§12 · Agent X2 · SL₂/SO₂ traces · 2026-10-17T20:22:44Z

Lemma X2.1 · $\sigma_n$ is translation by $h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)$ with $h^{-1}\mathrm{Fr}^n(h)=-I$

proved correction

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 $\overline{\F}_q$.

In both $\mathbb F_q$-forms of $\mathrm{SL}_2/\mathrm{SO}_2$, the automorphism $\sigma_n$ is defined over $\mathbb F_{q^n}$ by an element of $\mathrm{PGL}_2(\mathcal O_n)$ and is geometrically the action of $h=\operatorname{diag}(\sqrt u,\sqrt u^{-1})\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)$. Hence the trace of any equivariant Weil complex at $\sigma_n(y)$ equals the trace at $y$ twisted by the action of the central element $-I$.

Lemma X2.1.

Let $q$ be odd, $k=\overline{\mathbb F}_q$, $d\in\{1,-1\}$, and $X_d=\{A\in\mathrm{Sym}_2:\det A=d\}$ with $G=\mathrm{SL}_2$ acting by $g\cdot A=gAg^{\mathsf T}$. Then $X_1$ is the $\mathrm{SO}_2$-form ($\theta(g)=(g^{\mathsf T})^{-1}$, base point $I$, stabilizer $\mathrm{SO}(x^2+y^2)$), and $X_{-1}$ is the split form $\mathrm{SL}_2/T$ (base point $H=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, stabilizer the diagonal torus $T$). For $n\ge1$ put $\mathcal O_n=\mathbb F_{q^n}[\![t]\!]$, $F_n=\mathbb F_{q^n}(\!(t)\!)$, $K_n=\mathrm{SL}_2(\mathcal O_n)$. Fix a non-square $u\in\mathcal O_n^\times$, put $s=\operatorname{diag}(u,1)$, and $$\sigma_n(A)=u^{-1}\,sAs^{\mathsf T}.$$ (a) $\sigma_n$ is an automorphism of $LX_d$ defined over $\mathbb F_{q^n}$, with $\sigma_n(g\cdot A)=(sgs^{-1})\cdot\sigma_n(A)$. It preserves $X_d(\mathcal O_n)$ and commutes with every Hecke operator, so $\mathcal H_{K_n}\varphi_0\subseteq (C_{K_n})^{\sigma_n}$. On $X_{-1}=\mathrm{SL}_2/T$ it is the map $gT\mapsto sgs^{-1}T$ of Proposition J(2).

(b) Over $k$, $\sigma_n(A)=h\cdot A$ with $h=\operatorname{diag}(\sqrt u,\sqrt u^{-1})\in T(k[\![t]\!])\subseteq L^+\mathrm{SL}_2(k)$, and $h^{-1}\mathrm{Fr}^n(h)=-I$.

(c) For every $\mu\ne0$, $\sigma_n$ exchanges the two $K_n$-orbits of $\mathcal O_\mu(\mathbb F_{q^n})$; it preserves $\mathcal O_0(\mathbb F_{q^n})=K_n\cdot x_0$.

(d) Trace twist. Let $(M,\Phi)$ be an $L^+G$-equivariant Weil complex on $LX_d$ with $\mathbb F_{q^n}$-structure $\Phi^{(n)}$. For $y\in LX_d(k)$ let $\varepsilon_y$ be the action of $-I\in\operatorname{Stab}_{L^+G}(y)$ on the stalks $\mathcal H^i(M)_y$. Then $\varepsilon_y$ commutes with $\Phi^{(n)}_y$, and for every $y\in LX_d(\mathbb F_{q^n})$ $$\operatorname{tr}_n(M)(\sigma_ny)=\sum_i(-1)^i\operatorname{tr}\big(\Phi^{(n)}_y\circ\varepsilon_y\ \big|\ \mathcal H^i(M)_y\big).$$

