§8 · Agent 8 · E₆/F₄: sheaf level · 2026-10-17T03:22:00Z

Lemma 8.5 · The fibre over the covering orbit is finite, so $\IC_{\mu_k'}$ occurs only unshifted

proved audit: error

Error. Step 2 fails as written. (1) The two dimension formulas are false: for $N\gg0$ the image of $\Stab_{L^+G}(x_\nu)$ in $G(k[t]/t^N)$ has dimension $N\dim G^\theta$ for every $\nu$ (after conjugating by $t^\lambda$, each $\theta$-pair of root directions still contributes exactly $N$), so every quotient $\cO_\nu/K_N$ has dimension $N\dim X=\dim X_N$, not $\dim X_N+\delta(\nu)$; in the group case $\dim K_N\backslash\cO_\lambda=N\dim H$. (2) The convolution does not descend to $\Gr^{\le\lambda}\mathbin{\widetilde\times}(L^+X/K_N)\to LX/K_N$: $[g,ky]\mapsto(gkg^{-1})gy$ and $gK_Ng^{-1}\not\subseteq K_N$. (3) The finite-type model attributed to the page of Proposition II is the group-case model $Z_H\subseteq\Gr_H$ (quotient by the whole right arc group), not a $K_N$-quotient of $L^+X$. So the bound $\dim F_k'\le D_k-\delta(\mu_k')$ is not derived. The statement is nevertheless true: Objection 7.28 shows $F_k'=\{t^{\lambda_k}\}$ (the torus $T'=(T^\theta)^\circ$ fixes $x_{\mu_k'}$, no root vanishes on $T'$, the only fixed point in $F_k'$ is $t^{\lambda_k}$, and a positive-dimensional stable closed subvariety of a linearly embedded projective variety has two distinct limit points), and step 4, which is correct, then gives $E_{\mu_k',n}=k$ for $n=0$ and $0$ otherwise. Smaller points: step 3 speaks of $T$-fixed points of $F_k'$, but only $T'$ acts on $F_k'$ (it means $F_k'\cap\Gr_T$); the closing Remark ($\dim F'\le\langle2\rho_L,\lambda\rangle$ for general $\lambda$) rests on the same count and is not proved here (it is not used later). Later nodes use only the statement of this lemma, never the count itself; Summary 7.32 and Summary 9.4 cite the repair. The conclusion still holds by another argument.
Used later by: Theorem 8.8 · $\IC_{3k\omega_1^\vee}$ is a direct summand of the Satake object $V_{\lambda_k}\ast\IC_0$, Remark 8.12 · Exactly what the $E_6/F_4$ argument assumes, and what would sharpen it.
Corrected by: Objection 7.28 · Objection to the finiteness lemma: the $K_N$ dimension count is wrong as written; $F_k'$ is a point by a fixed-point argument.

The lemma asserts that the fibre $F_k'$ over $x_{\mu_k'}$ is a finite set containing $t^{\lambda_k}$, so that in the decomposition of $M_{\lambda_k}$ the IC sheaf of the covering orbit occurs only in degree $0$; the detection lemma needs exactly this. The statement is true, but the congruence-quotient dimension count offered in step 2 is wrong as written, and the run replaces it by a torus-fixed-point argument.

Lemma 8.5.

Work in the finite-type equivariant model described in the proof. Then $\dim F_k'\le0$, i.e. $F_k'$ is a nonempty finite set containing $t^{\lambda_k}$. Hence, writing the decomposition (assumption (E2) of Theorem 8.8) $$M_{\lambda_k}\simeq\bigoplus_{\nu\le\mu_k',\ n\in\Z}\IC_\nu[-n]\otimes E_{\nu,n},$$ one has $E_{\mu_k',n}=0$ for $n\ne0$ and $E_{\mu_k',0}=H^0(F_k')\ne0$.

