F)\subseteq H(F)\). So \(S'=(hm)T(hm)^{-1}\) with \(hm\in H(F)\).
7. **Data.** The hyperspecial vertex \(x\) of \(\mathcal A(T)\) has stabilizer \(G(\mathcal O)=K\) (\(G\) simply connected: stabilizer = parahoric, Bruhat–Tits II); \(N(F)=T(F)\cdot N(\mathcal O)\) with Tits lifts in \(G(\mathbb Z)\), so \(N(F)K=T(F)K\). Hence \(G(F)=H(F)T(F)K\); inverting, \(G(F)=KT(F)H(F)\).

**Status.** Proved, modulo Delorme–Sécherre (read in arXiv v1; the statement had been quoted in S4-12 without its proof being checked) and the building-theoretic references inside it (Bruhat–Tits I–II, Landvogt), whose standing hypotheses do not involve the characteristic of \(k\); those were not opened.
SOURCES: Delorme–Sécherre, An analogue of the Cartan decomposition for p-adic reductive symmetric spaces, arXiv:math/0612545v1 (2006): abstract; Lemmas 2.2, 2.3, Props 2.1, 2.4; Lemmas 3.6, 3.9; Lemmas 4.1, 4.3, 4.6, 4.7, Prop. 4.4, Thm 4.8 (read from the arXiv PDF; published Pacific J. Math. 251 (2011)); Helminck–Helminck, Trans. AMS 350 (1998), Lemma 2.2 (not opened; replaced); Bruhat–Tits I, IHÉS 41 (1972), 7.4; Landvogt, LNM 1619 (cited inside DS, not opened); S4-12
--- A3-6 · Proposition · Char $\ne2$: 45 maximal-rank Borel orbits $Bn_wx_0$, all pairs of type $G$ or $U$
STATUS CLAIMED: proved
PARENTS: A3-2, A3-4
STATEMENT:
Let \(k\) be algebraically closed with \(\mathrm{char}\,k\ne2\) (e.g. \(k=\bar F\), \(F=\mathbb F_q(\!(t)\!)\)); \(X=G_k/H_k\), \(n_w\in N_G(T)(\mathbb Z)\) Tits lifts, \(Y_w:=Bn_wx_0\).
1. (Rank.) Every \(B\)-orbit has the form \(Y=g\cdot(B_gx_0)\), \(B_g=g^{-1}Bg\), where \(B_g\) contains a \(\theta\)-stable maximal torus \(T'\). Then \(B_g\cap H=T'^\theta\ltimes R_u(B_g\cap\theta B_g)^\theta\) is smooth, \(\mathcal X(Y)=(1-\theta^*)X^*(T')\), and \(\mathrm{rk}\,Y=\dim T'^{-\theta}\le2\).
2. (Orbits of maximal rank.) The \(B\)-orbits of rank 2 are exactly the \(Y_w\), \(w\in W\), and \(Y_w=Y_{w'}\iff w^{-1}w'\in W(X):=N_W(\Phi_L)\), a group of order 1152. \(W(X)=W_X\ltimes W_L\) with \(W_L=W(D_4)\) and \(W_X:=\{w\in W(X):w\Phi_L^+=\Phi_L^+\}=\langle w_{\gamma_1},w_{\gamma_6}\rangle\simeq S_3\), \(w_{\gamma_1}=s_1s_3s_4s_2s_5s_4s_3s_1\), \(w_{\gamma_6}=s_6s_5s_4s_2s_3s_4s_5s_6\) (involutions of length 8), acting faithfully on \(\mathcal X(X)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\) by \(s_{\gamma_1},s_{\gamma_6}\). The products \(s_{\alpha_i}s_{\beta_i}\in w_{\gamma_i}W_L\) (\(i=1,6\)) generate a second complement of \(W_L\), induced by \(\dot n_i\in H(\mathbb Z)\). There are \(51840/1152=45\) maximal-rank orbits, each containing the point \(n_wx_0\in X(\mathbb Z[1/2])\).
3. (Types.) For \(w\in W\) and a simple root \(\alpha\), \(Y_wP_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2/\hat S\) with \(\hat S\) smooth; \(\hat S=\mathrm{PGL}_2\) (type \(G\)) if \(w^{-1}\alpha\in\Phi_L\), and \(\hat S\) is a Borel subgroup (type \(U\), \(S=T\), quotient \(\mathbb P^1\)) otherwise. No maximal-rank pair has type \(T\) or \(N\).
4. (Knop's action.) \(w'\cdot Y_w:=Y_{w'w}\) is the action characterized on p. 9 of Sakellaridis (\(w_\alpha\) fixes the maximal-rank orbit of \(YP_\alpha\) in type \(G\) and swaps the two in type \(U\)); it is transitive and the stabilizer of \(\mathring X=Y_1\) is \(W(X)\).
PROOF / BODY:
1. **\(\theta\)-stable tori in Borels.** For a Borel \(B'\), \(R=B'\cap\theta(B')\) is connected solvable, \(\theta\)-stable and contains a maximal torus of \(G\); \(\theta\) has order 2, prime to \(\mathrm{char}\,k\), so it is a semisimple automorphism and fixes a maximal torus of \(R\) (Steinberg, Mem. AMS 80 (1968), Thm 7.5; characteristic-free, cited).
2. **(1).** \(b\in B_g\) fixes \(x_0\) iff \(\theta(b)=b\), so \(B_g\cap H=R^\theta\) scheme-theoretically; \(R=T'\ltimes R_u\) with \(\theta\) preserving both factors gives \(R^\theta=T'^\theta\ltimes R_u^\theta\), smooth ([[A3-2]] step 2). Rational \(B_g\)-eigenfunctions on \(B_g/R^\theta\) have as weights the characters of \(T'\) trivial on \(T'^\theta=\ker(\theta-1)\), i.e. \((1-\theta^*)X^*(T')\), of rank \(\dim T'^{-\theta}\le2\) ([[A3-4]](a)).