(a), the map. $\det(u^{-1}sAs^{\mathsf T})=u^{-2}u^2\det A$, and the formula has coefficients in $\mathcal O_n$ with inverse of the same shape, so $\sigma_n$ is an $\mathbb F_{q^n}$-automorphism of $LX_d$ preserving $L^+X_d$. One checks $u^{-1}s(gAg^{\mathsf T})s^{\mathsf T}=(sgs^{-1})(u^{-1}sAs^{\mathsf T})(sgs^{-1})^{\mathsf T}$. In the split form, $s^{-1}Hs^{-\mathsf T}=u^{-1}H$, so $\sigma_n(gHg^{\mathsf T})=(sgs^{-1})H(sgs^{-1})^{\mathsf T}$: this is $gT\mapsto sgs^{-1}T$. In the $\mathrm{SO}_2$-form, conjugation by $s$ does not preserve $\mathrm{SO}(x^2+y^2)$, and $\sigma_n$ is the right replacement; it is the action of $[s]\in\mathrm{PGL}_2(\mathcal O_n)$ for $g\cdot A=gAg^{\mathsf T}/\det g$.

(a), Hecke operators. $\operatorname{Ad}(s)$ preserves $K_n$, the Haar measure with $\operatorname{vol}K_n=1$, and each $K_nt^{m\alpha^\vee}K_n=\{g\in\mathrm{SL}_2(F_n):\min\operatorname{val}(g_{ij})=-m\}$, because $s^{\pm1}$ are integral. For $K_n$-biinvariant $f$, substituting $g=sg's^{-1}$ in $(f\ast\varphi)(\sigma_nA)=\sum_{g\in G/K_n}f(g)\varphi(g^{-1}\sigma_nA)$ and using step 1 gives $(f\ast\varphi)\circ\sigma_n=f\ast(\varphi\circ\sigma_n)$. Since $\sigma_n(X_d(\mathcal O_n))=X_d(\mathcal O_n)$, $\varphi_0\circ\sigma_n=\varphi_0$.

(b). As $p$ is odd, $u$ has a square root in $k[\![t]\!]$ (the constant term in $\mathbb F_{q^{2n}}$, the rest by the binomial series). Then $u^{-1}sAs^{\mathsf T}=hAh^{\mathsf T}$ entrywise: $(A_{11}u,\,A_{12},\,A_{22}u^{-1})$. Since $(\mathrm{Fr}^n\sqrt u)^2=u$ and $\sqrt u\notin\mathcal O_n$, $\mathrm{Fr}^n(\sqrt u)=-\sqrt u$, so $\mathrm{Fr}^n(h)=-h$.

(c). For $y=gx_\mu\in\mathcal O_\mu(\mathbb F_{q^n})$, $g\in L^+G(k)$, the $K_n$-orbit of $y$ is the class of $c(y)=\mathrm{Fr}^n(g)^{-1}g$ in $H^1(\mathbb F_{q^n},\pi_0\operatorname{Stab}(x_\mu))$ (Lemma 4.2, Lang's theorem, D3). For $\mu\ne0$, $\pi_0\operatorname{Stab}(x_\mu)=\{\pm I\}$, generated by the central $-I$ (Computation 2.1 for $X_1$; for $X_{-1}$ through the isomorphism over $k$, which carries $-I$ to $-I$); Frobenius acts trivially on a group of order two, so $H^1=\mathbb Z/2$. Now $\sigma_n(y)=(hg)x_\mu$ and $c(\sigma_ny)=\mathrm{Fr}^n(g)^{-1}(-h)^{-1}hg=(-I)\,c(y)$: the class flips. For $\mu=0$ the stabilizer is connected and $\mathcal O_0(\mathbb F_{q^n})$ is one $K_n$-orbit.

