{"agent": "S4", "family": "Sakellaridis bridge; E6/F4 hypotheses; exact image of r", "nodes": [{"id": "S4-1", "t": "2026-10-17T21:13:58Z", "kind": "lemma", "status": "proved", "label": "Theorem (cited)", "name": "Sakellaridis's theorem with all hypotheses", "title": "Sakellaridis's Theorem 8.0.2 and Corollary 8.0.4 with every hypothesis in his own terms", "summary": "States Sakellaridis's description of the unramified Hecke module \\(C_c^\\infty(X)^K\\) and his principal-module criterion exactly as in arXiv:0905.4244v3, with page numbers, every standing assumption and the invariants \\(A_X,\\Lambda_X,W_X,\\delta(X)\\) and the restriction map \\(r\\).", "statement": "Page numbers are the printed (= PDF) pages of arXiv:0905.4244v3.\n\n**Standing setting (§1.1 p. 2, §1.7 pp. 6–7, §2.1 p. 8).**\n- (S1) \\(k\\) is a non-archimedean local field of characteristic \\(0\\), ring of integers \\(\\mathfrak o\\), residue field \\(\\mathbb F_q\\); \"to avoid complicated formulas\" \\(k/\\mathbb Q_p\\) is assumed unramified, \"the modifications needed to remove this assumption are trivial\" (p. 2).\n- (S2) \\(G\\) is split reductive over \\(\\mathfrak o\\) with connected fibres and simply connected derived group; \\(B\\supseteq A\\) a Borel and a maximal split torus; \\(K=G(\\mathfrak o)\\), \\(J\\) the Iwahori subgroup (preimage of \\(B(\\mathbb F_q)\\)).\n- (S3) \\(X\\) is a quasi-affine \\(G\\)-scheme over \\(\\mathfrak o\\) with an open \\(B_k\\)-orbit \\(\\mathring X_k\\) such that \\(\\mathring X:=X\\smallsetminus\\overline{(X\\smallsetminus\\mathring X_k)}\\) is smooth and surjective over \\(\\mathfrak o\\) with special fibre homogeneous under \\(B_{\\mathbb F_q}\\); \\(X\\) carries a \\(G\\)-eigenmeasure and \\(\\mathring X\\) a \\(B\\)-invariant measure; \\(x_0\\in\\mathring X(\\mathfrak o)\\), \\(H=\\) its stabilizer, \\(X_k=H\\backslash G\\) (right action).\n- (S4) **Axiom 2.2.1 (p. 9).** For every \\(B_k\\)-orbit \\(Y_k\\) of maximal rank and every simple root \\(\\alpha\\), the \\(\\mathfrak o\\)-scheme \\(Y P_\\alpha/R(P_\\alpha)\\) is isomorphic to \\(\\mathrm{PGL}_2\\backslash\\mathrm{PGL}_2\\), \\(SU\\backslash\\mathrm{PGL}_2\\) (\\(S\\subseteq N(U)\\)), \\(T\\backslash\\mathrm{PGL}_2\\) (\\(T\\) a non-trivial torus over \\(\\mathfrak o\\)) or \\(N(T)\\backslash\\mathrm{PGL}_2\\); and the complement of \\(YP_\\alpha\\) is the closure of the complement of \\((YP_\\alpha)_k\\). (\\(Y\\) is the complement of the closure of the complement of \\(Y_k\\) in its closure.)\n- (S5) **Axiom 2.4.1 (p. 10).** \\(A_X^+\\) contains a complete set of representatives of the \\(K\\)-orbits on \\(X\\); elements of \\(A_X^+\\) with distinct images in \\(\\Lambda_X^+\\) lie in distinct \\(K\\)-orbits.\n- (S6) **Axiom 2.4.2 (p. 11).** For \\(x\\in A_X^+\\): \\(xJ\\subseteq xB(\\mathfrak o)\\).\n- (S7) **§7 standing assumption (p. 58).** \\(X\\) has no spherical root of type \\(N\\), and for every \\(\\gamma\\in\\Delta_X\\) the Levi \\(L_\\gamma\\) is a classical group.\n- (S8) **Statement 6.3.1 (pp. 44–45).** For a Brion path of the simple spherical reflection \\(w_\\gamma\\), with \\(\\check\\theta\\) the valuation of its codimension-one orbit: if \\(\\gamma\\) has type \\(G\\) then \\(\\check\\theta=\\check\\gamma\\) and \\(B_{w_\\gamma}(\\tilde\\chi)=-e^{\\check\\gamma}\\dfrac{1-q^{-\\langle\\check\\gamma,\\rho_{P(X)}\\rangle}e^{-\\check\\gamma}}{1-q^{-\\langle\\check\\gamma,\\rho_{P(X)}\\rangle}e^{\\check\\gamma}}(\\chi)\\); formulas (6.4)–(6.6) for types \\(T\\) split, \\(T\\) non-split, \\((U,\\psi)\\).\n- (S9) **Statement 7.1.5 (p. 60).** Every \\((\\check\\theta,\\sigma,r)\\in\\Theta\\) has \\(\\check\\theta>0\\) or \\(\\check\\theta<0\\); for every \\(\\gamma\\in\\Delta_X\\), \\(\\{(\\check\\theta,\\sigma,r)\\in\\Theta^+:w_\\gamma\\check\\theta<0\\}\\) is exactly the set of virtual weighted colours belonging to \\(\\gamma\\).\n- (S10) **§8 standing assumption (p. 63).** \\(X\\) is homogeneous and affine, satisfies the hypotheses of Theorem 7.2.1, and \\(\\mathring X\\) is a single \\(B\\)-orbit.\n\n**Theorem 7.2.1 (p. 60)** assumes exactly (S8) for all simple spherical roots and (S9); inside §7, i.e. under (S1)–(S7).\n\n**Invariants (pp. 3, 7–9, 41–44, 59–60, 63).**\n- \\(P(X)=\\{p:\\mathring Xp=\\mathring X\\}\\); \\(L(X)\\) a Levi of \\(P(X)\\) with \\([L(X),L(X)]\\subseteq H\\); \\(A\\subseteq B\\cap L(X)\\); \\(\\delta(X)=|d(X)|\\), \\(d(X)\\) the modular character of \\(L(X)\\cap B\\); \\(\\rho_{P(X)}\\) the half-sum of the roots of the unipotent radical of \\(P(X)\\).\n- \\(A_X:=L(X)/(L(X)\\cap H)\\), a torus identified with the \\(L(X)\\)-orbit of \\(x_0\\), a quotient of \\(A\\).\n- \\(\\mathcal X(X)\\) the group of \\(B\\)-eigencharacters of \\(k(X)^{(B)}\\); \\(\\Lambda_X=X_*(A_X)=\\operatorname{Hom}(\\mathcal X(X),\\mathbb Z)\\); \\(\\Lambda_X^+\\) the monoid of \\(G\\)-invariant valuations, \\(\\mathcal V\\) its cone; \\(\\Lambda_X\\simeq A_X(k)/A_X(\\mathfrak o)\\) by \\(\\check\\lambda\\mapsto e^{\\check\\lambda}(\\varpi)\\), and \\(A_X^+\\) the preimage of \\(\\Lambda_X^+\\) (the \"\\(X\\)-anti-dominant\" points).\n- \\(A^*\\), \\(A_X^*\\) the complex dual tori (unramified characters); \\(A\\twoheadrightarrow A_X\\) induces \\(A_X^*\\to A^*\\), whose image is again written \\(A_X^*\\).\n- \\(W_X\\subseteq W\\), the little Weyl group: the stabilizer of \\(\\mathring X\\) for Knop's \\(W\\)-action on orbits of maximal rank is \\(W(X)=W_X\\ltimes W_{P(X)}\\); \\(W_X\\) acts faithfully on \\(\\mathcal X(X)\\) by reflections with fundamental domain \\(\\mathcal V\\).\n- \\(\\delta(X)^{1/2}\\) is a point of \\(A^*\\) (an unramified character of \\(A\\)); \\(\\delta(X)^{1/2}A_X^*\\subseteq A^*\\) is the translate, with \\(W_X\\) acting by transport from \\(A_X^*\\).\n- \\(\\Delta_X\\) the normalized spherical roots (p. 42): the positive multiple of each generator of an extremal ray of \\(-\\mathcal V^\\vee\\) that is a root of \\(G\\) or a sum of two orthogonal roots simple for some basis; type \\(G\\) iff not proportional to a root of \\(G\\), otherwise type \\(T\\) or \\(N\\) according as \\(\\gamma'\\in\\mathcal X(X)\\) or not (p. 44); \\(L_\\gamma\\) the standard Levi of \\(\\operatorname{supp}\\gamma\\).\n- Colours \\(D\\) with valuations \\(\\check v_D\\); virtual weighted colours \\((\\check v_D,\\sigma,\\langle\\check v_D,\\rho_{P(X)}\\rangle)\\) plus the type-\\(T\\) additions (pp. 59–60); \\(\\Theta\\) the smallest \\(W_X\\)-stable, negation-closed multiset containing them; \\(\\Theta^+\\) its elements in the cone of colour valuations.\n- \\(r\\): the restriction map \\((8.2)\\ \\mathbb C[A^*]^W\\to\\mathbb C[\\delta(X)^{1/2}A_X^*]^{W_X}\\); \\(\\mathcal H_X\\) the image of \\(\\mathcal H(G,K)\\simeq\\mathbb C[A^*]^W\\) in \\(\\mathbb C[\\delta(X)^{1/2}A_X^*]\\) (p. 63).\n\n**Theorem 8.0.2 (p. 64).** Under (S1)–(S10): there is a canonical isomorphism \\(C_c^\\infty(X)^K\\simeq\\mathbb C[\\delta(X)^{1/2}A_X^*]^{W_X}\\), compatible with the \\(\\mathcal H(G,K)\\)-structure and the Satake isomorphism, under which \\(\\mathbf 1_{X(\\mathfrak o)}\\mapsto1\\).\n\n**Corollary 8.0.4 (p. 65).** Under (S1)–(S10), if \\(r\\) is surjective (equivalently \\(\\mathcal H_X\\simeq\\mathbb C[\\delta(X)^{1/2}A_X^*]^{W_X}\\)), then (1) \\(C_c^\\infty(X)^K\\) is a principal \\(\\mathcal H(G,K)\\)-module generated by \\(\\mathbf 1_{X(\\mathfrak o)}\\); (2) every Hecke eigenspace in \\(C^\\infty(X)^K\\) with character in \\(\\operatorname{specM}\\mathcal H_X\\) is one-dimensional and \\(\\dim\\operatorname{Hom}_G(\\pi,C^\\infty(X))\\le1\\) for unramified \\(\\pi\\); (3) every non-zero \\(H\\)-invariant functional on an unramified \\(\\pi\\) is non-zero on \\(\\pi^K\\). The proof is \"straightforward and will be omitted\".\n\n**Example 8.0.1 (p. 64).** For \\(X=\\mathrm{SL}_2^3\\backslash\\mathrm{Sp}_4\\times\\mathrm{Sp}_4\\) one has \\(W_X=N_W(A_X^*)\\) and \\(\\delta(X)=1\\), but \\(W_X\\)-nonconjugate points of \\(A_X^*\\) in the diagonal \\(\\mathrm{SO}_4\\) are \\(W\\)-conjugate, so \\(r\\) is not surjective: the hypothesis of Corollary 8.0.4 is not automatic.", "parents": [], "corrects": [], "body": "1. **Where each item is stated.** (S1): §1.1, p. 2, which also states that \\(k\\) has characteristic zero. (S2), (S3): §1.7, pp. 6–7, which also requires (S4)–(S6) (\"We also assume that \\(X\\) satisfies Axioms 2.2.1, 2.4.1 and 2.4.2, which is always the case at almost every place if \\(X\\) and \\(G\\) are defined over a global field\"); \\(x_0\\) and \\(H\\): §2.1, p. 8. (S4): p. 9, where the author adds that it \"is clearly the case at almost every place if \\(X\\) is defined globally\". (S5), (S6): pp. 10–11, proved in [Sa12] \"under assumptions which hold at almost every place\", with Benoist–Oh and Delorme–Sécherre cited for \"similar statements\" for symmetric varieties. (S7): opening of §7, p. 58. (S8): pp. 44–45; by **Theorem 6.3.2 (p. 45)** it holds for every normalized spherical root and every Brion path as soon as (S7) holds. (S9): p. 60, conjectured by the author for affine homogeneous \\(X\\) and \"very easy to check in each particular case\". (S10): p. 63, which adds \"In particular, the strong form of the Cartan decomposition holds: \\(X/K\\simeq\\Lambda_X^+\\) and \\(A_X^*\\subseteq A^*\\)\".