SECOND AUDIT TASK for A5 (same setting, checklist, rules and return schema as your first brief). A different agent reran Sakellaridis's proof directly over F_q((t)) (the claims below, copied verbatim from its return; full return: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/sakellaridis-char-p/returns/A1.json; scripts: .../sakellaridis-char-p/scripts/A1/). Its result: Theorem 8.0.2 over F_q((t)) for symmetric X, p odd, under (A2), (A5), (A3b′), (A3c), Statements 6.3.1 and 7.1.5 read in characteristic p and cone(Θ+) ∩ C^- = {0}, without (A3a) or (A4)(i),(ii); for E6/F4 modulo (S6.3.1)_p. Audit adversarially, with priority: A1-5 (convergence, generic uniqueness, rationality of S_χ over F_q((t)) without [Sa08]), A1-6 (functional equations, the sum formula with W_X′ = N_W(a) ∩ [W/W_L], the claim that Knop's action is not needed), A1-2 (the last step is formal), A1-7 (recompute W_X′, β, the constant c, triangularity), A1-3, A1-4 (smooth stabilizers; finitely many B(F)-orbits over F_q((t)): H^1 with μ_2 factors, p odd), then A1-1, A1-8, A1-9 (the inventory and the formulation: are the listed hypotheses exactly what the proof uses? is (A3a) really not needed for the open-orbit route, given that Sakellaridis normalizes eigenmeasures on all maximal-rank orbits?). One fact for your information: [Sa08] (arXiv:math/0606130v2) works over "a locally compact non-archimedean local field in characteristic zero" (its §1.7 and §4.1), with no unramified assumption. Return check/objection nodes A5-14, A5-15, … (parents = the audited ids), and a final verdict node for this second task; write node files to scratchpad/solve-vi/A5/nodes2/, the assembled JSON (containing only the new nodes, with "agent":"A5") to scratchpad/solve-vi/A5/return2.json, and end with that JSON inside a ```json fence. Keep each response short (same working discipline). CLAIMS TO AUDIT --- A1-1 · Strategy · Every external input of Theorem 8.0.2 and its status over $\mathbb F_q((t))$ STATUS CLAIMED: proved PARENTS: STATEMENT: PROOF / BODY: **Method.** I read §§1–8 of the text extract line by line and listed every citation and every 'it is known', together with the step that uses it. Each input is classified as: char-free (the argument is valid over any non-archimedean local field), residue-characteristic only, a characteristic-0 citation that is replaced in a later node, avoided (the route below does not need it), or kept as a hypothesis. | # | input (place in arXiv:0905.4244v3) | used for | source / characteristic | status over F_q((t)), p odd | |---|---|---|---|---| | 1 | ψ of conductor O, dx(O)=1, 'k unramified over Q_p' (§1.1 p.2, §3.8 p.23, §5.3 p.32) | normalizations of all Fourier transforms | for F_q((t)) take ψ(x)=ψ_0(Tr_{F_q/F_p} Res(x dt)), conductor O; dx(O)=1 is self-dual. No 'unramified' issue | char-free | | 2 | x_0 ∈ X̊(O) via [Sa08, Prop. 3.2.1] + Hensel (§2.1 p.8) | choice of base point | Lang's theorem for the connected solvable B_{F_q} on the homogeneous X̊_{F_q}, then smooth lifting | residue field only; char-free | | 3 | [BLV86, Thm 3.4]: [L(X),L(X)] ⊆ H (§2.1) and (B_y)_u = L(X)∩U (§3.4, eq. (3.5)) | definition of A_X; δ_{P(X)} ∈ A_X^* | char 0 source | replaced for symmetric X, p odd: [[A1-3]] | | 4 | [Kn91, Cor. 1.8], [Kn91, §5]: Λ_X^+ ⊆ Λ_X, V = image of the antidominant chamber for symmetric X (§2.1) | labels of K-orbits in (A3b) | char 0 source | avoided: (A3b) is restated with antidominant labels (A3b′); not used in the proof | | 5 | [Kn95] Knop's W-action on B_{F̄}-orbits; W(X)=W_X⋉W_{P(X)} (§2.2) | definition of W_X; which T_w survive | char 0 source (not opened) | avoided: W_X′ := Stab_W(δ(X)^{1/2}A_X^*) ∩ [W/W_{P(X)}] defined analytically ([[A1-6]]); for M_θ computed ([[A1-7]]) | | 6 | [Kn94], [Kn96]: W_X a reflection group with fundamental domain V; root system (§2.2, §6.1) | triangularity in §8; Statement 6.3.1 | char 0 source | avoided as above; for M_θ, W_X′ = W(A_2) explicitly ([[A1-7]]) | | 7 | [Sa06, Prop. 3.6.1]: max-rank B_{F̄}-orbits are defined over F with F-points (§2.2) | intermediate orbits of Brion paths (§3.3, §6) | p-adic source (not opened) | not needed for the route of [[A1-5]]–[[A1-6]]; needed only to rerun §6 ([[A1-8]]) | | 8 | Axiom 2.2.1 consequences (p. 9): every max-rank orbit is reached from X̊ by type-U reflections | smoothness of all max-rank Y over O | uses Knop's transitivity (char 0) | not needed in the route below; needed for [[A1-8]] | | 9 | [Br01, Props 2, 4; Thm 4] (§2.3) | Prop. 3.3.6 (path independence of measures), §6.2 | char 0 source | not needed for Thm 8.0.2 in the route below (only the open orbit is normalized); needed for [[A1-8]] | | 10 | [Sa08, Thm 5.2.1] in Lemma 3.3.5 (p. 15) | a priori proportionality T_{w_α}S^Z ∼ S^Y | char 0 source | unnecessary: Fubini over Z×U_α ≅ Y gives the equality directly; char-free | | 11 | convergence and rationality of the integrals (3.2) [Sa08] (p. 12) | existence of S_χ, Δ_χ as rational families | char 0 source (not opened) | replaced: [[A1-5]] (convergence by eigenfunction bounds; rationality by Bernstein's principle) | | 12 | [Sa08] Thm 3.2.2 (multiplicity g·a) (p. 13) | spanning statements of Thm 4.2.2 | char 0 source | not needed for Thm 8.0.2; only generic uniqueness is used, proved in [[A1-5]] | | 13 | [Sa08] Thm 3.7.1 / Cor. 3.7.2 (pp. 21–22) | functional equations F_wΔ_χ = b_wΔ^{wX̊} | char 0 source | replaced: [[A1-6]] (uniqueness + rank-one kill for α ∈ Δ(X)) | | 14 | finiteness of B(F)-orbits on X(F) and of H^1 (implicit in [Sa08]) | Bruhat-filtration arguments | fails in general in char p (H^1(F,Z/p), H^1(F,μ_p) infinite) | proved for symmetric X, p odd: [[A1-4]] | | 15 | [BK98] Schwartz space S(U\G), F_wF_{w′}=F_{ww′}, basic vector (§3.8, §4.3) | cocycle (3.12), Prop. 4.2.1 | stated over p-adic fields (not opened) | replaced: F_w := normalized intertwining operator on I(χ) via (4.9); cocycle from T_w ([[A1-6]]) | | 16 | [Ca80, Thm 3.1] Gindikin–Karpelevich (4.8) (p. 28) | relation F_w vs T_w | not opened | rank-one SL_2 integral + cocycle; char-free ([[A1-6]]) | | 17 | [CS80, Thm 5.4] (4.5), Cor. 4.3.1 | normalization of Ω vs φ_{K,χ} | — | not needed for Thm 8.0.2 (a scalar independent of λ cancels; [[A1-2]]) | | 18 | Tate's local functional equation (§5.3) | rank-one constants b | any local field (Tate's thesis; Weil, Basic Number Theory VII) | char-free with ψ of conductor O | | 19 | [Lu01, §1] (type T split, p. 31); §5.6–5.8 types T, N (quadratic characters, F^×/F^{×2}) | rank-one