§12 · Agent X2 · SL₂/SO₂ traces · 2026-10-17T20:23:32Z

Lemma X2.2 · Weil inertia lemma: $\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}$

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 $\overline{\F}_q$.

The inertia automorphism of $-I$ is a scalar on each simple $\mathrm{IC}_{\chi,\nu}$, read off at the top stalk, so no sign local system occurs in any cohomology sheaf of $\mathrm{IC}_{\mathrm{triv},\mu}$, and $\mathrm{IC}_{\mathrm{sgn},\mu}$ vanishes on $L^+X$. With the trace twist this proves the σ-behaviour claimed in step 2 of Theorem 4.11 and in Correction 10.8, in both $\mathbb F_q$-forms; a second, $\mathrm{PGL}_2$-descent proof covers the trivial case.

Lemma X2.2.

Let $X_d$, $q$ odd, $\sigma_n$ and $h$ be as in Lemma X2.1. Assume D1 of Setup 4.1: equivariant Weil complexes; the perverse t-structure whose simples are $\mathrm{IC}_{\chi,\nu}=j_{!*}\mathcal L_\chi[\delta(\nu)]$, $\chi\in\operatorname{Irr}\pi_0\operatorname{Stab}(x_\nu)$; and $\operatorname{End}(\mathrm{IC}_{\chi,\nu})=\overline{\mathbb Q}_\ell$. Give $\mathrm{IC}_{\chi,\nu}$ its canonical Weil structure $\Phi$, normalised at a rational $x_\nu$. Put $\chi(-1):=1$ for $\nu=0$.

(a) For all $\nu,\chi,i$ and all $y\in\overline{\mathcal O_\nu}(k)$, $-I$ acts on $\mathcal H^i(\mathrm{IC}_{\chi,\nu})_y$ by the scalar $\chi(-1)$.

(b) For $\nu'<\nu$: $\mathcal H^i(\mathrm{IC}_{\mathrm{triv},\nu})|_{\mathcal O_{\nu'}}\cong\mathcal L_{\mathrm{triv}}^{\oplus r_i}$, so no sign local system occurs in any cohomology sheaf of $\mathrm{IC}_{\mathrm{triv},\nu}$ along any lower orbit. For $\nu'\ne0$, $\mathcal H^i(\mathrm{IC}_{\mathrm{sgn},\nu})|_{\mathcal O_{\nu'}}\cong\mathcal L_{\mathrm{sgn}}^{\oplus r'_i}$, and $\mathrm{IC}_{\mathrm{sgn},\nu}|_{\mathcal O_0}=0$.

(c) As $\mathbb F_{q^n}$-Weil complexes, $\sigma_n^*(\mathrm{IC}_{\chi,\nu},\Phi^{(n)})\cong(\mathrm{IC}_{\chi,\nu},\chi(-1)\Phi^{(n)})$. Thus $\sigma_n^*\mathrm{IC}\cong\mathrm{IC}$ always holds geometrically, but as Weil sheaves it holds iff $\chi=\mathrm{triv}$; for $\chi=\mathrm{sgn}$ the pullback is the quadratic twist. For all $n\ge1$ and $y\in LX_d(\mathbb F_{q^n})$, $$\operatorname{tr}_n(\mathrm{IC}_{\chi,\nu})(\sigma_ny)=\chi(-1)\,\operatorname{tr}_n(\mathrm{IC}_{\chi,\nu})(y).$$

(d) Second proof for $\chi=\mathrm{triv}$. $\mathrm{IC}_{\mathrm{triv},\nu}\cong p^*N_\nu$ as Weil complexes, for an $L^+\mathrm{PGL}_2$-equivariant Weil complex $N_\nu$ and $p:L^+\mathrm{SL}_2\backslash LX_d\to L^+\mathrm{PGL}_2\backslash LX_d$. Hence $\operatorname{tr}_n\mathrm{IC}_{\mathrm{triv},\nu}$ is $\mathrm{PGL}_2(\mathcal O_n)$-invariant, in particular $\sigma_n$-invariant.

