# Assessment of the pasted note "From the Steinberg representation to Iwahori generation on spherical varieties" (2026-10-11)

What of it is absorbed into `classical.html`, what was corrected, what was dropped.

## Correct and absorbed

| Note | Claim | Status | Page |
|---|---|---|---|
| I.1–2 | a nonzero quotient module killing `v` shows `Av ≠ M`; `e_K` projects `V^I` onto `V^K`; so `V^I ≠ 0 = V^K` gives `H_I e_K H_I ≠ H_I` | correct | 2, Lemma C |
| II.3–7 | `St = C^∞(P^1)/C`; `K` transitive on `P^1(F)`; `St^K = 0`; two `I`-orbits, `dim St^I = 1`; `e_K = (1+T_s)/(q+1)`; `T_s = −1` on `St^I` | correct (the fibre-transitivity of the first congruence subgroup, which the note omits, is supplied) | 4 |
| III.9 | `H_I φ_0 = C_I ⟺ H_K φ_0 = C_K and H_I C_K = C_I` | correct; the direction "Iwahori generation ⇒ spherical generation" is new relative to the earlier page | 2, Proposition B |
| III.10 | `Q_X = C_I / H_I C_K`, `e_K Q_X = 0`, generation iff `Q_X = 0` | correct; this is the module form of the idempotent lemma | 2 |
| IV.11 | `P^1`: `H_I φ_0 = Cφ_0 ⊊ C_I`, quotient `St^I` | correct | 5A |
| IV.12 | `A^2∖0`: spherical generation by shells; radial integration `R` is well defined, equivariant, onto, `R φ_0 = 1`; `C_I ↠ St^I` kills `φ_0` | correct | 5B |
| IV.13 | `A^2`: `1_L ∈ H_I φ_0` for `I`-stable `L = gO^2` via `1_{IgK}`; two `I`-orbits per shell; every `I`-invariant function is a finite combination | correct | 5C |
| V.16 | Sakellaridis Thm 8.0.2 / Cor 8.0.4 / Example 8.0.1 | verified earlier against arXiv:0905.4244v3 | 7 |
| V.17 | Shtotland, IMRN 2026 rnag042 = arXiv:2406.16070 | exists; the abstract promises a description of `S(X)^I` as an `H_I`-module generalising Iwahori–Matsumoto | 7 |
| V.17 | Shtotland, arXiv:2501.01701 | exists; v2 (March 2026) retitled "Distinction of the Steinberg representation and the dual group of a symmetric space" | 7 |

## Corrected or qualified

- **Affineness.** `P^1` is projective and `A^2∖{0}` is quasi-affine; neither is in the paper's class of smooth affine spherical varieties. `A^2` is, and it is a positive example. The page's Table 1 records the type of each.
- **`GL_2/T` is not open.** The note calls it a "suggested first computation" and says the `T`-invariant functional on `St` "does not, by itself, settle the generation question". It does: `Hom_G(S(X), St) ≅ Hom_T(St~, C) = Hom_T(St, C) ≠ 0` (Frobenius reciprocity, `St` self-contragredient), and `St^I ≠ 0 = St^K`, so Corollary D gives `H_I φ_0 ⊊ C_I`. Proposition F.
- **Two definitions of `Q_X`.** §10 divides by `H_I C_K`, Part VI by `H_I φ_0`. They agree under spherical generation (Proposition B); the page uses the first.
- **Finite generation.** The note says the IMRN paper "also proves finite generation of this module". The arXiv abstract does not claim it; the page does not use it. Where an irreducible quotient of `Q_X` is wanted, the page states finite generation as a hypothesis.
- **Borel's dictionary.** The note's "Hecke module" language is kept, but the representation form (Corollary D) is proved directly from exactness of `I`-invariants; Borel's equivalence is cited only for the converse description of the irreducible modules killed by `e_K`.
- **Normalisation.** `e_K H_I e_K = H_K` holds as algebras only after rescaling (`f ↦ f/[K:I]`); the page states the isomorphism.

## Dropped

- The favicon "citations" (MathOverflow, arXiv, OUP) are not references.
- Part VI's research programme is replaced by the status table (Table 2) and the literature list.
- "It also proves finite generation", "Shtotland's Section 9 example `T\SL_2`": not verified, not used.