§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:31:20Z

Lemma 4.2 · $K$-orbits on $X(\F_q(\!(t)\!))$ via Lang's theorem: $\Lambda_S^+$ exactly when $Z_{G^\theta}(S)$ is connected

proved audit: minor defect

Minor defect (second audit). The lemma is correct as stated (re-derived: cocycle bijection, Lang for L+G by levelwise Lang plus a compactness argument, compact-openness from smoothness of the orbit map). Defects, none affecting a claim's truth: (1) the title's 'exactly when' needs the converse — if Z_{G^θ}(S) is disconnected some O_μ(F_q) splits — which the body does not argue; it holds in one line: Z_{G^θ}(S) = Z_{G^θ}(S)°·S[2] (Proposition III, step 4, 'finite check'), so π0 Z_{G^θ}(μ) is a quotient of S[2] ⊂ S(F_q), Frobenius acts trivially and H^1(F_q, π0) = π0 ≠ 1 for regular μ. (2) K denotes G(O) and, in Z_K(μ), Z_K(S), the symmetric subgroup (the criterion page's K). (3) 'Lemma Step 4(a),(b),(c)' is the numbering of the Proposition III PDF, not of criterion.html. (4) The identification with Sakellaridis's Axiom 2.4.1 is loose: what is proved is the bijectivity K\X(F) → Λ_S^+ expected in his Remark after 2.4.2, and it needs Z_{G^θ}(S), not G^θ, connected — the node's own SL2/SO2 check is a case with G^θ connected where the map is 2-to-1.

Over $\F_q(\!(t)\!)$ the $K$-orbits inside each geometric orbit $\cO_\mu$ are counted by Frobenius-twisted classes in $\pi_0Z_{G^\theta}(\mu)$; when $Z_{G^\theta}(S)$ is connected there is one compact open $K$-orbit per label and the orbit indicators form a basis of $C_K$. This replaces the Borel–Serre step of Lemma A, which needs characteristic 0.

Lemma 4.2.

Assume D0 and D3 of Setup 4.1, and let $K=G(\cO)$. Then $$X(F)=\bigsqcup_{\mu\in\Lambda_S^+}\cO_\mu(\F_q),$$ each $\cO_\mu(\F_q)$ is compact open in $X(F)$, and $$K\backslash\cO_\mu(\F_q)\ \simeq\ H^1\big(\F_q,\Stab_{L^+G}(x_\mu)\big)\ \simeq\ \{\text{Frobenius-twisted conjugacy classes of }\pi_0Z_{G^\theta}(\mu)\}.$$ In particular, if $Z_{G^\theta}(S)$ is connected (condition (1) of Proposition III), then $K\backslash X(F)=\Lambda_S^+$ with $\cO_\mu(\F_q)=K\cdot x_\mu$, and $C_K$ has the basis $\{\one_{\cO_\mu(\F_q)}\}_{\mu\in\Lambda_S^+}$.

The centralizers $Z_{G^\theta}(\mu)$ and $Z_{G^\theta}(S)$ are taken in the symmetric subgroup $G^\theta$; $K$ is the arc group's $\F_q$-points.

$X(F)=LX(\F_q)\subseteq LX(k)=\bigsqcup_\mu\cO_\mu(k)$ (D0), and each $\cO_\mu$ is Frobenius-stable (Setup 4.1). Hence $X(F)=\bigsqcup_\mu\cO_\mu(\F_q)$.

Let $y\in\cO_\mu(\F_q)$ and write $y=gx_\mu$ with $g\in L^+G(k)$. Then $c(y)=\mathrm{Fr}(g)^{-1}g$ lies in $\Stab(x_\mu)(k)$; it is a cocycle whose class does not depend on $g$, and two points are $K$-conjugate exactly when their classes agree. This uses $H^1(\F_q,L^+G)=1$: each $G(\cO/t^n)$ is the group of $\F_q$-points of a connected smooth group (an iterated affine-space fibration over $G$), the transition maps are surjective with connected kernels, Lang's theorem holds at each level, and a compactness argument on the finite nonempty sets of solutions passes to the limit. Every class occurs: given a cocycle $c\in\Stab(k)$, Lang's theorem in $L^+G$ gives $g$ with $\mathrm{Fr}(g)^{-1}g=c$, and $y=gx_\mu$ is rational.