constants for T, N | residue char ≠ 2 only | not needed for M_θ (types G, U only) | | 20 | [Was96], [Ak83]: rank ≤ 2 classification; Lemma 6.2.2, Prop. 6.2.1, Thm 6.11.1, §§6.4–6.21 | Thm 6.3.2 ((A4)(i),(ii) ⇒ Statement 6.3.1) | char 0 classification | (A4)(iii) kept as a hypothesis; for M_θ open ([[A1-8]]) | | 21 | [SV]: normalized spherical roots form a root datum (§6.1) | Φ_X, β | combinatorics over F̄ | for M_θ explicit: Φ_X = A_2, Δ_X = {γ_1, γ_6} | | 22 | Lemmas 7.1.1–7.1.4 (case-by-case), colours and Θ (§7.1) | definition of β, Statement 7.1.5 | char 0 case analysis | not needed for M_θ: β defined directly from Φ̌_X^+ ([[A1-7]]) | | 23 | [Kn91, §6] affine ⟺ V and colour valuations separated (§7.2) | ω constant; T ∩ V = {0} | char 0 source | M_θ: T = cone(Φ̌_X^+), T ∩ V = {0} elementary; constant c computed ([[A1-7]]) | | 24 | Lemma 3.4.2, Lemma 5.1.1 (unfolding) | Δ_χ as an integral over U\BG_ξ | needs smooth orbit maps | char-free once H and B_y are smooth: [[A1-3]] | | 25 | Lemma 3.9.1, Lemma 4.2.3, Prop. 4.2.1, §8 proof | sum formula and last step | — | char-free given (A3c), invariant measure, rank-one Fourier on F^2 ([[A1-2]], [[A1-6]]) | | 26 | Axioms 2.4.1, 2.4.2 ((A3b), (A3c)); [LV83], [Sa12], [BO07], [DS11] | Cartan and Iwahori orbits | hypotheses of the theorem | assumed (verification is (Q3)) | | 27 | 'hold at almost every place' (§1.7 p.7, §2.2 p.9) | justification of the axioms | global statement | not usable at a given place; the axioms stay hypotheses | **Logical skeleton of Theorem 8.0.2 (p. 64), as the route used here.** 1. A rational family of $(B,\chi)$-equivariant functionals \(S_\chi\) on \(C_c^\infty(X(F))\), \(\chi\in\delta(X)^{1/2}A_X^*\), with generic uniqueness ([[A1-4]], [[A1-5]]). 2. Functional equations \(F_wS_\chi=b_w(\chi)S_{w\chi}\) for \(w\in W_X'\) and the sum formula (4.2)/(6.2) for \(\Omega_\chi(x_\lambda)\) ([[A1-6]]). 3. The cocycle values \(B_{w_\gamma}\) (Statement 6.3.1) and the coboundary \(\beta\) (Theorem 7.2.1) ([[A1-7]], [[A1-8]]). 4. The formal passage from (7.4) and the Cartan decomposition to the module isomorphism ([[A1-2]]). **Finding.** Every characteristic-0 citation of the paper enters either through the geometry of Borel orbits over \(\bar F\) (Knop, Brion, Luna, Wasserman, Akhiezer, [BLV86], [Kn91]) or through [Sa08] (rationality, multiplicity, composition with intertwining operators). The harmonic analysis on \(F\) itself (Tate, Fourier transform on \(F^2\), Gindikin–Karpelevich, Iwahori decomposition) is characteristic-free once \(\psi\) has conductor \(O\). SOURCES: sakellaridis-v3.pdf, §§1.1–8, pp. 2–65 (text extract sak.txt) --- A1-2 · Proposition · Triangular spherical transforms give $C_K\cong\mathbb C[D]^{W_X}$, any characteristic STATUS CLAIMED: proved PARENTS: A1-1 STATEMENT: Let \(F\) be any non-archimedean local field, \(G,K,\mathcal H_K,\mathcal S\) as in the setting, \(X\) a \(G\)-variety over \(F\). Let \(\Lambda\) be a lattice with a finite reflection group \(W_X\) acting on \(\Lambda_{\mathbb R}\), \(C^-\) a closed Weyl chamber, \(\Lambda^-:=\Lambda\cap C^-\), and \(\mathcal T\subset\Lambda_{\mathbb R}\) a closed strictly convex cone with \(\mathcal T\cap C^-=\{0\}\); write \(\mu\succeq\lambda\) iff \(\mu-\lambda\in\mathcal T\). Let \(D\subset A^*\) be a closed subvariety stable under a subgroup of \(W\) that acts on it through \(W_X\), with a \(W_X\)-equivariant isomorphism \(D\cong\operatorname{Spec}\mathbb C[\Lambda]\); put \(m_\mu=\sum_{\nu\in W_X\mu}e^\nu\). Assume: (a) (Cartan) \(X(F)=\bigsqcup_{\lambda\in\Lambda^-}x_\lambda K\), a disjoint union. (b) There are a Zariski-dense open \(D_0\subset D\) and \(G\)-morphisms \(S_\chi:C_c^\infty(X(F))\to I(\chi)\) (\(\chi\in D_0\)); write \(S_\chi(\Phi)=\hat S(\Phi)(\chi)\,\varphi_{K,\chi}\) for \(\Phi\in C_c^\infty(X(F))^K\). (c) \(\hat S(\mathbf 1_{x_\lambda K})=\kappa\,Q_\lambda\) on \(D_0\), with \(\kappa\) a rational function on \(D\), not identically zero and independent of \(\lambda\), and \(Q_\lambda=\sum_{\mu\in\Lambda^-}a_{\lambda\mu}m_\mu\in\mathbb C[D]^{W_X}\) with \(a_{\lambda,\sigma\lambda}\ne0\) and \(a_{\lambda\mu}\neq0\Rightarrow\mu\succeq\sigma\lambda\), for some bijection \(\sigma\) of \(\Lambda^-\). Then \(\Phi\mapsto\hat S(\Phi)/\kappa\) is a linear isomorphism \(C_c^\infty(X(F))^K\xrightarrow{\sim}\mathbb C[D]^{W_X}\) under which \(f\in\mathcal H_K\) acts as multiplication by \((\mathcal Sf)|_D\). If moreover \(Q_0\) is a non-zero constant, then after rescaling \(\kappa\) it sends \(\mathbf 1_{x_0K}\mapsto1\). PROOF / BODY: This is the last paragraph of the proof of Theorem 8.0.2 (p. 64), written out with the normalizations made explicit. Nothing in it depends on the characteristic of \(F\). **Proof.** 1. *Equivariance.* \(S_\chi\) is \(G\)-equivariant for \((g\Phi)(x)=\Phi(g^{-1}x)\), and \(f*\Phi=\int_G f(g)\,g\Phi\,dg\) (vol \(K=1\)). Hence \(S_\chi(f*\Phi)=\pi_\chi(f)S_\chi(\Phi)\). For \(f\in\mathcal H_K\), \(\pi_\chi(f)\varphi_{K,\chi}=(\mathcal Sf)(\chi)\varphi_{K,\chi}\) by the definition of the Satake transform. So \(\hat S(f*\Phi)=(\mathcal Sf)\cdot\hat S(\Phi)\) on \(D_0\), and therefore on \(D\), since both sides are rational and agree on a dense set. 2. *Basis of the target.* \(C^-\cap\Lambda\) meets every \(W_X\)-orbit in \(\Lambda\) exactly once (a closed chamber is a fundamental domain of a finite reflection group). So \(\{m_\mu\}_{\mu\in\Lambda^-}\) is a basis of \(\mathbb C[\Lambda]^{W_X}=\mathbb C[D]^{W_X}\). 3. *Finiteness.* For \(\lambda\in\Lambda^-\) put \(F_\lambda=\{\mu\in\Lambda^-:\mu\succeq\lambda\}\). The set \((\lambda+\mathcal T)\cap C^-\) is closed and convex with recession cone \(\mathcal T\cap C^-=\{0\}\). It is therefore bounded, so \(F_\lambda\) is finite. Since \(\mathcal T\) is a strictly convex cone, \(\succeq\) is a partial order. If \(\mu\in F_\lambda\setminus\{\lambda\}\), then \(F_\mu\subseteq F_\lambda\setminus\{\lambda\}\). 4. *Independence.* Suppose \(\sum c_\lambda Q_\lambda=0\) is a finite relation. Pick \(\lambda_0\) such that \(\sigma\lambda_0\) is minimal for \(\succeq\) among the \(\sigma\lambda\) with \(c_\lambda\ne0\). By (c), \(a_{\lambda,\sigma\lambda_0}\ne0\) forces \(\sigma\lambda_0\succeq\sigma\lambda\). So the coefficient of \(m_{\sigma\lambda_0}\) is \(c_{\lambda_0}a_{\lambda_0,\sigma\lambda_0}\ne0\), a contradiction. 5. *Spanning.