\n2. **The invariants.** \\(A_X\\), \\(\\Lambda_X\\), \\(\\Lambda_X^+\\), \\(A_X^+\\), \\(A_X^*\\): §2.1, p. 8. \\(W(X)=W_X\\ltimes W_{P(X)}\\) and the reflection action: §2.2, p. 9. \\(\\delta(X)\\): §1.7, p. 7. The translate and its \\(W_X\\)-action: p. 3, where the map \\(\\delta(X)^{1/2}A_X^*\\to\\operatorname{specM}\\mathcal H(G,K)\\) is said to factor through \\(\\delta(X)^{1/2}A_X^*/W_X\\). Spherical roots, normalization and types: §6.1–6.2, pp. 41–44. Virtual colours and \\(\\Theta\\): §7.1, pp. 59–60. \\(r\\) and \\(\\mathcal H_X\\): §8, pp. 63–65.\n3. **Role of Example 8.0.1.** It is the only place in the paper where the hypothesis of Corollary 8.0.4 is shown to fail. The mechanism is a gluing: two points of \\(\\delta(X)^{1/2}A_X^*\\) that are not \\(W_X\\)-conjugate become \\(W\\)-conjugate, so every element of \\(\\operatorname{im}r\\) takes equal values at them while \\(\\mathbb C[\\delta(X)^{1/2}A_X^*]^{W_X}\\) separates them. Its Remark on p. 65 records that it forces a correction to [Sa08, Theorem 6.3.2]. The author does not verify (S1)–(S10) for this example.\n4. **Status.** Everything above is quoted from the paper; nothing is re-proved here (cited).", "concept": "bridge", "key": true, "sources": ["sakellaridis-v3.pdf, §1.1 p. 2; §1.7 pp. 6–7; §2.1 p. 8; Axiom 2.2.1 p. 9; Axioms 2.4.1–2.4.2 pp. 10–11", "sakellaridis-v3.pdf, §6.1–6.2 pp. 41–44; Statement 6.3.1 pp. 44–45; Theorem 6.3.2 p. 45", "sakellaridis-v3.pdf, §7 p. 58; §7.1 pp. 58–60; Statement 7.1.5 and Theorem 7.2.1 p. 60", "sakellaridis-v3.pdf, §8 p. 63; Example 8.0.1 and Theorem 8.0.2 p. 64; Corollary 8.0.4 p. 65"]}, {"id": "S4-2", "t": "2026-10-17T21:14:44Z", "kind": "check", "status": "proved", "label": "Audit", "name": "Proof skeleton; pages' hypotheses audited", "title": "How Theorem 8.0.2 is proved, where each hypothesis enters, what the pages drop", "summary": "Lays out the proof of Theorem 8.0.2 in six steps, records which hypothesis each step uses, and checks the pages' list (A1)–(A5) (field and group; the \\(\\mathcal O\\)-model; Axioms 2.2.1, 2.4.1, 2.4.2; spherical roots and colours; homogeneous affine with one open \\(B\\)-orbit) against the paper: accurate, with six losses, the main one being that the pages never fix an \\(\\mathcal O\\)-model of \\(X\\).", "statement": "**(I) Proof skeleton of Theorem 8.0.2** (labels (S1)–(S10) as in [[S4-1]]).\n1. *Mackey morphisms* (§3). The open \\(B\\)-orbit gives \\(S_\\chi:C_c^\\infty(X)\\to I(\\chi)\\) and adjoints \\(\\Delta_\\chi\\); the eigenfunctions are \\(\\Omega_\\chi=\\Delta_\\chi(\\Phi_K)\\). Uses (S1)–(S4): eigenmeasures are absolute values of forms over \\(\\mathfrak o\\) on the \\(\\mathfrak o\\)-schemes of maximal-rank orbits.\n2. *Sum formula* (Theorem 4.2.2, p. 26; (6.2), p. 44). For generic \\(\\chi\\) and \\(x\\in A_X^+\\): \\(\\Omega_{\\delta(X)^{1/2}\\tilde\\chi}(x)=\\delta_{P(X)}^{-1/2}(x)\\sum_{w\\in W_X}B_w(\\tilde\\chi)\\,{}^w\\tilde\\chi(x)\\). Uses (S5) to know that \\(A_X^+\\) meets every \\(K\\)-orbit and (S6) to pass from the Iwahori-fixed vector to \\(B(\\mathfrak o)\\)-orbits.\n3. *Simple spherical reflections* (§6). (S8) evaluates \\(B_{w_\\gamma}\\); it holds under (S7) by Theorem 6.3.2 (p. 45).\n4. *The formula with \\(L\\)-values* (Theorem 7.2.1, p. 60). Under (S8) and (S9), \\(B_w(\\tilde\\chi)=\\beta(\\tilde\\chi)/\\beta({}^w\\tilde\\chi)\\) with \\(\\beta=\\prod_{\\check\\gamma\\in\\check\\Phi_X^+}(1-e^{\\check\\gamma})\\big/\\prod_{\\Theta^+}(1-\\sigma_{\\check\\theta}q^{-r_{\\check\\theta}}e^{\\check\\theta})\\), so \\(\\Omega(x_{\\check\\lambda})/\\beta\\) is the \\(W_X\\)-invariant Laurent polynomial (7.4); for affine \\(X\\), \\(\\Omega(x_0)=c\\,\\beta\\).\n5. *Theorem 8.0.2* (p. 64). \\(P_{\\check\\lambda}:=\\Omega(x_{\\check\\lambda})/L_X^{1/2}\\in\\mathbb C[\\delta(X)^{1/2}A_X^*]^{W_X}\\), \\(P_0=1\\); \\(\\mathbf 1_{x_{\\check\\lambda}K}\\mapsto P_{\\check\\lambda}\\) is \\(\\mathcal H(G,K)\\)-equivariant through Satake; by the Remark after Theorem 7.2.1 (p. 62) \\(\\check\\lambda\\) is the lowest weight of \\(P_{\\check\\lambda}\\) for the order defined by the cone \\(\\mathcal T\\) of colour valuations, and \\(\\mathcal T\\cap\\mathcal V=\\{0\\}\\) for affine \\(X\\); triangularity makes the \\(P_{\\check\\lambda}\\) a basis. Uses (S10) for \\(X/K\\simeq\\Lambda_X^+\\).\n6. *The criterion.* \\(\\mathcal H(G,K)\\) acts on the target by multiplication through \\(r\\) and \\(\\mathbf 1_{X(\\mathfrak o)}\\mapsto1\\), so \\(\\mathcal H(G,K)\\cdot\\mathbf 1_{X(\\mathfrak o)}\\) corresponds to \\(\\operatorname{im}r\\): the module is generated by \\(\\mathbf 1_{X(\\mathfrak o)}\\) iff \\(r\\) is onto (\\(\\Leftarrow\\) is Corollary 8.0.4(1)).\n\n**(II) Correspondence with the pages' list.**\n\n| pages | requires | paper |\n|---|---|---|\n| (A1) field and group | \\(F\\) \\(p\\)-adic; \\(G\\) split, connected fibres, simply connected derived group | (S1), (S2) |\n| (A2) the variety over \\(\\mathcal O\\) | \\(X\\) quasi-affine over \\(\\mathcal O\\), \\(\\mathring X\\) smooth surjective with \\(B\\)-homogeneous special fibre, eigenmeasures | (S3) |\n| (A3) Axioms 2.2.1, 2.4.1, 2.4.2 | rank-one quotients over \\(\\mathcal O\\); \\(K\\)-orbit representatives in \\(A_X^+\\); \\(xJ\\subseteq xB(\\mathcal O)\\) | (S4)–(S6) |\n| (A4) spherical roots and colours | no type \\(N\\), \\(L_\\gamma\\) classical, Statement 6.3.1, Statement 7.1.5 | (S7)–(S9) |\n| (A5) homogeneous, affine, one open orbit | \\(X=H\\backslash G\\) affine, \\(\\mathring X(F)\\) one \\(B(F)\\)-orbit | (S10) |\n\n**(III) Verdict on the summary.** Accurate as a list, with these losses:\n- L1. The simplification \"\\(k/\\mathbb Q_p\\) unramified\" (p. 2) is dropped; the paper calls its removal trivial (not checked here).\n- L2. **No \\(\\mathcal O\\)-model is fixed.** Sakellaridis's \\(X\\) is an \\(\\mathfrak o\\)-scheme with a chosen \\(x_0\\in\\mathring X(\\mathfrak o)\\); \\(K\\backslash X(F)\\), the axioms and \\(\\mathbf 1_{X(\\mathfrak o)}\\) depend on it, while the pages' \\(X=G/G^\\theta\\) is a variety over \\(\\mathbb C\\). Every verdict must name the model.\n- L3. The paraphrase of Axiom 2.2.1 drops its second clause (complement of \\(YP_\\alpha\\) = closure of the complement of \\((YP_\\alpha)_k\\)), the \\(\\mathfrak o\\)-scheme structure on \\(Y\\), and that \\(T\\) may be a non-split torus over \\(\\mathfrak o\\).\n- L4. In the requirement \"spherical roots and colours\" (A4), \"Statement 6.3.1\" is not independent: Theorem 6.3.2 derives it from \"no type \\(N\\), \\(L_\\gamma\\) classical\". Conversely those two conditions are what make \"type of a colour\" and \\(\\Theta\\) well defined (Lemma 7.1.4, p. 59, proved by inspection of Theorem 6.11.1); the hypotheses of Theorem 7.2.1 proper are only 6.3.1 and 7.1.5.\n- L5. Theorem E's \"(SG) iff \\(r\\) onto\" combines Theorem 8.0.2 (\\(\\Rightarrow\\)) with Corollary 8.0.4(1) (\\(\\Leftarrow\\)); correct. Its item on Example 8.0.1 is correct as stated under the hypotheses; the paper does not verify them for that example.\n- L6. The sentence \"none of them is verified on these pages for any family\" is now outdated for \\(E_6/F_4\\): see [[S4-3]], [[S4-4]], [[S4-5]].", "parents": ["S4-1"], "corrects": [], "body": "1. **Step 1–2.** Read off §3 (eigenmeasures from forms over \\(\\mathfrak o\\), §3.2) and Theorem 4.2.2 (p. 26), whose formula is stated \"for \\(x\\in A_X^+\\)\"; the passage from the Iwahori vector to \\(K\\)-invariant vectors uses \\(x_0J\\subseteq x_0B_0\\) \"by Axiom 2.4.2\" (pp. 17 and 24). Cited.\n2. **Steps 3–4.** Statement 6.3.1 is used in the proof of Theorem 7.2.1 exactly at \"By Statements 7.1.5 and 6.3.1, for \\(w=w_\\gamma\\) this is equal to \\(B_{w_\\gamma}\\)\" (p. 61); the affine case uses Knop's separation of \\(\\mathcal V\\) from the colour valuations (p. 61). Cited.\n3. **Step 5.** The proof of Theorem 8.0.2 on p. 64: definition (8.1) of \\(P_{\\check\\lambda}\\), equivariance, lowest weight for \\(\\succ_2\\), \"the usual inductive argument (which here uses the fact that \\(\\mathcal T\\cap\\mathcal V=\\{0\\}\\) to commence)\". Cited.\n4. **Step 6.** Under the isomorphism, \\(f\\cdot\\varphi\\leftrightarrow r(f)\\cdot P_\\varphi\\) and \\(\\mathbf 1_{X(\\mathfrak o)}\\leftrightarrow1\\), so \\(\\mathcal H(G,K)\\mathbf 1_{X(\\mathfrak o)}\\leftrightarrow r(\\mathbb C[A^*]^W)\\). Proved here (one line).