$\Stab_{L^+G}(x_\mu)=Z_G(\mu)^\theta\ltimes(\text{pro-unipotent})^\theta$ (Proposition III, step 4(a)). The pro-unipotent part is a limit of connected unipotent groups, so

$$H^1(\F_q,\Stab)=H^1\big(\F_q,Z_{G^\theta}(\mu)\big)=H^1\big(\F_q,\pi_0Z_{G^\theta}(\mu)\big)$$

by Lang's theorem for $Z_{G^\theta}(\mu)^\circ$. Finally $\pi_0Z_{G^\theta}(\mu)=1$ for all $\mu$ if and only if $Z_{G^\theta}(S)$ is connected (Proposition III, step 4(b),(c)).

Compactness and openness: $K$ is compact; the orbit map $G\to X$ is smooth, so $G(F)\to X(F)$ is open, and $K$ is open in $G(F)$. A $K$-invariant compactly supported function is therefore a finite sum of orbit indicators. $\square$

Check against Proposition J(2). For $\SL_2/\SO_2$: $Z_{G^\theta}(\mu)=\{\pm1\}$ for $\mu\ne0$ and $Z_{G^\theta}(0)=\SO_2$ is connected, so $\cO_\mu(\F_q)$ consists of two $K$-orbits for $\mu\ne0$ and of one for $\mu=0$ — the square classes of the function page, now derived from the sheaf-side stabilizers. For $\GL_{2n}/\Sp_{2n}$, $\GL_n/(\GL_p\times\GL_q)$, the spheres, $E_6/F_4$ and the group there is one orbit per label (the column $Z_{G^\theta}(S)$ of the criterion page's table).

Relation to Sakellaridis. This is Sakellaridis's Axiom 2.4.1, together with the bijectivity he expects in the remark after Axiom 2.4.2 ("we certainly expect that the second map is bijective when $H$ is connected"), for $F=\F_q(\!(t)\!)$, proved here under D0 and D3.

Builds on

Setup 4.1 · Set-up of the sheaf–function dictionary over $\F_q(\!(t)\!)$: Weil objects, trace functions, obligations D0–D5

Built on by

Lemma 4.4 · Expansion of $\tr(V_\lambda\ast\IC_0)$ in the IC basis: the coefficients are traces of Frobenius on multiplicity spaces
Theorem 4.11 · $\SL_2/\SO_2$: the sign-twisted IC sheaves are not in the Satake span, so $\Csat\ne\cC_K$ over $\overline{\F}_q$
Theorem 4.6 · Theorem: (SG) over $\F_q(\!(t)\!)$ implies $\Csat=\cC_K$ over $\overline{\F}_q$
Lemma 4.8 · Under the two conditions of Proposition III, (SG) holds over $\F_q(\!(t)\!)$ without Sakellaridis's hypotheses
Audit 7.3 · Audit of the orbit count, the trace dictionary and the IC expansion: they hold under D0–D3, and D4 is elementary
Correction 10.1 · Correction: the mechanism of the $E_6/F_4$ divergence, separated by field
Correction 10.8 · Correction: the square-class automorphism multiplies $\tr_n\IC_{\chi,\nu}$ by $\chi(-1)$
Lemma X2.1 · $\sigma_n$ is translation by $h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)$ with $h^{-1}\mathrm{Fr}^n(h)=-I$
Theorem (repair of A4-11) X2.4 · Repair of A4-11: $\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}$ in both $\mathbb F_q$-forms, $q$ odd
Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the $\mathrm{SO}_2$-form iff $q\equiv1\bmod4$; Lang’s count holds in both
Audit X2.7 · No $\mathrm{SL}_2/\mathrm{SO}_2$ conclusion of the run rests on the false §8.2 remark

Original record, as the agent wrote it at 02:31 UTC

K-orbits on X(F_q((t))) via Lang: Lambda_S^+ exactly when Z_K(S) is connected

Lemma A4-2. (proved modulo D0, D3.) $X(F)=\bigsqcup_{\mu\in\Lambda_S^+}\mathcal O_\mu(\mathbb F_q)$, each $\mathcal O_\mu(\mathbb F_q)$ is compact open in $X(F)$, and $$K\backslash\mathcal O_\mu(\mathbb F_q)\ \simeq\ H^1\big(\mathbb F_q,\mathrm{Stab}_{L^+G}(x_\mu)\big)\ \simeq\ \{\text{Fr-twisted conjugacy classes of }\pi_0Z_K(\mu)\}.$$ In particular, if $Z_K(S)$ is connected (condition (1) of Proposition III) then $K\backslash X(F)=\Lambda_S^+$ with $\mathcal O_\mu(\mathbb F_q)=K\cdot x_\mu$, and $C_K$ has the basis $\{\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}\}_{\mu\in\Lambda_S^+}$.