3. **(2), orbits.** If \(\mathrm{rk}\,Y=2\), \((T'^{-\theta})^\circ\) is a maximal \(\theta\)-split torus, so it equals \(hSh^{-1}\), \(h\in H\) ([[A3-4]](c)); \(h^{-1}T'h\subseteq Z_G(S)=L\) is a maximal torus \(S\cdot T_8''\) of \(L\), and \(T_8''=mT_8m^{-1}\) with \(m\in\mathrm{Spin}_8\subseteq H\). So \(T'=h'Th'^{-1}\), \(h'\in H\), \(h'^{-1}B_gh'=n_w^{-1}Bn_w\) for some \(w\), and \(Y=gB_gx_0=(gh'n_w^{-1})Bn_wx_0\) with \(gh'n_w^{-1}\in N_G(B)=B\): \(Y=Y_w\). Conversely \(Y_w=n_w(n_w^{-1}Bn_w)x_0\) with \(T\) \(\theta\)-stable in \(n_w^{-1}Bn_w\), so \(\mathrm{rk}\,Y_w=\dim T^{-\theta}=2\).
4. **(2), the group.** \(\mathrm{Stab}_T(n_wx_0)=w(T^\theta)\), while every \(y=uax_0\in\mathring X\simeq U_P\times A_X\) has \(\mathrm{Stab}_T(y)\subseteq T^\theta\) ([[A3-2]](4)). So \(n_wx_0\in\mathring X\) forces \(w(T_8)=T_8\), i.e. \(w\in N_W(\Phi_L)\); this group has order \(51840/45=1152\) (the \(W\)-orbit of the subsystem \(\Phi_L\) has 45 elements; computed). Conversely \(W_L\subseteq W(X)\) (Tits lifts of \(s_2,\dots,s_5\) lie in \(\mathrm{Spin}_8\subseteq H\)) and \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H(\mathbb Z)\) lifts \(s_{\alpha_i}s_{\beta_i}\) (S4-13, an identity in \(G(\mathbb Z)\)). Computed: \(\langle s_{\alpha_1}s_{\beta_1},s_{\alpha_6}s_{\beta_6}\rangle\) has order 6 and acts on \(\mathcal X(X)\) by \(\omega_1\mapsto\omega_6-\omega_1,\ \omega_6\mapsto\omega_6\) and \(\omega_1\mapsto\omega_1,\ \omega_6\mapsto\omega_1-\omega_6\), i.e. by \(s_{\gamma_1},s_{\gamma_6}\); \(W_L\) acts trivially, so the intersection is trivial and this complement times \(W_L\) (order 1152) is all of \(N_W(\Phi_L)\). As \(W_L\) is simply transitive on the positive systems of \(\Phi_L\), each coset of \(W_L\) in \(N_W(\Phi_L)\) contains exactly one element preserving \(\Phi_L^+\) (its element of minimal length); these form the canonical complement \(W_X\). Computed: \(w_{\gamma_1}=s_1s_3s_4s_2s_5s_4s_3s_1\) and \(w_{\gamma_6}\) preserve \(\Phi_L^+\), are involutions of length 8, act as \(s_{\gamma_1},s_{\gamma_6}\), and generate a group of order 6 (while \(s_{\alpha_1}s_{\beta_1}\), of length 14, does not preserve \(\Phi_L^+\)).
5. **(3).** Conjugating by \(n_w^{-1}\), \((Y_w,\alpha)\) becomes \((B^wx_0,\beta)\), \(B^w=n_w^{-1}Bn_w\), \(\beta=w^{-1}\alpha\) simple for \(B^w\). Let \(P_\beta\) be the minimal parabolic of \(B^w\) for \(\beta\) and \(R=P_\beta\cap\theta(P_\beta)\), smooth connected and generated by \(T\) and the \(U_\gamma\), \(\gamma\in\Psi=(\Phi_w^+\cup\{-\beta\})\cap\theta(\Phi_w^+\cup\{-\beta\})\). Then \(\mathrm{Stab}_{P_\beta}(x_0)=R^\theta\) is smooth and \(\hat S\) is its image in \(P_\beta/R(P_\beta)\simeq\mathrm{PGL}_2\), smooth. (a) \(\beta\in\Phi_L\): \(U_{\pm\beta}\subseteq\mathrm{Spin}_8\subseteq H\), so \(\hat S=\mathrm{PGL}_2\). (b) \(\beta\notin\Phi_L\): then \(\theta\beta\ne\pm\beta\) (computed: the \(\theta\)-fixed roots are \(\Phi_L\), and no root has \(\theta\beta=-\beta\)); exactly one of \(\pm\beta\), say \(\varepsilon\beta\), lies in \(\Psi\) (\(\beta\in\Psi\iff\theta\beta\in\Phi_w^+\)), so the image of \(R\) is the Borel \(\bar T\ltimes\bar U_{\varepsilon\beta}\) and \(\hat S\) lies in it; \(\beta|_{T^\theta}\ne1\) (otherwise \(\theta\beta=-\beta\)), so \(\hat S\supseteq\bar T\); \(\varepsilon\theta\beta\in\Psi\) and \(\beta+\theta\beta\) is never a root (computed), so \(\{u_{\varepsilon\beta}(c)\theta(u_{\varepsilon\beta}(c))\}\subseteq R^\theta\) maps onto \(\bar U_{\varepsilon\beta}\). Thus \(\hat S=\bar T\ltimes\bar U_{\varepsilon\beta}\). No classification of subgroups of \(\mathrm{PGL}_2\) in characteristic \(p\) is needed.
6. **(4).** In type \(G\), \(Y_wP_\alpha=Y_w\ni\dot s_\alpha n_wx_0\); in type \(U\), \(Y_wP_\alpha\) is the union of the two distinct orbits \(Y_w\) and \(Y_{s_\alpha w}\) (\(P_\alpha=B\cup Bs_\alpha B\); they map to the two \(\bar B\)-orbits of \(\mathbb P^1\)), both of maximal rank. So \(s_\alpha\cdot Y_w=Y_{s_\alpha w}\) is the reflection Sakellaridis describes, depends only on the orbit, and extends to the action \(w'\cdot Y_w=Y_{w'w}\); the stabilizer of \(Y_1\) is \(\{w:n_wx_0\in\mathring X\}=W(X)\) (step 4).

The types agree with S4-5(3); all stabilizers are smooth, so no rank-one quotient is inseparable (checklist item 5).