* Induct on \(|F_\nu|\). For \(\nu=\sigma\lambda\), \(m_{\nu}=a_{\lambda\nu}^{-1}\big(Q_\lambda-\sum_{\mu\in F_\nu\setminus\{\nu\}}a_{\lambda\mu}m_\mu\big)\), and each such \(m_\mu\) lies in the span of the \(Q\)'s by step 3. This is exactly where \(\mathcal T\cap C^-=\{0\}\) "commences" the induction (p. 64). 6. *Conclusion.* By (a), \(\{\mathbf 1_{x_\lambda K}\}_{\lambda\in\Lambda^-}\) is a basis of \(C_c^\infty(X(F))^K\). By steps 4–5, \(\hat S/\kappa\) maps it bijectively onto the basis \(\{Q_\lambda\}\), and it is equivariant by step 1. The last claim is the case \(\lambda=0\). **Remarks.** (i) Sakellaridis's map \(\mathbf 1_{x_\lambda K}\mapsto P_\lambda\) (8.1) is this map up to two factors. One is the positive constant \(\operatorname{vol}(x_\lambda K)\), which is absorbed into \(Q_\lambda\) without changing (c). The other is the substitution \(\chi\mapsto\chi^{-1}\) composed with \(w_L\), the longest element of \(W_{P(X)}\). It is needed because \(S_\chi\) lives on \(\delta(X)^{-1/2}A_X^*\) while \(\Delta_\chi\) lives on \(\delta(X)^{1/2}A_X^*\) ((3.3), (3.6)). Here \(w_L\) fixes \(A_X^*\) pointwise, inverts \(\delta(X)\) and commutes with \(W_X\), and \(\mathcal Sf\) is \(W\)-invariant. After this substitution the leading index of \(Q_\lambda\) becomes \(\sigma\lambda=-w_0^X\lambda\) and the cone becomes \(-w_0^X\mathcal T\), which still meets \(C^-\) only in \(0\). The weight set \(S_\lambda\subseteq\lambda+\mathcal T\) is \(W_X\)-stable, so \(-S_\lambda=-w_0^XS_\lambda\subseteq\sigma\lambda-w_0^X\mathcal T\). This is why (c) allows a bijection \(\sigma\). (ii) The scalar relating \(\Omega_\chi=\Delta_\chi(\Phi_K)\) to \(\Delta_\chi(\varphi_{K,\chi})\) ((4.5), which uses [CS80]) is part of \(\kappa\). So (4.5) is not needed. (iii) Theorem 3.2.2 (multiplicity) is not used. Only the existence of the family \(S_\chi\) and the values \(\hat S(\mathbf 1_{x_\lambda K})\) enter. SOURCES: sakellaridis-v3.pdf, proof of Theorem 8.0.2, p. 64; Remark after Theorem 7.2.1, p. 62; (3.3), (3.6), pp. 13, 18 --- A1-3 · Lemma · Local structure and smooth Borel stabilizers for $G/G^\theta$ in characteristic $\neq2$ STATUS CLAIMED: proved PARENTS: A1-1 STATEMENT: Let \(k\) be an algebraically closed field with \(\operatorname{char}k\neq2\) (here \(k=\bar F\), \(F=\mathbb F_q((t))\), \(p\) odd). Let \(G\) be connected reductive over \(k\), \(\theta\) an involution, \(H=G^\theta\), \(X=G/H\), \(x_0=eH\). Let \(S\) be a maximal \(\theta\)-split torus, \(T\supseteq S\) a \(\theta\)-stable maximal torus, \(L=Z_G(S)\), and \(B\supseteq T\) a Borel subgroup whose positive roots satisfy \(\alpha>0,\ \alpha|_S\neq0\Rightarrow\theta\alpha<0\). Let \(P=LB\). Then: (a) \(H\) is smooth and \(G\to X\) is smooth. (b) \(\theta\) is the identity on \([L,L]\), so \([L,L]\subseteq H\). (c) \(\mathfrak b+\mathfrak h=\mathfrak g\). Hence \(Bx_0\) is open, the orbit map \(B\to Bx_0\) is smooth, and \(B_{x_0}=B\cap H=T^\theta\ltimes(U\cap L)\) is smooth, with unipotent radical \(U\cap L\). (d) \(P(X)=P\), \(L(X)=L\), \(A_X=T/T^\theta\). The modular characters satisfy \(\delta_B|_{B_{x_0}}=\delta_{B_{x_0}}=\delta(X)|_{B_{x_0}}\), and \(\delta_{P(X)}\) is trivial on \(T^\theta\) (eq. (3.5)). (e) (Springer's lemma assumed, see body) Every \(B\)-stabilizer \(B_y\) (\(y\in X(k)\)) is smooth, of the form \(T_1\ltimes U_1\) up to \(B\)-conjugacy. Here \(T_1\) is diagonalizable with component group an elementary abelian 2-group, \(U_1\) is connected unipotent, and the weights of \(T_1^\circ\) on \(\operatorname{Lie}U_1\) are non-zero and lie in an open half-space. PROOF / BODY: This replaces [BLV86, Théorème 3.4] at both places where the paper uses it (§2.1, p. 8: \([L(X),L(X)]\subseteq H\); §3.4, p. 17: \((B_y)_u=L(X)\cap U\)). It also supplies the smoothness that the differential-form calculus of §3.3 and Lemmas 3.4.2, 5.1.1 needs (checklist 5). **Proof.** 1. (a) \(H=G^{\langle\theta\rangle}\) is the fixed-point scheme of a group of order 2, which is invertible on \(k\), acting on a smooth affine scheme. It is therefore smooth (fixed points of a linearly reductive group; [CGP, Prop. A.8.10(2)], cited, not opened). \(G\to G/H\) is an \(H\)-torsor, hence smooth. 2. *Roots of \(L\).* Since \(T=(T^\theta)^\circ S\) and \(\alpha-\theta\alpha\) is trivial on \(T^\theta\), we have \(\alpha|_S=0\iff\theta\alpha=\alpha\). So \(\Phi_L=\{\alpha:\theta\alpha=\alpha\}\). For \(\alpha\in\Phi_L\), \(\theta\) preserves \(U_\alpha\) and acts on it by a sign \(\varepsilon_\alpha\). 3. (b) Suppose \(\varepsilon_\alpha=-1\) for some \(\alpha\in\Phi_L\). On the rank-one group \(G_\alpha=\langle U_\alpha,U_{-\alpha}\rangle\cong SL_2\) or \(PGL_2\), \(\theta\) is then conjugation by \(\alpha^\vee(\sqrt{-1})\), and \(G_\alpha\) contains a one-dimensional \(\theta\)-split torus \(S'\) (the Cayley transform of \(\alpha^\vee(\mathbb G_m)\); here \(2\in k^\times\)). \(S'\) centralizes \(S\) and \(S'\not\subseteq S\), so \(SS'\) is a larger \(\theta\)-split torus, a contradiction. Hence \(\varepsilon_\alpha=1\) for all \(\alpha\in\Phi_L\), and \(\theta\) fixes \([L,L]=\langle U_\alpha:\alpha\in\Phi_L\rangle\). 4. (c) Let \(\beta<0\). If \(\beta\notin\Phi_L\), then \(\theta\beta>0\), and for \(Y\in\mathfrak g_\beta\) we have \(Y=(Y+\theta Y)-\theta Y\in\mathfrak h+\mathfrak b\). If \(\beta\in\Phi_L\), then \(\mathfrak g_\beta\subseteq\mathfrak h\) by step 3. Together with \(\mathfrak t\subseteq\mathfrak b\) this gives \(\mathfrak g=\mathfrak b+\mathfrak h\). So the differential of \(B\to X\) at \(x_0\) is onto: \(Bx_0\) is open, \(B\to Bx_0\) is smooth, and \(B\cap H\) is smooth with Lie algebra \(\mathfrak b\cap\mathfrak h\). If \(g\in B\cap H\), then \(g=\theta(g)\in B\cap\theta(B)=T\cdot\prod_{\beta>0,\theta\beta>0}U_\beta=T(U\cap L)\), since \(\beta>0,\ \theta\beta>0\) forces \(\beta\in\Phi_L\). With step 3 this gives \(B\cap H=T^\theta\ltimes(U\cap L)\). 5. (d) \(P\subseteq P(X)\): \(Px_0=U_PT[L,L]x_0=U_PTx_0\subseteq Bx_0\). Conversely, let \(\alpha\notin\Phi_L\) be simple with \(P_\alpha\subseteq P(X)\). Every unit on the \(P(X)\)-homogeneous \(X̊\) is a character of \(P(X)\) (Rosenlicht), hence orthogonal to \(\alpha^\vee\). But \(\mathcal X(X)_{\mathbb Q}=\{\chi:\theta\chi=-\chi\}\), and \(\alpha^\vee\) has non-zero projection to \(X_*(S)_{\mathbb Q}\), so some \(\chi\in\mathcal X(X)\) has \(\langle\alpha^\vee,\chi\rangle\ne0\). This is a contradiction, so \(P(X)=P\). \(L\) is a Levi of \(P\) with \([L,L]\subseteq H\), and \(L/(L\cap H)=T/T^\theta\). On \(T^\theta\): \(\delta_{B_{x_0}}=|e^{2\rho_L}|=\delta(X)\). Since \(\theta\Phi(U_P)=-\Phi(U_P)\), \(2\rho_P\) is \(\theta\)-anti-invariant and hence trivial on \(T^\theta\). So \(\delta_B=|e^{2\rho_L+2\rho_P}|\) agrees with \(\delta(X)\) on \(T^\theta\), and \(\delta_{P(X)}=|e^{2\rho_P}|\) is trivial on \(T^\theta\). 