\n5. **The table and L1–L6** compare the classical page (section \"Sakellaridis's theorem, with its hypotheses\", anchors hyp-a1 … hyp-a5 and sakellaridis-theorem) with [[S4-1]] item by item. For L2, the pages' (A2) (\"\\(X\\) a quasi-affine \\(G\\)-scheme over \\(\\mathcal O\\) …\") presupposes a model but none of the family pages chooses one; the \\(E_6/F_4\\) model is fixed in [[S4-4]].", "concept": "bridge", "key": false, "sources": ["sakellaridis-v3.pdf, Theorem 4.2.2 p. 26; (6.2) p. 44; Theorem 6.3.2 p. 45; Lemma 7.1.4 p. 59", "sakellaridis-v3.pdf, Theorem 7.2.1 and proof pp. 60–62; Theorem 8.0.2 and proof p. 64; Corollary 8.0.4 p. 65", "classical.html, section 4 (#s-hypotheses, #sakellaridis-theorem)"]}, {"id": "S4-3", "t": "2026-10-17T21:15:37Z", "kind": "computation", "status": "proved", "label": "Proposition", "name": "E6/F4: spherical roots, colours, Theta", "title": "\\(E_6/F_4\\): type-\\(G\\) spherical roots, two colours; Statements 6.3.1, 7.1.5 hold", "summary": "Computes the spherical roots, their types and Levis, the colours and the set \\(\\Theta\\) of \\(E_6/F_4\\) from the restricted root system, and checks that every part of Sakellaridis's spherical-root-and-colour hypotheses holds: no type \\(N\\), classical \\(L_\\gamma\\), Statement 6.3.1 (via his Theorem 6.3.2 and §6.19 with \\(n=5\\)) and Statement 7.1.5.", "statement": "Let \\(F\\) have characteristic \\(0\\), \\(G=E_6\\) split simply connected, \\(\\theta=-w_L\\) on \\(T\\) (Bourbaki labels; black nodes \\(2,3,4,5\\), \\(w_L\\) the longest element of \\(W(D_4)\\)), \\(H=G^\\theta=F_4\\), \\(B\\subseteq P\\) with \\(P\\) minimal \\(\\theta\\)-split and \\(L(X)=L=Z_G(S)\\). Write \\(\\gamma_i:=\\alpha_i-\\theta\\alpha_i\\) and \\(e_i:=\\varphi(\\alpha_i^\\vee)\\in\\Lambda_X\\) (\\(i=1,6\\)). Then:\n1. \\(\\mathcal X(X)=\\mathbb Z\\omega_1\\oplus\\mathbb Z\\omega_6\\), \\(\\Lambda_X=\\mathbb Ze_1\\oplus\\mathbb Ze_6\\) is its dual basis, and\n$$\\gamma_1=2\\alpha_1+\\alpha_2+2\\alpha_3+2\\alpha_4+\\alpha_5=\\alpha_1+w_L\\alpha_1=2\\omega_1-\\omega_6,\\qquad\\gamma_6=\\alpha_2+\\alpha_3+2\\alpha_4+2\\alpha_5+2\\alpha_6=-\\omega_1+2\\omega_6 ,$$\nwith \\((\\alpha_1,w_L\\alpha_1)=0\\). \\(\\Delta_X=\\{\\gamma_1,\\gamma_6\\}\\) are the normalized spherical roots, both of **type \\(G\\)**; there is no root of type \\(N\\).\n2. \\(L_{\\gamma_1}\\), \\(L_{\\gamma_6}\\) are the standard Levis of \\(\\{1,\\dots,5\\}\\) and \\(\\{2,\\dots,6\\}\\), of type \\(D_5\\): classical.\n3. \\(X\\) has exactly two colours \\(D_1,D_6\\), with \\(\\check v_{D_1}=e_1=\\check\\gamma_1\\), \\(\\check v_{D_6}=e_6=\\check\\gamma_6\\) (the matrix \\(\\langle e_i,\\gamma_j\\rangle\\) is the Cartan matrix of \\(A_2\\)); \\(\\langle\\check\\gamma_i,\\rho_{P(X)}\\rangle=4\\).\n4. The virtual weighted colours are \\((\\check\\gamma_1,+,4)\\), \\((\\check\\gamma_6,+,4)\\); \\(\\Theta=\\check\\Phi_X\\times\\{(+,4)\\}\\) (the six coroots of the restricted \\(A_2\\), multiplicity one) and \\(\\Theta^+=\\check\\Phi_X^+\\).\n5. **Statement 7.1.5 holds**, and **Statement 6.3.1 holds** with \\(B_{w_{\\gamma}}(\\tilde\\chi)=-e^{\\check\\gamma}\\dfrac{1-q^{-4}e^{-\\check\\gamma}}{1-q^{-4}e^{\\check\\gamma}}\\) for \\(\\gamma\\in\\Delta_X\\).\n6. Consequently \\(\\beta(\\tilde\\chi)=\\prod_{\\check\\gamma\\in\\check\\Phi_X^+}\\dfrac{1-e^{\\check\\gamma}}{1-q^{-4}e^{\\check\\gamma}}\\).\n\nThese are statements about \\(X_{\\bar F}\\); they hold for every residue characteristic and do not depend on an \\(\\mathcal O\\)-model. Statement 6.3.1 is a statement about the cocycles \\(B_w\\), which the paper defines under (S1)–(S6) of [[S4-1]].", "parents": ["S4-1", "A6-4"], "corrects": ["A6-9"], "body": "1. **Roots.** \\(\\theta=-w_L\\) and \\(w_L=-1\\) on the span of \\(\\alpha_2,\\dots,\\alpha_5\\). Projecting \\(\\alpha_1\\) onto that span (it pairs only with \\(\\alpha_3\\)) gives \\(w_L\\alpha_1=\\alpha_1+\\alpha_2+2\\alpha_3+2\\alpha_4+\\alpha_5\\), the highest root of the \\(D_5\\) on nodes \\(1,\\dots,5\\) (as on the family page). Hence \\(\\gamma_1=\\alpha_1+w_L\\alpha_1\\) and \\((\\alpha_1,w_L\\alpha_1)=2+(-1)\\cdot2=0\\). Pairing with the simple coroots gives \\(\\langle\\alpha_1^\\vee,\\gamma_1\\rangle=2\\), \\(\\langle\\alpha_6^\\vee,\\gamma_1\\rangle=-1\\) and \\(0\\) on the black coroots, i.e. \\(\\gamma_1=2\\omega_1-\\omega_6\\). \\(\\gamma_6\\) follows by \\(1\\leftrightarrow6\\), \\(3\\leftrightarrow5\\).\n2. **Lattices.** \\(G\\) is simply connected, so \\(\\mathcal X(T)\\) is the weight lattice. \\(T^\\theta\\) is the torus of \\(\\alpha_2^\\vee,\\dots,\\alpha_5^\\vee\\) (connected: \\(S[2]\\subseteq(T^\\theta)^\\circ\\), family page §4) and \\(B\\cap H=B_L\\cap L^\\theta\\) is a Borel subgroup of \\(\\mathrm{Spin}_8\\) ([[A6-9]]). \\(B\\)-eigenfunctions on \\(\\mathring X=B/(B\\cap H)\\) have weights trivial on \\(T\\cap H=T^\\theta\\), and every such weight occurs; so \\(\\mathcal X(X)=\\{\\chi:\\langle\\alpha_j^\\vee,\\chi\\rangle=0,\\ j=2,\\dots,5\\}=\\mathbb Z\\omega_1\\oplus\\mathbb Z\\omega_6\\). The cocharacter of \\(A_X=T/T^\\theta\\) induced by \\(\\lambda\\in\\Lambda_T\\) pairs with \\(\\chi\\in\\mathcal X(X)\\) as \\(\\langle\\lambda,\\chi\\rangle\\); so \\(\\langle e_i,\\omega_j\\rangle=\\delta_{ij}\\).\n3. **Spherical roots.** For a symmetric variety the valuation cone \\(\\mathcal V\\) is the image of the antidominant chamber (Sakellaridis p. 8, citing [Kn91, §5]); the extremal rays of \\(-\\mathcal V^\\vee\\) are spanned by the restricted simple roots, whose primitive vectors in \\(\\mathcal X(X)\\) are \\(\\gamma_1,\\gamma_6\\) (both primitive, since \\(2\\omega_1-\\omega_6\\) and \\(-\\omega_1+2\\omega_6\\) have coprime coordinates).\n4. **Normalization and type** (p. 42, p. 44). \\(\\gamma_1\\) is the sum of the orthogonal roots \\(\\alpha_1,w_L\\alpha_1\\). \\(W(E_6)\\) is transitive on ordered pairs of orthogonal roots: it is transitive on roots, and the roots orthogonal to a given root form an \\(A_5\\) on which its stabilizer, containing \\(W(A_5)\\), is transitive. So the pair is conjugate to \\((\\alpha_1,\\alpha_6)\\) and is simple for some basis; hence \\(\\gamma_1'=\\gamma_1\\). No root is proportional to \\(\\gamma_1\\) (its \\(\\alpha_1\\)-coefficient is \\(2\\) and \\(\\gamma_1/2\\) is not in the root lattice), so \\(\\gamma_1\\) has type \\(G\\). Type \\(N\\) roots are proportional to roots, so none occurs. Same for \\(\\gamma_6\\).\n5. **Levis.** \\(\\operatorname{supp}\\gamma_1=\\{1,2,3,4,5\\}\\), \\(\\operatorname{supp}\\gamma_6=\\{2,\\dots,6\\}\\): both \\(D_5\\).\n6. **Colours.** \\(\\mathrm{Pic}(X_{\\bar F})\\simeq\\mathcal X(F_4)=0\\) (\\(G\\) simply connected) and \\(\\bar F[X]^\\times=\\bar F^\\times\\) (\\(G\\) semisimple), so \\(\\bar F[X]\\) is factorial and every colour is the divisor of an irreducible \\(B\\)-eigenfunction. The weights of \\(\\bar F[X]^{(B)}\\) are the spherical weights \\(a\\omega_1+b\\omega_6\\), \\(a,b\\ge0\\) (Helgason's condition: \\(\\langle\\lambda,\\bar\\alpha_i\\rangle/\\langle\\bar\\alpha_i,\\bar\\alpha_i\\rangle\\) equals \\(a\\), resp. \\(b\\), for \\(\\bar\\alpha_i=\\gamma_i/2\\)), each with a one-dimensional eigenspace; this monoid is free on two generators, so there are exactly two colours, with \\(v_{D_i}(f_{\\omega_j})=\\delta_{ij}\\), i.e. \\(\\check v_{D_i}=e_i\\) (up to exchanging the names \\(1\\leftrightarrow6\\), which depends on left/right conventions and is harmless by the diagram symmetry). Since \\(\\langle e_i,\\gamma_j\\rangle\\) is the Cartan matrix of \\(A_2\\), \\(e_i=\\check\\gamma_i\\).\n7. **\\(r\\)-values.** \\(\\langle\\check\\gamma_1,\\rho_{P(X)}\\rangle=\\langle\\alpha_1^\\vee,\\rho-\\rho_L\\rangle=1-(-3)=4\\) (\\(\\langle\\rho_L,\\alpha_1^\\vee\\rangle=-3\\), [[A6-4]]); \\(\\rho-\\rho_L\\) vanishes on the black coroots, so it is a character of \\(A_X\\). Same for \\(\\gamma_6\\).\n8. **\\(\\Theta\\).** Type-\\(G\\) colours are defined over \\(F\\), so \\(\\sigma=+\\); the additions (2)–(3) of the definition on p. 59 concern type \\(T\\) only. The \\(W_X=S_3\\)-orbit of \\(\\check\\gamma_1\\) is the six coroots of \\(\\check\\Phi_X\\) and contains \\(\\check\\gamma_6\\); it is negation-closed. So \\(\\Theta=\\check\\Phi_X\\times\\{(+,4)\\}\\) with multiplicity one, and the cone of colour valuations \\(\\mathbb R_{\\ge0}e_1+\\mathbb R_{\\ge0}e_6\\) meets it in \\(\\check\\Phi_X^+\\).\n9. **Statement 7.1.5.** Each coroot is positive or negative. \\(w_{\\gamma_1}\\) is a simple reflection of \\(\\check\\Phi_X\\), so \\(\\{\\check\\theta\\in\\check\\Phi_X^+:w_{\\gamma_1}\\check\\theta<0\\}=\\{\\check\\gamma_1\\}\\); and the colours belonging to \\(\\gamma_1\\) are those with \\(\\langle\\check v_D,\\gamma_1\\rangle>0\\) (Lemma 7.1.2, p. 58), i.e. \\(D_1\\) only (\\(\\langle\\check\\gamma_6,\\gamma_1\\rangle=-1\\)). Same for \\(\\gamma_6\\).\n10. **Statement 6.3.1.** By steps 4–5 (no type \\(N\\), classical \\(L_\\gamma\\)) and Theorem 6.3.2 (p. 45) it holds. Cross-check: \\(\\mathring XP_{\\gamma_1}/R(P_{\\gamma_1})\\) is a spherical variety of rank one for the \\(D_5\\) group (the only spherical root in the span of \\(\\operatorname{supp}\\gamma_1\\) is \\(\\gamma_1\\)), with spherical root of full support and no type \\(N\\); in the list of Theorem 6.11.1 (pp. 51–52) the only such entry is \\(\\mathrm{Spin}_9\\backslash\\mathrm{Spin}_{10}\\), computed in §6.19 (p. 56) with \\(n=5\\): \\(B_{w_\\gamma}=-e^{\\check\\gamma}(1-q^{-n+1}e^{-\\check\\gamma})/(1-q^{-n+1}e^{\\check\\gamma})\\), and \\(n-1=4=\\langle\\check\\gamma,\\rho_{P(X)}\\rangle\\), matching formula (6.3).