Proposition VI as a paper · hypotheses, statement and the structure of the proof (PDF) →

Structure of the proof

Steps 1–7 outline Sakellaridis’s proof of (a), naming the hypotheses each step uses; nothing there is re-proved. Steps 8–12 prove (b) by comparing \(\mathbb F_q(\!(t)\!)\) with the totally ramified fields \(F'_e\). Steps 13–18 prove (c), and step 19 is (d).

Part (a): Sakellaridis’s proof

1. Method Mackey morphisms and Hecke eigenfunctions

   The open \(B\)-orbit gives intertwining maps to principal series; applied to the \(K\)-fixed vector they produce the Hecke eigenfunctions (§3, §4.2):

   $$\begin{gathered}S_\chi:\ C_c^\infty(X)\to I(\chi)=\operatorname{Ind}_B^G(\chi\delta^{1/2}),\\ \Delta_\chi\ \text{its adjoint},\\ \Omega_{\tilde\chi}=\Delta_{\tilde\chi}(\Phi_K)\in C^\infty(X)^K .\end{gathered}$$
   How, and which hypotheses enter

   • Morphisms. \(S_\chi\) integrates against a \(B\)-eigenmeasure on the open orbit (Mackey theory, (3.2)); \(\Delta_\chi\) is its adjoint for the pairing with a \(G\)-eigenmeasure on \(X\) (3.1). Uses A2.
   • Normalization. The eigenmeasures are normalized compatibly on every orbit of maximal rank through their \(\mathcal O\)-structure (§3.3), which uses A3a; on \(\mathring X\) by \(\mu(x_0J)=1\), possible because \(x_0J\subseteq x_0B(\mathcal O)\) by A3c (p. 17).
   • Eigenfunctions. \(\Phi_K\in I(\chi)^K\) is the \(K\)-invariant vector of Proposition 4.2.1 (p. 26), and \(\Omega_{\tilde\chi}=\Delta_{\tilde\chi}(\Phi_K)\) is an eigenfunction of \(\mathcal H(G,K)\).
2. Result The sum formula on \(A_X^+\)

   Theorem 4.2.2 (p. 26), in the form (6.2) (p. 44): for almost every \(\tilde\chi\) and every \(x\in A_X^+\),

   $$\Omega_{\delta(X)^{1/2}\tilde\chi}(x)=\delta_{P(X)}^{-1/2}(x)\sum_{w\in W_X}B_w(\tilde\chi){}^{w}\tilde\chi(x).$$
   How, and which hypotheses enter

   • Cocycles. \(B_w\) is defined in (6.1) from the scalars of the functional equations \(F_w\Delta_{\tilde\chi}=b_w(\tilde\chi)\Delta_{w\tilde\chi}\) (4.3), \(F_w\) the equivariant Fourier transforms of §3.8, which are diagonal when no pair of maximal rank has type \(N\) (Remark 2 after Theorem 4.2.2).
   • The input from [Sa08]. The functional equations are [Sa08, Theorem 5.2.1], quoted as Theorem 3.7.1 (p. 21). [Sa08] proves them over every local non-archimedean field of characteristic \(0\) (§1.7, p. 7; §4.1, p. 17), so this step needs no unramified assumption (step 11).
   • Iwahori orbits, A3c. Lemma 3.9.1 (p. 24) evaluates the Iwahori vector at \(x\in A_X^+\): only the open orbit contributes, because \(xJ\subseteq xB(\mathcal O)\).
   • Cartan decomposition, A3b. \(A_X^+\) meets every \(K\)-orbit, so the values at \(A_X^+\) determine a \(K\)-invariant function.
3. Result Simple spherical reflections

   Statement 6.3.1 evaluates \(B_{w_\gamma}\) for each \(\gamma\in\Delta_X\); Theorem 6.3.2 (p. 45) proves it under A4 (i) and (ii). For type \(G\):

   $$B_{w_\gamma}(\tilde\chi)=-e^{\check\gamma}\,\frac{1-q^{-\langle\check\gamma,\rho_{P(X)}\rangle}e^{-\check\gamma}}{1-q^{-\langle\check\gamma,\rho_{P(X)}\rangle}e^{\check\gamma}}(\chi).$$
   How

   • Reduction. The labels of a Brion path for \(w_\gamma\) lie in \(\mathrm{supp}\,\gamma\), so \(X\) may be replaced by \(\mathring XP_\gamma/R(P_\gamma)\), of rank at most two (Lemma 6.2.2).
   • Cases. Those varieties are listed in Theorem 6.11.1 (pp. 51–52); the formula is computed for each of them whose group is classical (§§6.5–6.21), and glued along Brion paths (§6.4).
   • Hypotheses. No pair of maximal rank of type \(N\) and \(\delta(X)^{-1/2}\tilde\chi\) unramified (opening of §6), and \(L_\gamma\) classical: A4 (i), (ii).
4. Result The formula with \(L\)-values

   Theorem 7.2.1 (pp. 60–62): under Statements 6.3.1 and 7.1.5 the cocycle is a coboundary, and the eigenfunctions become Schur-type polynomials:

   $$\begin{gathered}B_w(\tilde\chi)=\frac{\beta(\tilde\chi)}{\beta({}^w\tilde\chi)},\\ \beta(\tilde\chi)=\frac{\prod_{\check\gamma\in\check\Phi_X^+}(1-e^{\check\gamma})}{\prod_{\check\theta\in\Theta^+}(1-\sigma_{\check\theta}q^{-r_{\check\theta}}e^{\check\theta})}(\tilde\chi).\end{gathered}$$
   How, and which hypotheses enter

   • Coboundary. By the cocycle relation it suffices to check \(w=w_\gamma\); there \(\beta(\tilde\chi)/\beta(w_\gamma\tilde\chi)\) is \(-e^{\check\gamma}\) times a product over \(\check\theta\gt0\) with \(w_\gamma\check\theta\lt0\), which Statements 7.1.5 and 6.3.1 identify with \(B_{w_\gamma}\) (p. 61).
   • Formula (7.4). \(\Omega_{\delta(X)^{1/2}\tilde\chi}(x_{\check\lambda})/\beta(\tilde\chi)=\delta_{P(X)}^{-1/2}(x_{\check\lambda})\prod_{\Theta^+}(1-\sigma_{\check\theta}q^{-r_{\check\theta}}T_{\check\theta})s_{\check\lambda}(\tilde\chi)\), with \(s_{\check\lambda}\) the Schur polynomial of lowest weight \(\check\lambda\) and \(T_{\check\theta}s_{\check\lambda}=s_{\check\theta+\check\lambda}\): a \(W_X\)-invariant polynomial.
   • Affine \(X\). Knop’s criterion (p. 61): \(X\) is affine iff \(\mathcal V\) and the colour valuations are separated by a hyperplane. Hence \(\mathcal T\cap\mathcal V=\{0\}\) (p. 62), and the value at \(x_0\) is \(c\,\beta\) with \(c=\beta(\delta_{P(X)}^{1/2})^{-1}\) (7.3). Uses A5.
   • A correction on p. 62. The proof ends with the inequality (7.5), \(\check\rho-w\check\rho+w|I|\succ_2 0\) for \(w\in W_X\) and \(I\subseteq\Theta^+\), \(|I|\) the sum of the \(\check\theta\in I\). Sakellaridis derives it from \(|\Theta^+|-w|\Theta^+|=\check\rho-w\check\rho\), which is false by a factor \(2\) when \(\Theta^+=\check\Phi_X^+\) with multiplicity one, since then \(|\Theta^+|=2\check\rho\). When the virtual colours of each simple spherical root sum to its coroot, the correct identity is \(|\Theta^+|-w|\Theta^+|=2(\check\rho-w\check\rho)\), and (7.5) holds with \(\mathcal T=\operatorname{cone}(\Theta^+)\) (S1-1). In characteristic \(0\) that condition is Lemma 7.1.4. For \(E_6/F_4\) an inversion-set argument gives the same (A1-7, audited A5-17). With a doubled coroot, which Lemma 7.1.4 excludes, the printed conclusion fails (A5-20). This is a remark on the proof, not a change of status.
5. Result Theorem 8.0.2