The inertia automorphism. $-I\in L^+G$ is central, $\mathbb F_q$-rational, and acts trivially on $LX_d$. For an equivariant complex $M$, the equivariance isomorphism restricted to $\{-I\}\times LX_d$ is an automorphism $\varepsilon_M$ of $M$; its stalks are the $\varepsilon_y$ of Lemma X2.1(d). It is an endomorphism of equivariant objects, by centrality: in stalk form, $\alpha_{g(-I),y}=\varepsilon_y\circ\alpha_{g,y}$ and $\alpha_{(-I)g,y}=\alpha_{g,y}\circ\varepsilon_{gy}$ agree. This is the ℓ-adic form of the natural automorphism of the identity built in Lemma 2.2, step 1. That construction uses only the functoriality of sheaves for 2-morphisms of stacks, so it is field-independent; the second audit graded it with all claims true. By Lemma X2.1(d), last line, $\varepsilon$ commutes with every Weil structure.

(a). $\varepsilon_{\mathrm{IC}_{\chi,\nu}}\in\operatorname{End}(\mathrm{IC}_{\chi,\nu})=\overline{\mathbb Q}_\ell$ is a scalar $c$; this also holds for the underlying non-equivariant object, since $\mathcal L_\chi$ has rank one on the connected orbit $\mathcal O_\nu$ and $j_{!*}$ is fully faithful. On $\mathcal O_\nu$, $\mathrm{IC}_{\chi,\nu}=\mathcal L_\chi[\delta(\nu)]$ with $\mathcal L_\chi=L^+G\times^{\operatorname{Stab}(x_\nu)}\chi$, and the stabilizer acts on the stalk at $x_\nu$ through $\chi$. So $c=\chi(-I)=\chi(-1)$; for $\nu=0$ the stabilizer is connected and $c=1$. A scalar automorphism of a complex acts by the same scalar on every stalk of every cohomology sheaf. The argument uses only $\operatorname{End}=$ scalars and the top stalk, not the claim being proved.

(b). For $\nu'\ne0$, $\pi_0\operatorname{Stab}(x_{\nu'})=\{\pm I\}$ is generated by $-I$ (Computation 2.1). So an equivariant local system on $\mathcal O_{\nu'}$ is $\mathcal L_{\mathrm{triv}}^{a}\oplus\mathcal L_{\mathrm{sgn}}^{b}$, and $-I$ acts by $+1$ on the first summand and by $-1$ on the second. By (a), $b=0$ for $\chi=\mathrm{triv}$ and $a=0$ for $\chi=\mathrm{sgn}$. On $\mathcal O_0$ the stabilizer $L^+\mathrm{SO}_2$ is connected and contains $-I$, which therefore acts trivially on stalks of every equivariant local system there. For $\chi=\mathrm{sgn}$, (a) then gives $-1=+1$ on $\mathcal H^i(\mathrm{IC}_{\mathrm{sgn},\nu})_{x_0}$, which therefore vanishes.

(c). By Lemma X2.1(d) and (a), $\operatorname{tr}_n(\mathrm{IC}_{\chi,\nu})(\sigma_ny)=\sum_i(-1)^i\operatorname{tr}(\chi(-1)\Phi^{(n)}_y\mid\mathcal H^i_y)=\chi(-1)\operatorname{tr}_n(\mathrm{IC}_{\chi,\nu})(y)$. The same computation at the level of complexes runs as follows. Let $\beta_g:g^*M\to M$ be the equivariance isomorphisms, with cocycle $\beta_{gg'}=\beta_{g'}\circ g'^*\beta_g$. Then $\beta_h$ transports $\sigma_n^*\Phi^{(n)}$ to $\Phi'=\beta_h\circ\sigma_n^*\Phi^{(n)}\circ(\mathrm{Fr}^{n*}\beta_h)^{-1}$, and two facts determine it. First, compatibility of $\Phi^{(n)}$ with the equivariance at $g=h$, using that $h$ and $\mathrm{Fr}^n(h)=-h$ induce the same map $\sigma_n$ of $LX_d$, gives $\sigma_n^*\Phi^{(n)}=\beta_{\mathrm{Fr}^n(h)}^{-1}\circ\Phi^{(n)}\circ\mathrm{Fr}^{n*}\beta_h$. Second, $\beta_{\mathrm{Fr}^n(h)}=\beta_{h(-I)}=\varepsilon_M\circ\beta_h$. Together they give $\Phi'=\varepsilon_M^{-1}\circ\Phi^{(n)}=\chi(-1)\Phi^{(n)}$. For $\chi=\mathrm{sgn}$ the two Weil structures are not isomorphic: an isomorphism of Weil complexes preserves trace functions, but the trace at $x_\nu$ is $(-1)^{\delta(\nu)}\ne0$ for one and $-(-1)^{\delta(\nu)}$ for the other.

