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

Theorem 4.6 · Theorem: (SG) over $\F_q(\!(t)\!)$ implies $\Csat=\cC_K$ over $\overline{\F}_q$

proved audit: correct ★ key step

Second audit: correct. Each step checked: (SG) makes the span of the traces all of C_K ⊗ Q̄_ℓ; a nonzero coordinate functional is nonzero on some generator; zero multiplicity space ⇒ zero coefficient; Lemma 4.5(iii). No purity is needed beyond the decomposition. D5 is an explicit hypothesis at the given p; Bezrukavnikov–Finkelberg Prop. 5 supplies it only for p outside a finite set depending on the finite-type model, which suffices whenever (SG) is known for all q (as for GL4/Sp4 and, later, GL2m/Sp2m).

For a symmetric space with $G^\theta$ and $Z_{G^\theta}(S)$ connected, spherical generation of functions over $\F_q(\!(t)\!)$, for a single $q$, forces every $\IC_\mu$ to be a summand of a Satake object, hence $\Csat=\cC_K$ over $\overline{\F}_q$, modulo D0–D4. So the divergence "functions generate, sheaves do not" cannot occur in this setting.

Theorem 4.6.

Let $X=G/G^\theta$ be symmetric with $G^\theta$ and $Z_{G^\theta}(S)$ connected, defined over $\F_q$, $q$ odd, $T$ split, and assume D0–D4 of Setup 4.1. If (SG) holds for $F=\F_q(\!(t)\!)$, i.e. $$\cH_K\cdot\one_{X(\cO)}=C_c^\infty(X(F))^K,$$ then $\Csat_{\overline{\F}_q}=\cC_{\overline{\F}_q}$; under D5 also $\Csat_{\C}=\cC_{\C}$. The same holds if (SG) holds for $\F_{q^n}(\!(t)\!)$ for one $n$.

By Lemma 4.3, $\cH_K\varphi_0\otimes\overline{\Q}_\ell=\operatorname{span}\{\tr(V_\lambda\ast\IC_0)\}_\lambda$. By Lemma 4.4 (part 1), $\{\tr\IC_\nu\}$ is a basis of $C_K\otimes\overline{\Q}_\ell$; let $\pi_\mu$ be the coordinate functional of $\tr\IC_\mu$.
(SG) says that the span in step 1 is all of $C_K\otimes\overline{\Q}_\ell$. Hence $\pi_\mu$ is nonzero on some $\tr(V_\lambda\ast\IC_0)$, that is, $m^{(1)}_{\mu\lambda}\ne0$ by Lemma 4.4 (part 3).
By Lemma 4.4 (part 4(i)), $E_{\mu,i}(\lambda)\ne0$ for some $i$: a shift of $\IC_\mu$ is a direct summand of $V_\lambda\ast\IC_0$ (Lemma 4.4, part 2).
Lemma 4.5 (part 3) gives $\Csat=\cC_K$.
Over $\F_{q^n}$: the same argument with $\tr_n$. $\square$

What is and is not used. No positivity and no purity beyond the decomposition in D2: step 3 uses only "zero multiplicity space implies zero coefficient". Lang's theorem (Lemma 4.2) is used to know that $C_K$ is spanned by the $\one_{\cO_\mu(\F_q)}$, $\mu\in\Lambda_S^+$; otherwise the traces of the $\IC_\nu$ with constant coefficients would not span $C_K$. The hypothesis "$Z_{G^\theta}(S)$ connected" enters through H3 (constant coefficients) and through Lemma 4.2.

Consequence for the question. For symmetric $X$ on these pages with $Z_{G^\theta}(S)$ connected, "(SG) over $\F_q(\!(t)\!)$ holds but $\Csat\ne\cC_K$" cannot occur (modulo D0–D5).

Builds on

Lemma 4.2 · $K$-orbits on $X(\F_q(\!(t)\!))$ via Lang's theorem: $\Lambda_S^+$ exactly when $Z_{G^\theta}(S)$ is connected
Lemma 4.3 · The trace of $V_\lambda\ast\IC_0$ is $\tr(V_\lambda)\ast\varphi_0$, and the traces of the $V_\lambda$ form a basis of $\cH_K$
Lemma 4.4 · Expansion of $\tr(V_\lambda\ast\IC_0)$ in the IC basis: the coefficients are traces of Frobenius on multiplicity spaces
Lemma 4.5 · $\Csat=\cC_K$ if and only if every $\IC_\mu$ is a direct summand of a Satake object