   Divided by the \(L\)-value, the eigenfunction values at the labels are polynomials; sending each orbit indicator to its polynomial is the isomorphism of (a) (proof on p. 64):

   $$\begin{gathered}P_{\check\lambda}(\chi\delta(X)^{1/2})=\frac{\Omega_{\chi\delta(X)^{1/2}}(x_{\check\lambda})}{L_X^{1/2}(\chi)},\\ \mathbf 1_{x_{\check\lambda}K}\longmapsto P_{\check\lambda},\qquad P_0=1 .\end{gathered}$$
   How, and which hypotheses enter

   • Polynomials. \(L_X^{1/2}=c\beta\) (Definition 7.2.3, p. 62), so \(P_{\check\lambda}\) is \(c^{-1}\) times the right side of (7.4), an element of \(\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\).
   • Equivariance. \(\mathbf 1_{x_{\check\lambda}K}\mapsto P_{\check\lambda}\) is \(\mathcal H(G,K)\)-equivariant for the Satake isomorphism.
   • Triangularity. By the Remark after Theorem 7.2.1 (p. 62) the other weights of \(P_{\check\lambda}\) are \(\succ_2\check\lambda\), where \(\check\lambda\succ_2\check\mu\) means \(\check\lambda-\check\mu\in\mathcal T\); the proof on p. 64 states that \(\check\lambda\) is the lowest weight of \(P_{\check\lambda}\). The Remark rests on (7.5), corrected in step 4. It places the weights in \(\check\lambda+\mathcal T\) but says nothing about the coefficient of \(e^{\check\lambda}\), which must be non-zero. The paper does not prove that; comparing coefficients of the open-orbit integrals, which have non-negative terms, gives it (S1-3, S1-4). Induction, started by \(\mathcal T\cap\mathcal V=\{0\}\), shows that the \(P_{\check\lambda}\) generate \(\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\).
   • Basis. By A5, \(X/K\simeq\Lambda_X^+\), so the indicators \(\mathbf 1_{x_{\check\lambda}K}\) form a basis of \(C_c^\infty(X)^K\).
   • Conclusion versus proof. The theorem asserts the module isomorphism with \(\mathbf 1_{X(\mathcal O)}\mapsto1\). The image of each orbit indicator, \(P_{\check\lambda}\) with lowest weight \(\check\lambda\), comes from its proof; an application that uses it must cite p. 64.
6. Conclusion (SG) holds iff \(r\) is onto

   Under the isomorphism of (a), the submodule generated by the basic function is the image of \(r\):

   $$\mathcal H(G,K)\cdot\mathbf 1_{X(\mathcal O)}\ \longleftrightarrow\ \operatorname{im}(r)\cdot1=\operatorname{im}r .$$
   Proof

   • The correspondence. \(f\in\mathcal H(G,K)\) acts on the target by multiplication by \(r(\mathcal Sf)\), \(\mathcal S\) is onto \(\mathbb C[A^*]^W\), and \(\mathbf 1_{X(\mathcal O)}\mapsto1\).
   • (⇒) If \(\mathcal H(G,K)\cdot\mathbf 1_{X(\mathcal O)}=C_c^\infty(X)^K\), then \(\operatorname{im}r\) is the whole target, so \(r\) is onto.
   • (⇐) If \(r\) is onto, the module is generated by \(\mathbf 1_{X(\mathcal O)}\): Corollary 8.0.4(1).
   • Parts (2), (3). Their proof “is straightforward and will be omitted” (p. 65).
7. Result Example 8.0.1: \(r\) need not be onto

   If two points of the translate are \(W\)-conjugate but not \(W_X\)-conjugate, every element of \(\operatorname{im}r\) agrees at them, while \(W_X\)-invariant functions separate them; this happens in Example 8.0.1 (p. 64).

   Proof

   Proof and remarks

   • Separation. \(W_X\) is finite, so \(\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) separates \(W_X\)-orbits; a \(W\)-invariant function takes equal values at \(W\)-conjugate points.
   • The example. The normalized spherical roots are \(\gamma_1=\alpha_1+\alpha_2\), \(\gamma_2=\alpha_1+\alpha_1'\), \(\gamma_3=\alpha_1'+\alpha_2'\); the glued points lie in the diagonal \(\mathrm{SO}_4\) of the dual group \(\mathrm{SO}_5\times\mathrm{SO}_5\).
   • Consequences in the paper. The Remark on p. 65 corrects [Sa08, Theorem 6.3.2]; the remark after Corollary 8.0.4 (pp. 65–66) notes that at glued points the fibre multiplicity exceeds one although the generic multiplicity is one.
   • Hypotheses. The paper does not check A1–A5 for this \(X\); “(SG) fails” there is conditional on them.

Part (b): the transfer to \(\mathbb F_q(\!(t)\!)\)

1. Method Close fields and finite determinacy

   \(F\) and \(F'_e\) are \(e\)-close. Membership in Cartan and Iwasawa cells is read from valuations of matrix coefficients in the fundamental Weyl modules \(V_i\) (highest weight \(\omega_i\), lowest-weight coordinate \(\ell_i\)), so every count used below is decided modulo \(\varpi^D\), with \(D\) independent of the field (\(\nu\in\Lambda_T^+\), respectively \(\nu\in\Lambda_T\)):

   $$\begin{gathered}g\in Kt^\nu K\iff\\ \operatorname{val}\rho_i(g)=\langle\nu,w_0\omega_i\rangle\ \ (\forall i),\\ g\in U(E)\,t^\nu K\iff\\ \operatorname{val}\ell_i\rho_i(g)=\langle\nu,w_0\omega_i\rangle\ \ (\forall i),\end{gathered}$$

   Status: proved · this run, audited (A4-2; audit A5-1).