SOURCES: sakellaridis-v3.pdf, §2.2 p. 9 (types of pairs, characterization of Knop's action, W(X) = W_X ⋉ W_P(X)); Steinberg, Mem. AMS 80 (1968), Thm 7.5 (cited); S4-5 part 3, S4-10, S4-13; scripts e6_charp_checks.py, e6_canonical_wx.py
--- A3-7 · Proposition · Char $\ne2$: two colours, the valuation cone, spherical roots $\gamma_1,\gamma_6$
STATUS CLAIMED: proved
PARENTS: A3-2, A3-3, A3-6
STATEMENT:
Let \(k\) be algebraically closed, \(\mathrm{char}\,k\ne2\), \(X=G_k/H_k\). Let \(\Lambda_X^+\subseteq\Lambda_X=\mathrm{Hom}(\mathcal X(X),\mathbb Z)\) be the restrictions to \(B\)-eigenfunctions of the \(G\)-invariant \(\mathbb Z\)-valued valuations of \(k(X)\) trivial on \(k\), and \(\mathcal V\) the cone they span.
1. \(\mathcal X(X)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\).
2. \(k[X]^\times=k^\times\), \(\mathrm{Pic}\,X=0\), \(k[X]\) is factorial. \(X\) has exactly two colours \(D_1=\mathrm{div}(f_1)\), \(D_6=\mathrm{div}(f_6)\), with \(f_i\in k[X]\) a \(B\)-eigenfunction of weight \(\omega_i\); \(v_{D_i}=e_i\); the weights of regular \(B\)-eigenfunctions form \(\mathbb N\omega_1\oplus\mathbb N\omega_6\); \(V(\omega_1)^H\ne0\ne V(\omega_6)^H\). \(D_1\) is moved by \(P_{\alpha_1}\) and stable under \(P_{\alpha_6}\); \(D_6\) the reverse.
3. \(\mathcal V=C^-:=\{v:\langle v,\gamma_1\rangle\le0,\ \langle v,\gamma_6\rangle\le0\}\) and \(\Lambda_X^+=\Lambda_X\cap C^-\).
4. The spherical roots are \(\gamma_1=2\omega_1-\omega_6\) and \(\gamma_6=-\omega_1+2\omega_6\); both are normalized (\(\gamma_i=\alpha_i+\beta_i\), a sum of two orthogonal roots simple for some basis) and of type \(G\); \(\mathrm{supp}\,\gamma_1=\{1,\dots,5\}\), \(\mathrm{supp}\,\gamma_6=\{2,\dots,6\}\), so \(L_{\gamma_1},L_{\gamma_6}\) are of type \(D_5\); \(\check\gamma_i=e_i\), \(\langle e_i,\gamma_j\rangle\) is the Cartan matrix of \(A_2\); \(W_X=\langle s_{\gamma_1},s_{\gamma_6}\rangle\simeq S_3\) acts on \(\Lambda_X\otimes\mathbb Q\) with fundamental domain \(\mathcal V\).
PROOF / BODY:
1. **(1).** \(\mathring X=B/B_8\) ([[A3-2]]), so the weights of rational \(B\)-eigenfunctions are the characters of \(T\) trivial on \(T^\theta=T_8\), namely \((1-\theta^*)X^*(T)\); as the elementary divisors of \(1-\theta^*\) are \((1,1)\) (computed), this is the saturated lattice \(X^*(T)\cap(\mathbb Q\omega_1+\mathbb Q\omega_6)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\).
2. **Units, Pic.** A unit of \(k[G/H]\) pulls back to a unit of \(k[G]\), which is constant (\(G\) semisimple, Rosenlicht). \(\mathrm{Pic}\,G=0\) for simply connected \(G\) (Popov 1974, Fossum–Iversen 1973; characteristic-free), so line bundles on \(G/H\) correspond to characters of \(H\) (as \(k[G]^\times=k^\times\)), and \(X^*(H)=X^*(F_4)=0\). \(X\) is smooth and affine ([[A3-3]](4)), hence factorial.
3. **Colours.** \(X\smallsetminus\mathring X\) has pure codimension 1 (\(\mathring X\) affine; EGA IV 21.12.7), and its components are exactly the \(B\)-stable prime divisors, i.e. the colours (\(X\) is homogeneous). Each colour is \(\mathrm{div}(f_D)\), \(f_D\in k[X]\) a \(B\)-eigenfunction (factoriality, constant units) of weight \(\lambda_D\), dominant (a \(U\)-invariant weight vector of a rational \(G\)-module has dominant weight; Jantzen, RAG II.2). Units of \(k[\mathring X]=k[U_P\times A_X]\) are \(k^\times\cdot X^*(A_X)\); each is \(c\prod f_D^{n_D}\) (its divisor lies on \(X\smallsetminus\mathring X\)). So \(\{\lambda_D\}\) is a basis of \(\mathbb Z\omega_1\oplus\mathbb Z\omega_6\) made of dominant weights, i.e. \(\{\omega_1,\omega_6\}\): two colours, \(v_{D_i}(f_j)=\delta_{ij}\), \(v_{D_i}=e_i\). Frobenius reciprocity: \(f_6\) is a highest weight vector of weight \(\omega_6\) in \(k[X]=\mathrm{Ind}_H^Gk\), so \(0\ne\mathrm{Hom}_G(V(\omega_6),\mathrm{Ind}_H^Gk)=(V(\omega_6)^*)^H=V(\omega_1)^H\) (\(V(\omega_6)\) minuscule, irreducible); symmetrically \(V(\omega_6)^H\ne0\). A \(B\)-eigenfunction of weight \(\chi\) is \(P_\alpha\)-semi-invariant iff \(\langle\chi,\alpha^\vee\rangle=0\) (the \(L_\alpha\)-module it generates is a quotient of a Weyl module of \(L_\alpha\)), so \(D_6\) is \(P_{\alpha_1}\)-stable and \(D_1\) is not.
