§2 · Agent 2 · SL₂/SO₂: central inertia · 2026-10-17T02:39:50Z

Proposition 2.7 · The Satake span is exactly the even-label, constant-coefficient part, strictly inside the even part

proved audit: correct

Second audit: correct.

Thick spans of simple perverse sheaves are detected by composition factors; since every Satake object is a single $\IC_{2n}$, the Satake span consists of the objects whose perverse constituents are even-label IC sheaves with constant coefficients. Both conditions of Proposition III thus fail visibly: odd labels are missed, and sign local systems are missed.

Proposition 2.7.

Assume the standing input of Proposition III, step 6: on each closed finite union $Z$ of orbits, $\mathcal C_Z$ has a bounded perverse t-structure with finite-length heart whose simples are $\IC(\overline{\mathcal O_m},\underline k)$ and $\IC(\overline{\mathcal O_m},\mathcal L_m^-)$, $\mathcal O_m\subseteq Z$ (Beilinson–Bernstein–Deligne, 4.3.1).

Lemma. For a set $\Sigma$ of simple perverse sheaves let $\mathcal D_\Sigma=\{F\in\mathcal C_K:\text{every composition factor of every }{}^pH^i(F)\text{ lies in }\Sigma\}$. Then $\mathcal D_\Sigma=\Thick(\Sigma)$; in particular a simple $P\notin\Sigma$ is not in $\Thick(\Sigma)$.

Theorem. $$\Csat=\Thick\{\IC_{2n}:n\ge0\}=\mathcal D_{\{\IC(\overline{\mathcal O_{2n}},\underline k)\}}\ \subsetneq\ \mathcal C_K^+=\mathcal D_{\{\IC(\overline{\mathcal O_m},\underline k)\}}\ \subsetneq\ \mathcal C_K=\mathcal D_{\text{all simples}},$$ with witnesses $\IC(\overline{\mathcal O_1},\underline k)\in\mathcal C_K^+\setminus\Csat$ and $\IC(\overline{\mathcal O_1},\mathcal L_1^-)\in\mathcal C_K\setminus\mathcal C_K^+$. Moreover $\mathcal C_K^+=\Thick\{V^{\PGL_2}_{m\omega^\vee}\mathbin{\bar\ast}\IC_0\}$.

Proof of the lemma.

$\mathcal D_\Sigma$ is thick. Shifts are clear. Cofibres: in the long exact sequence of perverse cohomology, each ${}^pH^i$ of the middle term is an extension of a quotient by a subobject of objects with factors in $\Sigma$, and composition factors of subquotients are composition factors. Retracts: composition factors of a summand are among those of the object. It contains $\Sigma$, so $\Thick(\Sigma)\subseteq\mathcal D_\Sigma$.
Conversely, $F\in\mathcal D_\Sigma$ is a finite iterated cofibre of its ${}^pH^i(F)[-i]$ (bounded t-structure), each a finite iterated extension of objects of $\Sigma$; so $F\in\Thick(\Sigma)$.
A simple $P\notin\Sigma$ has itself as only composition factor, so $P\notin\mathcal D_\Sigma$. This is the converse of the pages' summand condition: for a category with connected stabilizers, generation forces every $\IC_\mu$ to be a retract of some Satake object.

Proof of the theorem.

First equality: by Lemma 2.6, $V_{n\alpha^\vee}\ast\IC_0\simeq\IC_{2n}$, so the generators are exactly the $\IC_{2n}$; the lemma identifies their thick span.
$\mathcal C_K^+=\mathcal D_{\{\IC(\cdot,\underline k)\}}$: $\varepsilon$ commutes with perverse truncation, and subobjects and quotients of an object with $\varepsilon=\mathrm{id}$ have $\varepsilon=\mathrm{id}$; so $F\in\mathcal C_K^+$ has all perverse composition factors in $\mathcal C_K^+$. Conversely split $F=F^+\oplus F^-$ by Lemma 2.2(c); the composition factors of $F^-$ have $\varepsilon=-1$, i.e. are of the form $\IC(\cdot,\mathcal L^-)$. On a simple object, $\varepsilon=\mathrm{id}$ exactly when its coefficient system is constant (Lemma 2.2, Computation 2.1).
Strictness: $\IC(\overline{\mathcal O_1},\underline k)$ is a simple in $\mathcal C_K^+$ not among the $\IC_{2n}$; $\IC(\overline{\mathcal O_1},\mathcal L_1^-)$ has $\varepsilon=-1$.
$\PGL_2$: $t^{m\omega^\vee}\cdot I=\operatorname{diag}(t^m,t^{-m})=x_m$, so the leading-term map of $(\PGL_2,X)$ is $m\omega^\vee\mapsto m\alpha^\vee$, onto, and by the $\PGL_2$ half of Lemma 2.6 the $\PGL_2$-Satake objects are the $\IC_m$. The odd $\PGL_2$-Satake sheaves, which live on the non-neutral component of $\Gr_{\PGL_2}$, are exactly what the $\SL_2$ Hecke category lacks.