   How

   • Truncated rings. \(\mathcal O'_e/\varpi'^m\cong\mathbb F_q[t]/(t^m)\) for \(m\le e\), with \(\varpi'\mapsto t\) (definitions).
   • Smooth schemes. For a smooth affine \(\Lambda\)-scheme \(Y\) (here \(G\), \(H\), \(X\)), \(Y(\mathcal O_E)\to Y(\mathcal O_E/\varpi^m)\) is onto by Hensel, and \(Y(\mathcal O_E/\varpi^m)\) depends only on the \(\Lambda\)-algebra \(\mathcal O_E/\varpi^m\); in particular \(K/K(m)=G(\mathcal O_E/\varpi^m)\).
   • Detection. The Weyl modules are free over \(\Lambda\) with a basis of \(T\)-weight vectors, so the Cartan and Iwasawa double cosets of \(g\in G(E)\) are determined by the displayed valuations; if \(g\in Kt^aK\), \(g'\in Kt^bK\) and \(gg'\in Kt^cK\), then \(c\preceq a+b\), and \(\{c\in\Lambda_T^+:c\preceq d\}\) is finite.
   • Finite determinacy. A finite Boolean combination \(\mathcal B\) of conditions \(\operatorname{val}M(z)\ge c\), \(\operatorname{val}M(z)=c\) on \(z\in\prod_jY_j(\mathcal O_E)\), \(Y_j\in\{G,H\}\), whose entries are polynomials over \(\Lambda\) in \(\varpi^{\pm1}\) and the matrix coefficients of \(\rho_i(z_j^{\pm1})\), \(\rho_i(\theta(z_j)^{\pm1})\), depends only on \(z\bmod\varpi^D\), \(D=D(\mathcal B)\) independent of \(E\). Over \(D\)-close fields the solution sets correspond, so counts, existence statements and equivalence relations defined this way agree.
   • Audit. A5-1 re-derives every part; \(D\) depends only on the formal shape of the conditions, never on the test field.
2. Result The Cartan decomposition over \(\mathbb F_q(\!(t)\!)\)

   Under T1–T4, counting \(K\)-orbits in each Cartan shell transfers the decomposition from \(F'_e\) to \(F\); the first two lines are equivalent:

   $$\begin{gathered}X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa,\\ G(F)=K\,T(F)\,H(F),\\ K\backslash X(F)\simeq\Lambda_S^+ .\end{gathered}$$

   Status: proved · this run, audited (A4-5; audit A5-3). For \(E_6/F_4\) also proved directly from Delorme–Sécherre in characteristic \(p\) (A3-4, A3-5, A3-8; audit A5-9).

   How, and which hypotheses enter

   • Shells. For a test field \(E\) and \(\lambda\in\Lambda_T^+\) put \(Z_\lambda(E)=Kt^\lambda X(\mathcal O_E)\). By T2 and the Cartan decomposition of \(G(E)\), \(X(E)=\bigcup_\lambda Z_\lambda(E)\).
   • Distinct labels. \(\tau(kx)=k\,\tau(x)\,\theta(k)^{-1}\) and \(\theta(K)=K\), so \(K\tau(x)K\) depends only on \(Kx\); as \(\tau(x_\kappa)=t^\kappa\), distinct labels give distinct orbits.
   • Counts. \(N_\lambda(E)=\#K\backslash Z_\lambda(E)\) and \(M_\lambda(E)=\{\kappa:\ x_\kappa\in Z_\lambda(E)\}\) are finite and computed in \(G(\mathcal O/\varpi^D)\) and \(H(\mathcal O/\varpi^D)\), with the Cartan decompositions of \(G\) and of \(H\) (T1) and T3, for \(D\ge D(\lambda)\) independent of \(E\) (step 8).
   • Transfer. Over \(F'_e\), \(e\ge D(\lambda)\), every \(K\)-orbit in \(Z_\lambda\) contains some \(x_\kappa\) by T4. The counts agree over \(F\), so the same holds there.
   • Audit. A5-3: sound, with a uniform \(D(\lambda)\); identifying the labels with \(\Lambda_X^+\), as A3b requires, also needs the valuation cone in characteristic \(p\) (A3-7(3)); the parametrization by \(\Lambda_S^+\) stands as proved.
   • Second proof for \(E_6/F_4\). Delorme–Sécherre’s decomposition \(G(F)=H(F)\,T(F)\,K\) is proved for every local field of residue characteristic \(\ne2\) (A3-5); the maximal \((\theta,F)\)-split tori form one \(H(F)\)-class by a direct argument (A3-4); with \(W_X\) realized in \(K\) this gives A3b over \(\mathbb F_q(\!(t)\!)\) (A3-8).
3. Result Structure constants and Satake coefficients agree over close fields

   For \(\lambda\in\Lambda_T^+\) and \(\mu,\nu\in\Lambda_S^+\), the structure constants of \(C_K\) and the Satake coefficients of \(T_\lambda\) are counts of the kind in step 8; with the Cartan decomposition over both fields (step 9 and T4) they agree over \(F\) and \(F'_e\) for \(e\ge D(\lambda,\mu)\):

   $$\begin{gathered}c^\nu_{\lambda\mu}(E):=(T_\lambda*\mathbf 1_{Kx_\mu})(x_\nu)\\ =\#\{gK\subseteq Kt^\lambda K:\ g^{-1}x_\nu\in Kx_\mu\},\\ \mathcal S(T_\lambda)=\sum_{\nu\in\Lambda_T}q^{-\langle\rho,\nu\rangle}\,n_{\lambda\nu}(E)\,e^\nu,\\ n_{\lambda\nu}(E)=\#\big((U(E)t^\nu K\cap Kt^\lambda K)/K\big).\end{gathered}$$

   Status: proved · this run, audited (A4-6; audit A5-4, which supplies the bound on \(\nu\)).

   How

   • Finite support. \(T_\lambda*\mathbf 1_{Kx_\mu}=\sum_{\nu\in N(\lambda,\mu)}c^\nu_{\lambda\mu}(E)\,\mathbf 1_{Kx_\nu}\), where \(N(\lambda,\mu)=\{\nu\in\Lambda_S^+:\ \nu\preceq\lambda+\mu+(-\theta\lambda)^+\}\) is finite and the same for every test field.
   • Each constant. By step 9 the \(K\)-orbit of a point is read from the Cartan double coset of its \(\tau\)-image (T3), so \(c^\nu_{\lambda\mu}\) counts the \(\bar k\in G(\mathcal O/\varpi^D)\) for which \(t^{-\lambda}k^{-1}t^\nu\theta(k)t^{\theta\lambda}\) has prescribed Weyl-module valuations: a finitely determined count (step 8).
   • Satake coefficients. \(n_{\lambda\nu}\) is an Iwasawa count of the same kind; no decomposition of \(X\) is needed. So \(\mathcal S(T_\lambda)\in\mathbb C[\Lambda_T]^W\) is the same element for \(F\) and \(F'_e\).
   • Audit. A5-4: the uniform bound needs a field-independent bound on the \(\nu\) with \(n_{\lambda\nu}\ne0\), which A4-6 asserted without proof; a primitive-row argument gives \(\langle\lambda,w_0\omega_i\rangle\le\langle\nu,w_0\omega_i\rangle\le\langle\lambda,\omega_i\rangle\).
4. Result Theorem 8.0.2 over ramified \(p\)-adic fields