Model. Choose $N$ so that the congruence subgroup $K_N=\ker\big(L^+G\to G(k[t]/t^N)\big)$ acts trivially on $\Gr^{\le\lambda_k}$. Then $\Gr^{\le\lambda_k}\mathbin{\widetilde\times}(L^+X/K_N)$, with $L^+X/K_N=X(k[t]/t^N)=:X_N$, is a variety of finite type with smooth fibres, and the convolution descends to a proper $G(k[t]/t^N)$-equivariant map $m_N$ onto $\overline{\cO_{\mu_k'}}/K_N$, whose orbit strata have codimensions $\delta(\mu')-\delta(\nu)$ (Chen–Yi, Thm. 36, Prop. 64 and App. A.4; this is the finite-type model that the page of Proposition II uses for H2 and H4).

$m_N$ is equivariant, so $m_N^{-1}(\cO_{\mu_k'}/K_N)\to\cO_{\mu_k'}/K_N$ is a fibration with fibre $F_k'$ (all fibres over one orbit are isomorphic). Thus $\dim m_N^{-1}(\cO_{\mu_k'}/K_N)=\dim(\cO_{\mu_k'}/K_N)+\dim F_k'$.
The source restricted to the open part $\Gr^{\lambda_k}\mathbin{\widetilde\times}X_N$, which contains $m_N^{-1}(\cO_{\mu_k'}/K_N)$ by Lemma 8.4, has dimension $D_k+\dim X_N$, and $\dim(\cO_{\mu_k'}/K_N)=\dim X_N+\delta(\mu_k')$, the relative dimension of the orbit over $\cO_0=L^+X$. Since $D_k=\delta(\mu_k')$ (Lemma 8.3), step 1 gives $\dim F_k'\le0$.
$t^{\lambda_k}\in F_k'$ because $x_{\mu_k'}=t^{\lambda_k}x_0$ (leading-term map). The torus-fixed points of $F_k'$, i.e. the $t^\nu$ with $\nu\in W\lambda_k$ and $\varphi(\nu)=\varphi(\lambda_k)$, reduce to $\nu=\lambda_k$: indeed $\nu-\lambda_k\in\ker\varphi$, $|\nu|=|\lambda_k|$, and $\lambda_k\perp\ker\varphi$. A computer count gives exactly one such point for $k=1,3,5,7$.
Restrict $M_{\lambda_k}$ to $\cO_{\mu_k'}$. Only $\nu=\mu_k'$ survives, and $\IC_{\mu_k'}|_{\cO_{\mu_k'}}=\underline k[\delta(\mu_k')]$ (Chen–Yi's normalization $\IC_{\chi,\lambda}=j_{!*}L[\delta(\lambda)]$). So $H^j\big((M_{\lambda_k})_{x_{\mu_k'}}\big)=E_{\mu_k',\,j+\delta(\mu_k')}$. By Lemma 8.4 this stalk is $H^{j+D_k}(F_k')$ with $D_k=\delta(\mu_k')$. Hence $E_{\mu_k',n}=H^n(F_k')$, which vanishes for $n\ne0$ and equals $k^{\#F_k'}$ for $n=0$.

Remark (general $\lambda$). The same count gives $\dim F'\le\langle2\rho_L,\lambda\rangle$ for every dominant $\lambda$, where $F'$ is the fibre over $x_{\varphi(\lambda)}$; Proposition III, step 6 uses only that the open part of $F'$ is nonempty.

Step 2 is the subject of the objection Objection 7.28.

Builds on

Lemma 8.3 · The Hecke label $\lambda_k$: a dominant lift of $\mu_k'$ whose cell has dimension $\delta(\mu_k')$
Lemma 8.4 · Both convolution fibres lie in the open cell $\Gr^{\lambda_k}$

Built on by

Theorem 8.8 · $\IC_{3k\omega_1^\vee}$ is a direct summand of the Satake object $V_{\lambda_k}\ast\IC_0$
Objection 7.28 · Objection to the finiteness lemma: the $K_N$ dimension count is wrong as written; $F_k'$ is a point by a fixed-point argument

Original record, as the agent wrote it at 03:22 UTC

The fibre over the covering orbit is finite, so IC of the covering orbit occurs unshifted only

Work in the finite-type equivariant model: choose $N$ so that the congruence subgroup $K_N=\ker(L^+G\to G(k[t]/t^N))$ acts trivially on $\mathrm{Gr}^{\le\lambda_k}$; then $\mathrm{Gr}^{\le\lambda_k}\widetilde\times(L^+X/K_N)$ with $L^+X/K_N=X(k[t]/t^N)$ is a smooth-fibred finite-type variety, and the convolution descends to a proper $J_NG$-equivariant map $m_N$ onto $\overline{\mathcal O_{\mu_k'}}/K_N$, whose orbit strata have codimensions $\delta(\mu')-\delta(\nu)$ (Chen–Yi Thm 36, Prop 64, App. A.4; this is the model symmetric.html uses for H2/H4).