(d), conventions. Work with the cohomology sheaves $\mathcal F=\mathcal H^i(M)$, which are equivariant constructible Weil sheaves. Equivariance gives stalk isomorphisms $\alpha_{g,y}:\mathcal F_{gy}\to\mathcal F_y$ for $g\in L^+G(k)$, with the cocycle identity $\alpha_{gg',y}=\alpha_{g',y}\circ\alpha_{g,g'y}$. The Weil structure $\Phi_y:\mathcal F_{\mathrm{Fr}(y)}\to\mathcal F_y$ is a morphism of equivariant sheaves, i.e. $\Phi_y\circ\alpha_{\mathrm{Fr}(g),\mathrm{Fr}(y)}=\alpha_{g,y}\circ\Phi_{gy}$. The same holds for the $n$-fold composite $\Phi^{(n)}$ with $\mathrm{Fr}^n$. Put $\varepsilon_y=\alpha_{-I,y}$.

(d), computation. Let $y$ be $\mathbb F_{q^n}$-rational and $z=hy=\sigma_n(y)$; then $\mathrm{Fr}^n(h)=h\cdot(-I)$ and $\mathrm{Fr}^n(z)=z$. The compatibility with $g=h$ gives $\Phi^{(n)}_y\circ\alpha_{h(-I),y}=\alpha_{h,y}\circ\Phi^{(n)}_z$, and the cocycle identity gives $\alpha_{h(-I),y}=\alpha_{-I,y}\circ\alpha_{h,(-I)y}=\varepsilon_y\circ\alpha_{h,y}$. Hence

$$\Phi^{(n)}_z=\alpha_{h,y}^{-1}\circ\big(\Phi^{(n)}_y\circ\varepsilon_y\big)\circ\alpha_{h,y},$$

so the traces on $\mathcal F_z$ and $\mathcal F_y$ agree after inserting $\varepsilon_y$. Taking the alternating sum over $i$ gives the formula. With $g=-I$, which is rational, the compatibility gives $\Phi_y\circ\varepsilon_{\mathrm{Fr}(y)}=\varepsilon_y\circ\Phi_y$, so $\varepsilon$ commutes with $\Phi^{(n)}$. $\square$

Status. Proved, given the standing formalism D1 of Setup 4.1: equivariant Weil complexes whose stalks carry the equivariance isomorphisms. This is the same input as the Lang-torsor trace computations of Computation 2.8(b), Proposition 1.7 and Correction 10.8, step 5. Answer to starting point (1): $\sigma_n$ is induced $\mathbb F_{q^n}$-rationally by $[s]\in\mathrm{PGL}_2(\mathcal O_n)$, a group normalizing $K_n$ and permuting the stabilizers, and geometrically by $h\in L^+\mathrm{SL}_2(k)$. If $u$ is a constant non-square, then $h\in T(\mathbb F_{q^{2n}})$ is even a constant loop. Only the cocycle $h^{-1}\mathrm{Fr}^n(h)=-I$ is used.

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 4.2 · $K$-orbits on $X(\F_q(\!(t)\!))$ via Lang's theorem: $\Lambda_S^+$ exactly when $Z_{G^\theta}(S)$ is connected
Computation 2.1 · Stabilizers for $\SL_2/\SO_2$: $\mu_2$ times a pro-unipotent group
Proposition 1.2 · $\SL_2/T$ versus $\GL_2/T$: one variety, two Hecke algebras
Correction 10.8 · Correction: the square-class automorphism multiplies $\tr_n\IC_{\chi,\nu}$ by $\chi(-1)$

Built on by

Lemma X2.2 · Weil inertia lemma: $\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}$
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
Proposition X2.6 · (SG) fails for $\mathrm{SL}_2/\mathrm{SO}(x^2+y^2)$ over every $F$, $q$ odd
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

classical.html Proposition J(2)
chen-yi-v4.pdf Prop. 17
Computation 2.1
Lemma 4.2

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 X2.1 · \(\sigma_n\) is translation by \(h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)\) with \(h^{-1}\mathrm{Fr}^n(h)=-I\)

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