Proof. 1. $X(F)=LX(\mathbb F_q)\subseteq LX(k)=\bigsqcup\mathcal O_\mu(k)$ and each $\mathcal O_\mu$ is Fr-stable (A4-1), so $X(F)=\bigsqcup_\mu\mathcal O_\mu(\mathbb F_q)$.

Let $y\in\mathcal O_\mu(\mathbb F_q)$, $y=gx_\mu$ with $g\in L^+G(k)$. Then $c(y):=\mathrm{Fr}(g)^{-1}g\in\mathrm{Stab}(x_\mu)(k)$ is a 1-cocycle whose class in $H^1(\mathbb F_q,\mathrm{Stab})$ is independent of $g$, and $y,y'$ are $K$-conjugate iff the classes agree (standard, using $H^1(\mathbb F_q,L^+G)=1$: $L^+G=\varprojlim G(\mathcal O/t^n)$, each $G(\mathcal O/t^n)$ is the group of $\mathbb F_q$-points of a connected smooth algebraic group (iterated affine-space fibration over $G$), transition maps surjective with connected kernels; Lang at each level and a compactness argument on the finite nonempty solution sets give the limit statement). Surjectivity onto $H^1$: given a cocycle $c\in\mathrm{Stab}(k)$, Lang in $L^+G$ gives $g$ with $\mathrm{Fr}(g)^{-1}g=c$; then $y=gx_\mu$ is rational.
$\mathrm{Stab}_{L^+G}(x_\mu)=Z_G(\mu)^\theta\ltimes(\text{pro-unipotent})^\theta$ (criterion.html Lemma Step 4(a)); the pro-unipotent part is a limit of connected unipotent groups, so $H^1(\mathbb F_q,\mathrm{Stab})=H^1(\mathbb F_q,Z_K(\mu))=H^1(\mathbb F_q,\pi_0Z_K(\mu))$ (Lang for $Z_K(\mu)^\circ$), and $\pi_0Z_K(\mu)=1$ for all $\mu$ iff $Z_K(S)$ is connected (Lemma Step 4(b),(c)).
Compact open: $K$ is compact; $G\to X$ is smooth, so $G(F)\to X(F)$ is open, and $K$ is open in $G(F)$. A $K$-invariant compactly supported function is a finite sum of orbit indicators. $\square$

Check (consistency with classical.html Prop. J(2)). For $\mathrm{SL}_2/\mathrm{SO}_2$: $Z_K(\mu)=\{\pm1\}$ for $\mu\ne0$ and $Z_K(0)=\mathrm{SO}_2$ connected, so $\mathcal O_\mu(\mathbb F_q)$ is two $K$-orbits for $\mu\neq0$ and one for $\mu=0$: the square classes of the function page, now derived from the sheaf-side stabilizers. For $\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$, $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$, spheres, $E_6/F_4$, the group: one orbit per label (Table 1 column "Z connected"). This is Sakellaridis's Axiom 2.4.1 ("we certainly expect the second map is bijective when $H$ is connected") for $F=\mathbb F_q(\!(t)\!)$, proved here under D0, D3.

Sources

/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/criterion.html Lemma Step 4
/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/classical.html Lemma A, Prop. J(2)
../two-definitions-iwahori/sources/sakellaridis-v3.pdf Axiom 2.4.1 and the Remark after 2.4.2
sources/chen-yi-v4.pdf Prop. 17

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

Lemma 4.2 · \(K\)-orbits on \(X(\F_q(\!(t)\!))\) via Lang's theorem: \(\Lambda_S^+\) exactly when \(Z_{G^\theta}(S)\) is connected

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K-orbits on X(F_q((t))) via Lang: Lambda_S^+ exactly when Z_K(S) is connected

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