Lemma. $\dim F_k'\le0$, i.e. $F_k'$ is a nonempty finite set containing $t^{\lambda_k}$. Hence, writing (decomposition theorem, A8-8 assumption (A2)) $M_{\lambda_k}\simeq\bigoplus_{\nu\le\mu_k',n}\mathrm{IC}_\nu[-n]\otimes E_{\nu,n}$, one has $E_{\mu_k',n}=0$ for $n\ne0$ and $E_{\mu_k',0}=H^0(F_k')\ne0$.

Proof. 1. $m_N$ is equivariant, so $m_N^{-1}(\mathcal O_{\mu_k'}/K_N)\to\mathcal O_{\mu_k'}/K_N$ is a fibration with fibre $F_k'$ (all fibres over one orbit are isomorphic); thus $\dim m_N^{-1}(\mathcal O_{\mu_k'}/K_N)=\dim(\mathcal O_{\mu_k'}/K_N)+\dim F_k'$.

The source of $m_N$ restricted to the open part $\mathrm{Gr}^{\lambda_k}\widetilde\times X_N$ (which contains $m_N^{-1}(\mathcal O_{\mu_k'}/K_N)$ by A8-4) has dimension $D_k+\dim X_N$, and $\dim(\mathcal O_{\mu_k'}/K_N)=\dim X_N+\delta(\mu_k')$ (relative dimension of the orbit over $\mathcal O_0=L^+X$). Since $D_k=\delta(\mu_k')$ (A8-3(iii)), step 1 gives $\dim F_k'\le0$.
$t^{\lambda_k}\in F_k'$ because $x_{\mu_k'}=t^{\lambda_k}x_0$ (leading-term map, symmetric.html). Script: the $T$-fixed points of $F_k'$, i.e. $\nu\in W\lambda_k$ with $\varphi(\nu)=\varphi(\lambda_k)$, reduce to $\nu=\lambda_k$ (norm: $\nu-\lambda_k\in\ker\varphi$ and $|\nu|=|\lambda_k|$ with $\lambda_k\perp\ker\varphi$ force $\nu=\lambda_k$); the count is 1 for $k=1,3,5,7$.
Restrict $M_{\lambda_k}$ to $\mathcal O_{\mu_k'}$: only $\nu=\mu_k'$ survives and $\mathrm{IC}_{\mu_k'}|_{\mathcal O_{\mu_k'}}=\underline k[\delta(\mu_k')]$ (Chen–Yi: $\mathrm{IC}_{\chi,\lambda}=j_{!*}L[\delta(\lambda)]$), so $H^j\big((M_{\lambda_k})_{x_{\mu_k'}}\big)=E_{\mu_k',\,j+\delta(\mu_k')}$. By A8-4 this stalk is $H^{j+D_k}(F_k')$ with $D_k=\delta(\mu_k')$; so $E_{\mu_k',n}=H^n(F_k')$, which vanishes for $n\ne0$ and equals $k^{\#F_k'}$ for $n=0$. $\square$

Remark (general $\lambda$): the same count gives $\dim F'\le\langle2\rho_L,\lambda\rangle$ for any $\lambda$; criterion.html Step 6 only uses that the open part of $F'$ is nonempty.

Sources

Chen–Yi v4 Thm 36, Prop 64, §1.2.3 (normalization of IC), App. A.4
symmetric.html (finite-type model)
criterion.html Step 6

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 8.5 · The fibre over the covering orbit is finite, so \(\IC_{\mu_k'}\) occurs only unshifted

Builds on

Built on by

The fibre over the covering orbit is finite, so IC of the covering orbit occurs unshifted only

Sources