§1 · Agent 1 · Literature and Table 1 audit · 2026-10-18T18:49:14Z

Objection 1.8 · Errors and unproved claims found on the Iwahori-generation pages

proved

Four corrections: \(L^+X\) is not one \(K\)-orbit for \(\mathbb A^n\); two \(\mathrm{GL}_p\times\mathrm{GL}_q\) failures need \(\operatorname{char}F=0\) but are not flagged; an averaging aside on the group page is unreferenced and unused; and the brief's hypothesis list must replace the single-orbit condition for \(\mathbb A^n\).

Objection 1.8.

  1. index.html, §2 Setting: "\(L^+X=X(\mathcal O)\) ... is the \(K\)-orbit of the base point" is false for the row \(\mathbb A^n\): \(K\backslash\mathcal O^n=\{0\}\sqcup\{X_m\}_{m\ge0}\). No proof on the pages uses it.
  2. summary.html Table 1, index.html Table 1 and classical.html Table 2 list (IG) as failing for \(\mathrm{GL}_{2p}/(\mathrm{GL}_p\times\mathrm{GL}_p)\) and \(\mathrm{GL}_3/(\mathrm{GL}_1\times\mathrm{GL}_2)\) with no field condition; the witnesses rest on Shtotland, Theorem 1.5 (\(\operatorname{char}F=0\), summary.html Table 2) or Matringe (\(p\)-adic \(F\)). They should carry "\(\operatorname{char}F=0\)" like the symplectic rows. Only \(\mathrm{GL}_2/T\) (classical.html, Prop. F) is proved for every \(F\), in particular for \(\mathbb F_q((t))\).
  3. group.html, definitions: "\(a_!\) and \(a_*\) agree up to a shift that is suppressed" for the \(\psi\)-averaging is asserted without proof or reference; the proof of Proposition II uses only \(\Phi=a_!(-\boxtimes\psi^*\exp)\), so the aside is unused.
  4. For this run: the hypothesis "\(L^+X\) a single \(K\)-orbit" fails for \(\mathbb A^n\) and must be replaced by "\(\mathrm{IC}_0\) is the constant sheaf on the closed pro-smooth \(L^+X\), pulled back from a finite level, with \(\operatorname{tr}\mathrm{IC}_0=\pm\varphi_0\)", which holds for every row.
  5. Not an error, a reframing: hypothesis.html and case-affine.html call Proposition IV.1 conditional on the first half of I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).; with \(\mathcal C_I:=\bigcup\mathcal C_I^{(N,M)}\) it is unconditional (Setup 1.1).
  6. Unverified: summary.html §1 says function-level (SG) holds unconditionally for \(\mathrm{GL}_2/T\); classical.html, Prop. F, proves it by citing "the paper's family" (a sheaf-level result). (IG) for \(\mathrm{GL}_2/T\) fails regardless (Corollary D needs no (SG)).

Checks.

  1. classical-affine.html, Lemma 1(a): the \(K\)-orbits on \(F^n\) are \(\{0\}\) and the shells; \(\mathcal O^n\) contains \(\{0\}\) and every \(X_m\), \(m\ge0\). The hub's sentence holds for homogeneous rows (\(G\to X\) a torsor under a smooth connected group, residue field algebraically closed; Proposition 1.5, (f)).
  2. classical-families.html, Proposition B(i): witnesses "\(\mathrm{St}_{2p}\otimes\chi_0\) (Theorem A) and the untwisted \(\mathrm{St}_{2p}\) ... (Tool 3, Shalika model)"; summary.html Table 2 records Theorem A's hypotheses ("\(\operatorname{char}F=0\)") and Matringe's ("\(G=\mathrm{GL}_n\) over a \(p\)-adic field"). The page header "char \(F=0\) where a cited theorem needs it" is a general disclaimer, not carried to the tables.
  3. group.html, "The affine Whittaker category and the averaging functor": the sentence is a parenthesis; steps 3–6 compute \(\Phi(c_s)\) as \(H_c^*(\mathbb A^1,\exp)\otimes(\cdots)\), i.e. with \(a_!\).
  4. Lemma 1.3, item 3. Where a single orbit is used in the spherical model (satake run, Lemma lem:lang) it serves to make \(\varphi_0=\mathbf 1_{K x_0}\) an orbit indicator; at Iwahori level only \(\operatorname{tr}\mathrm{IC}_0=\pm\varphi_0\) is needed (Lemma 1.6, D2).
  5. classical.html, §5, Proposition F, proof, first item.

