The Iwahori level
Does the basic function still generate when \(K\) is replaced by the Iwahori subgroup \(I\)? Usually not: the group, every quasi-split symmetric variety and the matrices fail, each because of a Steinberg-type representation. These pages prove the criterion that reduces the question to one obstruction module, and decide it on examples.
The questionWhat does (IG) ask beyond (SG)?
For the same basic function \(\phiz=\one_{X(\cO)}\), with \(\CK=\Sw(X)^K\subseteq\CI=\Sw(X)^I\):
\[\text{(SG)}\quad\HK\ast_K\phiz=\CK\qquad\text{against}\qquad\text{(IG)}\quad\HI\ast\phiz=\CI .\](IG) implies (SG), and the gap between them is the module \(\QX=\CI/\HI\ast\CK\), which \(\eK\) kills (the criterion). A representation with Iwahori-fixed vectors, no \(K\)-fixed vector and a nonzero map from \(\Sw(X)\) makes \(\QX\ne0\) (witnesses).
Map of the Iwahori pages
- criterion, \(Q_X\)
- group, \(\mathrm{GL}_2\)
- symmetric
- \(\mathbb A^n\): Iwahori
| page | contents |
|---|---|
| Criterion | convolution at two levels; the idempotent \(\eK\); the criterion; \(\QX\), \(Q_{X,\phiz}\); the obstruction principle; witnesses; \(\St\) of \(\GL_2\) |
| Group and \(\GL_2\) | the group and the sign character; \(\mathbb P^1\), \(\Aff^2\smallsetminus\{0\}\), \(\Aff^2\) |
| Symmetric varieties | \(\GL_2/T\); the Steinberg sweep; \(\GL_{2n}/\Sp_{2n}\) and Speh witnesses |
| 𝔸ⁿ: Iwahori | the lattice chain; (IG) for \(\Aff^n\), every \(n\); why the origin matters |
Status of (IG)Where does the basic function generate at the Iwahori level?
| \(X\) | (SG) | (IG) | witness or reason |
|---|---|---|---|
| group \(H\), \(W\ne1\) | holds | fails | \(\St_H\boxtimes\widetilde{\St}_H\), sign character (group) |
| \(\GL_2/\Sp_2=\Gm\) | holds | holds | \(I\backslash X=K\backslash X\) (\(n=1\)) |
| \(\GL_2/T\), \(\PGL_2/T\) | holds | fails | \(\St_2\), \(T\)-distinguished (\(\GL_2/T\)) |
| \(\PGL_2/T_u\), \(T_u\) unramified anisotropic | holds | holds | \(T_u\backslash G/I=K\backslash G/I\) (F2, §3) |
| \(\SL_2/T\), \(q\) odd | fails | fails | parity and square classes; sign summand (F2, D(ii)) |
| \(\GL_{2p}/(\GL_p\times\GL_p)\) | holds | fails | \(\St_{2p}\) (Shalika model), \(\St_{2p}\otimes\chi_0\) (F2, B(i)) |
| \(\GL_3/(\GL_1\times\GL_2)\) | holds | fails | \(\St_2\times\one\) (F2, B(ii)) |
| \(\GL_n/(\GL_p\times\GL_q)\), \(q\gt p\), \(n\ge4\) | holds | open | candidate \(\St_{2p}\times\one_{\GL_{q-p}}\) (F2, B(iii)) |
| \(\GL_{2n}/\Sp_{2n}\), \(n\ge2\) | holds | fails (\(\operatorname{char}F=0\)) | Speh \(U(\St_2,2)\times\one_{\GL_2}^{\times(n-2)}\) (symplectic) |
| \(\SL_{2n}/\Sp_{2n}\), \(n\ge2\) | holds | fails (\(\operatorname{char}F=0\)) | Speh restricted to \(\SL_{2n}\) (F2, Theorem C) |
| \(\Spin_{2n}/\Spin_{2n-1}\) | holds | \(n=2,3\) fail; \(n\ge4\) open | the group \(\SL_2\); \(\SL_4/\Sp_4\) (F2, §2) |
| \(E_6/F_4\), \(p\) odd | fails | fails | (IG) implies (SG) (criterion) |
| \(\Aff^n\) under \(\GL_n\) (not homogeneous) | holds | holds, \(\QX=0\) | lattice chain (𝔸ⁿ) |
| \(\Mat_n\) under \(\GL_n\times\GL_n\), \(n\ge2\) (not homogeneous) | \(n=2\) holds; \(n\ge3\) open | fails | \(\St_n\boxtimes\St_n\) via Godement–Jacquet [GJ] (F2, Prop. E) |
The (SG) column is that of computing r, where each entry is proved or cited. Pattern: wherever a Steinberg-, induced-Steinberg- or Speh-type representation is \(X\)-distinguished, the Iwahori-generated part is proper; for quasi-split split symmetric \(X\) the twisted Steinberg always is (the sweep). The positive cases are the torus-like \(\Gm\), the non-homogeneous \(\Aff^n\), and the non-split \(\PGL_2/T_u\).