4. **Lower bound \(C^-\subseteq\mathcal V\), and \(\Lambda_X\cap C^-\subseteq\Lambda_X^+\).** Let \(\kappa\in\Lambda_S\) be dominant and \(\mu\in X_*(T)\) with \(\varphi(\mu)=\kappa\) (\(\varphi(\Lambda_T)=\Lambda_S\), family page §2). Put \(v_\kappa(f):=\mathrm{ord}_t\,f(g\,\mu(t)\,x_0)\), via \(k(X)\hookrightarrow k(G)(\!(t)\!)\) induced by the dominant map \(G\times\mathbb G_m\to X\), \((g,t)\mapsto g\mu(t)x_0\): a \(G\)-invariant \(\mathbb Z\)-valued valuation trivial on \(k\). Over \(k(\!(s)\!)\), \(s^2=t\): \(\tau(\kappa(s)x_0)=\kappa(s)^2=\kappa(t)=\tau(\mu(t)x_0)\), so \(\mu(t)x_0=\kappa(s)x_0\). For \(g=bu^-\) generic (\(b\in B\), \(u^-\in U^-\)), \(\kappa(s)^{-1}u^-\kappa(s)\to u_0\in U^-\cap Z_G(\kappa)\) (\(\kappa\) dominant) and \(u_0x_0\in\mathring X\) for generic \(u^-\) (\(\mathring X\) is open and contains \(x_0\)); hence \(\kappa(s)^{-1}u^-\kappa(s)x_0=b(s)x_0\) with \(b(s)\in B(k(U^-)[[s]])\) (Hensel for the smooth surjection \(U_P\times T\to\mathring X\)), and \(f_\chi(bu^-\kappa(s)x_0)=(\text{unit})\cdot s^{-\langle\kappa,\chi\rangle}f_\chi(x_0)\). So \(v_\kappa(f_\chi)=-\langle\kappa,\chi\rangle/2=-\langle\mu,\chi\rangle\) (for \(\chi\in\mathcal X(X)\), \(\theta\chi=-\chi\) and \(\langle\varphi(\mu),\chi\rangle=2\langle\mu,\chi\rangle\)). As \(\kappa\) runs over \(\Lambda_S^+\), \(-\bar\mu\) runs over \(\Lambda_X\cap C^-\) (\(\langle-\bar\mu,\gamma_i\rangle=-\tfrac12\langle\kappa,\gamma_i\rangle\)). This needs no dominant lift of \(\kappa\): the wall labels \(3k\omega_1^\vee,3k\omega_6^\vee\) (\(k\) odd) are covered.
5. **Upper bound \(\Lambda_X^+\subseteq C^-\).** Let \(M_1\) be the standard Levi of \(\{1,\dots,5\}\), \(Q_1=M_1U_{Q_1}\). \(\theta\) negates \(\alpha_6\)-coefficients, so \(\theta(\Phi_{M_1})=\Phi_{M_1}\), \(\theta(U_{Q_1})=U_{Q_1}^-\), \(Q_1\cap H=M_1^\theta\), and as in [[A3-2]](4) \(U_{Q_1}\times X_1\to X\), \(X_1:=M_1x_0\simeq M_1/M_1^\theta\), is an open immersion (\(16+10=26\)). With \(\mathrm{pr}:Q_1x_0\to X_1\), a \(G\)-invariant valuation \(v\) gives an \(M_1\)-invariant valuation \(v'=v\circ\mathrm{pr}^*\) of \(k(X_1)\) (\(\mathrm{pr}\) is \(Q_1\)-equivariant), and \(f_\chi|_{Q_1x_0}=\mathrm{pr}^*(f_\chi|_{X_1})\) (\(U_{Q_1}\subseteq U\)), so \(\langle v,\chi\rangle=v'(f_\chi|_{X_1})\). Let \(\ell=f_1|_{X_1}\) and \(E=\mathrm{span}(M_1\ell)\). \([M_1,M_1]=\mathrm{Spin}_{10}\) and \(\omega_1\) restricts to its vector weight (node 1 ends the long arm of the \(D_5\) diagram \(1-3-4\{2,5\}\)), so \(E\) is a nonzero quotient of the Weyl module of highest weight \(\omega_1\), which is the natural 10-dimensional module (irreducible for \(p\ne2\)): \(E\simeq V_{10}\) up to a central character. Let \(Q^*\in\mathrm{Sym}^2E\) be the \(\mathrm{Spin}_{10}\)-invariant nondegenerate quadratic form on \(E^*\) and \(F_1\in k[X_1]\) its image (a sum of products of two elements of \(E\)). \(F_1\) is \(M_1\)-semi-invariant; its weight vanishes on \(\alpha_1^\vee,\dots,\alpha_5^\vee\) and lies in \(2\omega_1+\mathbb Z\{\alpha_1,\dots,\alpha_5\}\), so it is \(\omega_6\) (since \(\gamma_1=2\omega_1-\omega_6\) lies in that lattice). \(F_1\ne0\): the evaluation \(\iota(x_0)\in E^*\) is \(T^\theta\)-fixed, so supported on the weights of \(E\) lying in \(\mathcal X(X)\), which are \(\omega_1\) and \(\omega_1-\gamma_1=\omega_6-\omega_1\) (the weights of \(V(\omega_1)\) in \(\mathcal X(X)\) are \(\omega_1,\omega_6-\omega_1,-\omega_6\), computed, and \(-\omega_6=\omega_1-(\gamma_1+\gamma_6)\) has nonzero \(\alpha_6\)-coefficient); \(\dot n_1\in M_1^\theta\) (\(\alpha_1,\beta_1\in\Phi_{M_1}\)) fixes \(\iota(x_0)\) and exchanges the two weight lines, so both coordinates are nonzero; these are the highest and lowest weights of \(E\), paired by \(Q^*\), so \(F_1(x_0)=c\,\ell_{\omega_1}(x_0)\ell_{\omega_6-\omega_1}(x_0)\ne0\). Hence \(F_1=c'f_6|_{X_1}\), \(c'\ne0\) (eigenfunction spaces of \(B\cap M_1\) on the spherical \(X_1\) are lines). As \(v'(e)\ge v'(\ell)\) for \(e\in E\), \(v'(F_1)\ge2v'(\ell)\): \(\langle v,\omega_6\rangle\ge2\langle v,\omega_1\rangle\), i.e. \(\langle v,\gamma_1\rangle\le0\). The Levi \(M_6\) of \(\{2,\dots,6\}\) (\(\theta\) negates \(\alpha_1\)-coefficients; \(\omega_6\) is its vector weight; \(\dot n_6\in M_6^\theta\)) gives \(\langle v,\gamma_6\rangle\le0\).
