§12 · Agent X2 · SL₂/SO₂ traces · 2026-10-17T20:25:47Z

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

proved correction ★ key step

Correction. Corrects Theorem 4.11 · $\SL_2/\SO_2$: the sign-twisted IC sheaves are not in the Satake span, so $\Csat\ne\cC_K$ over $\overline{\F}_q$.

With σ replaced by the $\mathrm{PGL}_2(\mathcal O_n)$-action of Lemma X2.1 and the missing invariance supplied by Lemma X2.2, every step of Theorem 4.11 goes through. The theorem holds as stated, modulo D0–D2, and uses neither Chen–Yi §8.2 nor Thm. 81, so it covers the $\mathrm{SO}_2$-form also for $q\equiv3\bmod4$. Correction 10.8 is confirmed.

Theorem (repair of A4-11) X2.4.

Let $q$ be odd and $X=X_d$ either $\mathbb F_q$-form of $\mathrm{SL}_2/\mathrm{SO}_2$ (Lemma X2.1). Assume D0–D2 of Setup 4.1 in the form with disconnected stabilizers: simple perverse objects $\mathrm{IC}_{\chi,\mu}$ with $\chi\in\{\mathrm{triv},\mathrm{sgn}\}$ for $\mu\ne0$; $V_\lambda\ast\mathrm{IC}_0$ pure and geometrically $\simeq\bigoplus_b\mathrm{IC}_{\chi_b,\nu_b}[s_b]$. Then for all $n\ge1$, all $\lambda\in\Lambda_T^+$ and all $\mu\ne0$: $$m^{(n)}_{(\mathrm{sgn},\mu),\lambda}=0,\qquad E_{(\mathrm{sgn},\mu),i}(\lambda)=0\ \ \forall i,\qquad \mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C^{\mathrm{Sat}}_{K,\overline{\mathbb F}_q}.$$ Hence $\mathcal C^{\mathrm{Sat}}_{K,\overline{\mathbb F}_q}\ne\mathcal C_{K,\overline{\mathbb F}_q}$. Over $\mathbb C$ the same holds unconditionally by Theorem 2.4 (Betti), with no D5 needed.

The original Theorem 4.11 holds after two named changes: (i) σ is the map $A\mapsto u^{-1}sAs^{\mathsf T}$, which is conjugation by $s$ only on the split form; (ii) step 2's invariance of $\operatorname{tr}_n\mathrm{IC}_{\mathrm{triv},\mu}$ is Lemma X2.2(c) or (d). Its Remark (i) should cite Chen–Yi Remark 13 for the nontrivial coefficients; Remark 89 concerns non-semisimplicity.

Function side. By Lemma X2.1(a), $\mathcal H_{K_n}\varphi_0\subseteq(C_{K_n})^{\sigma_n}$, and by Lemma 4.3 (form-independent: it uses only $G=\mathrm{SL}_2$, split) $\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}_n(V_\lambda)\ast\varphi_0\in\mathcal H_{K_n}\varphi_0\otimes\overline{\mathbb Q}_\ell$.

Basis. $C_{K_n}\otimes\overline{\mathbb Q}_\ell$ has as basis the indicators of the $K_n$-orbits: one for $\mu=0$ and two, $O'_\mu\ni x_\mu$ and $O''_\mu$, for $\mu\ne0$ (Lemma 4.2, Lemma X2.1(c)). $\mathcal L_\chi$ is Frobenius-stable, since Frobenius acts trivially on $\{\pm I\}$, so $\mathrm{IC}_{\chi,\mu}$ has a canonical Weil structure normalised at $x_\mu$. Its trace is supported on $\overline{\mathcal O_\mu}(\mathbb F_{q^n})$ and equals $(-1)^{\delta(\mu)}(\mathbf 1_{O'_\mu}+\chi(-1)\mathbf 1_{O''_\mu})$ on $\mathcal O_\mu(\mathbb F_{q^n})$ (Lang torsor, or Lemma X2.2(c)). The labels $m\alpha^\vee$ are totally ordered and the $2\times2$ blocks $\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ are invertible, so the $\operatorname{tr}_n\mathrm{IC}_{\chi,\mu}$ form a basis.

σ-eigenbasis (the missing step). By Lemma X2.2(c), $\operatorname{tr}_n\mathrm{IC}_{\mathrm{triv},\mu}$ is $\sigma_n$-invariant and $\operatorname{tr}_n\mathrm{IC}_{\mathrm{sgn},\mu}$ is $\sigma_n$-anti-invariant on all of $X(F_n)$, lower orbits included. So $(C_{K_n})^{\sigma_n}\otimes\overline{\mathbb Q}_\ell=\operatorname{span}\{\operatorname{tr}_n\mathrm{IC}_{\mathrm{triv},\mu}\}$, and the $(-1)$-eigenspace is the span of the sign traces. Only the trivial half is needed: if $g=\sum_\mu a_\mu\operatorname{tr}_n\mathrm{IC}_{\mathrm{sgn},\mu}$ is σ-invariant and $\mu_0$ is the largest label with $a_{\mu_0}\ne0$, then on $\mathcal O_{\mu_0}(\mathbb F_{q^n})$ one has $g=\pm a_{\mu_0}(\mathbf 1_{O'}-\mathbf 1_{O''})$, which is anti-invariant; hence $a_{\mu_0}=0$. For the trivial half, Lemma X2.2(d) gives a second proof by descent to $\mathrm{PGL}_2$.

