§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:30:06Z

Lemma 6.9 · Status of Sakellaridis's hypotheses (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 \(E_6/F_4\) and the spheres

open audit: minor defect

Minor defect (second audit). All claims correct. The (A5)(A5) X = H\G is homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points. argument (theta(u) = u l theta(l)^{-1} lies in U_{P^-} cap P = 1) is valid; L_gamma = L_{supp(gamma)} (Sakellaridis 6.1) is D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p). for both spherical roots of E6/F4; the paraphrases of 1.7, 2.4, 7.1 and Appendix A are accurate (row 14 = SO_n backslash SO_n x SO_{n+1}); the sketched one-G(F)-orbit argument is right (open G(F)-orbits all meet the dense X°(F)). Citation slip in the node's source list: 'Theorem 6.4.1' is Proposition 6.4.1; the theorem deriving Statement 6.3.1 from 'no type N + L_gamma classical' is Theorem 6.3.2, so 6.3.1 is not an independent gap once type N is excluded. For the sphere, 'L_gamma classical' is also true (the spherical root 2 epsilon_1 has support all of D_n), though only implicit in the node.

Both function-level verdicts of this family hold only under Sakellaridis's hypotheses. Here (A1)(A1) Field and group: F is a local field of characteristic 0 (p-adic), and G is split over the integers, with connected fibres and simply connected derived group. Its characteristic-p counterpart is (A1ₚ). and (A5)(A5) X = H\G is homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points. are proved for \(E_6/F_4\) and the spheres, and the Levi subgroups \(L_\gamma\) of \(E_6/F_4\) are shown to be classical; (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit., (A3)(A3) Sakellaridis's three axioms over O, relabelled: (A3a) rank-one models over O (Axiom 2.2.1); (A3b) the Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2). For E6/F4 all three hold in both characteristics (VI run). at the given place and the rest of (A4)(A4) Spherical roots and colours: no spherical root of type N, every Levi L_γ of a simple spherical root classical, and his Statements 6.3.1 and 7.1.5 on the spherical roots and the virtual coloured weights. remain open.

Lemma 6.9.

Let \(X\) be \(E_6/F_4\) or \(\Spin_{2n}/\Spin_{2n-1}\), \(n\ge2\), with \(H=G^\theta\).

So for \(E_6/F_4\), (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.; for the spheres, (SG) holds 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.; in both cases the remaining obligations are (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit., (A3)(A3) Sakellaridis's three axioms over O, relabelled: (A3a) rank-one models over O (Axiom 2.2.1); (A3b) the Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2). For E6/F4 all three hold in both characteristics (VI run). at the given place, and the parts of (A4)(A4) Spherical roots and colours: no spherical root of type N, every Levi L_γ of a simple spherical root classical, and his Statements 6.3.1 and 7.1.5 on the spherical roots and the virtual coloured weights. other than "\(L_\gamma\) classical". Status: open.

The hypotheses are those listed on the classical page (section 4), quoted from Sakellaridis §1.7, §2.4, the opening of §6, §7.1–7.2 and Appendix A.

Proof of (A5)(A5) X = H\G is homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points..

  1. Let \(P\supseteq B\) be the minimal \(\theta\)-split parabolic, so \(P=U_P\rtimes L\) and \(B=U_P\rtimes B_L\). For \(g=ul\in B\cap H\) with \(u\in U_P\) and \(l\in B_L\), the equation \(\theta(g)=g\) gives \(\theta(u)=u\,l\,\theta(l)^{-1}\). The left side lies in \(\theta(U_P)=U_{P^-}\), the right side in \(P\), and \(U_{P^-}\cap P=1\); so \(u=1\) and \(\theta(l)=l\). Hence \(B\cap H=B_L\cap L^\theta\).
  2. For \(E_6/F_4\): \(L^\theta=S[2]\cdot\Spin_8=\Spin_8\), because \(S[2]\subseteq(T^\theta)^\circ\subseteq\Spin_8\) (family page, section 4). So \(B\cap H\) is a Borel subgroup of \(\Spin_8\), connected.
  3. For the sphere: \(L^\theta=S[2]\cdot\Spin(V_0)=\Spin(V_0)\), since \(S[2]=\{\pm1\}\) and \(-1\in\Spin(V_0)\). So \(B\cap H\) is a Borel subgroup of \(\Spin(V_0)\), connected.
  4. The remark on the classical page then gives \(H^1(F,B\cap H)=1\), so \(\mathring X(F)\) is a single \(B(F)\)-orbit.
  5. For \(E_6/F_4\), \(H^1(F,F_4)=1\) is not needed for Theorem E; that \(X(F)\) is a single \(G(F)\)-orbit would follow from \(\mathring X(F)\) being one \(B(F)\)-orbit together with the open-orbit argument of Lemma A. This is not pursued.

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Original record, as the agent wrote it at 02:30 UTC

Status of (A1)–(A5) for E6/F4 and the spheres: what is proved here and what remains

Both verdicts are 'under (A)'. Per hypothesis (classical.html §4 list; Sakellaridis §1.7, §2.4, §6 opening, §7.1–7.2, Appendix A):

So: E6/F4: (SG) fails under (A); the sphere: (SG) holds under (A); in both the remaining obligations are (A2), (A3) at the given place, and the parts of (A4) other than 'L_γ classical'.

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