(d). $\mathrm{PGL}_2$ acts on $X_d$ by $g\cdot A=gAg^{\mathsf T}/\det g$, extending the $\mathrm{SL}_2$-action. $\mathrm{SL}_2\to\mathrm{PGL}_2$ is an étale $\mu_2$-torsor ($p$ odd), and arc spaces of étale maps are base changes ($L^+\mathrm{SL}_2=L^+\mathrm{PGL}_2\times_{\mathrm{PGL}_2}\mathrm{SL}_2$). So $p$ is an étale $\mu_2$-gerbe; $p^*$ is t-exact for the $\delta$-normalised t-structures and commutes with $j_{!*}$. Over $k$, units of $k[\![t]\!]$ are squares, so the $L^+\mathrm{PGL}_2$-orbits are the $\mathcal O_\nu$. The constant sheaf on $\mathcal O_\nu$, with its canonical $L^+\mathrm{SL}_2$-equivariant structure, is $p^*$ of the constant $L^+\mathrm{PGL}_2$-equivariant sheaf. Hence $\mathrm{IC}_{\mathrm{triv},\nu}\cong p^*N_\nu$ with $N_\nu=j_{!*}\overline{\mathbb Q}_\ell[\delta(\nu)]$, compatibly with the canonical Weil structures, both normalised by eigenvalue $1$ at $x_\nu$. The trace function of $N_\nu$ is invariant under $L^+\mathrm{PGL}_2(\mathbb F_{q^n})=\mathrm{PGL}_2(\mathcal O_n)$, which contains $[s]$, and $p$ is the identity on points of $LX_d$. This is the $\mathbb F_q$-rational, ℓ-adic form of the descent to $\mathrm{PGL}_2$ in Lemma 2.3. It reproves (b) for $\chi=\mathrm{triv}$: each $\mathcal H^i(N_\nu)|_{\mathcal O_{\nu'}}$ is a $\mathrm{PGL}_2$-equivariant local system, whose pullback is a sum of $\mathcal L_{\mathrm{triv}}$, because $-I$ lies in the kernel. $\square$

Answer to starting points (1)–(2). $\sigma_n^*\mathrm{IC}_{\chi,\nu}\cong\mathrm{IC}_{\chi,\nu}$ geometrically, and as Weil complexes up to the scalar $\chi(-1)$. This confirms the scalar claimed by Correction 10.8, in both $\mathbb F_q$-forms. Sign local systems never occur in the cohomology sheaves of $\mathrm{IC}_{\mathrm{triv},\mu}$. Explicit small cases are in Computation X2.3.

Builds on

Lemma X2.1 · $\sigma_n$ is translation by $h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)$ with $h^{-1}\mathrm{Fr}^n(h)=-I$
Computation 2.1 · Stabilizers for $\SL_2/\SO_2$: $\mu_2$ times a pro-unipotent group
Lemma 2.2 · Central inertia splits $\mathcal C_K$ into even and odd parts; the even part is the $\PGL_2$-equivariant category
Lemma 2.3 · Hecke convolution preserves the central grading, so every Satake object is even
Setup 4.1 · Set-up of the sheaf–function dictionary over $\F_q(\!(t)\!)$: Weil objects, trace functions, obligations D0–D5
Correction 10.8 · Correction: the square-class automorphism multiplies $\tr_n\IC_{\chi,\nu}$ by $\chi(-1)$

Built on by

Computation X2.3 · Stalks for $\mu=\alpha^\vee,2\alpha^\vee$: $\mathcal L_{\mathrm{sgn}}$ is Kummer on each slice branch
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
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 2.2 (inertia construction)
Lemma 2.3 (descent to PGL2)
chen-yi-v4.pdf App. A (equivariant sheaves on placid stacks)
BBD 1.4 (intermediate extension)

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 X2.2 · Weil inertia lemma: \(\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\)

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Sources