6. **(4).** \(-\mathcal V^\vee=\mathrm{cone}(\gamma_1,\gamma_6)\) and \(\gamma_1,\gamma_6\) are primitive in \(\mathcal X(X)\). \(\gamma_1=\alpha_1+\beta_1\) with \((\alpha_1,\beta_1)=0\), and \(\alpha_1\pm\beta_1\) are not roots (computed); \(W(E_6)\) is transitive on ordered pairs of orthogonal roots (a statement about the finite root system, S4-3 step 4), so the pair is simple for some basis; a sum of two orthogonal roots has squared length 4, so no other positive multiple of \(\gamma_1\) is such a sum, and \(\gamma_1/2\) is not in the root lattice: normalized, type \(G\). Same for \(\gamma_6\). \(\langle e_1,\gamma_1\rangle=\langle\alpha_1^\vee,\gamma_1\rangle=2\), \(\langle e_1,\gamma_6\rangle=-1\), and symmetrically. \(W_X\) acts by \(s_{\gamma_1},s_{\gamma_6}\) ([[A3-6]](2)), and \(C^-\) is the antidominant chamber of the restricted \(A_2\).
SOURCES: sakellaridis-v3.pdf, §2.1 p. 8 (Λ_X^+, V); §6.1 pp. 41–42 (spherical roots, normalized spherical roots); §6.2 p. 44 (type G); families/e6-f4.html §2 (φ(Λ_T) = Λ_S) and §5 (wall labels); Jantzen, Representations of Algebraic Groups, II.2 (cited); Popov 1974 / Fossum–Iversen 1973 (Pic G = 0, cited); S4-3 steps 3, 4, 6; script e6_charp_checks.py
--- A3-8 · Theorem · (A3b) for $\mathcal M_\theta$ over $\mathbb F_q((t))$: $K\backslash X(F)\simeq\Lambda_X^+$
STATUS CLAIMED: proved
PARENTS: A3-3, A3-5, A3-6, A3-7
STATEMENT:
Let \(F=\mathbb F_q(\!(t)\!)\), \(p\) odd, \(K=G(\mathbb F_q[[t]])\). For \(\kappa\in\Lambda_S\) put \(x_\kappa:=\mu(t)x_0\) for any \(\mu\in X_*(T)\) with \(\varphi(\mu)=\kappa\) (well defined up to \(A_X(\mathcal O)=\mathrm{im}\,T(\mathcal O)\subseteq K\)). Then
1. \(X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}K\,x_\kappa\);
2. in Sakellaridis's right-action terms, \(A_X^+\) (points \(x_0\cdot a\), \(a\in A_X(F)\), with label in \(\Lambda_X^+\)) contains a complete set of representatives of \(X(F)/K\), and points of \(A_X^+\) with distinct labels lie in distinct \(K\)-orbits. So Axiom 2.4.1 holds, and \(X(F)/K\simeq\Lambda_X^+\simeq\Lambda_S^+\).
PROOF / BODY:
1. **Dictionary (checklist item 8).** Sakellaridis's point \(x_0\cdot e^{\lambda}(\varpi)\) is our \(e^{-\lambda}(t)x_0=\mu(t)x_0\) with \(\bar\mu=-\lambda\in\Lambda_X\). By [[A3-7]](3), \(\lambda\in\Lambda_X^+=\Lambda_X\cap C^-\) iff \(\bar\mu\in-C^-\) iff \(\kappa=\varphi(\mu)\in\Lambda_S^+\) (\(\langle\varphi(\mu),\gamma_i\rangle=2\langle\bar\mu,\gamma_i\rangle\)). So \(A_X^+\) corresponds to \(\{x_\kappa:\kappa\in\Lambda_S^+\}\) up to \(A_X(\mathcal O)\).
2. **Every \(K\)-orbit meets \(A_X^+\).** For \(x\in X(F)\): \(x=gx_0\) ([[A3-3]](2)), \(g=kt'h\) ([[A3-5]]), so \(x=kt'x_0\) with \(t'\in T(F)=T(\mathcal O)\cdot\{\mu(t)\}\), \(T(\mathcal O)\subseteq K\): \(x\in K\mu(t)x_0\). The elements \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H(\mathbb Z)\cap N_G(T)\subseteq K\) (S4-13) satisfy \(\dot n_i\mu(t)x_0=(s_{\alpha_i}s_{\beta_i}\mu)(t)x_0\) and \(\varphi(s_{\alpha_i}s_{\beta_i}\mu)=s_{\gamma_i}\varphi(\mu)\) ([[A3-6]](2)); the group they generate acts on \(\Lambda_S\) as \(W_X\simeq\langle s_{\gamma_1},s_{\gamma_6}\rangle\), which moves every \(\kappa\in\Lambda_S\) into \(\Lambda_S^+\).
3. **Distinct labels, distinct orbits.** \(\tau\) is injective on \(F\)-points and \(\tau(kx)=k\tau(x)\theta(k)^{-1}\) with \(\theta(K)=K\), so \(Kx\mapsto K\tau(x)K\) is well defined; \(\tau(\mu(t)x_0)=\varphi(\mu)(t)\), and \(\kappa\in\Lambda_S^+\) is \(G\)-dominant (\(\langle\alpha,\kappa\rangle=\langle\alpha|_S,\kappa\rangle\ge0\) for \(\alpha\in\Phi^+\)). The Cartan decomposition \(G(F)=\bigsqcup_{\nu\in\Lambda_T^+}Kt^\nu K\) holds over every non-archimedean local field (Tits, Corvallis 3.3.3; Bruhat–Tits I; for \(k(\!(t)\!)\), \(k\) algebraically closed, Zhu (2.1.2)), so distinct \(\kappa\) give distinct double cosets, hence distinct \(K\)-orbits. Points with the same label differ by \(A_X(\mathcal O)=\mathrm{im}\,T(\mathcal O)\subseteq K\) ([[A3-3]](3)).
4. Hence the \(K\)-orbits are exactly the \(Kx_\kappa\), \(\kappa\in\Lambda_S^+\).

Every step is characteristic-free given \(p\) odd; the characteristic-0 inputs of S4-12/S4-13 (Kneser, Vust/Richardson, Helminck–Wang) are replaced by Bruhat–Tits III ([[A3-3]]) and [[A3-4]].