\n11. **\\(\\beta\\).** Substitute into (7.2). (Heuristic remark: this is Macdonald's \\(c\\)-function for the restricted root system \\(A_2\\) with all multiplicities \\(8\\), as expected for \\(E_{6(-26)}\\).)\n\nChecked by exact computation in c1_a4.py: the roots and pairings of steps 1, 4, 7, the absence of any root \\(\\beta\\) with \\(\\theta\\beta=-\\beta\\), \\(\\Theta\\) and Statement 7.1.5.", "concept": "e6f4-functions", "key": true, "sources": ["sakellaridis-v3.pdf, §2.1 p. 8; §6.1 p. 42; §6.2 p. 44; Statement 6.3.1 pp. 44–45; Theorem 6.3.2 p. 45", "sakellaridis-v3.pdf, Theorem 6.11.1 pp. 51–52; §6.19 (6.18) p. 56; Lemma 7.1.2 p. 58; virtual weighted colours and Θ pp. 59–60; Statement 7.1.5 p. 60", "families/e6-f4.html §§1–4"]}, {"id": "S4-4", "t": "2026-10-17T21:16:25Z", "kind": "lemma", "status": "proved", "label": "Proposition", "name": "E6/F4: the O-model; field, group, X̊ checks", "title": "\\(E_6/F_4\\), \\(p\\) odd: the \\(\\mathcal O\\)-model meets §1.7 and §8", "summary": "Fixes the integral model of \\(E_6/F_4\\) that every function-level verdict needs and proves, for residue characteristic \\(p\\ne2\\), Sakellaridis's conditions on the group (S2), on the \\(\\mathcal O\\)-scheme \\(X\\) and its open \\(B\\)-orbit (S3), and his §8 conditions (S10); \\(X(\\mathcal O)\\) is a single \\(K\\)-orbit, so \\(\\varphi_0=\\mathbf 1_{Kx_0}\\).", "statement": "**The model \\(\\mathcal M_\\theta\\).** \\(F\\) a \\(p\\)-adic field (characteristic \\(0\\)) with \\(p\\ne2\\); \\(G_{\\mathcal O}\\) the split simply connected Chevalley group of type \\(E_6\\); \\(\\theta\\) an automorphism of order \\(2\\) of the \\(\\mathcal O\\)-group scheme \\(G_{\\mathcal O}\\) (equivalently an \\(F\\)-involution with \\(\\theta(K)=K\\)) of type E IV, stabilizing a split maximal torus \\(T_{\\mathcal O}\\) on which it acts by \\(-w_L\\), with \\(B_{\\mathcal O}\\supseteq T_{\\mathcal O}\\) contained in the minimal \\(\\theta\\)-split parabolic \\(P_{\\mathcal O}=U_P\\rtimes L_{\\mathcal O}\\), \\(L=Z_G(S)\\). Put \\(H_{\\mathcal O}:=G_{\\mathcal O}^\\theta\\), \\(X_{\\mathcal O}:=G_{\\mathcal O}/H_{\\mathcal O}\\), \\(x_0=eH\\), \\(K=G(\\mathcal O)\\). (Existence of such \\((G_{\\mathcal O},\\theta,T_{\\mathcal O},B_{\\mathcal O})\\) for \\(p\\ne2\\), e.g. from the split Albert algebra over \\(\\mathcal O\\), is cited, not checked: Springer–Veldkamp, *Octonions, Jordan algebras and exceptional groups*, ch. 7.)\n\nThen, with (S1)–(S10) as in [[S4-1]]:\n1. (S2) holds for every \\(p\\).\n2. \\(H_{\\mathcal O}\\) is smooth over \\(\\mathcal O\\) with connected reductive fibres of type \\(F_4\\); \\(X_{\\mathcal O}\\) is smooth, separated, and quasi-affine (cited inputs (C1)–(C2) below).\n3. \\(B\\cap H=B_L\\cap L^\\theta\\), a Borel subgroup scheme of \\(L^\\theta=\\mathrm{Spin}_8\\), and the orbit map \\(B_{\\mathcal O}/(B\\cap H)\\to X_{\\mathcal O}\\) is an open immersion with affine image \\(U\\simeq U_P\\times A_X\\).\n4. \\(\\mathring X:=X_{\\mathcal O}\\smallsetminus\\overline{(X_F\\smallsetminus\\mathring X_F)}\\) equals \\(U\\); so \\(\\mathring X\\) is smooth and surjective over \\(\\mathcal O\\) with special fibre homogeneous under \\(B_{\\mathbb F_q}\\).\n5. \\(X(F)\\) carries a \\(G(F)\\)-invariant measure and \\(\\mathring X(F)\\) a \\(B(F)\\)-invariant measure. With 3–5: (S3) holds.\n6. \\(X(\\mathcal O)=K\\cdot x_0\\), so \\(\\varphi_0=\\mathbf 1_{X(\\mathcal O)}=\\mathbf 1_{Kx_0}\\).\n7. (S10) holds (any \\(p\\)): \\(X_F\\) is homogeneous and affine, and \\(\\mathring X(F)\\) is one \\(B(F)\\)-orbit.\n\nCited inputs: (C1) fixed points of a finite group of order invertible on the base acting on a smooth scheme are smooth (Edixhoven, Compositio 81 (1992), Prop. 3.4); (C2) Steinberg, *Endomorphisms of linear algebraic groups*, Mem. AMS 80 (1968), Thm. 8.1: \\(G^\\sigma\\) is connected for \\(G\\) simply connected and \\(\\sigma\\) semisimple; and quasi-affineness of \\(G_{\\mathcal O}/H_{\\mathcal O}\\) for reductive \\(H_{\\mathcal O}\\) (Alper, *Adequate moduli spaces and geometrically reductive group schemes*, Algebraic Geometry 1 (2014), generalized Matsushima theorem; result number not checked). (C3) EGA IV\\(_4\\), Cor. 21.12.7: in a locally noetherian normal scheme, the complement of an open subset whose inclusion is affine has pure codimension one.", "parents": ["S4-1", "S4-2", "A6-9"], "corrects": ["A6-9"], "body": "1. **Group.** The Chevalley group scheme is split reductive with connected fibres and simply connected derived group over any \\(\\mathcal O\\). (S1) requires characteristic \\(0\\), which a \\(p\\)-adic \\(F\\) has.\n2. **\\(H_{\\mathcal O}\\).** \\(\\theta\\) generates a \\(\\mu_2\\)-action with \\(2\\in\\mathcal O^\\times\\), so \\(H_{\\mathcal O}\\) is smooth by (C1). On each fibre \\(\\theta\\) is a semisimple automorphism (order \\(2\\), characteristic \\(\\ne2\\)) of a simply connected group, so the fibres of \\(H\\) are connected by (C2); the generic fibre is \\(F_4\\), and fixed points of an involution of a reductive group in characteristic \\(\\ne2\\) are reductive, so \\(H_{\\mathcal O}\\) is reductive. \\(X_{\\mathcal O}=G_{\\mathcal O}/H_{\\mathcal O}\\) is then smooth over \\(\\mathcal O\\) of relative dimension \\(78-52=26\\), and quasi-affine by the cited Matsushima-type theorem.\n3. **Stabilizer of \\(x_0\\) in \\(B\\).** For \\(g=ul\\in B\\cap H\\), \\(u\\in U_P\\), \\(l\\in B_L\\): \\(\\theta(u)=u\\,l\\theta(l)^{-1}\\in U_{P^-}\\cap P=1\\) (the big cell \\(U_{P^-}\\times P\\to G\\) is an open immersion over \\(\\mathcal O\\)), so \\(u=1\\) and \\(l\\in L^\\theta\\): \\(B\\cap H=B_L\\cap L^\\theta\\) scheme-theoretically. \\(L^\\theta=S[2]\\cdot\\mathrm{Spin}_8=\\mathrm{Spin}_8\\) on each fibre with \\(p\\ne2\\) (family page §4: \\(S[2]\\subseteq(T^\\theta)^\\circ\\)), so \\(B\\cap H\\) is a Borel subgroup scheme of \\(\\mathrm{Spin}_8\\): smooth with connected fibres, relative dimension \\(16\\). Fibrewise, \\(B/(B\\cap H)\\to X\\) is a monomorphism of smooth varieties of the same dimension \\(42-16=26\\), hence an open immersion; a morphism of flat finite-type \\(\\mathcal O\\)-schemes that is an open immersion on both fibres is flat (fibrewise criterion), étale and universally injective, hence an open immersion (EGA IV 17.9.1). Its image \\(U\\simeq U_P\\times A_X\\) (via \\(B=U_P\\rtimes B_L\\) and \\(B_L/(B_L\\cap L^\\theta)=L/L^\\theta=A_X\\)) is affine.\n4. **\\(\\mathring X=U\\).** Let \\(Z:=X_{\\mathcal O}\\smallsetminus U\\) (closed). Then \\(\\overline{Z_F}\\subseteq Z\\). Suppose an irreducible component \\(C\\) of \\(Z_{\\mathbb F_q}\\) is not contained in \\(\\overline{Z_F}\\). \\(Z_{\\mathbb F_q}\\) is the complement of the affine open \\(U_{\\mathbb F_q}\\) in the smooth \\(X_{\\mathbb F_q}\\), so by (C3) \\(\\dim C=25\\), and \\(C\\) is then an irreducible component of \\(Z\\) of codimension \\(2\\) in the regular \\(27\\)-dimensional scheme \\(X_{\\mathcal O}\\). But \\(U\\) is affine and \\(X_{\\mathcal O}\\) is separated, so \\(U\\hookrightarrow X_{\\mathcal O}\\) is affine and (C3) forces every component of \\(Z\\) to have codimension \\(1\\): contradiction. Hence \\(Z=\\overline{Z_F}\\) and \\(\\mathring X=X_{\\mathcal O}\\smallsetminus Z=U\\). \\(U\\simeq B/(B\\cap H)\\) is smooth, has the section \\(x_0\\), and its special fibre is one \\(B_{\\mathbb F_q}\\)-orbit.\n5. **Measures.** \\(G\\) and \\(H\\) are unimodular, so \\(X(F)=G(F)/H(F)\\) has an invariant measure. \\(\\mathring X(F)=B(F)/(B\\cap H)(F)\\) (step 7) has a \\(B(F)\\)-invariant measure iff \\(\\delta_B=\\delta_{B\\cap H}\\) on \\(B\\cap H\\). On the maximal torus \\((T^\\theta)^\\circ\\) of \\(B_{\\mathrm{Spin}_8}\\), \\(\\delta_B=|e^{2\\rho}|\\) and \\(\\delta_{B\\cap H}=|e^{2\\rho_L}|\\), and \\(\\langle2\\rho-2\\rho_L,\\alpha_j^\\vee\\rangle=2-2=0\\) for \\(j=2,\\dots,5\\); both characters are trivial on unipotent elements.\n6. **\\(X(\\mathcal O)\\).** \\(G_{\\mathcal O}\\to X_{\\mathcal O}\\) is an \\(H_{\\mathcal O}\\)-torsor, and \\(H^1(\\mathcal O,H_{\\mathcal O})\\simeq H^1(\\mathbb F_q,H_{\\mathbb F_q})=1\\) (Hensel for the smooth \\(H_{\\mathcal O}\\); Lang for the connected \\(H_{\\mathbb F_q}\\)). So every \\(x\\in X(\\mathcal O)\\) lifts to \\(G(\\mathcal O)\\): \\(X(\\mathcal O)=Kx_0\\).\n7. **(S10).** \\(X_F\\) is homogeneous, and affine since \\(F_4\\) is reductive (Matsushima). \\(B\\cap H\\) is connected solvable with split maximal torus, so \\(H^1(F,B\\cap H)=1\\) and \\(\\mathring X(F)=(B/B\\cap H)(F)\\) is one \\(B(F)\\)-orbit ([[A6-9]], via the remark after Theorem E on the classical page).\n\n**Which \\(p\\).