Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide

Decided classification and the cells where a divergence can still hide

Classification (SH over \(\overline{\mathbb F}_q\), ℓ-adic, with Betti via the pages' open item 1; FN over the stated \(F\)):

Answer to the question. Among the decided cells there is no divergence: wherever both levels are decided they agree, including the two cells decided here (\(\mathrm{SL}_2/\mathrm{SO}_2\): both fail; \(\mathrm{GL}_4/\mathrm{Sp}_4\): both hold, the first case with H5 failing and \(\mathcal C^{\mathrm{Sat}}=\mathcal C\)). One implication is now proved in the only comparable setting (A1-6): over \(\mathbb F_q(\!(t)\!)\), with connected stabilizers, FN ⟹ SH; so "FN holds, SH fails" cannot occur there under H3. The reverse divergence "SH holds, FN fails" is not excluded in general (A1-6 Remark (a)).

Cells where a divergence can still hide, with the exact obligation:

1. \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge3\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\) (SH open, FN under (A) in char 0). To exclude "FN holds, SH fails": prove (SG) over \(\mathbb F_q(\!(t)\!)\) for one odd \(q\) — either extend Theorem E/Prop F to \(\operatorname{char}F=p\), or, label by label, find \(f_\mu\in\operatorname{span}\{T_\lambda:\varphi(\lambda)\ge\mu\text{ in the component}\}\) with \(f_\mu\ast\varphi_0\) vanishing on the orbits above \(\mu\) and not on \(\mathcal O_\mu\) (a finite lattice count as in A1-4); then A1-6 gives SH. To produce "SH fails": exhibit an uncovered \(\mu\) with \(\mathrm{IC}_\mu\) a summand of no \(V_\lambda\ast\mathrm{IC}_0\) (equivalently, by A1-5, \(\mathcal C^{\mathrm{Sat}}\ne\mathcal C\)); by A1-6 this would also refute (SG) over \(\mathbb F_q(\!(t)\!)\), hence contradict a char-\(p\) extension of Prop F.
2. \(E_6/F_4\) (both open): (a) prove \(K\backslash X(F)\cong\Lambda_S^+\) via \(\tau\) over \(F\) with \(q\) odd (Sakellaridis Axiom 2.4.1 for this \(X\)); (b) decide \(\mathbf 1_{\mathcal O_{3k\omega_1^\vee}}\in\mathcal H_K\cdot\varphi_0\), \(k\) odd (cancellation among \(T_\lambda\) with \(\varphi(\lambda)\ge3k\omega_1^\vee\)); then A1-6 decides SH in the positive case; in the negative case decide whether \(\mathrm{IC}_{3k\omega_1^\vee}\) is a Satake summand (A1-5).
3. \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \((p,q)\ne(1,1)\): SH holds; FN known only under (A) in char 0. To exclude "SH holds, FN fails" over \(\mathbb F_q(\!(t)\!)\): prove \(K\backslash X(F)\cong\Lambda_S^+\) (pairs of complementary subspaces versus a lattice) and apply Corollary C with the support bound of A1-4 step 1 (coverage holds, so no cancellation is needed). Gap: the orbit classification over \(F\) is not on the pages.
4. Spheres with \(q\) even and \(\mathrm{SL}_2/\mathrm{SO}_2\) in residue characteristic 2: outside Chen–Yi's \(p\ne2\); no sheaf statement to compare.
5. Betti versus ℓ-adic: every sheaf verdict above is ℓ-adic; the Betti \(\mathcal C\) of the pages inherits it through the comparison of decomposition multiplicities (open item 1). A divergence between Betti and ℓ-adic sheaf statements would be a divergence inside the sheaf level, not between the levels.

Blocked route (recorded): the function-level parity obstruction for \(\mathrm{SL}_2/T\) (\(\mathbb C[y^2]\subsetneq\mathbb C[y]\)) does not lift to a sheaf-level obstruction, because thick spans contain retracts and \(\mathrm{IC}_0\) is a summand of the "odd" object \(V_{(1,0)}\ast\mathrm{IC}_0\) (A1-2 step 4); whether the odd \(\mathrm{IC}_{(2j+1)\alpha^\vee}\) are Satake summands for \(\mathrm{SL}_2/\mathrm{SO}_2\) stays open and would need Chen–Yi's relative Kostka–Foulkes multiplicities or [CN24, Cor. 13.8] with the Hecke action identified as restriction along \(\check G_X\to\check G\).