§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:40:19Z

Objection 4.12 · Obstacles: \(\C\) versus \(\overline{\F}_q\), characteristic \(0\) in (A1)(A1) Field and group: F is a local field of characteristic 0 (p-adic), and G is split over the integers, with connected fibres and simply connected derived group. Its characteristic-p counterpart is (A1ₚ). versus \(\F_q(\!(t)\!)\), Weil versus geometric, eigenvalue cancellation

open audit: minor defect

Minor defect (second audit). Items 1, 3, 4, 5, 6 and the gap statement of item 2 are correct (item 4's 'iff' holds because Frobenius acts trivially on π0, a quotient of S[2] ⊂ S(F_q)). Two imprecisions in an open list that nothing depends on: (1) the parenthetical 'its axioms 2.4.1–2.4.2 hold there by Lemma 4.2 and Chen–Yi's Iwahori results' is unsupported — Lemma 4.2 parametrizes K-orbits only for Z_{G^θ}(S) connected and does not identify the x_μ with Sakellaridis's A_X^+, nothing in Chen–Yi or the run proves Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points. (xJ ⊆ xB(O)) over F_q((t)), and (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit. contain further hypotheses (A2, A4, A5); (2) Chen–Yi's §8.2 assumption is that T0 is split for a θ-stable Borel pair, not the torus T ⊇ S (as in Setup 4.1).

Lists, with status, what separates the dictionary's theorems from the pages' statements: the passage from \(\overline{\F}_q\) to \(\C\) (D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p).), the absence of any transfer of Sakellaridis's theorem from \(p\)-adic fields to \(\F_q(\!(t)\!)\), the role of Weil structures over all \(\F_{q^n}\), and the orbit count when \(Z_{G^\theta}(S)\) is disconnected.

Objection 4.12.

  1. \(\C\) versus \(\overline{\F}_q\) (D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p).). The pages define \(\cC_K\) over \(\C\); the criterion page already imports steps 3 and 5 from Chen–Yi's \(\ell\)-adic theory and lists the transfer as its open item 1. The dictionary's results Theorem 4.6, Theorem 4.9, Theorem 4.11 are proved over \(\overline{\F}_q\). Passage to \(\C\) needs only that "\(\IC_\mu\) is a summand of \(V_\lambda\ast\IC_0\)" — a statement on the finite-type equivariant model \(\Gr_{\le\lambda}\widetilde\times\overline{\cO_0}\to\overline{\cO_{\varphi(\lambda)}}\), both spread out over a localisation of \(\Z[1/2]\) — is invariant under BBD 6.1.9 / Bezrukavnikov–Finkelberg Prop. 5. Gap, the same as the pages'.
  2. \(p\)-adic versus \(\F_q(\!(t)\!)\). Theorem E assumes \(\operatorname{char}F=0\) (A1)(A1) Field and group: F is a local field of characteristic 0 (p-adic), and G is split over the integers, with connected fibres and simply connected derived group. Its characteristic-p counterpart is (A1ₚ)., so the "(SG) under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies." rows of Table 1 feed nothing into the dictionary, and no comparison of \(C_K\) as an \(\cH_K\)-module between \(\Q_p\) and \(\F_p(\!(t)\!)\) is available here. For \(\GL_{2n}/\Sp_{2n}\) the lattice counts of Theorem 4.9 are field-independent (they count \(\F_q\)-points of fixed schemes), and the proof of Sakellaridis's Theorem 8.0.2 does not visibly use characteristic \(0\) beyond the axioms; but this is unverified, so "(SG) under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies. for \(\GL_{2m}/\Sp_{2m}\), \(m\ge3\)" is not transported. Gap: either (i) prove Theorem E over \(\F_q(\!(t)\!)\) for these \(X\) (its axioms 2.4.1–2.4.2 hold there by Lemma 4.2 and Chen–Yi's Iwahori results), or (ii) run the finite lattice computation of Dead end 4.10.
  3. Weil structure versus geometric sheaf. \(\Csat=\cC_K\) is a geometric statement; the trace needs Weil structures. All objects involved (\(V_\lambda\), \(\IC_0\), \(\IC_{\chi,\mu}\), the convolution) carry canonical ones because \(G,\theta,X,T\) are defined over \(\F_q\) with \(T\) split — Chen–Yi's standing assumption in §8.2, which holds for all Table 1 rows since split forms exist over \(\F_q\). The subtle point is the Frobenius on multiplicity spaces (Lemma 4.4, part 3): a nonzero multiplicity space can have trace zero over \(\F_q\), so detection uses all \(\F_{q^n}\) (Lemma 4.4, part 4), or Chen–Yi's Tate-type assertion (cited only).
  4. Orbits over \(\F_q\) versus \(\Lambda_S^+\). By Lemma 4.2, Lang's theorem gives \(K\backslash X(F)=\Lambda_S^+\) iff \(Z_{G^\theta}(S)\) is connected; otherwise \(H^1(\F_q,\pi_0)\) splits orbits and the simple objects acquire characters \(\chi\) (Theorem 4.11). This replaces the pages' Lemma A, which used Borel–Serre in characteristic \(0\).
  5. Linear independence of the \(\tr\IC_\nu\). Unitriangular with nonzero diagonal with respect to the orbit indicators (Lemma 4.4, part 1), by the support bound D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers. and finiteness of lower sets.
  6. \(\mathcal D\)-modules. The Summary notes that the \(\mathcal D\)-module setting has no Frobenius; the \(\ell\)-adic route is the only dictionary used.