   Let \(F'\supseteq\mathbb Q_q\) have residue field \(\mathbb F_q\), of any ramification. With \(\psi'\) of conductor \(\mathcal O'\) and \(dx(\mathcal O')=1\), every formula of the proof of (a) holds as printed, so Theorem 8.0.2 holds over \(F'\) under A1–A5, and the image of each orbit indicator is the polynomial of steps 4 and 5, which depends only on \(q\):

   $$\begin{gathered}\mathbf 1_{Kx_\kappa}\longmapsto P_\kappa=c^{-1}\,\delta_{P(X)}^{-1/2}(x_\kappa)\\ \cdot\prod_{\check\theta\in\Theta^+}\big(1-\sigma_{\check\theta}q^{-r_{\check\theta}}T_{\check\theta}\big)\,s_{\check\lambda(\kappa)},\\ c=\beta(\delta_{P(X)}^{1/2})^{-1}.\end{gathered}$$

   Status: proved · this run, audited (A4-7; audit A5-5); the inputs from [Sa08] checked by the root against [Sa08], §1.7 and §4.1.

   How, and which hypotheses enter

   • Where “unramified” enters. §1.1 states the simplification (p. 2). It is used for the measure in the Fourier transform of §3.8, “the correct choice only under our simplifying assumption” (p. 23), and in Tate’s formulas of §5.3, “we are using our simplifying assumption” (p. 32). In both places it only makes \(\psi\) of conductor \(\mathcal O\) compatible with \(dx(\mathcal O)=1\), and §1.7 already fixes \(\psi\) with conductor \(\mathcal O\).
   • Ramified fields. \(\psi'(x)=\psi_{\mathbb Q_p}(\operatorname{Tr}(cx))\) with \(c\mathcal O'=\mathfrak d_{F'/\mathbb Q_p}^{-1}\) has conductor \(\mathcal O'\), and then \(dx(\mathcal O')=1\) is the self-dual measure: (3.10), (4.1), Lemma 3.9.1, (4.5), Tate’s formulas, (5.2) and (5.3) hold over \(F'\) verbatim (A4-7(a)).
   • Inputs from [Sa08]. The Mackey morphisms and their functional equations, [Sa08, Theorem 5.2.1] (quoted as Theorem 3.7.1, p. 21), are proved in [Sa08] for every local non-archimedean field of characteristic \(0\): “We will be working throughout over fields of characteristic zero” (§1.7, p. 7), “\(k\) will always denote a locally compact non-archimedean local field in characteristic zero” (§4.1, p. 17). This was the one chain A5-5 left unchecked.
   • Independence of \(e\). \(P_\kappa\) is given by (7.4) and (8.1) through \(q\), \(\Theta^+\), \(\delta_{P(X)}\) and Sakellaridis’s label \(\check\lambda(\kappa)\), none of which depends on \(F'\); the statement of Theorem 8.0.2 involves no choice beyond \(\operatorname{vol}K=1\) (A4-7(c), (d)).
   • For \(\mathcal M_\theta\). A1–A5 hold over every \(\mathcal O'\) (steps 13–17, A3a included), so (a) gives Theorem 8.0.2 over every \(p\)-adic \(F\), and T5 holds.
5. Conclusion The characteristic-\(p\) theorem: (b)

   Under T1–T5, over \(F=\mathbb F_q(\!(t)\!)\), the map \(\Psi\) defined by the images over \(F'_e\) is the isomorphism of (b); for \(\mathcal M_\theta\) it is the only one:

   $$\begin{gathered}\Psi:\ C_K\to\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X},\\ \Psi(\mathbf 1_{Kx_\kappa})=P_\kappa,\qquad\Psi(\mathbf 1_{X(\mathcal O)})=1,\\ \Psi(f*\eta)=r(\mathcal Sf)\cdot\Psi(\eta)\\ (f\in\mathcal H_K,\ \eta\in C_K).\end{gathered}$$

   Status: proved · this run, audited (A4-8, A4-9; audits A5-6, A5-7). A4-8 was conditional on A3a over \(\Lambda\) and on the [Sa08] check; both now hold (steps 17 and 11).

   How, and which hypotheses enter

   • Bases. By step 9 and T4 the indicators \(\mathbf 1_{Kx_\kappa}\), \(\kappa\in\Lambda_S^+\), form a basis of \(C_K\) over \(F\) and over \(F'_e\); over \(F'_e\) their images \(P_\kappa\) form a basis of the target (T5, step 5). So \(\Psi\) is a linear bijection.
   • Equivariance, one pair at a time. Given \(\lambda\), \(\mu\), take \(e\ge D(\lambda,\mu)\). By step 10 and Theorem 8.0.2 over \(F'_e\), $$\begin{aligned}\Psi(T_\lambda*\mathbf 1_{Kx_\mu})&=\sum_\nu c^\nu_{\lambda\mu}(F)\,P_\nu\\ &=\sum_\nu c^\nu_{\lambda\mu}(F'_e)\,P_\nu\\ &=r(\mathcal ST_\lambda)\,P_\mu,\end{aligned}$$ with \(\mathcal S(T_\lambda)\) the same over both fields. The \(T_\lambda\) span \(\mathcal H_K\).
   • The basic function. \(X(\mathcal O)=Kx_0\) by T2 and T3, and \(P_0=1\).
   • Uniqueness for \(\mathcal M_\theta\). An \(\mathcal H_K\)-linear self-map \(h\) of the target with \(h(1)=1\) fixes \(\operatorname{im}r\). As \(\operatorname{im}r\) has codimension one and \(\operatorname{im}r\) acts on the quotient by a character (Proposition VII), for any \(s\) there is \(a\in\operatorname{im}r\smallsetminus0\) with \(as\in\operatorname{im}r\), so \(a\,h(s)=as\) and \(h(s)=s\) in the domain \(\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\).
   • The target. \(A_X^*\), \(W_X\) and \(\delta(X)\) are read over \(F'_e\), where they depend only on \((G,\theta,T)\); identifying them with the invariants of \(X_F\) uses structure theory over \(\bar F\) (A5-6), done for \(E_6/F_4\) in A3-7 and A3-10.
   • Audits. A5-6: the module transfer, uniqueness and the consequences for (SG) are correct. A5-7: A4-9 is a correct conditional theorem whose hypotheses T4 and T5 live at \(p\)-adic places.
   • A second route holds for \(E_6/F_4\) · audited (A5-21). A1 reruns Sakellaridis’s proof directly over \(\mathbb F_q(\!(t)\!)\). Its open-orbit analytic half holds with characteristic-\(\ne2\) inputs only: convergence, generic uniqueness and rationality of \(S_\chi\) (A1-5), the functional equations and the sum formula (A1-6), the formal last step (A1-2) and the algebra of \(\mathcal M_\theta\) (A1-7); audits A5-14, A5-15, A5-16, A5-17. Its one remaining input, Statement 6.3.1 in characteristic \(p\), follows from A3a through Theorem 8.0.2 at the ramified fields \(F'_e\) (A1-8, route B; audit A5-19). So for \(E_6/F_4\) both routes rest on A3a, which step 17 proves. The general form of this route is the remark under (b).