** Steps 2–6 use \\(p\\ne2\\). Steps 1 and 7 hold for all \\(p\\). The model \\(\\mathcal M_\\theta\\) matters: \\(K\\backslash X(F)\\) and \\(\\varphi_0\\) are those of this model.", "concept": "e6f4-functions", "key": false, "sources": ["sakellaridis-v3.pdf, §1.1 p. 2; §1.7 pp. 6–7; §2.1 p. 8; §8 p. 63", "families/e6-f4.html §4 (L^θ = Spin_8, S[2] ⊆ (T^θ)°)", "Edixhoven 1992 Prop. 3.4; Steinberg 1968 Thm 8.1; EGA IV 17.9.1 and 21.12.7; Alper 2014 (not checked against local sources)"]}, {"id": "S4-5", "t": "2026-10-17T21:17:13Z", "kind": "lemma", "status": "proved", "label": "Proposition", "name": "E6/F4: Axiom 2.4.2, half of 2.4.1, types", "title": "\\(E_6/F_4\\), \\(p\\) odd: Axiom 2.4.2, half of Axiom 2.4.1, types in 2.2.1", "summary": "Proves that Iwahori orbits equal \\(B(\\mathcal O)\\)-orbits at every point of \\(A_X^+\\) (Axiom 2.4.2), using a dominant lift at covered labels and, at the uncovered labels \\(3k\\omega_1^\\vee,3k\\omega_6^\\vee\\) (\\(k\\) odd), a wall parabolic with Levi \\(D_5\\) and a rational central half-translation; proves that distinct labels give distinct \\(K\\)-orbits; and shows that maximal-rank \\(B\\)-orbits only meet rank-one quotients of type \\(G\\) or \\(U\\).", "statement": "Work in the model \\(\\mathcal M_\\theta\\) of [[S4-4]] with \\(p\\ne2\\), left actions (Sakellaridis's \\(x_0a\\), \\(a=t^{-\\mu}\\) antidominant, is our \\(t^\\mu x_0\\) with \\(\\varphi(\\mu)\\) dominant, and his \\(xJ\\subseteq xB(\\mathfrak o)\\) is our \\(Jx\\subseteq B(\\mathcal O)x\\)). For \\(\\kappa\\in\\Lambda_S^+\\) let \\(x_\\kappa\\in A_X(F)x_0=T(F)x_0\\) be any point with \\(\\tau(x_\\kappa)\\in t^\\kappa S(\\mathcal O)\\), \\(\\tau(gx_0)=g\\theta(g)^{-1}\\).\n1. **(Axiom 2.4.1, second clause, and more.)** \\(Kx_\\kappa=Kx_{\\kappa'}\\) iff \\(\\kappa=\\kappa'\\); points of \\(A_X(F)x_0\\) with the same label lie in one \\(K\\)-orbit. Only \\(\\theta(K)=K\\) is used.\n2. **(Axiom 2.4.2.)** \\(Jx_\\kappa\\subseteq B(\\mathcal O)x_\\kappa\\) for every \\(\\kappa\\in\\Lambda_S^+\\).\n3. **(Axiom 2.2.1, field part.)** There are \\(|W|/|W_X||W_{P(X)}|=51840/(6\\cdot192)=45\\) Borel orbits of maximal rank in \\(X_{\\bar F}\\), and for each of them \\(Y\\) and each simple root \\(\\alpha\\) the pair \\((Y,\\alpha)\\) has type \\(G\\) or \\(U\\), never \\(T\\) or \\(N\\).", "parents": ["S4-4", "S4-3", "A6-4"], "corrects": ["A6-9"], "body": "**Proof of 1.** (a) \\(\\tau:X\\to G\\) is injective and \\(\\tau(kx)=k\\tau(x)\\theta(k)^{-1}\\); since \\(\\theta(K)=K\\), the double coset \\(K\\tau(x)K\\) depends only on \\(Kx\\). (b) \\(\\tau(t^\\mu x_0)=t^{\\varphi(\\mu)}\\), and \\(\\kappa\\in\\Lambda_S^+\\subseteq\\Lambda_T^+\\) is dominant, so by the Cartan decomposition \\(G(F)=\\bigsqcup_{\\nu\\in\\Lambda_T^+}Kt^\\nu K\\) distinct labels give distinct double cosets, hence distinct \\(K\\)-orbits. (c) Two points of \\(A_X(F)x_0\\) with the same label differ by \\(A_X(\\mathcal O)\\), which is the image of \\(T(\\mathcal O)\\subseteq K\\) because the kernel \\(T^\\theta\\) of \\(T\\to A_X\\) is a split torus (\\(H^1(\\mathcal O,T^\\theta)=0\\)).\n\n**Proof of 2, covered labels** (\\(\\kappa=\\varphi(\\mu)\\) with \\(\\mu\\in\\Lambda_T^+\\); by the family-page Lemma these are all labels except \\(3k\\omega_1^\\vee,3k\\omega_6^\\vee\\), \\(k\\) odd). By 1(c) and \\(T(\\mathcal O)\\subseteq J\\cap B(\\mathcal O)\\) it suffices to take \\(x_\\kappa=t^\\mu x_0\\).\n1. Iwahori factorization: \\(J=U_P(\\mathcal O)\\,(J\\cap L)\\,U_P^-(\\mathfrak p)\\).\n2. \\(\\langle\\beta,\\mu\\rangle\\ge0\\) for \\(\\beta\\in\\Phi(U_P)\\), so \\(t^{-\\mu}U_P^-(\\mathfrak p)t^\\mu\\subseteq U_P^-(\\mathfrak p)\\) and \\(U_P^-(\\mathfrak p)t^\\mu x_0\\subseteq t^\\mu U_P^-(\\mathfrak p)x_0\\).\n3. Open cell over \\(\\mathcal O\\): \\(U_P\\times A_X\\to X\\), \\((u,a)\\mapsto uax_0\\), is an open immersion (as in [[S4-4]], step 3, with \\(P\\cap H=L^\\theta\\)). A point \\(u'x_0\\), \\(u'\\in U_P^-(\\mathfrak p)\\), reduces to \\(\\bar x_0\\), so it lies in the cell: \\(u'x_0=uax_0\\) with \\(u\\in U_P(\\mathfrak p)\\) and \\(a\\in A_X(\\mathcal O)=\\operatorname{im}T(\\mathcal O)\\). Hence \\(U_P^-(\\mathfrak p)x_0\\subseteq U_P(\\mathfrak p)T(\\mathcal O)x_0\\).\n4. \\(t^\\mu U_P(\\mathfrak p)t^{-\\mu}\\subseteq U_P(\\mathfrak p)\\), so \\(U_P^-(\\mathfrak p)t^\\mu x_0\\subseteq U_P(\\mathfrak p)T(\\mathcal O)t^\\mu x_0\\).\n5. \\(L^\\theta=\\mathrm{Spin}_8=[L,L]\\) is normal in \\(L\\), so \\(L\\) acts on \\(Lx_0=L/L^\\theta\\) through the torus \\(L/[L,L]\\) and \\(J\\cap L\\) acts on \\(T(F)x_0\\) through \\(T(\\mathcal O)\\).\n6. \\(Jt^\\mu x_0\\subseteq U_P(\\mathcal O)(J\\cap L)U_P(\\mathfrak p)T(\\mathcal O)t^\\mu x_0=U_P(\\mathcal O)U_P(\\mathfrak p)(J\\cap L)T(\\mathcal O)t^\\mu x_0\\subseteq U_P(\\mathcal O)T(\\mathcal O)t^\\mu x_0\\subseteq B(\\mathcal O)t^\\mu x_0\\).\n\n**Why a second argument is needed.** A representative \\(\\mu\\) with \\(\\varphi(\\mu)=\\kappa\\) and \\(\\langle\\beta,\\mu\\rangle\\ge0\\) on \\(\\Phi(U_P)\\) exists iff \\(\\kappa\\) has a dominant lift, since \\(W_L\\) permutes \\(\\Phi(U_P)\\) and fixes \\(\\Lambda_S\\); exact search over \\(a,b\\le18\\) (c2b_iwahori.py) finds the failures exactly at \\(3k\\omega_1^\\vee,3k\\omega_6^\\vee\\), \\(k\\) odd.\n\n**Proof of 2, uncovered labels** \\(\\kappa=3k\\omega_1^\\vee\\), \\(k\\) odd (the labels \\(3k\\omega_6^\\vee\\) follow with nodes \\(1\\leftrightarrow6\\), \\(3\\leftrightarrow5\\); representatives found by the same computation).\n7. Let \\(M\\) be the standard Levi of nodes \\(2,\\dots,6\\) (type \\(D_5\\)) and \\(Q=MU_Q\\supseteq P\\); \\(\\Phi(U_Q)\\) is the \\(16\\) positive roots with \\(\\alpha_1\\)-coefficient \\(>0\\). \\(\\theta=-w_L\\) negates \\(\\alpha_1\\)-coefficients and preserves \\(\\Phi_M\\), so \\(\\theta(U_Q)=U_Q^-\\), \\(Q\\cap H=M^\\theta\\), and \\(U_Q\\times M/M^\\theta\\to X\\) is an open immersion over \\(\\mathcal O\\) (dimensions \\(16+(46-36)=26\\); same argument as step 3). Likewise \\(U_{P\\cap M}\\times A_X\\to M/M^\\theta\\).\n8. \\(\\zeta:=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\\) is the primitive cocharacter of \\(Z(M)^\\circ\\subseteq S\\): \\(\\langle\\beta,\\zeta\\rangle=0\\) on \\(\\Phi_M\\), \\(=3\\) on \\(\\Phi(U_Q)\\), and \\(\\varphi(\\zeta)=2\\zeta\\). Put \\(\\mu_1:=2\\alpha_1^\\vee+\\alpha_3^\\vee+\\alpha_4^\\vee+\\alpha_5^\\vee+\\alpha_6^\\vee\\); then \\(\\varphi(\\mu_1)=3\\omega_1^\\vee\\), \\(\\min_{\\Phi(U_Q)}\\langle\\beta,\\mu_1\\rangle=0\\) and \\(\\min_{\\Phi(U_{P\\cap M})}\\langle\\beta,\\mu_1\\rangle=-1\\). For \\(\\mu_k:=\\mu_1+\\tfrac{k-1}2\\zeta\\): \\(\\varphi(\\mu_k)=3k\\omega_1^\\vee\\) and the same two bounds hold.\n9. **A rational half-translation.** Choose \\(z\\in Z(M)^\\circ(\\bar F)\\) with \\(z^2=\\zeta(\\varpi)\\) and define \\(R_z(mx_0):=mzx_0\\) for \\(m\\in M(F)\\). It is well defined (\\(z\\) is central in \\(M\\)), \\(M(F)\\)-equivariant, and lands in \\(X(F)\\): for \\(\\sigma\\in\\operatorname{Gal}\\), \\(z^{-1}\\sigma(z)\\in\\zeta(\\mu_2)\\subseteq S[2]\\subseteq H\\). Since \\(\\theta(z)=z^{-1}\\), \\(\\tau(z^kx_0)=z^{2k}=t^{k\\zeta}=t^{\\varphi(\\mu_k)}=\\tau(t^{\\mu_k}x_0)\\), so \\(t^{\\mu_k}x_0=R_z^k(x_0)\\).\n10. **The Levi part.** \\((J\\cap M)t^{\\mu_k}x_0=R_z^k\\big((J\\cap M)x_0\\big)\\subseteq R_z^k\\big(B_M(\\mathcal O)x_0\\big)=B_M(\\mathcal O)t^{\\mu_k}x_0\\), the inclusion being the covered case \\(\\kappa=0\\) inside \\(M/M^\\theta\\) (steps 1–6 for \\(P\\cap M\\), \\(\\mu=0\\)).\n11. **The unipotent part.** By step 8, \\(U_Q^-(\\mathfrak p)t^{\\mu_k}x_0\\subseteq t^{\\mu_k}U_Q^-(\\mathfrak p)x_0\\subseteq t^{\\mu_k}U_Q(\\mathfrak p)\\,\\mathcal Y\\subseteq U_Q(\\mathfrak p)\\,t^{\\mu_k}\\mathcal Y\\), where \\(\\mathcal Y=\\{y\\in(M/M^\\theta)(\\mathcal O):\\bar y=\\bar x_0\\}\\) (open cell of step 7). Each \\(y\\in\\mathcal Y\\) is \\(uax_0\\) with \\(u\\in U_{P\\cap M}(\\mathfrak p)\\), \\(a\\in\\operatorname{im}T(\\mathcal O)\\), and \\(t^{\\mu_k}ut^{-\\mu_k}\\in\\prod_\\beta U_\\beta(\\varpi^{\\langle\\beta,\\mu_k\\rangle}\\mathfrak p)\\subseteq U_{P\\cap M}(\\mathcal O)\\) because \\(\\langle\\beta,\\mu_k\\rangle\\ge-1\\). So \\(t^{\\mu_k}\\mathcal Y\\subseteq B_M(\\mathcal O)t^{\\mu_k}x_0\\).\n12. With \\(J=U_Q(\\mathcal O)(J\\cap M)U_Q^-(\\mathfrak p)\\): \\(Jt^{\\mu_k}x_0\\subseteq U_Q(\\mathcal O)(J\\cap M)U_Q(\\mathfrak p)B_M(\\mathcal O)t^{\\mu_k}x_0=U_Q(\\mathcal O)(J\\cap M)t^{\\mu_k}x_0\\subseteq U_Q(\\mathcal O)B_M(\\mathcal O)t^{\\mu_k}x_0=B(\\mathcal O)t^{\\mu_k}x_0\\).\n\n**Proof of 3.** (a) \\(W(X)=W_X\\ltimes W_{P(X)}\\) has order \\(6\\cdot|W(D_4)|=1152\\), and Knop's action is transitive on maximal-rank orbits (Sakellaridis p. 9), giving \\(45\\). (b) If \\((Y,\\alpha)\\) had type \\(T\\) or \\(N\\), then \\(Y\\) is the open \\(B\\)-orbit of \\(YP_\\alpha\\) and the rank-one quotient pulls back a \\(B\\)-eigenfunction of weight \\(\\alpha\\) or \\(2\\alpha\\), so \\(\\alpha\\in\\mathbb Q\\otimes\\mathcal X(Y)=w(\\mathbb Q\\omega_1\\oplus\\mathbb Q\\omega_6)\\) for the \\(w\\) with \\(Y=w\\mathring X\\) (Knop's action transports the weight lattices: Knop, *On the set of orbits for a Borel subgroup*, Comment. Math. Helv. 70 (1995); cited, not checked). But \\(W\\) preserves roots and no root lies in \\(\\mathbb Q\\omega_1\\oplus\\mathbb Q\\omega_6\\): \\((a\\omega_1+b\\omega_6)^2=\\tfrac43(a^2+ab+b^2)\\ne2\\) for integers, and indeed no root \\(\\beta\\) has \\(\\theta\\beta=-\\beta\\) (c1_a4.py).