Built on by

Theorem 4.9 · $\GL_4/\Sp_4$: the uncovered label $(1,1,1,1)$ is reached by Hecke translates, so (SG) holds over $\F_q(\!(t)\!)$ and $\Csat=\cC_K$
Objection 4.12 · Obstacles: $\C$ versus $\overline{\F}_q$, characteristic $0$ in (A1) versus $\F_q(\!(t)\!)$, Weil versus geometric, eigenvalue cancellation
Summary 4.13 · Verdict: one direction is a theorem, the other fails formally; Table 1 rows decided under D0–D5
Proposition 4.7 · The converse fails in general: $\Csat=\cC_K$ gives detection by Hecke translates, not spherical generation
Audit 7.14 · Audit: the three proofs that (SG) implies $\Csat=\cC_K$ are one proof in three notations; inputs listed and compared
Audit 7.4 · Audit of the thick-span criterion and the one-way transfer theorem; the converse is correctly refused
Audit 7.23 · Audit of the $\GL_{2m}/\Sp_{2m}$ theorem: the induction holds (directly, not through Proposition B as stated); the $\SL$ corollary and the sheaf consequence hold
Re-derivation X1.5 · (SG) for GL₂ₘ/Sp₂ₘ re-derived from Lemmas 2–3 alone; A4-13's open cells now decided
Re-derivation X1.6 · The verdict inputs downstream of A1-4, A4-10, A5-6, A7-11: what changes
Summary 9.4 · Synthesis: $E_6/F_4$ is the one divergence (categorical generation holds, functions fail under (A)); no reverse case

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

Theorem: spherical generation over F_q((t)) implies thick generation of C over F_q-bar

Theorem A4-6 ((SG) $\Rightarrow$ $\mathcal C^{\mathrm{Sat}}=\mathcal C$). Let $X=G/G^\theta$ be symmetric with $G^\theta$ and $Z_{G^\theta}(S)$ connected, defined over $\mathbb F_q$, $q$ odd, $T$ split. Assume D0–D4. If (SG) holds for $F=\mathbb F_q(\!(t)\!)$, i.e. $\mathcal H_K\cdot\mathbf 1_{X(\mathcal O)}=C_c^\infty(X(F))^K$, then $\mathcal C^{\mathrm{Sat}}_{\overline{\mathbb F}_q}=\mathcal C_{\overline{\mathbb F}_q}$; under D5 also $\mathcal C^{\mathrm{Sat}}_{\mathbb C}=\mathcal C_{\mathbb C}$. The same holds if (SG) holds for $\mathbb F_{q^n}(\!(t)\!)$ for one $n$.

Proof. 1. By A4-3, $\mathcal H_K\varphi_0\otimes\overline{\mathbb Q}_\ell=\operatorname{span}\{\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)\}_\lambda$. By A4-4.1, $\{\operatorname{tr}\mathrm{IC}_\nu\}$ is a basis of $C_K\otimes\overline{\mathbb Q}_\ell$; let $\pi_\mu$ be the coordinate functional of $\operatorname{tr}\mathrm{IC}_\mu$. 2. (SG) says the span in 1 is all of $C_K\otimes\overline{\mathbb Q}_\ell$, so $\pi_\mu$ is nonzero on some $\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)$, i.e. $m^{(1)}_{\mu\lambda}\ne0$ by A4-4.3. 3. By A4-4.4(i), $E_{\mu,i}(\lambda)\ne0$ for some $i$: $\mathrm{IC}_\mu$ is a direct summand of $V_\lambda\ast\mathrm{IC}_0$ (A4-4.2). 4. A4-5(iii). 5. Over $\mathbb F_{q^n}$: same with $\operatorname{tr}_n$. $\square$

What is and is not used. No positivity and no purity beyond D2's decomposition: step 3 uses only "zero multiplicity space $\Rightarrow$ zero coefficient". Lang (A4-2) is used to know that $C_K$ is spanned by the $\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}$ indexed by $\Lambda_S^+$ (else the $\operatorname{tr}\mathrm{IC}_\nu$ with constant coefficients would not span $C_K$). The hypothesis "Z connected" enters through H3 (constant coefficients) and through A4-2. Consequence for the team question: on these pages, for symmetric $X$ with $Z_{G^\theta}(S)$ connected, "(SG) over $\mathbb F_q(\!(t)\!)$ holds but $\mathcal C^{\mathrm{Sat}}\ne\mathcal C$" cannot occur (modulo D0–D5).

Sources

/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/classical.html (SG), Lemma A
/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/criterion.html Proposition III

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Theorem 4.6 · Theorem: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) over \(\overline{\F}_q\)

Builds on

Built on by

Theorem: spherical generation over F_q((t)) implies thick generation of C over F_q-bar

Sources