Part (c): the hypotheses for \(E_6/F_4\)

1. Result \(E_6/F_4\): spherical roots of type \(G\), two colours; A4 holds in every characteristic \(\ne2\)

   In every characteristic \(\ne2\), with statements over \(\bar F\) for \(F\) of characteristic \(0\) or \(p\), the two normalized spherical roots are sums of orthogonal roots, of type \(G\), with Levi subgroups of type \(D_5\):

   $$\begin{gathered}\gamma_1=2\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5\\ =\alpha_1+w_L\alpha_1=2\omega_1-\omega_6,\\ \gamma_6=\alpha_2+\alpha_3+2\alpha_4+2\alpha_5+2\alpha_6\\ =-\omega_1+2\omega_6,\end{gathered}$$ $$\begin{gathered}B_{w_\gamma}(\tilde\chi)=-e^{\check\gamma}\,\frac{1-q^{-4}e^{-\check\gamma}}{1-q^{-4}e^{\check\gamma}},\\ \beta(\tilde\chi)=\prod_{\check\gamma\in\check\Phi_X^+}\frac{1-e^{\check\gamma}}{1-q^{-4}e^{\check\gamma}} .\end{gathered}$$
   Proof

   Proof (S4-3; A3-7, A3-10)

   • Roots. \(\theta=-w_L\) and \(w_L\alpha_1=\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5\), the highest root of the \(D_5\) on nodes \(1,\dots,5\); so \(\gamma_1=\alpha_1-\theta\alpha_1=\alpha_1+w_L\alpha_1\) with \((\alpha_1,w_L\alpha_1)=2-2=0\). \(\mathcal X(X)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\), dual to \(\Lambda_X=\mathbb Ze_1\oplus\mathbb Ze_6\).
   • Type. \(\gamma_1\) and \(\gamma_6\) are primitive in \(\mathcal X(X)\) and span the extremal rays (for symmetric \(X\), \(\mathcal V\) is the image of the antidominant chamber, p. 8). \(W(E_6)\) is transitive on ordered pairs of orthogonal roots, so \((\alpha_1,w_L\alpha_1)\) is simple for some basis and \(\gamma_1\) is normalized; no root is proportional to it, so its type is \(G\). Hence no spherical root of type \(N\), and by S4-5, item 3, no pair \((Y,\alpha)\) of maximal rank has type \(T\) or \(N\), which is the opening assumption of §6 (S4-10; Knop’s transport of weight lattices, cited): A4 (i). In every characteristic \(\ne2\), A3-6 proves this directly (audited A5-10), and step 17 proves more.
   • Levi subgroups. \(\mathrm{supp}\,\gamma_1=\{1,\dots,5\}\), \(\mathrm{supp}\,\gamma_6=\{2,\dots,6\}\): type \(D_5\), classical. A4 (ii).
   • Colours. \(\bar F[X]\) is factorial with \(B\)-eigenweights \(a\omega_1+b\omega_6\), \(a,b\ge0\), so there are exactly two colours (in characteristic \(\ne2\) the weights \(\omega_1,\omega_6\) are forced by the diagram automorphism commuting with \(\theta_0\): A3-7 with the repair of A5-11), with \(\check v_{D_i}=e_i=\check\gamma_i\) (\(\langle e_i,\gamma_j\rangle\) is the Cartan matrix of \(A_2\)), and \(\langle\check\gamma_i,\rho_{P(X)}\rangle=\langle\alpha_i^\vee,\rho-\rho_L\rangle=1-(-3)=4\).
   • \(\Theta\) and Statement 7.1.5. Both colours are of type \(G\), so \(\sigma=+\); the \(S_3\)-orbit of \(\check\gamma_1\) is the six coroots, so \(\Theta=\check\Phi_X\times\{(+,4)\}\) and \(\Theta^+=\check\Phi_X^+\). Each coroot is \(\gt0\) or \(\lt0\); \(w_{\gamma_1}\) sends only \(\check\gamma_1\) to a negative coroot, and by Lemma 7.1.2 (p. 58) only \(D_1\) belongs to \(\gamma_1\) since \(\langle e_6,\gamma_1\rangle=-1\). A4 (iv).
   • Statement 6.3.1. By Theorem 6.3.2. Cross-check: \(\mathring XP_{\gamma_1}/R(P_{\gamma_1})\) has rank one with spherical root of full support, so it is \(\mathrm{Spin}_9\backslash\mathrm{Spin}_{10}\) of Theorem 6.11.1, and (6.18) in §6.19 (p. 56) with \(n=5\) gives \(q^{-n+1}=q^{-4}\), as (6.3) with \(\langle\check\gamma,\rho_{P(X)}\rangle=4\) requires. A4 (iii), in characteristic \(0\). Over \(\mathbb F_q(\!(t)\!)\) the Brion-path combinatorics hold (A3-10, audited A5-12) and the analytic part is open (A3-11); steps 8–12 do not use it.
2. Result \(E_6/F_4\), \(p\) odd: the model \(\mathcal M_\theta\) satisfies A1 or A1ₚ, A2, A5

   The involution \(\theta_0\) is defined over \(\mathbb Z\). Over every complete DVR \(\mathcal O\) with \(2\in\mathcal O^\times\), \(p\)-adic or \(\mathbb F_q[\![t]\!]\), \(\mathcal M_\theta\) satisfies A1 (respectively A1ₚ), A2 and A5, and the integral points form one \(K\)-orbit:

   $$\begin{gathered}\mathring X=B_{\mathcal O}/(B\cap H)\simeq U_P\times A_X,\\ B\cap H=B_L\cap L^\theta\subseteq L^\theta=\mathrm{Spin}_8,\\ X(\mathcal O)=Kx_0,\qquad\varphi_0=\mathbf 1_{Kx_0}.\end{gathered}$$
   Proof

