Proposition VI over \(\mathbb F_q(\!(t)\!)\)
Sakellaridis proves his Theorem 8.0.2 only in characteristic \(0\) (§1.1). This page states it over \(\mathbb F_q(\!(t)\!)\) for \(E_6/F_4\), with its proof in three steps, as proved by a /solve-maths run and checked by a fresh audit. The hypotheses and the full proof are on VI · Statement and proof.
\(p\) is odd, \(F=\mathbb F_q(\!(t)\!)\), \(\mathcal O=\mathbb F_q[\![t]\!]\) and \(K=G(\mathcal O)\), with \(G\) the split simply connected group of type \(E_6\). \(X=E_6/F_4\) has the model \(\mathcal M_\theta=(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0})\otimes\mathcal O\). \(C_K=C_c^\infty(X(F))^K\) is a module over the Hecke algebra \(\mathcal H_K\), and \(\varphi_0=\mathbf 1_{X(\mathcal O)}\). Finally \(S=\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), and \(r:\mathbb C[A^*]^W\to S\) is the restriction (definitions).
Proposition VIb proved · audited
Over \(F=\mathbb F_q(\!(t)\!)\), Sakellaridis’s Theorem 8.0.2 holds for \(E_6/F_4\): there is a unique isomorphism of \(\mathcal H_K\)-modules
$$C_K\ \xrightarrow{\ \sim\ }\ S,\qquad\varphi_0\longmapsto1,$$with \(f\in\mathcal H_K\) acting on \(S\) as multiplication by \(r(\mathcal Sf)\). Since \(r\) is not onto (Proposition VII), \(\varphi_0\) does not generate:
$$C_K=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{Kx_{3\omega_1^\vee}} .$$Proof
$$\underbrace{\text{(A3a) over }\mathbb Z[\tfrac12]}_{1}\ \Longrightarrow\ \underbrace{\text{Thm 8.0.2 over }\mathbb Q_q(p^{1/e})}_{2}\ \xrightarrow[\ e\ \ge\ D\ ]{\ \bmod\ \varpi^{D}\ }\ \underbrace{\text{Thm 8.0.2 over }\mathbb F_q(\!(t)\!)}_{3}$$- Rank-one models over \(\mathcal O\). Sakellaridis’s Axiom 2.2.1, relabelled (A3a), holds for \(\mathcal M_\theta\) over every discrete valuation ring with \(2\) invertible. The \(45\) Borel orbits are the smooth schemes \(B/B_x\) over \(\mathbb Z[1/2]\), and each rank-one quotient is \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\) (90 pairs) or \(\mathbb P^1_{\mathcal O}\) (180 pairs). Theorem 2.8 →
- The ramified fields \(F'_e=\mathbb Q_q(p^{1/e})\). With step 1, every hypothesis of Theorem 8.0.2 holds over \(F'_e\). The paper’s assumption “unramified over \(\mathbb Q_p\)” only fixes the additive character (conductor \(\mathcal O\)) and the self-dual measure. Its source [Sa08] assumes only characteristic \(0\). So the theorem holds over \(F'_e\), and the image \(P_\kappa\) of each \(\mathbf 1_{Kx_\kappa}\) depends only on \(q\). Check 4.7 → · [Sa08] →
- Close fields. \(\mathcal O'_e/\varpi'^m\cong\mathbb F_q[t]/(t^m)\) for \(m\le e\). Each of the following is decided modulo \(\varpi^D\), with \(D\) independent of the field: the count \(\#K\backslash Kt^\lambda X(\mathcal O)\), the structure constants \((T_\lambda\ast\mathbf 1_{Kx_\mu})(x_\nu)\), and the Satake coefficients of \(T_\lambda\). So \(K\backslash X(F)\simeq\Lambda_S^+\). And \(\mathbf 1_{Kx_\kappa}\mapsto P_\kappa\) is \(\mathcal H_K\)-linear over \(F\), checked one \((\lambda,\mu)\) at a time against \(F'_e\), \(e\ge D\). Proposition 4.5 · 4.6 · Theorem 4.8 →
Audit. Eight fresh reviewers re-derived all 76 steps of the run; steps 1–3 and everything they rest on held. The three errors they found are in the general hypothesis list of the direct route below. One salvage agent repaired them, and the reviewers re-checked the repairs. Verdict after the audit →
Beyond \(E_6/F_4\) over \(\mathbb F_q(\!(t)\!)\)
- Every \(p\)-adic field. The same holds over every finite extension of \(\mathbb Q_p\), \(p\) odd, of any ramification. Step 1 settles the one hypothesis the \(p\)-adic check had left open, and the others hold there (S4, S4b).
- Every symmetric \(X\), by the transfer. Steps 2–3 prove the theorem over \(\mathbb F_q(\!(t)\!)\) for a symmetric \(X=G/G^\theta\) over \(W(\mathbb F_q)\) satisfying (T1)–(T5). These five conditions are: \(G^\theta\) split with connected fibres; one \(G\)-orbit of points over \(F\) and over \(F'_e\); \(\tau(x)=x\theta(x)^{-1}\) detects integral points; the Cartan decomposition over \(F'_e\); and the theorem over \(F'_e\). The target \(S\) is built from the invariants of \(X\) over \(F'_e\); for \(E_6/F_4\) they agree with those over \(\mathbb F_q(\!(t)\!)\). Proposition 4.9 →
- The hypotheses over \(\mathbb F_q[\![t]\!]\). For \(E_6/F_4\), Sakellaridis’s other hypotheses also hold over \(\mathbb F_q[\![t]\!]\), by characteristic-\(p\) arguments: the field and group, the integral model, the Cartan decomposition, the Iwahori orbits, the spherical roots and colours, and the one open Borel orbit ((A1ₚ)–(A5)). Summary 3.12 →
- A direct route. Rerunning Sakellaridis’s proof itself over \(\mathbb F_q(\!(t)\!)\) gives the theorem for a symmetric \(X\) under the hypotheses of Theorem S1.5, which do not include (A3a). Beyond the integral model, the Cartan and Iwahori conditions, Statements 6.3.1 and 7.1.5 in characteristic \(p\) and a cone condition, it asks three things: one set of colours for each simple spherical root, adapted data over \(F\), and no spherical root of type \(N\). Its last step needs non-zero leading coefficients, which hold by positivity (Lemma S1.3). For \(E_6/F_4\) its one open input comes from step 1 through the transfer, so it confirms the proposition but is not an independent proof.
- Two gaps in the published proof. On p. 62, Sakellaridis derives his triangularity (7.5) from an identity that is off by a factor \(2\). (7.5) holds whenever each simple spherical root has one set of colours, which his Lemma 7.1.4 gives in characteristic \(0\) (Lemma S1.1). His last step also needs non-zero leading coefficients, which the paper does not prove; positivity gives them. So his theorem stands as stated (Remark S1.4).
- A finding about another page. Chen–Yi prove their Prop. 17(i) by citing results over \(\mathbb C\). That result is the orbit parametrization behind the pages’ input D0, the orbit geometry over \(\overline{\mathbb F}_q\) (Check 4.3).