6. (e) **Cited input.** Springer [Some results on algebraic groups with involutions, 1985], char \(\neq2\), not opened locally: every \(B\)-orbit contains \(y=gH\) such that \(\theta_y:=\operatorname{Int}(g)\theta\operatorname{Int}(g)^{-1}\) stabilizes a maximal torus \(T'\subseteq B\). Then \(B_y=B\cap G^{\theta_y}=(B\cap\theta_yB)^{\theta_y}\) and \(B\cap\theta_yB=T'\ltimes U''\) with \(U''=\prod_{\beta\in\Phi^+\cap\theta_y\Phi^+}U_\beta\). Since \(H^1(\mathbb Z/2,V)=0\) for every \(k\)-vector space \(V\), filtering \(U''\) by \(\theta_y\)-stable subgroups with vector-group quotients shows that \((T'\ltimes U'')^{\theta_y}=T'^{\theta_y}\ltimes U''^{\theta_y}\), with \(U_1:=U''^{\theta_y}\) smooth and connected. \(T_1:=T'^{\theta_y}\) has component group of exponent 2. A weight of \((T'^{\theta_y})^\circ\) on \(\mathfrak g_\beta\) (\(\beta,\theta_y\beta>0\)) is \(\beta|\), which vanishes only if \(\theta_y\beta=-\beta\); that is impossible. The cocharacter \(\lambda=\check\rho+\theta_y\check\rho\) of \((T'^{\theta_y})^\circ\) satisfies \(\langle\lambda,\beta\rangle=\langle\check\rho,\beta\rangle+\langle\check\rho,\theta_y\beta\rangle>0\). So all weights lie in an open half-space. **Remark (where \(p\ne2\) and smoothness enter).** For \(p=2\), step 3 and the splitting in step 6 fail (\(H\) can be non-reduced and \(B\cap H\) can contain infinitesimal or étale 2-groups). Then \(B(F)\)-orbits can be infinite in number ([[A1-4]]). SOURCES: sakellaridis-v3.pdf, §2.1 p. 8, §3.4 p. 17 (uses of [BLV86, Thm 3.4]); §3.3 pp. 13–15; Springer, Some results on algebraic groups with involutions (1985), cited from memory (not opened); Conrad–Gabber–Prasad, Pseudo-reductive groups, Prop. A.8.10, cited (not opened) --- A1-4 · Lemma · Finiteness of $B(F)\backslash X(F)$ for symmetric $X$ over $\mathbb F_q((t))$, $p$ odd STATUS CLAIMED: proved PARENTS: A1-3 STATEMENT: Let \(F=\mathbb F_q((t))\) with \(p\) odd, \(G\) split reductive over \(F\), \(B=T\ltimes U\) a split Borel subgroup, \(\theta\) an involution of \(G\) over \(F\), \(X=G/G^\theta\). Assume Springer's results in char \(\neq2\): \(B_{\bar F}\) has finitely many orbits on \(X_{\bar F}\), and the lemma of [[A1-3]](e). Then for every \(y\in X(F)\), with \(Y=By\): (a) \(B\to Y\), \(b\mapsto by\), is smooth. Every \(B(F)\)-orbit in \(Y(F)\) is open and closed in \(Y(F)\). (b) \(Y(F)/B(F)\hookrightarrow H^1(F,M_y)\), where \(M_y:=B_y/(B_y\cap U)\) is a closed subgroup of the split torus \(T\) of the form \(\mathbb G_m^a\times\mu_2^b\). Hence \(|Y(F)/B(F)|\le4^b\), and \(Y(F)\) is a single \(B(F)\)-orbit when \(M_y\) is connected. (c) \(B(F)\backslash X(F)\) is finite, and \(X(F)\) has a finite filtration by \(B(F)\)-stable open subsets whose successive differences are single \(B(F)\)-orbits. (d) For \(X=M_\theta\): \(B\cap H=T^\theta\ltimes(U\cap L)\) with \(T^\theta\) connected, so \(X̊(F)=x_0B(F)\). The argument needs the stabilizers to have no \(p\)-torsion. For instance, a \(B_{\bar F}\)-orbit with stabilizer \(\mathbb Z/p\) (resp. \(\mu_p\)) would carry infinitely many \(B(F)\)-orbits, since \(H^1(F,\mathbb Z/p)=F/\wp(F)\) and \(H^1(F,\mu_p)=F^\times/F^{\times p}\) are infinite. PROOF / BODY: This replaces the implicit finiteness of Borel orbits and of Galois cohomology in [Sa08], used for Theorems 3.2.2 and 3.7.1 (checklist 6). **Proof.** 1. *Torsor description.* \(Y\) is the orbit of an \(F\)-point, a locally closed \(F\)-subvariety. \(B_y\) is smooth: smoothness is checked over \(\bar F\), where it is [[A1-3]](e). So \(B/B_y\xrightarrow{\sim}Y\) and \(B\to Y\) is smooth. By [Serre, Galois Cohomology, I.5.4, Cor. 1], \(Y(F)/B(F)\) is in bijection with \(\ker(H^1(F,B_y)\to H^1(F,B))\). Since \(H^1(F,T)=1\) (Hilbert 90, \(T\) split) and \(H^1(F,U)=1\) (\(U\) split unipotent, \(H^1(F,\mathbb G_a)=0\)), \(H^1(F,B)=1\). Hence \(Y(F)/B(F)\cong H^1(F,B_y)\). 2. *Unipotent part.* Let \(U_y:=B_y\cap U\). Over \(\bar F\), after \(B(\bar F)\)-conjugation, \(B_y=T_1\ltimes U_1\) with \(U_1=B_y\cap U\) ([[A1-3]](e)). So \(U_y\) is smooth, connected and unipotent. \(M_y=B_y/U_y\hookrightarrow T\) is a closed subgroup of a split torus. It is therefore split diagonalizable, and it is smooth because its geometric component group has exponent 2. 3. *\(U_y\) and its twists are split.* Let \(c\) be a cocycle with values in \(B_y(F_s)\). The inner twist \({}_cB_y^\circ\) is a smooth connected solvable \(F\)-group. Let \(T''\) be a maximal \(F\)-torus of it. Since \(M_y\) is commutative, inner twisting does not change it, so \(T''\hookrightarrow M_y\subseteq T\) and \(T''\) is split. Over \(\bar F\), \(T''\) is conjugate to \(T_1^\circ\). By [[A1-3]](e) its weights on \(\operatorname{Lie}({}_cU_y)\) are non-zero and lie in an open half-space. Choose \(\lambda\in X_*(T'')\), defined over \(F\) since \(T''\) is split, positive on all of them. Then \({}_cU_y=U_{ {}_cB_y^\circ}(\lambda)\), which is \(F\)-split unipotent by [Conrad–Gabber–Prasad, Prop. 2.1.10] (cited, not opened). Hence \(H^1(F,{}_cU_y)=1\). 4. *Injectivity.* By [Serre, I.5.5, Cor. 2], the fibres of \(H^1(F,B_y)\to H^1(F,M_y)\) are quotients of the sets \(H^1(F,{}_cU_y)\), which are trivial by step 3. So the map is injective. Write \(M_y\cong\mathbb G_m^a\times\mu_2^b\). Then \(H^1(F,M_y)=(F^\times/F^{\times2})^b\), and \(|F^\times/F^{\times2}|=4\) for \(p\) odd, with classes \(1,\varepsilon,t,\varepsilon t\). This proves (b). 5. *(a), openness.* A smooth morphism of smooth \(F\)-varieties induces a submersion of \(F\)-analytic manifolds, so \(B(F)\to Y(F)\) is open (implicit function theorem over a complete non-discretely valued field, valid in every characteristic). Hence each of the finitely many \(B(F)\)-orbits in \(Y(F)\) is open, and therefore also closed, in \(Y(F)\). 6. *(c).* There are finitely many \(F\)-rational \(B_{\bar F}\)-orbits (Springer). Order them by decreasing dimension: \(\overline{Y}(F)\setminus Y(F)\) lies in orbits of smaller dimension. Refining by the finitely many \(B(F)\)-orbits of step 5 gives the filtration. 7. *(d).* For \(M_\theta\), \(T^\theta\) is the torus generated by \(\alpha_2^\vee,\dots,\alpha_5^\vee\), which is connected (S4-3 step 2; family page §4). With [[A1-3]](c), \(M_{x_0}=T^\theta\) is connected, so \(b=0\) in (b). 