   Proof (S4-11, S4-4; A3-2, A3-3) and the cited inputs

   • The involution over \(\mathbb Z\). \(C\) fixes every \(\sigma_{\alpha_i}\), hence \(\sigma_L\), so \(\theta_0^2=\operatorname{Int}(\sigma_L^2)\); an exact computation in \(\mathfrak{so}_8\) shows that \(\theta_0\) is the identity on \(\mathrm{Spin}_8\), whence \(\sigma_L^2\), central in \(\mathrm{Spin}_8\), acts trivially in the three eight-dimensional representations and \(\sigma_L^2=1\). \(\theta_0\) acts on \(\mathcal X(T)\) by \(-w_L\notin W(E_6)\), so it is outer, and \(\dim\mathfrak g^{\theta_0}=4+24+24=52\): type E IV.
   • A1, A1ₚ. \(G_{\mathcal O}\) is split reductive with connected fibres and simply connected derived group over any \(\mathcal O\); \(F\) has characteristic \(0\), or is \(\mathbb F_q(\!(t)\!)\).
   • \(H_{\mathcal O}\) and \(X_{\mathcal O}\). \(2\in\mathcal O^\times\), so \(H_{\mathcal O}=G_{\mathcal O}^{\theta_0}\) is smooth; its fibres are connected (simply connected \(G\), semisimple automorphism) and reductive of type \(F_4\). \(X_{\mathcal O}\) is smooth of relative dimension \(78-52=26\), separated, and quasi-affine.
   • Open orbit. For \(g=ul\in B\cap H\), \(u\in U_P\), \(l\in B_L\): \(\theta(u)=u\,l\,\theta(l)^{-1}\in U_{P^-}\cap P=1\), so \(B\cap H=B_L\cap L^\theta\), a Borel subgroup scheme of \(\mathrm{Spin}_8\) of relative dimension 16. The orbit map \(B_{\mathcal O}/(B\cap H)\to X_{\mathcal O}\) is an open immersion on both fibres (\(42-16=26\)), hence an open immersion (EGA IV 17.9.1), with affine image \(U\simeq U_P\times A_X\). The complement of an affine open in the regular \(X_{\mathcal O}\) has pure codimension one (EGA IV 21.12.7), which forces \(\mathring X=U\).
   • Measures. \(G\) and \(H\) are unimodular; on \(B\cap H\), \(\delta_B=\delta_{B\cap H}\) since \(\langle2\rho-2\rho_L,\alpha_j^\vee\rangle=0\) for \(j=2,\dots,5\).
   • Integral points. \(H^1(\mathcal O,H_{\mathcal O})=H^1(\mathbb F_q,H_{\mathbb F_q})=1\) (Hensel for the smooth \(H_{\mathcal O}\), Lang for the connected special fibre), so every \(x\in X(\mathcal O)\) lifts to \(G(\mathcal O)\).
   • A5. \(X_F\) is affine since \(F_4\) is reductive; \(B\cap H\) is connected solvable with split torus, so \(H^1(F,B\cap H)=1\) and \(\mathring X(F)\) is one \(B(F)\)-orbit.
   • Characteristic \(p\). A3-2 and A3-3 (audited A5-8) prove all of this over \(\mathbb F_q[\![t]\!]\) using only \(2\in\mathcal O^\times\): quasi-affineness from Conrad’s Theorem 2.3.1 in place of Alper, and \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\) from Bruhat–Tits III, Theorem 4.7, in place of Kneser.
   • Cited, not re-proved. The Chevalley involution over \(\mathbb Z\) (SGA 3, Exp. XXIII); smoothness of fixed points of a finite group of invertible order (Edixhoven 1992, Prop. 3.4); connectedness of \(G^\theta\) for simply connected \(G\) (Steinberg 1968, Thm. 8.1); quasi-affineness of \(G_{\mathcal O}/H_{\mathcal O}\) (Alper 2014, result number not checked); EGA IV 17.9.1 and 21.12.7. The ramified \(p\)-adic case of A1 is step 11.
3. Result \(E_6/F_4\), \(p\) odd: A3c, distinct labels, the types of the rank-one quotients

   With left actions, labels \(\kappa\in\Lambda_S^+\) and points \(x_\kappa\in T(F)x_0\) with \(\tau(x_\kappa)\in\varpi^\kappa S(\mathcal O)\), \(\tau(gx_0)=g\theta(g)^{-1}\):

   $$\begin{gathered}Kx_\kappa=Kx_{\kappa'}\iff\kappa=\kappa',\\ Jx_\kappa\subseteq B(\mathcal O)x_\kappa\ \ (\kappa\in\Lambda_S^+),\\ \text{every }(Y,\alpha)\text{ of maximal rank}\\ \text{has type }G\text{ or }U .\end{gathered}$$
   Proof

   Proof (S4-5; A3-6, A3-9)

   • Distinct labels. \(\tau\) is injective and \(\tau(kx)=k\tau(x)\theta(k)^{-1}\) with \(\theta(K)=K\), so \(K\tau(x)K\) depends only on \(Kx\); \(\tau(\varpi^\mu x_0)=\varpi^{\varphi(\mu)}\) and the Cartan decomposition of \(G(F)\) separate the labels. Points of \(T(F)x_0\) with the same label differ by \(\operatorname{im}T(\mathcal O)\subseteq K\).
   • A3c, labels with a dominant lift. Iwahori factorization \(J=U_P(\mathcal O)(J\cap L)U_P^-(\mathfrak p)\); conjugation by \(\varpi^\mu\) with \(\mu\) dominant shrinks \(U_P^-(\mathfrak p)\); the open cell \(U_P\times A_X\) over \(\mathcal O\) absorbs \(U_P^-(\mathfrak p)x_0\) into \(U_P(\mathfrak p)T(\mathcal O)x_0\); and \(L\) acts on \(L/L^\theta\) through the torus \(L/[L,L]\).
   • A3c, the uncovered labels \(3k\omega_1^\vee\), \(3k\omega_6^\vee\), \(k\) odd. No single cocharacter works there. The wall parabolic \(Q=MU_Q\) with Levi \(M\) of type \(D_5\) on nodes \(2,\dots,6\), the representative \(\mu_1=2\alpha_1^\vee+\alpha_3^\vee+\alpha_4^\vee+\alpha_5^\vee+\alpha_6^\vee\), and a rational central half-translation \(R_z\) (\(z^2=\zeta(\varpi)\), \(\zeta=3\omega_1^\vee\)) reduce the Levi part to the label \(0\) inside \(M/M^\theta\), and the bound \(\langle\epsilon,\mu_k\rangle\ge-1\) on the roots \(\epsilon\) of \(U_{P\cap M}\) keeps the unipotent part integral.
   • Types. There are \(|W|/(|W_X|\,|W_{P(X)}|)=51840/(6\cdot192)=45\) Borel orbits of maximal rank. A pair of type \(T\) or \(N\) would put a root in \(w(\mathbb Q\omega_1\oplus\mathbb Q\omega_6)\) (Knop 1995, transport of weight lattices; cited, not checked), and no root lies there: \(\tfrac43(a^2+ab+b^2)\ne2\) for integers. In every characteristic \(\ne2\), A3-6 replaces Knop’s result by a direct argument (audited A5-10, which supplies the converse \(Y_w=Y_{w'}\Rightarrow w^{-1}w'\in W(X)\)).
   • Characteristic \(p\). A3-9 re-checks each step over \(\mathbb F_q[\![t]\!]\): Iwahori factorizations, the open cells \(U_P\times A_X\) and \(U_Q\times M/M^\theta\), and \(R_z\), defined over \(F(\sqrt t)\), separable since \(p\) is odd (audited A5-12).
4. Result \(E_6/F_4\), \(p\) odd: A3b holds in full, in both characteristics

