§5 · Agent 5 · Functions without (A1)–(A5) · 2026-10-17T03:32:17Z

Dead end 5.9 · \(E_6/F_4\) at function level: what the lattice method would need

blocked audit: minor defect

Minor defect (second audit). The list of missing inputs is accurate, the parametrisation of \(\Lambda_S^+\) and the uncovered labels match the families page, and the status 'blocked' is correct. The closing expectation is stale: Theorem 6.6, recorded earlier in the run, shows under (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit. for \(p\)-adic \(F\) that the restriction map is not onto at \(3\omega_1^\vee\), so \((\mathrm{SG})\) fails under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.; the cell should read 'fails under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies., \(p\)-adic; open over \(\F_q(\!(t)\!)\)'. A heuristic expectation, not a proved claim.

The method that settles the other symmetric families would need, for \(E_6/F_4\), an integral classification of the orbits on the norm-one hypersurface of the split Albert algebra and a triangular Hecke row at the uncovered labels \(3k\omega_1^\vee\), \(3k\omega_6^\vee\). Neither is available, and the bilinear duality behind the symplectic argument has no direct cubic analogue; the row stays open.

Dead end 5.9.

Open. The method of Theorem 5.3–Theorem 5.7 applies to \(E_6/F_4\) once the following is proved: for the norm-one hypersurface \(Q_N=\{N=1\}\) of the split Albert algebra, the \(E_6(\cO)\)-orbits on \(Q_N(F)\) are parametrised by \(\Lambda_S^+=\{a\omega_1^\vee+b\omega_6^\vee:\ a,b\ge0,\ a\equiv b\bmod3\}\), and the counts \((T_\lambda\ast\varphi_0)(x_\mu)\) vanish unless \(\mu\) lies in an explicit finite set above a label computable from \(\lambda\).

  1. An integral model and its orbits. \(X=E_6/F_4\) is the norm-one hypersurface of the split Albert algebra \(\mathbb A\), and \(X(F)=E_6(F)/F_4(F)\) is a single orbit because every Albert algebra over a local field is split (cited). What is missing is a classification \(K\backslash X(F)\leftrightarrow\Lambda_S^+\) by invariants of the lattice \(\mathbb A(\cO)\) relative to \(x\), the analogue of the elementary-divisor type.
  1. A triangular row at the uncovered labels. For \(3k\omega_1^\vee\) and \(3k\omega_6^\vee\) with \(k\) odd (families page for \(E_6/F_4\), section 5) one needs a Hecke row with a two-sided bound as in Lemma 5.6 — for instance some \(\lambda\) with \(\varphi(\lambda)=3\omega_1^\vee+(\text{something})\) — and a module-theoretic inequality replacing Lemma 3 there. The symplectic proof used the dual lattice \(\Lambda_0^\#\) of a bilinear form; for \(E_6/F_4\) the natural substitute is the cubic norm pairing, which is not a bilinear duality, so that argument does not transfer as it stands.
  1. Expectation. No counterexample is in sight either: nothing on the workspace pages or in the mechanism of Theorem 5.7 suggests that \((\mathrm{SG})\) fails for \(E_6/F_4\).

Builds on

  • Theorem 5.7 · \((\mathrm{SG})\) for \(\GL_{2m}/\Sp_{2m}\) (all \(m\)) and \(\SL_{2n}/\Sp_{2n}\) (all \(n\)), every non-archimedean local field

Built on by

Original record, as the agent wrote it at 03:32 UTC

E_6/F_4 at function level: what is missing

Open. Needed for the method of nodes 3–7: (1) an \(\mathcal O\)-model of \(X=E_6/F_4\) (norm-one elements of the split Albert algebra \(\mathbb A\), \(X(F)=E_6(F)/F_4(F)\) is one orbit since every Albert algebra over a local field is split, cited) with a classification \(K\backslash X(F)\leftrightarrow\Lambda_S^+=\{a\omega_1^\vee+b\omega_6^\vee:a\equiv b\ (3)\}\) by invariants of the lattice \(\mathbb A(\mathcal O)\) relative to \(x\) (analogue of the elementary-divisor type); (2) for the uncovered labels \(3k\omega_1^\vee,3k\omega_6^\vee\) (\(k\) odd; families/e6-f4.html §5) a Hecke row with a two-sided sandwich as in node 6, e.g. \(\lambda\) with \(\varphi(\lambda)=3\omega_1^\vee+(\text{something})\) and a module-theoretic inequality replacing Lemma 3 (the symplectic proof used the self-dual lattice \(\Lambda_0^\#\); for \(E_6/F_4\) the natural substitute is the cubic norm pairing, not a bilinear duality, so the argument does not transfer as is). Theorem-strength lemma required: for the norm-one hypersurface \(Q_N=\{N=1\}\subset\mathbb A\), the \(E_6(\mathcal O)\)-orbits on \(Q_N(F)\) are parametrised by \(\Lambda_S^+\) and the Hecke counts \((T_\lambda\ast\varphi_0)(x_\mu)\) vanish unless \(\mu\) lies in an explicit finite set above a label computable from \(\lambda\). No counterexample is in sight either: nothing on these pages or in the mechanism of node 7 suggests (SG) fails for \(E_6/F_4\).

Sources