8. *Why no \(p\)-torsion matters.* If \(B_y=\mathbb Z/p\), step 1 gives \(Y(F)/B(F)\cong H^1(F,\mathbb Z/p)\). Artin–Schreier theory gives \(|F/\wp F|=\infty\) for \(F=\mathbb F_q((t))\), and \(F^\times/F^{\times p}\) is infinite since \(t^{1/p}\notin F\). For symmetric varieties with \(p\) odd these stabilizers do not occur, by [[A1-3]](e). **Consequence for (A5).** Part (d) proves the clause "\(X̊(F)\) is a single \(B(F)\)-orbit" of (A5) for \(M_\theta\) over \(\mathbb F_q((t))\), independently of characteristic-0 inputs. SOURCES: Serre, Galois Cohomology, I.5.4 Cor. 1 and I.5.5 Cor. 2 (standard); Conrad–Gabber–Prasad, Pseudo-reductive groups, Prop. 2.1.10 (cited, not opened); Springer 1985 (finiteness of B-orbits on G/G^θ, char ≠ 2; cited from memory); S4b.json node S4-3 step 2 (T^θ connected for E6/F4) --- A1-5 · Proposition · $S_\chi$ over $\mathbb F_q((t))$: convergence, generic uniqueness, rationality STATUS CLAIMED: proved PARENTS: A1-3, A1-4 STATEMENT: Let \(F=\mathbb F_q((t))\), \(p\) odd, \(X=G/G^\theta\) affine with the data of [[A1-3]] (\(S\) chosen of maximal dimension), and assume \(X̊(F)=x_0B(F)\) (true for \(M_\theta\) by [[A1-4]](d)). Put \(D':=\delta(X)^{-1/2}A_X^*\subset A^*\), the parameters allowed by condition (3.3) for the open orbit. For \(\chi\in D'\), let \(S_\chi\) be Sakellaridis's integral (3.2) over \(X̊(F)\). (a) *Convergence.* There is a non-empty open subset \(R\subseteq D'\) for the analytic topology (a translated open cone in the real parts) on which (3.2) converges absolutely for every \(\varphi\in C_c^\infty(X(F))\). There it defines \(G\)-morphisms \(S_\chi:C_c^\infty(X(F))\to I(\chi)\) depending holomorphically on \(\chi\). (b) *Generic uniqueness.* There is a dense Zariski-open \(D_1'\subseteq D'\) such that for \(\chi\in D_1'\) the space \(\operatorname{Hom}_G(C_c^\infty(X(F)),I(\chi))\) has dimension at most 1. More precisely, a morphism whose functional \(\mathrm{ev}_1\circ(\cdot)\) vanishes on \(C_c^\infty(X̊(F))\) is zero. (c) *Rationality.* For every \(\varphi\), \(\chi\mapsto S_\chi(\varphi)(1)\) extends from \(R\) to a rational function on \(D'\). On a dense Zariski-open \(D_0'\subseteq D_1'\) the extension is the unique (up to scalar) non-zero \(G\)-morphism \(C_c^\infty(X(F))\to I(\chi)\). PROOF / BODY: This replaces the convergence and rationality of (3.2) and the multiplicity statement of Theorem 3.2.2, all taken from [Sa08], at the level needed for Theorem 8.0.2. The argument is the classical one: orbit filtration plus Bernstein's principle. Its two characteristic-sensitive inputs are supplied by [[A1-3]] and [[A1-4]]. **Proof of (a).** 1. Regular \(B\)-eigenfunctions generate \(F(X)^{(B)}\): the denominators of a \(B\)-eigen rational function form a non-zero \(B\)-stable ideal, which contains a \(B\)-eigenvector by Lie–Kolchin; this holds in every characteristic. Choose \(f_1,\dots,f_r\in F[X]^{(B)}\) whose weights \(\eta_i\) span \(\mathcal X(X)_{\mathbb R}\). 2. Each \(f_i\) is \(U\)-invariant, \(B\)-eigen and nowhere zero on \(X̊\). Write \(a(x)\in A_X(F)\) for the image of \(x\in X̊(F)\) under \(X̊\to X̊/U\cong A_X\), and \(\lambda(x)\in\Lambda_X\) for its class. Then \(|f_i(x)|=|f_i(x_0)|\,q^{-\langle\lambda(x),\eta_i\rangle}\). 3. On the compact set \(\operatorname{supp}\varphi\) we have \(|f_i|\le C_i\). Hence \(\lambda(x)\) lies in \(P_\varphi=\{\lambda:\langle\lambda,\eta_i\rangle\ge-c_i\ \forall i\}\), a translate of the strictly convex cone \(\mathcal T'=\{\lambda:\langle\lambda,\eta_i\rangle\ge0\}\). 4. On \(X̊\), the integrand of (3.2) is \(\varphi\) times an unramified character \(\psi_\chi(a(x))\) times the \(G\)-invariant measure \(|\omega_X|\) (the eigenmeasures differ by unramified characters, §3.2). So \(|S_\chi(\varphi)(1)|\le\|\varphi\|_\infty\sum_{\lambda\in P_\varphi}|\psi_\chi(e^\lambda)|\,v_\lambda\) with \(v_\lambda=\operatorname{vol}(\operatorname{supp}\varphi\cap\{\lambda(x)=\lambda\})\) and \(\sum v_\lambda\le\operatorname{vol}(\operatorname{supp}\varphi)<\infty\). 5. \(|\psi_\chi(e^\lambda)|=q^{-\langle\lambda,s(\chi)\rangle}\) with \(s(\chi)\) affine in \(\operatorname{Re}\chi\). This is bounded on \(P_\varphi\) when \(s(\chi)\) lies in the interior of the dual cone \(\operatorname{cone}(\eta_i)\). That interior is a non-empty open set \(R\), independent of \(\varphi\), and convergence is uniform on compacta. Smoothness and \(B\)-equivariance of \(g\mapsto S_\chi(g\varphi)(1)\) are clear, so \(S_\chi(\varphi)\in I(\chi)\). **Proof of (b).** 6. Take the filtration \(X(F)=O_1\sqcup\dots\sqcup O_N\) of [[A1-4]](c), with \(O_1=X̊(F)\) and each \(U_j=O_1\cup\dots\cup O_j\) open. The Bernstein–Zelevinsky sequences \(0\to C_c^\infty(U_j)\to C_c^\infty(U_{j+1})\to C_c^\infty(O_{j+1})\to0\) hold for any \(\ell\)-space ([BZ76, 1.8]; characteristic-free). By [[A1-4]](a), \(O_{j+1}\cong B(F)/B_y(F)\) is a homeomorphism. So a \((B,\chi)\)-equivariant functional on \(C_c^\infty(O_{j+1})\) exists only if \(\chi\) satisfies the modular condition (3.3) on \(B_y(F)\). Call the set of such \(\chi\) \(E_y\). It is Zariski-closed in \(D'\): the condition fixes \(e^\mu(\chi)\) for \(\mu\) in the image \(\Gamma_y\) of \(B_y(F)\) in \(T(F)/T(O)\), whose rational span is \(X_*(T_y)_{\mathbb Q}\) for a maximal (split, [[A1-4]] step 3) \(F\)-torus \(T_y\subseteq B_y\). 7. *Claim: \(E_y\ne D'\) for \(y\notin X̊\).* If \(E_y=D'\), every \(e^\mu\) with \(\mu\in\Gamma_y\) is constant on the translate \(D'\) of \(A_X^*\). So \(X_*(T_y)_{\mathbb Q}\subseteq X_*(T^\theta)_{\mathbb Q}\), i.e. \(T_y\subseteq(T^\theta)^\circ\) inside \(T\). On the other hand, \(\dim T_y=\dim T-\dim(T')^{-\theta_y}\ge\dim T-\dim S=\dim T^\theta\). Here \(T'\), \(\theta_y\) are as in [[A1-3]](e), and \((T')^{-\theta_y}\), a \(\theta_y\)-split torus, is conjugate by \(g\) to a \(\theta\)-split torus, which has dimension \(\le\dim S\) by the choice of \(S\). Hence \(T_y=(T^\theta)^\circ\). Transport \(\theta_y|_{T'}\) to an involution \(\theta'_T\) of the root datum of \((G,T)\) via \(T'\cong B/U\cong T\). It has the same \(+1\)-eigenspace as \(\theta|_T\). Both preserve a \(W\)-invariant inner product, so their \(-1\)-eigenspaces are the orthogonal complement of that space, and \(\theta'_T=\theta|_T\). By [[A1-3]](e), \(\dim B_y=\dim T^\theta+\#\{\beta>0:\theta\beta\neq\beta,\ \theta\beta>0\}/2+\#\{\beta>0:\theta\beta=\beta,\ \theta_y|_{\mathfrak g_\beta}=1\}\). For the \(\theta\)-order positive system the middle term is 0 (if \(\beta>0\) and \(\beta\notin\Phi_L\) then \(\theta\beta<0\)), and the last term is at most \(|\Phi_L^+|\). The same count with all signs \(+1\) gives \(\dim B_{x_0}\) ([[A1-3]](c)). So \(\dim By\ge\dim X\). Then \(By\) is open, hence equal to \(X̊\), a contradiction. 