Checked and correct (no change needed): case-affine.html steps 1–3 (\(I\subseteq t^\lambda Kt^{-\lambda}\) entry by entry; the isomorphism \(\overline{It^\lambda I}\times^I\mathcal O^n\simeq\) cell \(\times\Lambda_{m,j}\)); case-gm.html (the stabilizer is \((I\cap\mathrm{SL}_2(\mathcal O))\rtimes(1+t^M\mathcal O)\)); classical.html Proposition E (\(\varepsilon\) respects \(T_s^2=(q-1)T_s+q\): \(1=-(q-1)+q\)); classical-affine.html Table 3 (\(\mathbf 1_{gK}\ast\varphi_0=[K:I]\mathbf 1_{g\mathcal O^n}\) with \(\operatorname{vol}(I)=1\), matching classical.html 4C's \((q+1)\) for \(n=2\)).

Builds on

  • Lemma 1.3 · \(\mathbb A^n\) under GL(n): hypotheses hold, every lattice sheaf is a summand
  • Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument

Built on by

  • Summary 1.10 · Verdict of the audit: rows 1–3 agree, the rest reduce to one module computation
Original record, as the agent wrote it at 18:49 UTC

Errors and unproved claims found on the Iwahori-generation pages

Checks.

  1. classical-affine.html, Lemma 1(a): the \(K\)-orbits on \(F^n\) are \(\{0\}\) and the shells; \(\mathcal O^n\) contains \(\{0\}\) and every \(X_m\), \(m\ge0\). The hub's sentence holds for homogeneous rows (\(G\to X\) a torsor under a smooth connected group, residue field algebraically closed; Proposition 1.5, (f)).
  2. classical-families.html, Proposition B(i): witnesses "\(\mathrm{St}_{2p}\otimes\chi_0\) (Theorem A) and the untwisted \(\mathrm{St}_{2p}\) ... (Tool 3, Shalika model)"; summary.html Table 2 records Theorem A's hypotheses ("\(\operatorname{char}F=0\)") and Matringe's ("\(G=\mathrm{GL}_n\) over a \(p\)-adic field"). The page header "char \(F=0\) where a cited theorem needs it" is a general disclaimer, not carried to the tables.
  3. group.html, "The affine Whittaker category and the averaging functor": the sentence is a parenthesis; steps 3–6 compute \(\Phi(c_s)\) as \(H_c^*(\mathbb A^1,\exp)\otimes(\cdots)\), i.e. with \(a_!\).
  4. Lemma 1.3, item 3. Where a single orbit is used in the spherical model (satake run, Lemma lem:lang) it serves to make \(\varphi_0=\mathbf 1_{K x_0}\) an orbit indicator; at Iwahori level only \(\operatorname{tr}\mathrm{IC}_0=\pm\varphi_0\) is needed (Lemma 1.6, D2).
  5. classical.html, §5, Proposition F, proof, first item.

Checked and correct (no change needed): case-affine.html steps 1–3 (\(I\subseteq t^\lambda Kt^{-\lambda}\) entry by entry; the isomorphism \(\overline{It^\lambda I}\times^I\mathcal O^n\simeq\) cell \(\times\Lambda_{m,j}\)); case-gm.html (the stabilizer is \((I\cap\mathrm{SL}_2(\mathcal O))\rtimes(1+t^M\mathcal O)\)); classical.html Proposition E (\(\varepsilon\) respects \(T_s^2=(q-1)T_s+q\): \(1=-(q-1)+q\)); classical-affine.html Table 3 (\(\mathbf 1_{gK}\ast\varphi_0=[K:I]\mathbf 1_{g\mathcal O^n}\) with \(\operatorname{vol}(I)=1\), matching classical.html 4C's \((q+1)\) for \(n=2\)).

Sources