   The existence half of A3b is a Cartan decomposition for \(G/H\); it follows from Delorme–Sécherre’s theorem, whose index set has one element here. With step 15:

   $$\begin{gathered}G(F)=K\,T(F)\,H(F),\\ X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa,\qquad K\backslash X(F)\simeq\Lambda_S^+ .\end{gathered}$$

   Status: proved for \(p\) odd, in both characteristics: over \(\mathbb F_q(\!(t)\!)\) by A3-4, A3-5, A3-8 (audited A5-9) and by the transfer of step 9; over \(p\)-adic \(F\) from Delorme–Sécherre, Pacific J. Math. 251 (2011), Theorem 1.1 (arXiv:math/0612545, Theorem 0.4 = 4.8), with one conjugacy class of maximal \(\theta\)-split tori (Kneser, \(H^1(F,\mathrm{Spin}_8)=1\)) and \(W_X\) realized in \(H(\mathcal O)\cap N_G(T)\).

   Proof

   Proof (S4-12, S4-13) and the cited inputs

   • Delorme–Sécherre. For \(F\) of odd residue characteristic, \(G\) split, \(\sigma\) an involution, \(H\) open in \(G^\sigma\), \(A\) a maximal \((\sigma,F)\)-split torus inside a \(\sigma\)-stable maximal split torus, \(\{A_j\}_{j\in J}\) the \(H(F)\)-classes of maximal \((\sigma,F)\)-split tori with \(y_jAy_j^{-1}=A_j\), and \(K\) the stabilizer of a special point in the apartment: \(G(F)=\bigcup_jH(F)\,y_j\,T(F)\,K\). Here \(A=S\), the torus \(T\), and \(K=G(\mathcal O)\) the stabilizer of the hyperspecial vertex (Bruhat–Tits).
   • One class, \(|J|=1\). A maximal \((\theta,F)\)-split torus \(A'\) is \(G(F)\)-conjugate to \(S\) (Helminck–Wang, Prop. 10.3), so \(A'=hSh^{-1}\) with \(h\in H(\bar F)\) (Vust; Richardson). Then \(\tau\mapsto h^{-1}\tau(h)\) is a cocycle in \(Z_H(S)=L^\theta=\mathrm{Spin}_8\), and \(H^1(F,\mathrm{Spin}_8)=1\) (Kneser) lets one choose \(h\in H(F)\). In characteristic \(p\), A3-4 proves the single class directly and Bruhat–Tits III gives \(H^1=1\); Delorme–Sécherre’s proof uses no characteristic-\(0\) input (A3-5).
   • From \(G(F)\) to \(X(F)\). Inverting gives \(G(F)=K\,T(F)\,H(F)\); and \(X(F)=G(F)x_0\) because \(H^1(F,F_4)=1\). So every \(K\)-orbit meets \(T(F)x_0\).
   • Into \(A_X^+\). \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H(\mathcal O)\cap N_G(T)\) (\(i=1,6\)) lifts \(s_{\alpha_i}s_{w_L\alpha_i}\), which acts on \(\Lambda_S\) as the simple reflection \(s_{\gamma_i}\) of \(W_X\); so \(\dot n_i\varpi^\mu x_0=\varpi^{s_{\alpha_i}s_{w_L\alpha_i}\mu}x_0\) moves any label to a dominant one inside its \(K\)-orbit. With distinct labels (step 15), A3b holds.
   • Cited, not re-proved. Delorme–Sécherre, Theorem 1.1, with Helminck–Wang, Prop. 10.3, inside it; conjugacy of maximal \(\theta\)-split tori over \(\bar F\) (Vust; Richardson); Kneser’s \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\); Bruhat–Tits for \(K=\operatorname{Stab}(x)\).
5. Result \(E_6/F_4\): A3a holds over every \(\mathcal O\) with \(2\) invertible

   For \(\mathcal M_\theta\) and every DVR \(\mathcal O\) with \(2\in\mathcal O^\times\), mixed or equal characteristic, \(p=3\) included: the \(B_F\)-orbits are the \(45\) orbits \(B\dot wx_0\), all of maximal rank; Sakellaridis’s \(\mathcal O\)-scheme of each is the smooth orbit scheme \(Y_x\simeq B/B_x\); and, \(P_\alpha\)-equivariantly,

   $$\begin{gathered}B_x=(B\cap\theta_xB)^{\theta_x},\\ \alpha\in\Phi_x\ (90\ \text{pairs}):\\ Y_xP_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2\backslash\mathrm{PGL}_2,\\ \alpha\notin\Phi_x\ (180\ \text{pairs}):\\ Y_xP_\alpha/R(P_\alpha)\simeq B_2\backslash\mathrm{PGL}_2=\mathbb P^1_{\mathcal O},\end{gathered}$$

   with the complement clause for all \(270\) pairs. Here \(x=ww_Lw^{-1}\) runs over the \(45\) conjugates of \(w_L\) in \(W\) (\(|C_W(w_L)|=1152\)), \(\theta_x\) is the conjugate involution and \(\Phi_x\) the \(D_4\) of its fixed roots. So A3a holds.

   Status: proved · this run, audited (A2-8; audit A6-9, with the repairs A6-3, A6-5, A6-6; the \(45\times6\) table recomputed independently in A6-1).