8. Put \(D_1'=D'\setminus\bigcup_{y\notin X̊}E_y\), a finite union by [[A1-4]](c). For \(\chi\in D_1'\), induction along the filtration shows that a functional vanishing on \(C_c^\infty(O_1)\) vanishes. On \(C_c^\infty(O_1)=C_c^\infty(B(F)/B_{x_0}(F))\) the equivariant functionals form a line. **Proof of (c).** 9. *Bernstein's principle* (pure linear algebra over \(\mathbb C(D')\); see Banks, Math. Res. Lett. 5 (1998); cited, not opened). Let \(V\) be a countable-dimensional complex vector space, \(D'\) an irreducible variety, and \(\{\ell(v_i)=c_i(\chi)\}_{i\in I}\) a system with \(c_i\in\mathbb C[D']\). If the system has a unique solution for \(\chi\) in a non-empty analytic-open set, then it has a unique solution on a Zariski-dense open set, and that solution is rational in \(\chi\). 10. Take \(V=C_c^\infty(X(F))\), which is countable-dimensional. The equations are \(\ell(b\varphi)=c_\chi(b)\ell(\varphi)\) for \(b\) in a countable set of coset representatives of \(B(F)/B(O)\) together with \(b\in B(O)\); here \(c_\chi(b)\) is a monomial in the coordinates of \(A^*\). Add the normalization \(\ell(\mathbf 1_{x_0J_0})=\operatorname{vol}\) for a small compact open \(x_0J_0\subseteq X̊(F)\). By (b) the solution is unique on \(D_1'\), and on \(R\cap D_1'\neq\emptyset\) the integral (3.2) is a solution. Step 9 gives the rational extension and its uniqueness on a dense open \(D_0'\). SOURCES: sakellaridis-v3.pdf, (3.2)–(3.3) pp. 12–13, Theorem 3.2.2 p. 13; Bernstein–Zelevinsky 1976, §1 (ℓ-spaces; cited, not opened); Banks 1998 (Bernstein's principle; cited, not opened) --- A1-6 · Proposition · $F_wS_\chi=b_wS_{w\chi}$ and the sum formula (6.2) over $\mathbb F_q((t))$ STATUS CLAIMED: proved PARENTS: A1-5 STATEMENT: Setting of [[A1-5]], with (A2) and (A3c): \(xJ\subseteq xB(O)\) for \(x\in A_X^+\), where \(A_X^+\) is the set of points with antidominant label. Let \(D=\delta(X)^{1/2}A_X^*\), \(D'=\delta(X)^{-1/2}A_X^*=w_LD\), and \(\mathfrak a=X_*(S)_{\mathbb Q}\). Define $$W_X':=\{w\in W:\ w(D)=D\}.$$ (a) \(W_X'=N_W(\mathfrak a)\cap[W/W_L]\). It is a group mapping isomorphically onto \(N_W(\mathfrak a)/W_L\), it commutes with \(w_L\), and it fixes \(\delta(X)\). (b) Let \(F_w:=\prod_{\check\alpha>0,\,w\check\alpha<0}\frac{1-e^{\check\alpha}}{1-q^{-1}e^{-\check\alpha}}(\chi)\,T_w:I(\chi)\to I(w\chi)\) (eq. (4.9)). Then \(F_{w_1w_2}=F_{w_1}F_{w_2}\) for all \(w_1,w_2\), and \(F_w^*=F_{w^{-1}}\) for the pairing of \(I(\chi)\) with \(I(\chi^{-1})\). The vector \(\Phi_{K,\chi}:=\operatorname{vol}(J)^{-1}\sum_{w\in W}\prod_{\alpha>0,w^{-1}\alpha<0}(-e^{\check\alpha}(\chi))\,F_w\varphi_{J,w^{-1}\chi}\) is \(K\)-invariant (Proposition 4.2.1). (c) For generic \(\chi\in D'\): if \(w\in W_X'\) then \(F_w\circ S_\chi=b_w(\chi)S_{w\chi}\), with \(b_w\) rational on \(D'\) and \(b_{w_1w_2}(\chi)=b_{w_1}(w_2\chi)b_{w_2}(\chi)\). If \(w\notin W_X'\), the functional \(\mathrm{ev}_1\circ F_wS_\chi\) vanishes on \(C_c^\infty(X̊(F))\). (d) (Theorem 4.2.2 / eq. (6.2), formula part) Let \(\Delta_\chi\) be the adjoint of \(S_{\chi^{-1}}\) and \(\Omega_\chi:=\Delta_\chi(\Phi_{K,\chi})\) for \(\chi\in D\). For \(x\in A_X^+\) and generic \(\chi=\delta(X)^{1/2}\tilde\chi\), $$\Omega_\chi(x)=\delta_{P(X)}^{-1/2}(x)\sum_{w\in W_X'}B_w(\tilde\chi)\,{}^w\tilde\chi(x),\qquad B_w(\tilde\chi)=\prod_{\check\alpha>0,\,w\check\alpha<0}(-e^{\check\alpha}(\chi))\,b_w,$$ with \(b_w\) transported from \(D'\) to \(D\) through the adjoint (\(F_w^*=F_{w^{-1}}\)). \(\Omega_\chi\) is a \(K\)-invariant Hecke eigenfunction with eigencharacter \(\chi\). PROOF / BODY: This replaces Theorem 3.7.1/Corollary 3.7.2 of [Sa08], the Braverman–Kazhdan formalism [BK98] and [Ca80, Thm 3.1] on the route to (4.2). All arguments are valid over any non-archimedean local field \(F\) once \(\psi\) has conductor \(O\) and \(dx(O)=1\). **Proof.** 1. *(a).* \(w(D)=D\) forces \(w\) to normalize the subtorus \(A_X^*\), i.e. \(w\in N_W(\mathfrak a)\). Such \(w\) permutes \(\Phi_L=\Phi\cap\mathfrak a^\perp\). Also \(w(\delta(X)^{1/2})\delta(X)^{-1/2}=|e^{w\rho_L-\rho_L}|\), and \(w\rho_L-\rho_L\in\mathbb Q\Phi_L\), which restricts injectively to \(T^\theta\). So \(w(\delta(X)^{1/2})\in\delta(X)^{1/2}A_X^*\) iff \(w\rho_L=\rho_L\) iff \(w\Phi_L^+=\Phi_L^+\) iff \(w\) is the shortest element of \(wW_L\). The pointwise stabilizer of \(\mathfrak a\) is the reflection subgroup generated by the roots orthogonal to \(\mathfrak a\), namely \(W_L\) (Chevalley–Steinberg). So \(W_X'\cong N_W(\mathfrak a)/W_L\). Elements of \(W_X'\) normalize \(\Phi_L^+\), hence commute with \(w_L\) and fix \(\rho_L\). For \(G\) split and \(\operatorname{char}\ne2\), \(N_W(\mathfrak a)/Z_W(\mathfrak a)\) is the restricted Weyl group (Richardson 1982; cited). For \(M_\theta\) it is computed directly in [[A1-7]]. 2. *(b), rank one.* The function \(u\mapsto\varphi_{K,\chi}(\tilde w_\alpha u_\alpha(x))\) equals 1 for \(|x|\le1\). For \(|x|=q^n\ge q\) the \(SL_2\) identity \(\tilde w u(x)\in u'\check\alpha(x^{-1})K\) gives the value \(e^{-n\check\alpha}(\chi\delta^{1/2})\). Summing with \(\operatorname{vol}\{|x|=q^n\}=q^n(1-q^{-1})\) gives \(T_\alpha\varphi_K=c_\alpha(\chi)\varphi_K\) with \(c_\alpha=\frac{1-q^{-1}e^{\check\alpha}}{1-e^{\check\alpha}}\), the rank-one case of (4.8). \(T_{w_1w_2}=T_{w_1}T_{w_2}\) when lengths add (Fubini). For generic \(\chi\), \(\operatorname{End}_G I(\chi)=\mathbb C\) (Bruhat filtration / geometric lemma of Bernstein–Zelevinsky, valid for \(\ell\)-groups). Hence \(T_\alpha^2=c_\alpha(\chi)c_\alpha(w_\alpha\chi)\), and with the normalization (4.9), \(F_\alpha^2=1\). So the \(F_w\) satisfy the relations of \(W\), and \(F_{w_1w_2}=F_{w_1}F_{w_2}\) always. \(T_w^*=T_{w^{-1}}\) and the symmetric normalizing factor give \(F_w^*=F_{w^{-1}}\). The \(K\)-invariance of \(\Phi_{K,\chi}\) is Sakellaridis's proof of Proposition 4.2.1 projected to \(I(\chi)\). It uses only \(F_{ww_\beta}=F_wF_{w_\beta}\) and the \(SL_2\) identity \(\mathcal F\mathbf 1_{\mathfrak p\times O}=q^{-1}\mathbf 1_{O\times\mathfrak p^{-1}}\) for the symplectic Fourier transform (3.10) on \(F^2\). That identity is a one-line computation with \(\psi\) of conductor \(O\). 