The six items answer the question of what separates the dictionary's theorems from the pages' statements. Items 1 and 2 concern fields: the sheaf statement of the pages is Betti over \(\C\), Sakellaridis's Theorem E is \(p\)-adic, and the dictionary links \(\ell\)-adic sheaves over \(\overline{\F}_q\) with functions over \(\F_q(\!(t)\!)\); a comparison across a row of Table 1 needs both bridges. Item 3 explains why the detection direction (Proposition 4.7, Theorem 4.11) uses all finite extensions of \(\F_q\). Item 4 records where Lang's theorem enters. Items 5 and 6 are settled within the track.

Builds on

Built on by

Original record, as the agent wrote it at 02:40 UTC

Obstacles: C over C vs F_q-bar, char 0 in (A1) vs F_q((t)), Weil vs geometric, eigenvalue cancellation

Answer to (e), each item with its status.

  1. \(\mathbb C\) vs \(\overline{\mathbb F}_q\) (D5). The pages define \(\mathcal C\) over \(\mathbb C\); criterion.html already imports Steps 3 and 5 from Chen–Yi's \(\ell\)-adic theory and lists the transfer as open item 1. The dictionary statements A4-6, A4-9, A4-11 are proved over \(\overline{\mathbb F}_q\); passage to \(\mathbb C\) needs only that "\(\mathrm{IC}_\mu\) is a summand of \(V_\lambda\ast\mathrm{IC}_0\)" (a statement on the finite-type equivariant model \(\mathrm{Gr}_{\le\lambda}\tilde\times\overline{\mathcal O_0}\to\overline{\mathcal O_{\varphi(\lambda)}}\), both spread out over a localisation of \(\mathbb Z[1/2]\)) is invariant under BBD 6.1.9 / Bezrukavnikov–Finkelberg Prop. 5. Gap, same as the pages'.
  2. \(p\)-adic vs \(\mathbb F_q(\!(t)\!)\). Theorem E assumes \(\operatorname{char}F=0\) (A1), so the "(SG) under (A)" rows of Table 1 feed nothing into the dictionary. No general comparison of \(C_K\) as \(\mathcal H_K\)-module between \(\mathbb Q_p\) and \(\mathbb F_p(\!(t)\!)\) is available here. For \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\) the lattice counts of A4-9 are field-independent (they count \(\mathbb F_q\)-points of fixed schemes), and the proof of Sakellaridis 8.0.2 does not visibly use \(\operatorname{char}0\) beyond the axioms; but unverified, so "(SG) under (A) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge3\)" is not transported. Gap: either (i) prove Theorem E over \(\mathbb F_q(\!(t)\!)\) for these \(X\) (its axioms 2.4.1–2.4.2 hold there by A4-2 and Chen–Yi's Iwahori results), or (ii) run the finite lattice computation of A4-10.
  3. Weil structure vs geometric sheaf. \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) is a geometric statement; the trace needs Weil structures. All objects involved (\(V_\lambda\), \(\mathrm{IC}_0\), \(\mathrm{IC}_{\chi,\mu}\), the convolution) carry canonical ones because \(G,\theta,X,T\) are defined over \(\mathbb F_q\) with \(T\) split (Chen–Yi §8.2 standing assumption, which holds for all Table 1 rows: split forms exist over \(\mathbb F_q\)). The subtle point is the Frobenius on multiplicity spaces (A4-4.3): a nonzero multiplicity space can have trace zero over \(\mathbb F_q\), so the detection direction uses all \(\mathbb F_{q^n}\) (A4-4.4), or Chen–Yi's Tate-type assertion (cited only).
  4. Orbits over \(\mathbb F_q\) vs \(\Lambda_S^+\). A4-2: Lang gives \(K\backslash X(F)=\Lambda_S^+\) iff \(Z_K(S)\) connected; otherwise \(H^1(\mathbb F_q,\pi_0)\) splits orbits and the simple objects acquire characters \(\chi\) (A4-11). This is where the pages' Lemma A (which used Borel–Serre in \(\operatorname{char}0\)) is replaced.
  5. Linear independence of \(\operatorname{tr}\mathrm{IC}_\nu\). Unitriangular with nonzero diagonal w.r.t. orbit indicators (A4-4.1), by the support bound D0 and finiteness of lower sets.
  6. \(\mathcal D\)-modules. The Summary notes the \(\mathcal D\)-module setting has no Frobenius; the \(\ell\)-adic route above is the only dictionary used.

Sources