\n\nField and model: 1 uses only \\(\\theta(K)=K\\) (any \\(p\\) for which \\(\\theta\\) is defined over \\(\\mathcal O\\)); 2 uses \\(p\\ne2\\) and \\(\\mathcal M_\\theta\\); 3 is a statement over \\(\\bar F\\).", "concept": "e6f4-functions", "key": true, "sources": ["sakellaridis-v3.pdf, Axiom 2.2.1 p. 9 (Knop action, W(X) = W_X ⋉ W_P(X)); Axioms 2.4.1–2.4.2 pp. 10–11", "families/e6-f4.html §5 (dominant lifts; uncovered labels)", "Knop 1995 (weight lattices under the W-action; cited, not checked)"]}, {"id": "S4-6", "t": "2026-10-17T21:18:09Z", "kind": "blocked", "status": "open", "label": "Gap", "name": "E6/F4: what remains of the axioms", "title": "\\(E_6/F_4\\): Cartan decomposition and Axiom 2.2.1 over \\(\\mathcal O\\) stay open", "summary": "After [[S4-3]]–[[S4-5]], exactly two hypotheses of Theorem 8.0.2 remain unverified for the model \\(\\mathcal M_\\theta\\): the existence half of Axiom 2.4.1 (every \\(K\\)-orbit on \\(X(F)\\) meets \\(T(F)x_0\\)) and the integral form of Axiom 2.2.1 for the 45 maximal-rank Borel orbits. Both are stated exactly, with the routes that look closest.", "statement": "For \\(\\mathcal M_\\theta\\) (model of [[S4-4]]), \\(p\\ne2\\), \\(F\\) \\(p\\)-adic, the hypotheses (S1)–(S10) of [[S4-1]] hold except possibly for:\n\n**(E) Existence half of Axiom 2.4.1 (p. 10).**\n$$G(F)=K\\cdot T(F)\\cdot H(F),\\qquad\\text{equivalently}\\qquad X(F)=\\bigcup_{\\kappa\\in\\Lambda_S^+}K\\,x_\\kappa,\\qquad\\text{equivalently}\\qquad K\\backslash X(F)\\xrightarrow{\\ \\sim\\ }\\Lambda_S^+ .$$\n(The three forms agree because \\(A_X(F)x_0=T(F)x_0\\), \\(H^1(F,T^\\theta)=1\\), and by [[S4-5]](1).)\n\n**(I) Integral half of Axiom 2.2.1 (p. 9).** For each of the \\(45\\) Borel orbits \\(Y_F\\) of maximal rank, with \\(Y\\subseteq X_{\\mathcal O}\\) the \\(\\mathcal O\\)-scheme \"complement of the closure of the complement of \\(Y_F\\) in its closure\", and each simple root \\(\\alpha\\): if \\((Y_F,\\alpha)\\) has type \\(U\\) then \\(YP_\\alpha/R(P_\\alpha)\\simeq SU\\backslash\\mathrm{PGL}_2\\) over \\(\\mathcal O\\) with \\(S\\subseteq N(U)\\); if it has type \\(G\\) then \\(YP_\\alpha/R(P_\\alpha)\\simeq\\mathrm{PGL}_2\\backslash\\mathrm{PGL}_2\\); and the complement of \\(YP_\\alpha\\) is the closure of the complement of \\((YP_\\alpha)_F\\). (Types \\(T,N\\) do not occur: [[S4-5]](3).)\n\nNothing in this run or in the local sources proves (E) or (I) at a given place; Sakellaridis proves the axioms only at almost every place of a global field (pp. 9–11).", "parents": ["S4-1", "S4-4", "S4-5"], "corrects": [], "body": "1. **Why (E) is the remaining content of Axiom 2.4.1.** Its second clause is [[S4-5]](1). Its first clause says that \\(A_X^+x_0\\) meets every \\(K\\)-orbit; since \\(A_X(F)x_0=T(F)x_0\\) and points with the same label are \\(K\\)-conjugate, this is (E). Without (E), \\(C_K\\) could contain indicators of \\(K\\)-orbits not of the form \\(Kx_\\kappa\\), and Theorem 8.0.2 would not apply.\n2. **What is proved towards (E).** Iwasawa \\(G(F)=K\\,P(F)\\) and \\(L(F)\\twoheadrightarrow A_X(F)\\) (\\(H^1(F,\\mathrm{Spin}_8)=1\\)) give \\(X(F)=K\\cdot U_P(F)\\cdot T(F)x_0\\); the missing step is to remove \\(U_P(F)\\). (Proved here; elementary.)\n3. **Route (i), Albert algebra (heuristic).** With \\(G\\) the norm-preserving group of the split Albert algebra \\(J\\) and \\(H=\\operatorname{Stab}(e)\\), \\(X(F)\\simeq\\{x\\in J(F):N(x)=1\\}\\) (one \\(G(F)\\)-orbit since \\(H^1(F,F_4)=1\\)) and \\(T(F)x_0\\) is the set of diagonal elements of norm \\(1\\). (E) would follow from an elementary-divisor theorem: every \\(x\\in J(F)\\) is \\(G(\\mathcal O)\\)-equivalent to a diagonal element. Krutelevich (J. Algebra 314 (2007)) proves the analogue over \\(\\mathbb Z\\); the version over \\(\\mathcal O\\) with \\(p\\ne2\\), and the identification of \\(G(\\mathcal O)\\) with the structure group of \\(J(\\mathcal O)\\), are not checked here.\n4. **Route (ii), buildings (gap).** Benoist–Oh (2007) and Delorme–Sécherre (2011), which Sakellaridis cites on p. 11 for \"similar statements\", give decompositions \\(G(F)=K\\,A^+\\,\\mathcal F\\,H(F)\\) with a finite set \\(\\mathcal F\\); (E) needs \\(\\mathcal F=\\{1\\}\\) for \\(\\mathcal M_\\theta\\). Their exact hypotheses and the form of \\(\\mathcal F\\) were not checked (not in the local sources).\n5. **(I).** Every maximal-rank orbit is \\(w\\cdot\\mathring X\\) for Knop's action, reached from \\(\\mathring X\\) by type-\\(U\\) moves (p. 9). The open-orbit argument of [[S4-4]] (steps 3–4: fibrewise open immersion plus EGA IV 21.12.7) would prove (I) once each \\(Y_F\\) has a representative \\(y\\in X(\\mathcal O)\\) with \\(B_y\\) smooth over \\(\\mathcal O\\) and \\(Y=B\\cdot y\\) on both fibres. This is a finite verification, \\(45\\times6\\) pairs, not done here.\n6. **Blocked route recorded.** Proving Axiom 2.4.2 at the uncovered labels with a single \\(P\\)-dominant cocharacter is impossible ([[S4-5]]); the wall-parabolic argument replaces it. No analogous trick is known here for (E).", "concept": "e6f4-functions", "key": false, "sources": ["sakellaridis-v3.pdf, Axiom 2.2.1 p. 9; Axioms 2.4.1–2.4.2 and the remark on [Sa12], Benoist–Oh, Delorme–Sécherre, pp. 10–11"]}, {"id": "S4-7", "t": "2026-10-17T21:18:34Z", "kind": "lemma", "status": "proved", "label": "Proposition", "name": "im r is exactly the codimension-one ring K", "title": "\\(\\operatorname{im}r=\\{g:g(y_1)=g(\\sigma y_1)\\}\\), of codimension one, every \\(q\\)", "summary": "Proves, by exact identities in \\(q^{1/2}\\), that the image of Sakellaridis's restriction map for \\(E_6/F_4\\) is precisely the subring of invariants that agree at the two glued points, of codimension one; the target has only constant units, so it is cyclic over the Hecke algebra iff \\(r\\) is onto, which fails, and it needs exactly two generators. This is pure algebra about \\(r\\) and uses none of Sakellaridis's hypotheses.", "statement": "Notation of [[A6-4]] and [[A6-6]]: \\(S:=\\mathbb C[\\Lambda_S]^{W_X}\\) (the target of \\(r\\) after translating \\(\\delta(X)^{1/2}A_X^*\\) to \\(A_X^*\\)), \\(r(f)(y)=f(q^{\\rho_L}y)\\), \\(R:=\\operatorname{im}r\\), \\(M_\\kappa\\) the \\(W_X\\)-orbit sums, \\(\\sigma\\) the diagram automorphism (\\(M_{(a,b)}\\mapsto M_{(b,a)}\\)), \\(P:=M_{(3,0)}+M_{(0,3)}\\), \\(y_1=(q^{-5},q)\\), \\(\\xi:=W\\cdot q^{\\rho_L}y_1=W\\cdot q^{\\rho-\\omega_4}\\in A^*/W\\), and\n$$\\mathcal K:=\\{g\\in S:\\ g(y_1)=g(\\sigma y_1)\\}.$$\nFor every real \\(q>1\\):\n1. \\(S^\\sigma=\\mathbb C[M_{(1,1)},P]\\), a polynomial ring, and \\(S=S^\\sigma\\oplus S^\\sigma\\,(M_{(3,0)}-M_{(0,3)})\\).\n2. \\(S^\\sigma\\subseteq R\\).\n3. \\(R=\\mathcal K=S^\\sigma\\oplus\\mathfrak n\\,(M_{(3,0)}-M_{(0,3)})\\), where \\(\\mathfrak n=(M_{(1,1)}-M_{(1,1)}(y_1),\\,P-P(y_1))\\) is the maximal ideal of \\(S^\\sigma\\) at \\(y_1\\). Hence \\(S=R\\oplus\\mathbb C\\,M_{(3,0)}\\), and \\(S/R\\simeq\\mathbb C\\) with \\(\\mathbb C[A^*]^W\\) acting by \\(f\\mapsto f(q^{\\rho-\\omega_4})\\).\n4. \\(S^\\times=\\mathbb C^\\times\\). Consequently \\(S\\) is cyclic as a \\(\\mathbb C[A^*]^W\\)-module (acting through \\(r\\)) iff \\(r\\) is onto, which is false; \\(S=R\\cdot1+R\\cdot M_{(3,0)}\\), so the minimal number of generators is exactly \\(2\\); and \\(\\dim_{\\mathbb C}S/\\mathfrak m_\\xi S=2\\).", "parents": ["A6-1", "A6-2", "A6-4", "A6-5", "A6-6"], "corrects": ["A6-8"], "body": "Write \\(t=q^{1/2}\\) and \\(c_8:=q^{-3}+q^{-2}+q^{-1}+2+q+q^2+q^3\\).\n1. **\\(S^\\sigma\\).** \\(\\sigma\\) acts on \\(\\Lambda_S\\) as \\(-w_0\\) ([[A6-4]](5)), so \\(\\langle W_X,\\sigma\\rangle=W_X\\times\\{\\pm1\\}\\), the Weyl group of \\(G_2\\) acting on \\(\\Lambda_S\\) with short roots \\(W_X(1,1)\\) and long roots \\(\\pm(3,0),\\pm(0,3),\\pm(3,-3)\\). \\(\\Lambda_S\\) is spanned by the short roots, so it is the root lattice of \\(G_2\\), which equals its weight lattice; the fundamental weights are \\((1,1)\\) and \\((3,0)\\), with \\(W(G_2)\\)-orbit sums \\(M_{(1,1)}\\) and \\(P\\). By Bourbaki, Lie VI §3.4, Thm. 1, \\(S^\\sigma=\\mathbb C[M_{(1,1)},P]\\) with \\(M_{(1,1)},P\\) algebraically independent.\n2. **\\(S\\) over \\(S^\\sigma\\).** \\(S\\) is generated by \\(M_{(1,1)},M_{(3,0)},M_{(0,3)}\\) ([[A6-2]], [[A6-4]](4)). With \\(M_{(0,3)}=P-M_{(3,0)}\\), \\(M_{(3,0)}M_{(0,3)}=M_{(3,3)}+3\\in S^\\sigma\\) and \\(M_{(3,0)}^2=P\\,M_{(3,0)}-M_{(3,0)}M_{(0,3)}\\), we get \\(S=S^\\sigma+S^\\sigma M_{(3,0)}=S^\\sigma+S^\\sigma(M_{(3,0)}-M_{(0,3)})\\); splitting into \\(\\sigma\\)-eigenspaces gives the direct sum, with \\(S^-=S^\\sigma(M_{(3,0)}-M_{(0,3)})\\).\n3. **\\(S^\\sigma\\subseteq R\\).