\(E_6/F_4\): (SG) fails [satake-two-levels, Prop. VII; computing r], so (IG) fails by the criterion.
The workspace pagesWhere are the longer arguments?
The Iwahori-generation workspace keeps the family-by-family proofs; these pages cite them by tab name.
| page | contents |
|---|---|
| F1 · Tools and the sweep | witnesses, the Steinberg, \(\GL_n\) and Speh sweeps, Theorem A, the Steinberg of \(\GL_2\) explicitly; checked literature |
| F2 · Family by family | \(\GL_n/(\GL_p\times\GL_q)\), \(\SL_{2n}/\Sp_{2n}\) and the spheres, rank one, \(\Mat_n\) |
| 𝔸ⁿ under GLₙ (formerly F3) | \(\Aff^n\) in valuation coordinates, with interactive pictures, and the sheaf-level sketch |
| Summary | every variety at both levels, functions and sheaves |
| Sheaves | the categorical statements, Propositions I–IV |
The former function-level hub of the workspace is archived; its content is on the Iwahori pages here.
LiteratureWhich results do these pages rest on?
- Spherical generation. [Sak, Thm. 8.0.2, Cor. 8.0.4]: for affine homogeneous \(X\) under Sakellaridis's hypotheses (A1)–(A5) on the field, the group and \(X\) (hypotheses), \(\CK\simeq\C[\dXh\AXs]^{\WX}\) with \(\phiz\mapsto1\), and (SG) holds iff \(\Res\) is onto (the Theorem, computing r); Ex. 8.0.1 is a failure. The hypotheses are assumed, not verified, wherever these pages use the Theorem.
- Iwahori modules of symmetric spaces. [Shtotland 2024] presents \(\Sw(G/H)^I\) as an \(\HI\)-module through \(I\)-orbits (Thms. 5.3, 7.13, 8.5), proves finite generation (Thm. 5.9), and treats \(T\backslash\SL_2\) (Props. 9.1–9.2) and \(\SL_{2n}/\Sp_{2n}\) (Prop. 9.3). It computes \(\QX\) for symmetric \(X\); it does not ask whether \(\phiz\) is cyclic.
- Steinberg distinction. [Shtotland 2025]: for split symmetric \(H\subseteq G\), \(H\)-distinction of \(\St_G\) reduces to a harmonic-function condition on a hypergraph attached to \(G/H\). Since \(\St_G^I\ne0=\St_G^K\) [Borel 1976, 3.5], each such distinction is a failure of (IG). Steinberg is not the only witness: for \(\GL_{2n}/\Sp_{2n}\) a Speh representation is distinguished.
- Representations. [Borel 1976, Thm. 4.10, Cor. 4.11, 3.5, 5.7, Lemma 4.7, 4.9 (∗)]; [Casselman 1980, Props. 2.6–2.7]; [Heumos–Rallis, Thms. 2.4.2, 11.1, 11.5], whose Thms. 3.1 and 3.2.2 rest on a false proposition of Klyachko, corrected in [Offen–Sayag, JFA 2008, Thm. 1, Cor. 1]; [Offen–Sayag, JNT 2007, Thm. 1, Prop. 2; IMRN 2008, Thms. 7–8, Cor. 2]; [Tadić 1986, Thm. D, 7.4–7.5, Thm. A.8]; [HKP, §4.1]. Located on the note's functions page or in the workspace's literature report.
- The two pasted notes. Rank-one note: its criterion and obstruction modules are those of the criterion page; its \(T\backslash\SL_2\), \(F^{1\times2}\) and unramified-torus results are absorbed into F2 and F3; its finite-generation claim is verified (Thm. 5.9). Representation note: its obstruction principle, Hecke computation, Steinberg construction, criterion and examples are correct and reproduced on the criterion page and the \(\GL_2\) examples; its examples \(\mathbb P^1\) and \(\Aff^2\smallsetminus\{0\}\) are not affine, so only \(\Aff^2\) lies in the Theorem's class; \(\GL_2/T\) is decided by the functional it exhibits (\(\GL_2/T\)); its attribution of finite generation to the IMRN paper is not used.