SOURCES: sakellaridis-v3.pdf, Axiom 2.4.1 p. 10; §2.1 p. 8 (A_X^+); §8 p. 63; zhu-satake-v2.pdf (2.1.2) (local extract functions-zhu-cartan.txt); Tits, Corvallis (1979) 3.3.3 (cited); S4-5 part 1; S4-13
--- A3-9 · Audit · Axiom 2.4.2 for $\mathcal M_\theta$ over $\mathbb F_q((t))$: S4-5 re-checked
STATUS CLAIMED: proved
PARENTS: A3-2, A3-3, A3-8
STATEMENT:
For every \(\kappa\in\Lambda_S^+\) and every point \(x\in A_X(F)x_0\) with label \(\kappa\): \(Jx\subseteq B(\mathcal O)x\). Equivalently, with right actions, \(xJ\subseteq xB(\mathfrak o)\) for \(x\in A_X^+\): Axiom 2.4.2 holds for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\), \(p\) odd.
PROOF / BODY:
0. **Reduction.** \(x=ax_\kappa\) with \(a\in\mathrm{im}\,T(\mathcal O)\subseteq J\cap B(\mathcal O)\), and \(Jax_\kappa=aJx_\kappa\subseteq aB(\mathcal O)x_\kappa=B(\mathcal O)ax_\kappa\); so \(x=x_\kappa=\mu(t)x_0\) suffices.
1. \(J=U_P(\mathcal O)(J\cap L)U_P^-(\mathfrak p)\): the big cell \(U_P^-\times L\times U_P\to G\) is an open immersion over \(\mathbb Z\); an element of \(J\) reduces into \(B(\mathbb F_q)\), inside the big cell, so lies in its \(\mathcal O\)-points with \(U_P^-\)-component \(\equiv1\bmod t\). Characteristic-free.
2. \(t^{-\mu}u_{-\beta}(c)t^{\mu}=u_{-\beta}(t^{\langle\beta,\mu\rangle}c)\). Formal.
3. \(U_P\times A_X\to X\) is an open immersion over \(\mathcal O\) ([[A3-2]](4)); \(u'x_0\) with \(u'\in U_P^-(\mathfrak p)\) reduces to \(\bar x_0\in\mathring X(\mathbb F_q)\), so lies in \(\mathring X(\mathcal O)\) and equals \(uax_0\) with \(u\in U_P(\mathfrak p)\), \(a\in A_X(\mathcal O)=\mathrm{im}\,T(\mathcal O)\).
4. \(t^\mu U_P(\mathfrak p)t^{-\mu}\subseteq U_P(\mathfrak p)\) for \(\langle\beta,\mu\rangle\ge0\) on \(\Phi(U_P)\). Formal.
5. \(L^\theta=\mathrm{Spin}_8=[L,L]\) ([[A3-2]]) is normal in \(L\), so \(L\) acts on \(Lx_0\simeq L/[L,L]\) through a torus and \(J\cap L\subseteq L(\mathcal O)\) acts on \(T(F)x_0\) through \(A_X(\mathcal O)\).
6. So \(J\mu(t)x_0\subseteq B(\mathcal O)\mu(t)x_0\) when \(\kappa\) has a dominant lift \(\mu\).
7–12. **Labels without dominant lift** (\(3k\omega_1^\vee\), \(3k\omega_6^\vee\), \(k\) odd; family page §5). \(Q=MU_Q\), \(M\) the Levi of \(\{2,\dots,6\}\); \(\theta\) negates \(\alpha_1\)-coefficients, so \(\theta(U_Q)=U_Q^-\), \(Q\cap H=M^\theta\) scheme-theoretically, and \(U_Q\times M/M^\theta\to X\), \(U_{P\cap M}\times A_X\to M/M^\theta\) are open immersions over \(\mathcal O\) (monomorphisms of smooth \(\mathcal O\)-schemes of equal relative dimensions \(16+10\) and \(8+2\); \(M^\theta\) smooth since \(2\in\mathcal O^\times\)). Recomputed exactly: \(\varphi(\mu_1)=3\omega_1^\vee=4\alpha_1^\vee+3\alpha_2^\vee+5\alpha_3^\vee+6\alpha_4^\vee+4\alpha_5^\vee+2\alpha_6^\vee\) for \(\mu_1=2\alpha_1^\vee+\alpha_3^\vee+\alpha_4^\vee+\alpha_5^\vee+\alpha_6^\vee\); \(\min_{\Phi(U_Q)}\langle\beta,\mu_1\rangle=0\) (16 roots), \(\min_{\Phi(U_{P\cap M})}\langle\beta,\mu_1\rangle=-1\) (8 roots); \(\langle\beta,3\omega_1^\vee\rangle=0\) on \(\Phi_M\), \(=3\) on \(\Phi(U_Q)\). **Half-translation:** \(z:=\zeta(s)\), \(s^2=t\), \(\zeta=3\omega_1^\vee\), lies in \(Z(M)^\circ(F')\), \(F'=F(\sqrt t)\); as \(p\) is odd, \(F'/F\) is a separable quadratic extension with \(\sigma(s)=-s\), and \(z^{-1}\sigma(z)=\zeta(-1)\in S[2]\subseteq T_8\subseteq H\). So \(R_z(mx_0):=mzx_0\) is a well-defined \(M(F)\)-equivariant map with values in \(X(F')^{\sigma}=X(F)\). \(\tau(z^kx_0)=z^{2k}=\varphi(\mu_k)(t)\), \(\mu_k=\mu_1+\tfrac{k-1}2\zeta\), so \(\mu_k(t)x_0=R_z^k(x_0)\). The Levi part \((J\cap M)\mu_k(t)x_0=R_z^k((J\cap M)x_0)\subseteq B_M(\mathcal O)\mu_k(t)x_0\) (label 0 inside \(M/M^\theta\), steps 1–6 for \(M\)); the unipotent part uses \(\langle\beta,\mu_k\rangle\ge-1\) on \(\Phi(U_{P\cap M})\); \(J=U_Q(\mathcal O)(J\cap M)U_Q^-(\mathfrak p)\) gives \(J\mu_k(t)x_0\subseteq B(\mathcal O)\mu_k(t)x_0\). The labels \(3k\omega_6^\vee\) follow from the pinned diagram automorphism \(\delta\) (\(1\leftrightarrow6\), \(3\leftrightarrow5\)), which commutes with \(C\), fixes \(\sigma_L\) and hence commutes with \(\theta_0\), and preserves \(T,B,K,J\).

