The agent was asked to audit the literature: what the Iwahori-generation pages prove at each level, in which setting, and whether the hypotheses the other agents would need hold row by row. It started with the settings. The pages' coherent \(\mathcal D\)-modules and this run's constructible sheaves give the same \(\mathcal C_I\), Hecke action and \(\mathrm{IC}_0\) level by level, and with \(\mathcal C_I\) defined as the union of the finite levels, the first half of the pages' hypothesis 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). becomes a definition ( (Setup 1.1)).
It then checked the group, \(\mathbb A^n\) and \(\mathbb G_m\) one by one: finitely many pieces, connected stabilizers, rational orbits, a finite-length heart and the decomposition with parity ( (Lemma 1.2), (Lemma 1.3), (Lemma 1.4)). For the group it recorded the expected summand set, the lowest two-sided cell, which misses the unit. Its status table ( (Audit 1.7)) reduces both levels, for every row, to Chen–Yi's affine Hecke module and the class of \(\mathrm{IC}_0\): sheaves generate when the classes \(C'_w m_0\) together involve every basis element, functions when their specializations span. So over \(\mathbb F_q(\!(t)\!)\) the only possible divergence is "sheaves generate, functions do not".
It also listed errors and unproved asides on the pages ( (Objection 1.8)): \(L^+\mathbb A^n\) is not one \(K\)-orbit, and two function failures in Table 1 lack the condition \(\operatorname{char}F=0\). Its locations of the standard inputs ( (Setup 1.9)) fed the literature of the proposition pages.
Result as the agent reported it partial
For every row of Table 1 the Iwahori-level status is recorded with setting and argument (Audit 1.7): group (and Spin_4/Spin_3) sheaves false (D-modules/C, Whittaker; l-adic same) and (IG) fails for every F; A^n and G_m sheaves true in all settings (A^n unconditional for C_I = union of finite levels) and (IG) holds for every F; GL_2/T (IG) fails for every F, sheaves open; the other symmetric rows have sheaves open and (IG) failures only in char F = 0 (or open). No divergence is proved. All hypotheses of the Iwahori summand criterion and of the trace bridge are verified for rows 1-3 over F_q-bar and C (finite levels, connected stabilizers, rational orbits, finite-length heart, decomposition with parity), with M_I = lowest two-sided cell (missing e) for the group and M_I = all simples for A^n and G_m; L^+X is not one K-orbit for A^n (only tr IC_0 = +-phi_0 is needed). For the symmetric rows (over F_q-bar, p odd) 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)., heart, decomposition and local systems are Chen-Yi theorems, I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant. holds for the GL_n/(GL_p x GL_q), GL_2n/Sp_2n, SL_2n/Sp_2n and Spin_2n/Spin_2n-1 families via connectedness of T_0^{psi_v}. Both levels are statements about Chen-Yi's module M and m_0 = [IC_0]: sheaves = support of the C'_w m_0 is everything, (IG) over F_q((t)) = their specializations span; hence a divergence over F_q((t)) can only be 'sheaves hold, (IG) fails', and for GL_2/T it occurs iff the support is full. Citations for every standard input are located (Setup 1.9), with the locally checked ones distinguished from recalled ones.
Gap
(1) Frobenius-stability of every I-orbit for the given q on the symmetric rows (only K-orbits are proved stable; Chen-Yi and Mars-Springer enlarge q). (2) I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant. for E_6/F_4 (needs the image of p: V -> twisted involutions). (3) The cell results for the group (Lusztig 1985 Thm 5.4, Thm 7.2; Shi 1987) and Kazhdan-Lusztig 1979/1980, Bernstein-Lunts, JMW are recalled, not re-read; BBD numbers not re-read (scan). (4) The support set M_I is not computed for any symmetric row; char-0 (IG) witnesses are not transferred to F_q((t)).
Blocked routes
- Frobenius-stability of all I-orbits with n = 1 via rational representatives of twisted involutions: needs that every twisted L^+T_0-conjugacy class in tau(V) over a twisted involution contains an F_q((t))-point, i.e. vanishing of a class in H^1(F_q, pi_0 T_0^{inv o psi}); not proved.
- Deciding I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant. for E_6/F_4 by H^1(<-u>, X_*(T)) for involutions u in W(E_6): needs the set of twisted involutions realized by I_0-orbits (Chen-Yi Lemmas 25-26); -1 itself has H^1 = L/2L != 0.
- Transfer of the char-0 (IG) failures (Shtotland Thm A, Matringe, Offen-Sayag) to F_q((t)): needs an H_I-module isomorphism C_c^infty(X(F))^I = C_c^infty(X(F'))^I with phi_0 -> phi_0 for F, F' with the same residue field (close local fields); Shtotland's presentation might give it, not checked.
- Sheaves => functions at Iwahori level: needs a spanning statement for the specialized classes C'_w m_0, not the support statement that C_I^Iwa = C_I gives; fails in general without unitriangularity (Laumon 1.1.1.5).
Brief as sent
Target: For every row of Table 1 (proof_graphs/iwahori-generation/index.html Table 1 and summary.html), establish exactly what is proved at Iwahori level at both levels (sheaves: C_I^{Iwa} = C_I; functions: (IG)), in which setting (D-modules, Betti, ℓ-adic; which F), and by which argument. Check, row by row, the hypotheses that the Iwahori summand criterion and the functions ⇒ sheaves bridge will need: a level presentation with finitely many pieces per level (I1), connected stabilizers at each level (I2), every finite-level orbit defined over F_q (Frobenius-stable) for the split forms, L^+X a single K-orbit with IC_0 the constant sheaf on a smooth closed subscheme. Locate exact citations (theorem numbers, in the local PDFs where possible) for the standard inputs the other agents will need.
Approach: Literature and status audit. Starting points: (1) read summary.html and index.html Table 1; read classical.html (Proposition B, Corollary D, E the group case) and classical-affine.html (F3) and record exactly which (IG) statements are proved, for which F, by which method; (2) check that the pages' coherent D-module setting and this run's constructible setting give the same C_I, H_{c,I}, convolution and IC_0 (Riemann–Hilbert at each finite level with finitely many orbits; regular holonomicity of equivariant D-modules); (3) for each row decide whether I1, I2 and Frobenius-stability of the finite-level orbits hold, fail or are open, with reasons; (4) locate the citations: Kazhdan–Lusztig categorification of the (affine) Hecke algebra by IC sheaves on flag varieties and its multiplicativity; parity sheaves (Juteau–Mautner–Williamson); the decomposition theorem (BBD 5.4.5, 6.2.5; equivariant form, Bernstein–Lunts); the lowest two-sided cell of an affine Weyl group (Shi; Lusztig, Cells in affine Weyl groups I–II) and the a-function; Lang's theorem for connected groups acting on orbits; the sheaf–function dictionary (Laumon 1987, local); (5) record any error or unproved claim you find in the pages.