** Exact expansions: \\(r(m_{\\omega_2^\\vee})=c_8M_{(1,1)}+c_0\\) ([[A6-4]](6)); \\(M_{(1,1)}^2=M_{(2,2)}+2P+2M_{(1,1)}+6\\); and \\(r(m_{\\omega_1^\\vee+\\omega_6^\\vee})=M_{(2,2)}+c_8P+d_1M_{(1,1)}+d_0\\) with Laurent polynomials \\(c_0,d_1,d_0\\) in \\(t\\). So \\(r(m_{\\omega_1^\\vee+\\omega_6^\\vee})-M_{(1,1)}^2=(c_8-2)P+(d_1-2)M_{(1,1)}+(d_0-6)\\) with \\(c_8-2=q^{-3}+q^{-2}+q^{-1}+q+q^2+q^3>0\\). As \\(R\\) is a ring, \\(M_{(1,1)},P\\in R\\), hence \\(S^\\sigma\\subseteq R\\).\n4. **Odd part.** \\(r\\circ\\sigma^*=\\sigma^*\\circ r\\) and \\(\\sigma^*\\) preserves \\(\\mathbb C[A^*]^W\\) ([[A6-4]](5)), so \\(R\\) is \\(\\sigma\\)-stable and \\(R=S^\\sigma\\oplus(R\\cap S^-)\\), \\(R\\cap S^-=I\\,(M_{(3,0)}-M_{(0,3)})\\) for the ideal \\(I:=\\{a\\in S^\\sigma:a(M_{(3,0)}-M_{(0,3)})\\in R\\}\\).\n5. **\\(I\\subseteq\\mathfrak n\\).** By [[A6-6]], \\(R\\subseteq\\mathcal K\\). For \\(a\\in S^\\sigma\\), \\(g=a(M_{(3,0)}-M_{(0,3)})\\) has \\(g(y_1)-g(\\sigma y_1)=2a(y_1)(M_{(3,0)}-M_{(0,3)})(y_1)\\), and \\((M_{(3,0)}-M_{(0,3)})(y_1)=q^6+q^3+q^{-9}-q^9-q^{-3}-q^{-6}\\ne0\\) for \\(q>1\\). So \\(a(y_1)=0\\).\n6. **\\(\\mathfrak n\\subseteq I\\).** Exact identities in \\(t\\) (c6_image.py):\n$$r(m_{\\omega_1^\\vee+\\omega_3^\\vee})-\\sigma^*r(m_{\\omega_1^\\vee+\\omega_3^\\vee})=(M_{(3,0)}-M_{(0,3)})\\,(c_8M_{(1,1)}-d),$$\n$$r(m_{3\\omega_1^\\vee})-\\sigma^*r(m_{3\\omega_1^\\vee})=(M_{(3,0)}-M_{(0,3)})\\,(P+2-c),$$\nwith \\(c=q^{-9}+q^{-6}+q^{-3}+2+q^3+q^6+q^9\\) (the coefficient in \\(r(m_{3\\omega_1^\\vee})=M_{(6,0)}+c\\,M_{(0,3)}\\), [[A6-5]]) and \\(d\\) the coefficient of \\(M_{(0,3)}\\) in \\(r(m_{\\omega_1^\\vee+\\omega_3^\\vee})\\); they use \\(M_{(1,1)}M_{(3,0)}=M_{(4,1)}+M_{(2,2)}+M_{(1,1)}\\) and \\(M_{(3,0)}^2=M_{(6,0)}+2M_{(0,3)}\\). Both left sides lie in \\(R\\), so \\(c_8M_{(1,1)}-d\\) and \\(P+2-c\\) lie in \\(I\\subseteq\\mathfrak n\\). Evaluating at \\(y_1\\) gives \\(d=c_8M_{(1,1)}(y_1)\\) and \\(c-2=P(y_1)\\), so \\(M_{(1,1)}-M_{(1,1)}(y_1)\\in I\\) and \\(P-P(y_1)\\in I\\). Since \\(S^\\sigma\\) is a polynomial ring in \\(M_{(1,1)},P\\), these generate \\(\\mathfrak n\\): \\(I=\\mathfrak n\\).\n7. **\\(R=\\mathcal K\\).** For \\(g=a+b(M_{(3,0)}-M_{(0,3)})\\), \\(g\\in\\mathcal K\\iff b(y_1)=0\\). So \\(\\mathcal K=S^\\sigma\\oplus\\mathfrak n(M_{(3,0)}-M_{(0,3)})=R\\). As \\(P\\in R\\), \\(S=R\\oplus\\mathbb C(M_{(3,0)}-M_{(0,3)})=R\\oplus\\mathbb CM_{(3,0)}\\). The functional \\(g\\mapsto g(y_1)-g(\\sigma y_1)\\) identifies \\(S/R\\) with \\(\\mathbb C\\), and \\(r(f)g\\mapsto f(q^{\\rho_L}y_1)\\,(g(y_1)-g(\\sigma y_1))\\) because \\(r(f)(y_1)=r(f)(\\sigma y_1)\\); \\(q^{\\rho_L}y_1=q^{\\lambda_1^*}\\) is \\(W\\)-conjugate to \\(q^{\\rho-\\omega_4}\\) ([[A6-6]]).\n8. **Units and cyclicity.** A unit of \\(S\\) is a unit of \\(\\mathbb C[\\Lambda_S]\\), i.e. \\(cy^\\kappa\\); \\(W_X\\)-invariance forces \\(\\kappa\\) to be \\(S_3\\)-fixed, so \\(\\kappa=0\\). If \\(S=R\\,s\\), then \\(1=r(f)s\\) makes \\(s\\) a unit, \\(s\\in\\mathbb C^\\times\\), and \\(S=R\\): contradiction with 7. \\(S=R+RM_{(3,0)}\\) gives two generators. **Fibre:** \\(y_1\\) and \\(\\sigma y_1\\) are not \\(W_X\\)-conjugate ([[A6-6]]), so they give distinct maximal ideals of \\(S\\) containing \\(\\mathfrak m_\\xi S\\) (\\(\\mathfrak m_\\xi\\subseteq\\mathbb C[A^*]^W\\) the ideal of \\(\\xi\\)); by the Chinese remainder theorem \\(\\dim S/\\mathfrak m_\\xi S\\ge2\\), and \\(\\le2\\) because \\(S=R+RM_{(3,0)}\\) and \\(R/\\mathfrak m_\\xi R=\\mathbb C\\).\n\nThis proves the conjecture of [[A6-8]] (\\(\\operatorname{im}r=\\mathcal K\\)) without the unramifiedness check it left open. Only the algebra of \\(r\\) is used: no hypothesis of Sakellaridis enters.", "concept": "e6f4-functions", "key": true, "sources": ["Bourbaki, Lie VI §3.4 Thm 1 (invariants of W on the weight lattice)", "families/e6-f4.html; data.json steps A6-1 … A6-6"]}, {"id": "S4-8", "t": "2026-10-17T21:19:25Z", "kind": "lemma", "status": "proved", "label": "Corollary", "name": "C_K needs exactly two generators", "title": "Under Theorem 8.0.2, \\(C_K\\) is not cyclic and needs exactly two generators", "summary": "Transports [[S4-7]] through Sakellaridis's isomorphism: for \\(E_6/F_4\\) no single function generates \\(C_K\\) over \\(\\mathcal H_K\\), the basic function together with the indicator of the orbit \\(3\\omega_1^\\vee\\) does, and the cokernel of \\(\\mathcal H_K\\varphi_0\\subseteq C_K\\) is one-dimensional with Satake parameter \\(q^{\\rho-\\omega_4}\\). It also records the little that holds without his hypotheses.", "statement": "Let \\(F\\) be \\(p\\)-adic, \\(p\\ne2\\), \\(X=\\mathcal M_\\theta\\) ([[S4-4]]), and **assume the conclusion of Sakellaridis's Theorem 8.0.2 for \\(X\\)** (it holds if (E) and (I) of [[S4-6]] hold, by [[S4-3]]–[[S4-5]]). Then:\n1. \\(C_K\\) is not a cyclic \\(\\mathcal H_K\\)-module: no \\(\\varphi\\in C_K\\) has \\(\\mathcal H_K\\varphi=C_K\\).\n2. \\(C_K=\\mathcal H_K\\varphi_0\\oplus\\mathbb C\\,\\mathbf 1_{O_{3\\omega_1^\\vee}}=\\mathcal H_K\\varphi_0\\oplus\\mathbb C\\,\\mathbf 1_{O_{3\\omega_6^\\vee}}\\) as vector spaces; in particular \\(C_K=\\mathcal H_K\\varphi_0+\\mathcal H_K\\mathbf 1_{O_{3\\omega_1^\\vee}}\\), and the minimal number of generators is \\(2\\).\n3. \\(C_K/\\mathcal H_K\\varphi_0\\simeq\\mathbb C\\), on which \\(\\mathcal H_K\\) acts by the unramified character with Satake parameter \\(q^{\\rho-\\omega_4}\\in A^*/W\\).\n4. \\(\\dim\\operatorname{Hom}_{\\mathcal H_K}(C_K,\\chi_\\xi)=2\\) for \\(\\xi=q^{\\rho-\\omega_4}\\): the \\(K\\)-invariant eigenfunctionals \\(\\varphi\\mapsto P_\\varphi(y_1)\\) and \\(\\varphi\\mapsto P_\\varphi(\\sigma y_1)\\) are independent.\n\n**Without Sakellaridis's hypotheses** the following hold and nothing more is proved here: (a) [[S4-7]] is unconditional algebra about \\(r\\); (b) [[S4-5]](1),(2) and [[S4-4]] hold for \\(\\mathcal M_\\theta\\), \\(p\\ne2\\); (c) the criterion \"two linearly independent \\(\\mathcal H_K\\)-eigenfunctionals on \\(C_K\\) with the same character \\(\\Rightarrow\\) \\(C_K\\) not cyclic\" is unconditional, since a cyclic module \\(\\mathcal H_K/\\mathfrak a\\) has \\(\\dim\\operatorname{Hom}(\\mathcal H_K/\\mathfrak a,\\chi)\\le1\\). Over \\(\\mathbb F_q(\\!(t)\\!)\\) the paper gives nothing (it assumes characteristic \\(0\\)).", "parents": ["S4-7", "S4-1", "S4-3", "S4-6", "A6-6"], "corrects": ["A6-6"], "body": "1. **Transport.** Theorem 8.0.2 gives an \\(\\mathcal H_K\\)-isomorphism \\(C_K\\simeq S\\) with \\(\\varphi_0\\mapsto1\\) and \\(\\mathcal H_K\\simeq\\mathbb C[A^*]^W\\) acting through \\(r\\) (after the translation \\(\\delta(X)^{1/2}A_X^*\\to A_X^*\\); the left/right and sign conventions change \\(r\\) by automorphisms of source and target and change none of the statements, [[A6-1]], [[A6-7]]; the character \\(q^{\\rho-\\omega_4}\\) is fixed by \\(-w_0\\), so inversion does not move it). Then 1 and 3 are [[S4-7]](4),(3), and \\(\\mathcal H_K\\varphi_0\\leftrightarrow R\\).\n2. **The indicator of \\(O_{3\\omega_1^\\vee}\\).** In the proof of Theorem 8.0.2 (p. 64) the indicator of the \\(K\\)-orbit with label \\(\\check\\lambda\\) maps to \\(P_{\\check\\lambda}\\), whose lowest weight for the order defined by the cone \\(\\mathcal T\\) of colour valuations is \\(\\check\\lambda\\), with non-zero coefficient (Remark after Theorem 7.2.1, p. 62). By [[S4-3]], \\(\\mathcal T=\\mathbb R_{\\ge0}\\check\\gamma_1+\\mathbb R_{\\ge0}\\check\\gamma_6\\), so in dominant coordinates \\(P_{3\\omega_1^\\vee}=c\\,M_\\kappa+c_1M_{(1,1)}+c_0\\) with \\(c\\ne0\\), where \\(\\kappa\\in\\{(3,0),(0,3)\\}\\) depends on the left/right convention: the dominant \\(\\kappa'\\) with \\(\\kappa-\\kappa'\\in\\mathbb N\\check\\gamma_1+\\mathbb N\\check\\gamma_6\\) are \\(\\kappa,(1,1),(0,0)\\). \\(M_{(1,1)},1\\in R\\) and \\(M_{(3,0)},M_{(0,3)}\\notin R\\) ([[S4-7]]: \\(P\\in R\\), \\(M_{(3,0)}\\notin R\\)), so \\(P_{3\\omega_1^\\vee}\\notin R\\) and \\(S=R\\oplus\\mathbb CP_{3\\omega_1^\\vee}\\) by \\(\\dim S/R=1\\). This gives 2; the same for \\(3\\omega_6^\\vee\\). It also settles the stronger reading flagged in the audit of [[A6-6]]: \\(\\mathbf 1_{O_{3\\omega_1^\\vee}}\\notin\\mathcal H_K\\varphi_0\\).\n3. **Point 4.** Evaluation at a point \\(y\\) of \\(\\operatorname{Spec}S\\) is an algebra map \\(S\\to\\mathbb C\\); composed with \\(C_K\\simeq S\\) it is an \\(\\mathcal H_K\\)-eigenfunctional with character \\(f\\mapsto f(q^{\\rho_L}y)\\). At \\(y_1,\\sigma y_1\\) the characters coincide (both \\(\\xi\\)) and the functionals differ (on \\(M_{(3,0)}\\)). Every eigenfunctional with character \\(\\xi\\) kills \\(\\mathfrak m_\\xi S\\), and \\(\\dim S/\\mathfrak m_\\xi S=2\\) ([[S4-7]](4)), so the dimension is exactly \\(2\\). By the criterion (c) this re-proves 1.\n4. **What the hypotheses buy.