3. *(c), \(w\in W_X'\).* \(F_wS_\chi\in\operatorname{Hom}_G(C_c^\infty(X(F)),I(w\chi))\) and \(w\chi\in D'\) (since \(W_X'\) commutes with \(w_L\), it also stabilizes \(D'\)). By [[A1-5]](b), for generic \(\chi\) this space is the line spanned by \(S_{w\chi}\). By [[A1-5]](c), \(b_w\) is rational. The cocycle relation follows from \(F_{w_1w_2}=F_{w_1}F_{w_2}\). 4. *(c), \(w\notin W_X'\).* The restriction of \(\mathrm{ev}_1\circ F_wS_\chi\) to \(C_c^\infty(X̊(F))=C_c^\infty(B(F)/B_{x_0}(F))\) is a \((B,w\chi)\)-equivariant functional. It is non-zero only if \(w\chi\) satisfies (3.3) for \(x_0\), i.e. \(w\chi\in D'\). Since \(w(D')\neq D'\) and both are irreducible of the same dimension, \(w\chi\notin D'\) for generic \(\chi\). 5. *(d).* For \(f\in I(\chi)^J\) and \(x\in A_X^+\), \(J\)-invariance gives \(\Delta_\chi(f)(x)=\langle f,S_{\chi^{-1}}(\mathbf 1_{xJ}|\omega_X|)\rangle/|\omega_X|(xJ)\). By (A3c), \(\mathbf 1_{xJ}\in C_c^\infty(X̊(F))\). By steps 3–4 and \(F_w^*=F_{w^{-1}}\), the term of \(w\) in \(\Delta_\chi(\Phi_{K,\chi})(x)\) is \(\langle\varphi_{J},F_{w^{-1}}S_{\chi^{-1}}(\mathbf 1_{xJ})\rangle\). This is 0 unless \(w^{-1}\in W_X'\), in which case it equals \(b_{w^{-1}}(\chi^{-1})\langle\varphi_J,S_{w^{-1}\chi^{-1}}(\mathbf 1_{xJ})\rangle\). Lemma 3.9.1 (whose proof uses only \(xJ\subseteq xB(O)\)) evaluates the last pairing as \(\operatorname{vol}(J)\,|\omega_X|(xJ)\) times the character \({}^{w}\tilde\chi\delta^{-1/2}\) at \(x\). Collecting the factors \(\prod(-e^{\check\alpha})\) as in the passage from Proposition 4.2.1 to Theorem 4.2.2 and to (6.1)–(6.2) gives the formula. \(\Phi_{K,\chi}\) is \(K\)-invariant and \(\Delta_\chi\) is a \(G\)-morphism, so \(\Omega_\chi\) is a \(K\)-invariant eigenfunction. It is non-zero because the term \(w=1\) contributes the character \(\tilde\chi\delta_{P(X)}^{-1/2}\), which is linearly independent of the others for generic \(\chi\) (the remark at the end of §3.10). **What is not used.** Knop's action, the orbits \(wX̊\ne X̊\), Brion's paths and Proposition 3.3.6 do not enter: only the open orbit is normalized. Theorem 3.2.2 (multiplicity \(g\cdot a\)) does not enter either. For symmetric \(X\), \(W_X'\cap W_L=1\): the type-\(G\) vanishing \(T_\alpha S_\chi=0\) (\(\alpha\in\Delta(X)\)) of Theorem 3.7.1 is not needed, because those \(w\) already fail \(w(D)=D\). SOURCES: sakellaridis-v3.pdf, §3.7–3.10 pp. 21–25, Proposition 4.2.1 and Theorem 4.2.2 pp. 26–27, (4.8)–(4.9) p. 28, Lemma 3.9.1 p. 24, (6.1)–(6.2) p. 44; Richardson 1982 (restricted Weyl group, char ≠ 2; cited, not opened) --- A1-7 · Computation · $M_\theta$: $W_X'\cong S_3$, coboundary $\beta$, formula (7.4), triangularity STATUS CLAIMED: proved PARENTS: A1-6, A1-2 STATEMENT: Let \(X=M_\theta\) (\(E_6/F_4\)) over \(F=\mathbb F_q((t))\), \(p\) odd, with the data of the setting: \(\gamma_1=\alpha_1-\theta\alpha_1\), \(\gamma_6=\alpha_6-\theta\alpha_6\), \(\Lambda_X=\mathbb Ze_1\oplus\mathbb Ze_6\), \(\langle e_i,\gamma_j\rangle\) the Cartan matrix of \(A_2\), \(\langle\check\gamma_i,\rho_{P(X)}\rangle=4\). (a) \(W_X'=N_W(\mathfrak a)\cap[W/W_L]\) has order 6. It acts faithfully on \(\mathfrak a=\mathbb Q\gamma_1\oplus\mathbb Q\gamma_6\) as the Weyl group of the root system \(\Phi_X=\{\pm\gamma_1,\pm\gamma_6,\pm(\gamma_1+\gamma_6)\}\) of type \(A_2\), with simple reflections \(w_{\gamma_1},w_{\gamma_6}\). Moreover \(N_W(\mathfrak a)=W_X'\ltimes W_L\), of order \(1152\). On \(\Lambda_X\), \(W_X'\) acts as \(W(A_2)\) with simple coroots \(e_1=\check\gamma_1\), \(e_6=\check\gamma_6\). The closed antidominant chamber is \(C^-=\{\lambda:\langle\lambda,\gamma_1\rangle\le0,\ \langle\lambda,\gamma_6\rangle\le0\}\). (b) Assume \((\mathrm{S6.3.1})_p\): for \(\gamma\in\{\gamma_1,\gamma_6\}\) the cocycle of [[A1-6]](d) satisfies $$B_{w_\gamma}(\tilde\chi)=-e^{\check\gamma}\,\frac{1-q^{-4}e^{-\check\gamma}}{1-q^{-4}e^{\check\gamma}}(\tilde\chi).$$ Put \(\beta(\tilde\chi)=\prod_{\check\gamma\in\check\Phi_X^+}\frac{1-e^{\check\gamma}}{1-q^{-4}e^{\check\gamma}}(\tilde\chi)\). Then \(B_w=\beta/(\beta\circ w)\) for all \(w\in W_X'\), and for \(\lambda\in\Lambda_X\cap C^-\) $$\frac{\Omega_\chi(x_\lambda)}{\beta(\tilde\chi)}=\delta_{P(X)}^{-1/2}(x_\lambda)\prod_{\check\gamma\in\check\Phi_X^+}(1-q^{-4}T_{\check\gamma})\,s_\lambda(\tilde\chi)\qquad\text{(eq. (7.4))},$$ with \(s_\lambda\) the Weyl character with lowest weight \(\lambda\). In particular \(\Omega_\chi(x_0)=c\,\beta(\tilde\chi)\) with \(c=(1+q^{-4})(1+q^{-4}+q^{-8})=\beta(\delta_{P(X)}^{1/2})^{-1}\). (c) \(P_\lambda:=\Omega_\chi(x_\lambda)/(c\beta)\) lies in \(\mathbb C[\Lambda_X]^{W_X'}\), \(P_0=1\), all weights of \(P_\lambda\) lie in \(\lambda+\mathcal R\) (\(\mathcal R\) the cone of positive coroots), the coefficient of \(m_\lambda\) is non-zero for every \(q\ge3\), and \(\mathcal R\cap C^-=\{0\}\). Hence the family \(\{P_\lambda\}\) satisfies hypothesis (c) of [[A1-2]] with \(\mathcal T=\mathcal R\). PROOF / BODY: This is the combinatorial half of §§6–7 for \(M_\theta\), done without colours, Knop's criterion [Kn91, §6], or the case analysis of Lemmas 7.1.1–7.1.4. All of it is characteristic-free algebra. The only analytic input is \((\mathrm{S6.3.1})_p\), the char-\(p\) form of the hypothesis (A4)(iii) for \(M_\theta\) (see [[A1-8]]). **Proof.** 1. *(a), computation.* I generated \(W(E_6)\) (51840 elements) as integer matrices on the root lattice. I formed \(w_L\) (longest element of \(W(D_4)\), nodes 2–5) and \(\theta=-w_L\) on \(X(T)\). This gives \(\gamma_1=2\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5\) and \(\gamma_6=\alpha_2+\alpha_3+2\alpha_4+2\alpha_5+2\alpha_6\), as in S4-3. The elements preserving \(\mathfrak a=\operatorname{span}(\gamma_1,\gamma_6)\) number \(1152=6\cdot192\). Those that are minimal in their \(W_L\)-coset (\(w\alpha_i>0\) for \(i=2,\dots,5\)) number 6 and are closed under products. Their actions on \((\gamma_1,\gamma_6)\) are: \(\mathrm{id}\); \((-\gamma_1,\gamma_1+\gamma_6)\); \((\gamma_1+\gamma_6,-\gamma_6)\); \((\gamma_6,-\gamma_1-\gamma_6)\); \((-\gamma_1-\gamma_6,\gamma_1)\); \((-\gamma_6,-\gamma_1)\). These are exactly the six elements of \(W(A_2)\) for the simple system \(\{\gamma_1,\gamma_6\}\). They are distinct, so the action is faithful and \(N_W(\mathfrak a)/W_L\cong S_3\), with no extra element. So the generic degree \(g\) of \(D/W_X'\to A^*/W\) equals 1, consistent with [[A1-5]](b). The action on \(\Lambda_X\) is the contragredient one, with simple coroots \(e_1,e_6\) because \(\langle e_i,\gamma_j\rangle\) is the Cartan matrix. 