**Characteristic.** Beyond \(2\in\mathcal O^\times\), the residue characteristic enters only through the separability of \(F(\sqrt t)/F\), which fails exactly for \(p=2\).
SOURCES: sakellaridis-v3.pdf, Axiom 2.4.2 p. 11; Lemma 3.9.1 p. 24 (where the axiom is used); S4-5 (proof of part 2); families/e6-f4.html §5 (dominant lifts; uncovered labels); script e6_charp_checks.py
--- A3-10 · Proposition · (A4) in char $p$: (i), (ii), Statement 7.1.5, and the combinatorics of 6.3.1
STATUS CLAIMED: proved
PARENTS: A3-6, A3-7
STATEMENT:
Over an algebraic closure of \(\mathbb F_q(\!(t)\!)\), \(p\) odd (indeed over any algebraically closed field of characteristic \(\ne2\)):
- (i) No pair \((Y,\alpha)\) with \(Y\) of maximal rank has type \(N\) or \(T\). The condition '\(\delta(X)^{-1/2}\tilde\chi\) unramified' constrains only the characters used, which in Theorem 8.0.2 are unramified.
- (ii) \(L_{\gamma_1}\) and \(L_{\gamma_6}\) are of type \(D_5\), classical.
- (iv) \(D_1\) and \(D_6\) are relevant; \(D_1\) belongs to \(\gamma_1\) and not to \(\gamma_6\), and \(D_6\) the reverse. The virtual weighted colours are \((e_1,+,4)\) and \((e_6,+,4)\), with \(\rho_{P(X)}=\rho-\rho_L=4\omega_1+4\omega_6\). \(\Theta=\{\pm e_1,\pm e_6,\pm(e_1+e_6)\}\times\{(+,4)\}\), multiplicity one; \(\Theta^+=\{e_1,e_6,e_1+e_6\}\); Statement 7.1.5 holds.
- (iii, combinatorial part) For \(w=w_{\gamma_1}=s_1s_3s_4s_2s_5s_4s_3s_1\in W_X\) (length 8, acting on \(\mathcal X(X)\) as \(s_{\gamma_1}\), [[A3-6]](2)), the path
$$\mathring X=Y_1\xrightarrow{\alpha_1}Y_{s_1}\xrightarrow{\alpha_3}Y_{s_3s_1}\xrightarrow{\alpha_4}Y_{s_4s_3s_1}\overset{\alpha_2,\ \alpha_5}{\rightrightarrows}Y_{s_2s_4s_3s_1}=Y_{s_5s_4s_3s_1}$$
is a Brion path for \(w_{\gamma_1}\) in the sense of §6.2 (all edges of type \(U\), orbit dimensions \(26,25,24,23,22\)); its codimension-one orbit has closure \(D_1\), so \(\check\theta=e_1=\check\gamma_1\) as Statement 6.3.1 requires for type \(G\); the rank-two varieties \(ZP_{\delta\epsilon}/R(P_{\delta\epsilon})\) at its vertices are \(\mathbb P^2=P_\alpha\backslash\mathrm{SL}_3\) at the three \(A_2\)-vertices and \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\times\mathrm{PGL}_2\) at the bottom, so the path satisfies the hypothesis of Prop. 6.4.1 (as in §6.19 with \(n=5\)). The same holds for \(\gamma_6\) by the diagram automorphism \(\delta\).
PROOF / BODY:
1. **(i)** is [[A3-6]](3). The unramifiedness condition is on \(\tilde\chi\): in the proof of Theorem 8.0.2 only unramified characters occur (S4-10, item 4).
2. **(ii)** is [[A3-7]](4).
3. **(iv), relevant colours.** \(D\) belongs to \(\gamma\) if \(\bar D\) is the closure of the codimension-one orbit of a Brion path for \(w_\gamma\) (p. 58). The labels of a Brion path for \(w_{\gamma_1}\) spell a reduced word of \(w_{\gamma_1}=s_1s_3s_4s_2s_5s_4s_3s_1\in W_{\{1,\dots,5\}}\) ([[A3-6]](2)), so they avoid \(\alpha_6\). The first edge starts at \(\mathring X\) with a label \(\alpha\) of type \(U\) for \(\mathring X\), i.e. \(\alpha\notin\Phi_L\) by [[A3-6]](3); so \(\alpha=\alpha_1\), and the codimension-one orbit is \(Y_{s_1}=\mathring XP_{\alpha_1}\smallsetminus\mathring X\), whose closure is the colour moved by \(P_{\alpha_1}\), namely \(D_1\) ([[A3-7]](2)). The path in (iii) exists, so \(D_1\) belongs to \(\gamma_1\), and only \(D_1\); symmetrically \(D_6\) and \(\gamma_6\). \(D_1\) is defined over \(F\) (it is the component of \(X\smallsetminus\mathring X\) moved by the \(F\)-group \(P_{\alpha_1}\), so Galois-stable), hence \(\sigma=+\).
4. **(iv), \(\Theta\).** \(\langle\rho_L,\alpha_j^\vee\rangle=(-3,1,1,1,1,-3)\) (computed) gives \(\langle\rho-\rho_L,\alpha_j^\vee\rangle=(4,0,0,0,0,4)\), so \(\rho_{P(X)}=4\omega_1+4\omega_6\) and \(r=\langle e_i,\rho_{P(X)}\rangle=4\). Both spherical roots are of type \(G\), so there are no additions (2), (3) of p. 59. With \(s_{\gamma_i}(v)=v-\langle v,\gamma_i\rangle e_i\): \(s_{\gamma_1}e_1=-e_1\), \(s_{\gamma_6}e_1=e_1+e_6\), \(s_{\gamma_1}(e_1+e_6)=e_6\); the orbit of \((e_1,+,4)\) is the six coroots of \(A_2\) and contains \((e_6,+,4)\), so \(\Theta\) is this orbit with multiplicity one. \(\Theta^+\) (inside the cone of the colour valuations \(e_1,e_6\)) is \(\{e_1,e_6,e_1+e_6\}\); every element of \(\Theta\) is \(>0\) or \(<0\); \(\{\check\theta\in\Theta^+:s_{\gamma_1}\check\theta<0\}=\{e_1\}\) since \(s_{\gamma_1}e_6=e_1+e_6\) and \(s_{\gamma_1}(e_1+e_6)=e_6\); this is the virtual colour of \(D_1\), the one belonging to \(\gamma_1\). Same for \(\gamma_6\). So Statement 7.1.5 holds.