** Every statement 1–4 passes through Theorem 8.0.2; for \\(\\mathcal M_\\theta\\) the only unverified inputs of that theorem are (E) and (I) of [[S4-6]].", "concept": "e6f4-functions", "key": true, "sources": ["sakellaridis-v3.pdf, Theorem 8.0.2 and proof p. 64; Remark after Theorem 7.2.1 p. 62; Corollary 8.0.4 p. 65"]}, {"id": "S4-9", "t": "2026-10-17T21:19:54Z", "kind": "verdict", "status": "proved", "label": "Summary", "name": "E6/F4 function cell, re-derived", "title": "\\(E_6/F_4\\) function cell: two generators, modulo (E) and (I)", "summary": "Re-derives the verdict's \\(E_6/F_4\\) function cell: what is unconditional, what holds under Sakellaridis's Theorem 8.0.2, which of its hypotheses are now verified for an explicit \\(\\mathcal O\\)-model with \\(p\\) odd, and which two remain; the cell stays a \\(p\\)-adic statement.", "statement": "**\\(E_6/F_4\\), functions.** \\(F\\) a \\(p\\)-adic field (characteristic \\(0\\)), residue characteristic \\(p\\ne2\\), \\(X\\) the model \\(\\mathcal M_\\theta=G_{\\mathcal O}/G_{\\mathcal O}^\\theta\\) (split simply connected \\(E_6\\) over \\(\\mathcal O\\), \\(\\theta\\) an \\(\\mathcal O\\)-involution of type E IV, \\(\\theta(K)=K\\)), \\(\\varphi_0=\\mathbf 1_{X(\\mathcal O)}=\\mathbf 1_{Kx_0}\\).\n\n| claim | status | where |\n|---|---|---|\n| \\(\\operatorname{im}r\\) is the codimension-one subring \\(\\{g:g(y_1)=g(\\sigma y_1)\\}\\) | proved, unconditional algebra, every \\(q\\) | [[S4-7]] |\n| (SG) fails; \\(C_K\\) is not cyclic for any generator; \\(C_K=\\mathcal H_K\\varphi_0\\oplus\\mathbb C\\mathbf 1_{O_{3\\omega_1^\\vee}}\\); \\(C_K/\\mathcal H_K\\varphi_0\\) is the character \\(q^{\\rho-\\omega_4}\\) | proved under Theorem 8.0.2 for \\(\\mathcal M_\\theta\\) | [[S4-8]] |\n| §1.1/§1.7 conditions on \\(F\\) and \\(G\\) (characteristic \\(0\\); split, simply connected derived group) | verified, every \\(p\\) (with the paper's \"unramified over \\(\\mathbb Q_p\\)\" simplification) | [[S4-4]] |\n| §1.7 conditions on \\(X\\) over \\(\\mathcal O\\) (quasi-affine, \\(\\mathring X\\) smooth surjective and \\(B\\)-homogeneous, eigenmeasures) | verified for \\(p\\ne2\\), modulo cited integral-model facts | [[S4-4]] |\n| §7: no type \\(N\\), \\(L_\\gamma\\) classical; Statements 6.3.1 and 7.1.5 | verified (statements over \\(\\bar F\\)) | [[S4-3]] |\n| §8: homogeneous, affine, one open \\(B(F)\\)-orbit | verified, every \\(p\\) | [[S4-4]] |\n| Axiom 2.4.2 (\\(xJ\\subseteq xB(\\mathcal O)\\) on \\(A_X^+\\)) | verified, \\(p\\ne2\\) | [[S4-5]] |\n| Axiom 2.4.1, distinct labels give distinct \\(K\\)-orbits | verified | [[S4-5]] |\n| Axiom 2.4.1, every \\(K\\)-orbit meets \\(T(F)x_0\\) | **open** | [[S4-6]] (E) |\n| Axiom 2.2.1: only types \\(G,U\\) | verified over \\(\\bar F\\) | [[S4-5]] |\n| Axiom 2.2.1 as \\(\\mathcal O\\)-schemes for the 45 maximal-rank orbits | **open** | [[S4-6]] (I) |\n\n**Cell.** \"Fails, and \\(C_K\\) needs exactly two generators (\\(\\varphi_0\\) and \\(\\mathbf 1_{O_{3\\omega_1^\\vee}}\\)); \\(p\\)-adic \\(F\\), \\(p\\) odd, model \\(\\mathcal M_\\theta\\); conditional only on the Cartan decomposition \\(X(F)=\\bigcup_\\kappa Kx_\\kappa\\) and Axiom 2.2.1 over \\(\\mathcal O\\).\" Over \\(\\mathbb F_q(\\!(t)\\!)\\) the cell is undecided: Sakellaridis assumes characteristic \\(0\\), and the algebra of [[S4-7]] does not by itself compute \\(C_K\\) there.", "parents": ["S4-2", "S4-3", "S4-4", "S4-5", "S4-6", "S4-7", "S4-8", "A6-6", "A7-32"], "corrects": ["A7-32", "R-4"], "body": "1. **Inputs from the run.** [[A6-6]] proves that \\(q^{\\rho_L}y_1\\) and \\(q^{\\rho_L}\\sigma y_1\\) are \\(W(E_6)\\)-conjugate while \\(y_1,\\sigma y_1\\) are not \\(W_X\\)-conjugate; this is pure algebra about \\(r\\) and holds as stated. [[A7-32]] records the \\(E_6/F_4\\) function cell as \"(SG) fails under (A1)–(A5) (Sakellaridis's field/group, \\(\\mathcal O\\)-model, axioms, spherical-root/colour and affine-one-orbit hypotheses), with (A2) (the \\(\\mathcal O\\)-model), (A3) (the axioms) at the place and (A4) (spherical roots and colours) beyond '\\(L_\\gamma\\) classical' unverified\".\n2. **Sharpening of the conclusion.** [[S4-7]] identifies \\(\\operatorname{im}r\\) exactly; with Theorem 8.0.2, [[S4-8]] turns \"\\(\\varphi_0\\) does not generate\" into \"nothing generates\", gives the second generator, and the cokernel.\n3. **Sharpening of the hypotheses.** [[S4-2]] fixes what Theorem 8.0.2 needs and that a model must be named. For \\(\\mathcal M_\\theta\\): the spherical-root and colour hypotheses hold in full ([[S4-3]]), contrary to \"unverified\" in [[A6-9]] and [[A7-32]]; the \\(\\mathcal O\\)-model conditions hold for \\(p\\ne2\\) ([[S4-4]]); of the three axioms, 2.4.2 and half of 2.4.1 hold ([[S4-5]]). What is left is (E) and (I) of [[S4-6]].\n4. **Fields.** Everything uses Sakellaridis's characteristic-\\(0\\) theorem, so the cell is \\(p\\)-adic. A comparison with the sheaf side over \\(\\overline{\\mathbb F}_q\\) remains a comparison across fields, exactly as [[A7-32]] says.\n5. **Effect on the team verdict.** The divergence \"categorical generation holds, function-level generation fails\" at \\(E_6/F_4\\) is unchanged in direction and strengthened in content; its function half now rests on two named statements, (E) and (I), instead of three families of unverified hypotheses.", "concept": "synthesis", "key": true, "sources": ["data.json: A6-6, A6-9, A7-32, R-4", "sakellaridis-v3.pdf, Theorem 8.0.2 p. 64; Corollary 8.0.4 p. 65"], "inputs": {"A6-6": "holds as stated (the gluing of y_1 and σy_1 and M_(3,0) ∉ im r are pure algebra about r); its conclusion '(SG) fails under (A1)–(A5)' (Sakellaridis's field/group, O-model, axioms, spherical-root/colour, affine-one-orbit hypotheses) holds with change: under Theorem 8.0.2 C_K is not cyclic at all, needs exactly two generators, and im r equals the codimension-one ring K ([[S4-7]], [[S4-8]])", "A7-32": "holds with change: the E6/F4 function cell is 'fails, C_K needs exactly two generators, p-adic F, p odd, model M_θ', and the unverified hypotheses shrink from '(A2) the O-model, (A3) the axioms at the place, (A4) the spherical-root and colour hypotheses beyond L_γ classical' to two statements: the Cartan decomposition (existence half of Axiom 2.4.1) and Axiom 2.2.1 as O-schemes ([[S4-6]]); (A4), the spherical-root and colour hypotheses, now hold in full ([[S4-3]])"}}], "result": {"status": "partial", "statement": "Sakellaridis's Theorem 8.0.2 and Corollary 8.0.4 are stated with all hypotheses, and the pages' summary of them is accurate apart from six losses, chiefly that no \\(\\mathcal O\\)-model is fixed. For \\(X=E_6/F_4\\) over a \\(p\\)-adic \\(F\\), \\(p\\) odd, with the model \\(\\mathcal M_\\theta=G_{\\mathcal O}/G_{\\mathcal O}^\\theta\\): (i) unconditionally, \\(\\operatorname{im}r\\) is the codimension-one subring \\(\\{g:g(y_1)=g(\\sigma y_1)\\}\\) of \\(S=\\mathbb C[\\delta(X)^{1/2}A_X^*]^{W_X}\\), and \\(S\\) has only constant units; (ii) under Theorem 8.0.2, \\(C_K\\) is not cyclic for any generator, \\(C_K=\\mathcal H_K\\varphi_0\\oplus\\mathbb C\\,\\mathbf 1_{O_{3\\omega_1^\\vee}}\\) needs exactly two generators, and \\(C_K/\\mathcal H_K\\varphi_0\\) is the character \\(q^{\\rho-\\omega_4}\\); (iii) every hypothesis of Theorem 8.0.2 is verified for \\(\\mathcal M_\\theta\\) (spherical roots of type \\(G\\), classical \\(D_5\\) Levis, Statements 6.3.1 and 7.1.5, the \\(\\mathcal O\\)-model conditions, one open \\(B(F)\\)-orbit, Axiom 2.4.2 at every label, distinctness in Axiom 2.4.1, types in Axiom 2.2.1) except two.", "gap": "(E) the existence half of Axiom 2.4.1: \\(G(F)=K\\,T(F)\\,H(F)\\), i.e. every \\(K\\)-orbit on \\(X(F)\\) meets \\(T(F)x_0\\); (I) Axiom 2.2.1 as \\(\\mathcal O\\)-schemes for the 45 maximal-rank Borel orbits. Also cited, not re-checked: smoothness and connectedness of \\(G_{\\mathcal O}^\\theta\\) for \\(p\\ne2\\) (Edixhoven, Steinberg), quasi-affineness of \\(G_{\\mathcal O}/G_{\\mathcal O}^\\theta\\) (Alper), existence of the \\(\\mathcal O\\)-involution (Albert-algebra model), Knop's transport of weight lattices, and the paper's claim that dropping \\(F/\\mathbb Q_p\\) unramified is trivial. Over \\(\\mathbb F_q(\\!(t)\\!)\\) nothing is decided."}, "blocked_routes": ["Axiom 2.4.2 at the uncovered labels \\(3k\\omega_1^\\vee,3k\\omega_6^\\vee\\) (\\(k\\) odd) via a single cocharacter \\(\\mu\\) with \\(\\langle\\beta,\\mu\\rangle\\ge0\\) on \\(\\Phi(U_P)\\): impossible, since such \\(\\mu\\) is \\(W_L\\)-conjugate to a dominant lift; replaced by the wall parabolic with \\(D_5\\) Levi and the rational half-translation \\(R_z\\) ([[S4-5]]).", "Bounding the number of generators of \\(C_K\\) by Forster–Swan gives only \\(\\le3\\); the exact value \\(2\\) needed the identification \\(\\operatorname{im}r\\supseteq S^\\sigma\\) ([[S4-7]])."], "scripts": ["/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/e6core.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c1_a4.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c2b_iwahori.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c3_wall.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c4_cyclic.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c4_run.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c5_codim.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c6_image.py"]}