2. *(b), cocycle.* \(w\mapsto\prod_{\check\alpha>0,w\check\alpha<0}(-e^{\check\alpha})=\sigma(w)e^{\check\rho-w^{-1}\check\rho}\) is a cocycle and \(b_w\) is a cocycle ([[A1-6]](c)), so \(B\) is a cocycle. So is \(w\mapsto\beta/(\beta\circ w)\). For a simple \(\gamma\), \(w_\gamma\) permutes \(\check\Phi_X^+\setminus\{\check\gamma\}\) and negates \(\check\gamma\). Hence \(\beta/(\beta\circ w_\gamma)=\frac{(1-e^{\check\gamma})(1-q^{-4}e^{-\check\gamma})}{(1-e^{-\check\gamma})(1-q^{-4}e^{\check\gamma})}=-e^{\check\gamma}\frac{1-q^{-4}e^{-\check\gamma}}{1-q^{-4}e^{\check\gamma}}=B_{w_\gamma}\). Two cocycles that agree on generators agree everywhere. 3. *(b), symmetrization.* By [[A1-6]](d), \(\Omega_\chi(x_\lambda)/\beta=\delta_{P(X)}^{-1/2}(x_\lambda)\sum_{w}{}^w\!\big(e^{\lambda}/\beta\big)\), and \(1/\beta=\prod(1-q^{-4}e^{\check\gamma})/\prod(1-e^{\check\gamma})\). Expanding the numerator and using \(\sum_w{}^w(e^\nu/\prod_{\check\gamma>0}(1-e^{\check\gamma}))=s_\nu:=\sum_w(-1)^we^{\check\rho-w\check\rho+w\nu}/\prod(1-e^{\check\gamma})\) (Weyl's formula in lowest-weight form, \(\check\rho\) the half-sum) gives the displayed product, with \(T_{\check\gamma}s_\nu=s_{\nu+\check\gamma}\). 4. *The constant.* Since \(s_{w\star\mu}=(-1)^ws_\mu\) for \(w\star\mu=w(\mu-\check\rho)+\check\rho\), and \(s_0=1\), we get \(s_{|I|}=(-1)^{\ell(w)}\) when \(I\subseteq\check\Phi^+_X\) is the inversion set of \(w\). For the two remaining subsets \(\{\check\gamma_1+\check\gamma_6\}\) and \(\{\check\gamma_1,\check\gamma_6\}\) one has \(|I|=\check\rho\), so the numerator \(e^{\check\rho}\sum_w(-1)^w\) vanishes and \(s_{|I|}=0\). In \(A_2\) the inversion sets have sizes 0, 1, 1, 2, 2, 3. So \(c=\prod(1-q^{-4}T)s_0=1+2q^{-4}+2q^{-8}+q^{-12}=(1+q^{-4})(1+q^{-4}+q^{-8})\). Independently, \(e^{\check\gamma}(\delta_{P(X)}^{1/2})=q^{-\langle\check\gamma,\rho_{P(X)}\rangle}\) equals \(q^{-4}\) for the two simple coroots and \(q^{-8}\) for their sum. This gives \(\beta(\delta_{P(X)}^{1/2})=\frac{(1-q^{-4})^2}{(1-q^{-8})(1-q^{-12})}=c^{-1}\), in agreement with Lemma 4.2.3 (\(\Omega_{\delta^{1/2}}=1\)), whose proof uses only the invariant measure. A numerical check of \(\sum_{w}{}^w(1/\beta)=(1+t)(1+t+t^2)\) at random points, and of \(\beta/(\beta\circ w_{\gamma_1})=B_{w_{\gamma_1}}\), confirms steps 2–4. 5. *(c), weights.* The weights of the numerator of \(s_{\lambda+|I|}\) are \((\check\rho-w\check\rho+w|I|)+w\lambda\). First, \(w\lambda-\lambda\in\mathcal R\) for antidominant \(\lambda\). Second, write \(I=I_+\sqcup I_-\) according to the sign of \(w\check\theta\). The map \(\check\theta\mapsto-w\check\theta\) injects \(I_-\) into the inversion set \(\{\check\gamma>0:w^{-1}\check\gamma<0\}\), whose sum is \(\check\rho-w\check\rho\). So \(\check\rho-w\check\rho+w|I|\) is a sum of positive coroots, i.e. lies in \(\mathcal R\). (This corrects the intermediate identity \(|\Theta^+|-w|\Theta^+|=\check\rho-w\check\rho\) on p. 62, which is off by a factor 2 when \(\Theta^+=\check\Phi_X^+\); the inequality (7.5) itself holds.) Dividing by \(\prod(1-e^{\check\gamma})\) keeps the weights in \(\lambda+\mathcal R\). 6. *(c), leading coefficient.* The coefficient of \(m_\lambda\) in \(\prod(1-q^{-4}T)s_\lambda\) is \(1+\sum_{\emptyset\ne I}(\pm1)(-q^{-4})^{|I|}[s_{\lambda+|I|}=\pm s_\lambda]\). Its absolute value is at least \(1-((1+q^{-4})^3-1)>0\) for \(q\ge3\). The factors \(\delta_{P(X)}^{-1/2}(x_\lambda)\) and \(c^{-1}\) are non-zero constants. \(P_0=1\) by step 4. 7. *Cone.* If \(v=a\check\gamma_1+b\check\gamma_6\in\mathcal R\cap C^-\), then \(2a-b\le0\) and \(-a+2b\le0\), so \(a+b\le0\) and \(a=b=0\). SOURCES: sakellaridis-v3.pdf, Statement 6.3.1 pp. 44–45, Theorem 7.2.1 and proof pp. 60–62, Lemma 4.2.3 p. 26; S4b.json node S4-3 (γ_i, e_i, ⟨γ̌_i,ρ_{P(X)}⟩ = 4); scripts we6.py, cprime.py --- A1-8 · Dead end · $B_{w_\gamma}$ for $M_\theta$ over $\mathbb F_q((t))$: two routes, missing lemmas STATUS CLAIMED: blocked PARENTS: A1-6, A1-7 STATEMENT: Over \(F=\mathbb F_q((t))\), \(p\) odd, \(X=M_\theta\), the following is **not** established here: \((\mathrm{S6.3.1})_p\): \(B_{w_\gamma}(\tilde\chi)=-e^{\check\gamma}\frac{1-q^{-4}e^{-\check\gamma}}{1-q^{-4}e^{\check\gamma}}\) for \(\gamma=\gamma_1,\gamma_6\). What is proved: (i) *Reduction to one generating function.* Put \(Z_F(\chi):=S_\chi(\mathbf 1_{X(O)})(1)\), a rational function by [[A1-5]]. Then \(b_w(\chi)=c'_w(\chi)Z_F(\chi)/Z_F(w\chi)\) with \(c'_w=\prod_{\check\alpha>0,w\check\alpha<0}\frac{1-q^{-1}e^{\check\alpha}}{1-q^{-1}e^{-\check\alpha}}\). So \((\mathrm{S6.3.1})_p\) holds if \(Z_F\) equals the rational function \(R_q\) given by the characteristic-0 formula. The constants \(c'_w\) are the same in both characteristics. (ii) *Level lemma.* Write \(\mu\in\Lambda_X\) as \((m_1,m_6)\) via \(\langle\mu,\eta_i\rangle\) for the two \(B\)-eigenfunctions \(f_1,f_6\) of the colours. The coefficient \(z_\mu\) of \(Z_F\) is \(q^{-N\dim X}\,\#\{\bar x\in X(O/\varpi^N):\operatorname{ord}f_i(\bar x)=m_i\}\) for any \(N>\max m_i\). It therefore depends only on the ring \(O/\varpi^N\). For a totally ramified \(E/W(\mathbb F_q)[1/p]\) of degree \(e\ge N\), \(O_E/\varpi_E^N\cong\mathbb F_q[t]/(t^N)\). The two remaining obligations are stated as Lemmas M1–M2 (route A) and M4 (route B) in the body. PROOF / BODY: **Proof of (i).** \(\mathbf 1_{X(O)}\) is \(K\)-invariant, so \(S_\chi(\mathbf 1_{X(O)})=Z_F(\chi)\varphi_{K,\chi}\). The rank-one computation of [[A1-6]] step 2 gives \(F_w\varphi_{K,\chi}=c'_w(\chi)\varphi_{K,w\chi}\). Apply \(F_wS_\chi=b_w(\chi)S_{w\chi}\) ([[A1-6]](c), \(w\in W_X'\)) to \(\mathbf 1_{X(O)}\). Then \(Z_F(\chi)c'_w(\chi)=b_w(\chi)Z_F(w\chi)\). Here \(Z_F\not\equiv0\), since on the region \(R\) it is an integral of a positive function for real \(\chi\). **Proof of (ii).** By [[A1-5]] step 2, for \(x\in X̊(F)\), \(\lambda(x)\) is determined by \((\operatorname{ord}f_1(x),\operatorname{ord}f_6(x))\), because \(\eta_1,\eta_6=\omega_1,\omega_6\) form a basis of \(\mathcal X(X)\) (S4-3). On \(\{\lambda(x)=\mu\}\) the integrand of (3.2) is constant times \(|\omega_X|\). \(X\) is smooth over \(O\) (S4-4/S4-11: \(\theta_0\) and the quotient are defined over \(\mathbb Z\), smooth after inverting 2). So the fibres of \(X(O)\to X(O/\varpi^N)\) are balls of volume \(q^{-N\dim X}\) for the residually non-vanishing invariant form. The condition \(\operatorname{ord}f_i=m_i