# Assessment of the pasted note "Cyclic Generation of Iwahori-Invariant Functions on Spherical Varieties" (2026-10-11)

A rank-one investigation (T\SL_2 negative, F^{1×2} positive, unramified torus positive, ramified torus open), written with right modules on H\G. What of it is absorbed into `classical.html`, `classical-cases.html` and `classical-affine.html`, what was verified against the sources, what was corrected.

## Verified against the papers (texts extracted from arXiv PDFs on 2026-10-11)

| Note | Claim | Source check | Page |
|---|---|---|---|
| §1.1 | right action, `f∗h(x) = ∫ f(xg⁻¹)h(g)dg`, `S(X)^I` compactly supported | Shtotland IM, Definition 2.37, verbatim | cases, notation paragraph |
| §1.2 | criterion `φ₀A = M ⟺ φ₀H_K = Me and (Me)A = M`; `Q_X`, `Q_{X,φ₀}`, exact sequence; `Q_X ≠ 0 ⟹ no K-invariant vector generates` | elementary; same as Proposition B of the classical page | cases §1, classical §2 |
| §1.2 Prop 2 | finitely generated `M` with `Q_X ≠ 0` has a simple quotient killed by `e` | elementary | cases, Tool 1 |
| §3.1 | Shtotland IM Theorem 5.9: `S(X)^I` finitely generated over `H(G,I)` | **verified** (statement and proof present; Remark 5.10 gives the alternative via Delorme and AGS) — this corrects the earlier assessment, which had found no such claim in the abstract | cases, Tool 1; classical §2, §7 |
| §3.1 | Theorems 1.1, 1.2, 3.4, 5.3, 7.13 (`γ = (q−1)/n_max δ_max + 2/n_min δ_min`), 8.5 | verified | cases §8 |
| §3.1, §4 | Propositions 9.1, 9.2 for `T\SL_2`: decomposition `C_sgn ⊕ H/H(T_{s0}+1) ⊕ H/H(T_{s1}+1) ⊕ H`; orbits `z₀,z₁,x_w,y_w`; lengths; relations (4.1), (4.2) and the `(q−1)/2` relations; `4 | q−3` assumed (five torus orbits when `4 | q−1`) | verified verbatim | cases §5 |
| §3.3 | Shtotland St paper: Theorem 1.5 (sufficient conditions), Proposition 1.11 (harmonic functions), Proposition 2.7 (sign action, `T_o ↦ sgn(o)`) | verified; Prop 2.7 is attributed there to Broussous [Bro18] §3.3 | cases, Tool 2 |
| §3.2 | Broussous, arXiv:2406.17459, orbits of `H_F` on chambers, odd residue characteristic | abstract verified; "Theorem 3.6" not checked | cases §8 |
| §3.4 | Sakellaridis Theorem 8.0.2 / Corollary 8.0.4 | verified earlier (two-definitions note) | classical §7 |
| §4 | the sign line `C(z₀ − z₁)` is a submodule and the functional `λ` is a quotient | correct; both follow from Prop 9.1 (direct summand) | cases §5 |
| §4.6 | SG fails for `T\SL_2` | correct; reproved on the tree: `K_{SL}`-orbits are `(d, k mod 2, ε)`, Hecke translates of `φ₀` live on `k ≡ d (mod 2)` and are invariant under `diag(u,1)`, `u` a non-square unit | cases, Proposition D |
| §5 | `φ₊A = C_c^∞(F^{1×2})^I` and `φ₊H_K = C_c^∞(F^{1×2})^K` via the lattice chain | correct; it is the case `n = 2` (after transposing) of Theorem 3 of the affine page, which does every `n` | affine page |
| §7.1 | unramified anisotropic torus: `T_u\G/I = K\G/I`, generation holds | correct; verified (units of `O_E/ϖⁿ` are the primitive vectors; pullback along the proper map `T_u\G → K\G`) | cases §5 |
| §7.2 | ramified torus: open question | recorded as open | cases §5, §7 |

## Corrected or qualified

- **Group matters.** The note's Theorem 3 is for `G = SL_2`, where SG fails. The classical page's Proposition F is for `GL_2/(GL_1 × GL_1)` (same function spaces as `PGL_2/T`), where SG holds (one `K`-orbit per distance; the recursion for `T_1`). There is no contradiction: the `K_{SL}`-orbits refine the `K_{GL}`-orbits by parity of the projection and by the square class of the branch. The paper's table has the same distinction at sheaf level: `Z_{G^θ}(S) = {±1}` is disconnected for `SL_2/SO_2` and (H5) fails ("misses α̌"), which is the parity half of the function-level failure; the square-class half has no sheaf counterpart.
- **Finite generation.** The earlier assessment (`representation-note-assessment.md`) recorded the claim as unverified; it is Theorem 5.9 of arXiv:2406.16070v3. Consequently the representation form of the obstruction (Corollary D of the classical page) is both necessary and sufficient for the paper's symmetric families (Tool 1 of the cases page).
- **Steinberg sweep.** The note suggests harmonic functions on Shtotland's hypergraph as a strategy. The cases page uses his Theorem 1.5 / Remark 1.6 directly: for split symmetric `X`, `St ⊗ χ₀` is `H`-distinguished iff `X` is quasi-split and non-exceptional, which decides the group case, `GL_{2p}/(GL_p × GL_p)`, and `SL_2/T`, `PGL_2/T`, `GL_2/T`, and is silent for the non-quasi-split families (`GL_{2n}/Sp_{2n}`, spheres `n ≥ 3`, `E_6/F_4`) and for the exceptional `GL_{2n+1}/(GL_n × GL_{n+1})`.
- **Conventions.** Right modules on `H\G` (the note, Shtotland) and left modules on `G/H` (the pages) are exchanged by `x ↦ x⁻¹` and the anti-involution `f ↦ f(g⁻¹)` of `H(G,I)`, which fixes every `T_w`; the "left-ideal notation" remark in the note's §4.3 is this point.
- **The favicon "citations"** (Google favicon images) are not references and are dropped.

## Dropped

- The note's §6.1–6.3 programme (finite generator sets from Theorem 5.9, signed functionals on the hypergraph as a computational strategy) is summarized in one sentence of the cases page (§7, open items); the caveat that a finite-depth search is not a proof is kept.