   How (A2-3 to A2-7), in five lines

   • \(\theta\)-stable tori. Over an algebraically closed \(k\), \(\operatorname{char}k\ne2\), every \(\theta\)-stable maximal torus \(T'\) is \(H(k)\)-conjugate to \(T\). With \(a=\dim T'_+\) (the \(\theta\)-fixed part), \(r+c\) the number of real and complex roots and \(i_c\), \(i_n\) those of imaginary roots on which \(\theta=\pm1\), the eigenspace dimensions give $$\begin{gathered}52=a+\tfrac{r+c}2+i_c,\\ 26=(6-a)+\tfrac{r+c}2+i_n,\end{gathered}$$ so \(i_c-i_n=32-2a\); for \(a\le3\) the imaginary roots form a simply laced system of rank \(\le3\) with \(i_c\le12\), impossible, so \(a=4\) and \(T'_+\) is \(H\)-conjugate to \(T^\theta\) (A2-3).
   • The \(45\) orbits. For \(z=gx_0\), the Borel subgroups \(g^{-1}Bg\) and its \(\theta\)-image share a maximal torus (Conrad, Theorem 1.4.12, with the line supplied in A6-3) that \(\theta\) stabilizes, so every orbit meets \(N_G(T)x_0\); the Bruhat cell of \(\tau(z)\) separates the \(45\) orbits \(By_x\), all of rank \(2\). Their stabilizers \(B_x=T^{\theta_x}\ltimes U^{\theta_x}_{\Phi^+\cap\theta_x\Phi^+}\) are smooth over \(\mathbb Z[1/2]\) (Conrad, Theorem 5.1.13 and Proposition 5.1.16).
   • Orbit immersions through \(\tau\). For \(y=nx_0\), \(b\mapsto b\,\theta_n(b)^{-1}\) is flat onto an open \(\mathcal U_n\) of a fixed locus in the locally closed \(B\,\theta_n(B)\subseteq G\), so \(Y_x=\tau_n^{-1}(\mathcal U_n)\) is locally closed in \(X\) and \(B/B_x\simeq Y_x\), smooth with fibres the orbits (A2-4). This is Sakellaridis’s scheme \(Y\), complements read inside closures (A6-7).
   • Rank-one quotients. If \(\alpha\in\Phi_x\) then the \(\mathrm{SL}_2\) of \(\alpha\) lies in \(G^{\theta_x}\), \(P_\alpha Y_x=Y_x\) and the quotient is a point (type \(G\)). Otherwise \(P_\alpha Y_x=Y_x\sqcup Y_{s_\alpha xs_\alpha}\) is smooth, and \(py\mapsto\pi(p)[0:1]\) identifies its quotient with \(\mathrm{PGL}_2/B_2=\mathbb P^1_{\mathcal O}\) (type \(U\), \(S=T_2\)); types \(T\), \(N\) never occur (A2-5, A2-6; categorical quotient in A6-5).
   • Closures over \(\mathcal O\). By induction along type-\(U\) raisings from the closed orbit \(x=w_0\), the closure of \(Y_{x,F}\) in \(X_{\mathcal O}\) has special fibre the closure of \(Y_{x,\kappa}\), with the Bruhat order on the conjugates of \(w_L\) as closure order on every fibre (\(821\) relations); this gives the complement clause (A2-7; \(P_\alpha Z_{x'}\) closed since \(P_\alpha/B\simeq\mathbb P^1_{\mathcal O}\) is proper, A6-6).
   • Caveat. The triviality of \(\theta_0\) on \(\mathrm{Spin}_8\) uses the sign convention \(C(x_{\pm\alpha_i}(c))=x_{\mp\alpha_i}(-c)\) of S4-11; with the opposite sign the same formula gives an involution of type E I (A6-2).
6. Result \(E_6/F_4\): \(\mathcal M_\theta\) satisfies T1–T5

   For \(\mathcal M_\theta\), \(p\) odd, the hypotheses of part (b) hold, so step 12 applies over \(\mathbb F_q(\!(t)\!)\).

   Status: proved · this run, audited (A4-9’s reading; audit A5-7).

   How

   • T1. \(H_\Lambda\) is smooth with connected fibres of type \(F_4\) (S4-4, step 14), and \((T^\theta)^\circ\) is a split maximal torus of every fibre (A4-4).
   • T2. From \(\mathring X(E)=B(E)x_0\) by the density argument (E3) of A4-5, or from \(H^1(E,F_4)=1\) (step 14).
   • T3. \(\tau:X_\Lambda\to G_\Lambda\) is a closed immersion (A4-3).
   • T4. Delorme–Sécherre over every \(p\)-adic field of odd residue characteristic (S4-12, S4-13; step 16).
   • T5. Step 11, since A1–A5 hold over every \(\mathcal O'_e\), A3a by step 17; Sakellaridis’s label of \(Kx_\kappa\) is \(\kappa\in\Lambda_S^+=\Lambda_X^+\), the same for every \(e\).

Part (d): the criterion for \(E_6/F_4\)

1. Conclusion \(E_6/F_4\): the criterion in both characteristics, and (SG) fails

   For \(\mathcal M_\theta\), \(p\) odd, over \(\mathbb F_q(\!(t)\!)\) and over every \(p\)-adic \(F\), of any ramification, \(C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) with \(\varphi_0\mapsto1\), so (SG) holds iff \(r\) is onto. Since \(r\) is not onto (Proposition VII), (SG) fails and \(C_K\) needs exactly two generators:

   $$\begin{gathered}\operatorname{im}r=\{g\in\mathbb C[\Lambda_S]^{W_X}:\ g(y_1)=g(\sigma y_1)\},\\ C_K=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{Kx_{3\omega_1^\vee}} .\end{gathered}$$

   Status: proved · from (a)–(c); “(SG) fails” with Proposition VII.

   What is used, and where it is proved

   • \(p\)-adic \(F\). A1–A5 hold for \(\mathcal M_\theta\) over every \(p\)-adic \(\mathcal O\) (steps 13–17; the ramified case by step 11), so (a) applies.
   • \(\mathbb F_q(\!(t)\!)\). T1–T5 hold (step 18), so (b) applies (step 12), with the images \(P_\kappa\) of the \(p\)-adic fields.
   • \(r\) is not onto (S4-7), for every \(q\): after translating the target to \(\mathbb C[\Lambda_S]^{W_X}\), \(y_1\) and \(\sigma y_1\) are not \(W_X\)-conjugate while their shifts by \(q^{\rho_L}\) are \(W(E_6)\)-conjugate, as in step 7, and \(\operatorname{im}r\) is exactly the subring of functions agreeing at them.
   • Consequences (S4-8; over \(\mathbb F_q(\!(t)\!)\), A4-8). The isomorphism alone gives that (SG) fails, that \(C_K\) is not cyclic, and that \(C_K/\mathcal H_K\varphi_0\) is the character of Satake parameter \(q^{\rho-\omega_4}\). That \(\mathbf 1_{Kx_{3\omega_1^\vee}}\) is a second generator uses the image of an orbit indicator: from the proof of Theorem 8.0.2 (p. 64; step 5) over \(p\)-adic \(F\), and from \(\Psi\) (step 12) over \(\mathbb F_q(\!(t)\!)\).
   • Scope. \(p\) odd and the model \(\mathcal M_\theta\); residue characteristic \(2\) is not covered.
   • Where. Proposition VII proves the algebra and the module statements; Proposition VIII compares the function level with the sheaf level, where every object is Satake.

Other families on these pages

• \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) and \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\). \(r\) is onto, so (SG) holds under A1–A5 (Proposition F of the classical page). Sakellaridis’s table (rows 1 and 2, p. 76) lists them among examples satisfying the assumptions of Theorem 7.2.1; A3a–A3c he proves at almost every place. For \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), Proposition IV proves (SG) without A1–A5.
• The spheres \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\). Proposition V proves (SG) directly and finds \(r\) onto as a consistency check; there A1 and A5 hold and A2, A3a–A3c, A4 are not verified.
• \(\mathrm{SL}_2/T\). A5 fails (square classes), so the theorem does not apply; Proposition III decides it directly.
• \(\mathbb F_q(\!(t)\!)\). The paper assumes characteristic \(0\) (§1.1, p. 2); part (b) now covers symmetric \(X\) with T1–T5, and (d) covers \(E_6/F_4\). Proposition II works over \(\mathbb F_q(\!(t)\!)\).