Coefficients. Expand $\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\sum m^{(n)}_{(\chi,\nu),\lambda}\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}$ with $m^{(n)}=\sum_i(-1)^i\operatorname{tr}(A^n_{(\chi,\nu),i})$, as in Lemma 4.4, part 3. That argument is valid verbatim for the $\mathrm{IC}_{\chi,\nu}$: $\operatorname{End}=\overline{\mathbb Q}_\ell$, Hom between non-isomorphic simples is zero, and each is Frobenius-stable. By steps 1 and 3, every sign coefficient vanishes: $m^{(n)}_{(\mathrm{sgn},\mu),\lambda}=0$ for all $n$.

Weights. ${}^pH^i(V_\lambda\ast\mathrm{IC}_0)$ is pure of weight $w+i$ (D2; BBD 5.4.1). $\mathrm{IC}_{\chi,\nu}$ is pure of weight $\delta(\nu)$: $\mathcal L_\chi$ has Frobenius eigenvalues $\pm1$ at rational points, and Gabber's theorem (BBD 5.3.2) applies on finite-type models. So the eigenvalues of $A_{(\chi,\nu),i}$ have absolute value $q^{(w+i-\delta(\nu))/2}$, distinct for distinct $i$, and the Vandermonde argument of Lemma 4.4, part 4(ii), gives $E_{(\mathrm{sgn},\mu),i}(\lambda)=0$. Chen–Yi's pointwise purity (Thm. 81), proved under the §8.2 hypotheses, is not used.

Non-generation. Lemma 4.5, part 2, holds verbatim with the simples $\mathrm{IC}_{\chi,\mu}$, as in Lemma 1.5. With $\Sigma$ all simples except $\mathrm{IC}_{\mathrm{sgn},\mu}$, it gives $\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}$.

Form-independence. Steps 1–6 use only: $G=\mathrm{SL}_2$ split; rational $x_\mu$; $\pi_0\operatorname{Stab}=\{\pm I\}$ with trivial Frobenius action; and $\sigma_n$ from Lemma X2.1. These hold in both forms for every odd $q$. So the theorem holds for the $\mathrm{SO}_2$-form also when $q\equiv3\bmod4$, where Chen–Yi's §8.2 hypothesis fails (Correction (repair of A7-16) X2.5). The conclusion is geometric over $\overline{\mathbb F}_q$, so it does not depend on the form.

Cross-check without traces. The ℓ-adic form of Lemma 2.6 gives $V_{n\alpha^\vee}\ast\mathrm{IC}_0\cong\mathrm{IC}(\overline{\mathcal O_{2n}},\overline{\mathbb Q}_\ell)$ in every odd characteristic; the second audit verified that the convolution map is an isomorphism over $\mathcal O_{2n}$ there. With Lemma 4.5, part 2, this already excludes every $\mathrm{IC}_{\mathrm{sgn},\mu}$. The two routes agree.

Correction 10.8 checked. Steps 1–6 of Correction 10.8 are correct for the split form. In step 3, $\Phi'=c\,\Phi$ needs $\operatorname{Aut}=$ scalars, which holds already for the underlying object of $\mathrm{IC}_{\chi,\nu}$ (rank-one local system on a connected orbit); and step 5 is Lemma X2.2(a). Lemma X2.2 gives the same identity in both forms, and a second proof for $\chi=\mathrm{triv}$. $\square$

Builds on

Theorem 4.11 · $\SL_2/\SO_2$: the sign-twisted IC sheaves are not in the Satake span, so $\Csat\ne\cC_K$ over $\overline{\F}_q$
Lemma X2.1 · $\sigma_n$ is translation by $h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)$ with $h^{-1}\mathrm{Fr}^n(h)=-I$
Lemma X2.2 · Weil inertia lemma: $\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}$
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
Lemma 1.5 · Summand criterion: $\Csat=\mathcal C_K$ if and only if every simple IC sheaf is a summand of a Satake object
Correction 10.8 · Correction: the square-class automorphism multiplies $\tr_n\IC_{\chi,\nu}$ by $\chi(-1)$
Lemma 2.6 · Each Satake object is a single IC sheaf: $V_{n\alpha^\vee}\ast\IC_0\simeq\IC(\overline{\mathcal O_{2n}})$

Built on by

Audit X2.7 · No $\mathrm{SL}_2/\mathrm{SO}_2$ conclusion of the run rests on the false §8.2 remark
Summary X2.8 · Re-derivation of Summary 4.13, Summary 7.19, Summary 7.25, Summary 7.32 from the repaired steps
Re-check V2 · Re-check by V2
Summary 9.6 · Verdict after the repairs: every input held or repaired; $E_6/F_4$ still the one divergence

Sources

Lemma 4.3
Lemma 4.4 parts 3-4
Lemma 4.5 part 2
BBD 5.3.2, 5.4.1
chen-yi-v4.pdf Remarks 13 and 89
Lemma 2.6 and its second-audit note on odd characteristic

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 (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

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Sources