5. **(iii), the path.** Orbit dimensions from [[A3-6]](1): \(\dim Y_v=42-\big(4+\#(R_v\cap\Phi_L)+\tfrac12\#(R_v\smallsetminus\Phi_L)\big)\), \(R_v=v^{-1}\Phi^+\cap\theta(v^{-1}\Phi^+)\). Computed: dimensions \(26,25,24,23,22,22\); every edge of type \(U\); \(Y_{s_2s_4s_3s_1}=Y_{s_5s_4s_3s_1}\) since \((s_2s_4s_3s_1)^{-1}s_5s_4s_3s_1=w\in W(X)\); \(\ell(w)=8=2\cdot3+2\) (reduced); \(w\) stabilizes \(\Phi_L^+\) and maps \(\omega_1\mapsto\omega_6-\omega_1\), \(\omega_6\mapsto\omega_6\), so it is the element \(w_{\gamma_1}\) of the canonical \(W_X\); this proves Sakellaridis's Prop. 6.2.1 for \(\mathcal M_\theta\) in characteristic \(p\).
6. **(iii), gluing.** At \(Z=Y_v\) with adjacent edge labels \(\delta,\epsilon\), the Levi \(L^v_{\delta\epsilon}\) has roots \(\pm\delta',\pm\epsilon',\pm(\delta'+\epsilon')\) (\(\delta'=v^{-1}\delta\), \(\epsilon'=v^{-1}\epsilon\)). Computed at each of the three \(A_2\)-vertices: \(\theta\) fixes exactly one pair \(\pm\gamma_0\) of these (a black root, so \(U_{\pm\gamma_0}\subseteq H\)) and moves the others out of the Levi; the roots of \(P^v_{\delta\epsilon}\cap\theta P^v_{\delta\epsilon}\) in the Levi are \(\pm\gamma_0\) and two others, forming a maximal parabolic subset; the image of \(T^\theta\) in the adjoint torus has rank 2. As in [[A3-6]] step 5, the stabilizer of the rank-two quotient is that maximal parabolic of \(\mathrm{PGL}_3\), so the quotient is \(\mathbb P^2\). At the bottom (\(\alpha_2,\alpha_5\) orthogonal), \(\theta\delta'=-\epsilon'\) and \(\theta\epsilon'=-\delta'\) (computed): the Levi \(A_1\times A_1\) is \(\theta\)-stable with the two factors exchanged, the stabilizer contains a twisted diagonal \(\mathrm{PGL}_2\), and it is proper with identity component of Lie algebra a submodule of \(\mathfrak{sl}_2\oplus\mathfrak{sl}_2\) under \(\Delta(\mathfrak{sl}_2)\) (two irreducible summands for \(p\ne2\)), hence equal to it: the quotient is \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\times\mathrm{PGL}_2\).
7. **Left for (iii):** the analytic identities of Prop. 6.5.1 over \(\mathbb F_q(\!(t)\!)\) ([[A3-11]]).
SOURCES: sakellaridis-v3.pdf, opening of §6 p. 41; §6.2 pp. 43–44 (Brion paths, types of spherical roots); Statement 6.3.1 pp. 44–45; Props 6.4.1–6.4.2, 6.5.1 pp. 45–48; §6.19 p. 56; §7.1 pp. 58–60 (relevant and virtual colours, Θ, Statement 7.1.5); S4-3 steps 8–10; S4-10; scripts e6_charp_checks.py, e6_brion_glue.py, e6_canonical_wx.py

YOUR TASK (adversarial audit).
Part 1 (priority): the claimed proof that Sakellaridis's Theorem 8.0.2 holds for M_θ over F_q((t)) given (A3a) over W(F_q), by transfer from totally ramified p-adic fields: claims A4-2, A4-4, A4-5, A4-6, A4-7, A4-8 (main line), then A4-3 and A4-9 (side claims). Part 2: the claimed verification over F_q((t)) of (A1), (A2), (A3b), (A3c), (A4)(i),(ii),(iv), (A5) for M_θ: claims A3-1 … A3-10.
For each claim, re-derive every step yourself and test it against the CHECKLIST above, in particular: (1) characteristic-0 theorems used in characteristic p (Bruhat–Tits, Steinberg, Edixhoven, Delorme–Sécherre, Kneser, Helminck–Wang, Richardson, Knop: check the standing hypotheses of each, and say when you could not open a source); (2) the close-rings isomorphism and whether every condition used really is "of finite type" in the sense of A4-2(e) with a bound D independent of the field — including the quantifiers over H(O) and the Cartan decomposition of H used in A4-5 steps 4–6; (3) whether A4-5's counting argument needs (H-Cartan) over F'_e for every e and whether S4-12/S4-13 really give it for ramified F'; (4) whether Theorem 8.0.2 over ramified F' (A4-7) is correctly reduced to the printed formulas (ψ of conductor O', self-dual measure; §3.3 normalizations; whether the labels λ̌(κ) and the polynomials P_κ are really independent of e); (5) the uniqueness step and the use of S4-7/S4-8 in A4-8; (6) for A3: each step re-checked over F_q((t)) (inseparability, non-reduced stabilizers, H^1 over F_q((t)), the valuation cone and colours in characteristic p, p = 3).
Return nodes of kind check or objection only, one per audited claim or step group, each naming the claim and step audited and the outcome: holds · holds with a repair (give it) · needs hypothesis H · fails at step k (with counterexample or missing lemma). Use ids A5-1, A5-2, …; parents = the audited node ids (e.g. ["A4-5"]). End with one verdict node (kind verdict) listing every audited claim with its outcome. Same RULES, timestamps, node format and return schema as above; your files go in scratchpad/solve-vi/A5/.

WORKING DISCIPLINE.
- Never produce one response longer than about 12,000 words, thinking included. Work in short steps with frequent tool calls.
- As soon as a node is finished, write it to its own file /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve-vi/A5/nodes/A5-<n>.json (one node per tool call, valid JSON). Keep notes in files.
- At the end, assemble return.json from the node files with a short script, check that it parses, and end with the JSON inside a ```json fence.