Reading. Both obstructions of Proposition III are realised by concrete non-generated objects: the coverage failure ($\varphi(\Lambda_T^+)=2\Lambda_S^+$) is the gap $\mathcal C_K^+\setminus\Csat$ (odd orbits, constant coefficients), and the failure of H3 ($Z_{G^\theta}(S)=\{\pm I\}$, central) is the gap $\mathcal C_K\setminus\mathcal C_K^+$. This matches Chen–Yi, Remark 13, and Chen–Nadler (2024), Cor. 13.8: the subcategory $\Rep(\check G_X)$ of the perverse heart consists of the IC sheaves with trivial coefficients on the orbits labelled $\mu-\theta\mu$, here the even labels, and nothing else is reached.

Builds on

Lemma 2.3 · Hecke convolution preserves the central grading, so every Satake object is even
Theorem 2.4 · $\Csat\ne\mathcal C_K$ for $\SL_2/\SO_2$: the sign local system on the first orbit is not generated
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

Computation 2.8 · Function level for $\SL_2/T$: Lang count of $K$-orbits, the sign-class dictionary, and a Hecke dimension count for every $q$
Summary 2.9 · Verdict for $\SL_2/\SO_2$: both levels fail, and the sheaf argument is the geometric form of Proposition J(2)
Audit 7.2 · Audit: the $\SL_2/\SO_2$ verdict holds at both levels; recursions and fibre counts re-derived
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 02:39 UTC

Exact Satake span: the even-label constant-coefficient part, strictly inside the even part

Standing input. On each closed finite union of orbits $Z$, $\mathcal C_Z$ carries the perverse t-structure, bounded, with finite-length heart whose simple objects are $\mathrm{IC}(\overline{\mathcal O_m},\underline k)$ and $\mathrm{IC}(\overline{\mathcal O_m},\mathcal L_m^-)$, $m\in I(Z)$ (finitely many orbits, finite $\pi_0$, BBD 4.3.1) — the same input the pages use in Prop III step 6.

Lemma (proved). For a set $\Sigma$ of simple perverse sheaves, let $\mathcal D_\Sigma=\{F\in\mathcal C:\text{every composition factor of every }{}^pH^i(F)\text{ lies in }\Sigma\}$. Then $\mathcal D_\Sigma=\operatorname{Thick}(\Sigma)$.

$\mathcal D_\Sigma$ is thick: shifts (clear); cofibres (long exact sequence of ${}^pH^\bullet$: ${}^pH^i(B)$ is an extension of a quotient of ${}^pH^i(A)$ by a subobject of ${}^pH^i(C)$, and composition factors of subquotients are composition factors); retracts (composition factors of a summand are among those of the object). It contains $\Sigma$. So $\operatorname{Thick}(\Sigma)\subseteq\mathcal D_\Sigma$.
Conversely $F\in\mathcal D_\Sigma$ is a finite iterated cofibre of its ${}^pH^i(F)[-i]$ (bounded t-structure), each a finite iterated extension of objects of $\Sigma$: $F\in\operatorname{Thick}(\Sigma)$.
In particular a simple $P\notin\Sigma$ is not in $\operatorname{Thick}(\Sigma)$. (This is also the converse of the pages' 'summand condition': generation of a connected-stabilizer category forces every $\mathrm{IC}_\mu$ to be a retract of some Satake object.)

Theorem (proved from A2-3, A2-4, A2-6). $$\mathcal C^{\mathrm{Sat}}=\operatorname{Thick}\{\mathrm{IC}_{2n}:n\ge0\}=\mathcal D_{\{\mathrm{IC}(\overline{\mathcal O_{2n}},\underline k)\}}\ \subsetneq\ \mathcal C^+=\mathcal D_{\{\mathrm{IC}(\overline{\mathcal O_m},\underline k)\}}\ \subsetneq\ \mathcal C=\mathcal D_{\text{all simples}}.$$

First equality: A2-6 gives $V_{n\alpha^\vee}\ast\mathrm{IC}_0\simeq\mathrm{IC}_{2n}$, so the generators are exactly the $\mathrm{IC}_{2n}$.
$\mathcal C^+=\mathcal D_{\{\mathrm{IC}(\cdot,\underline k)\}}$: $F\in\mathcal C^+$ iff all perverse composition factors have $\varepsilon=\mathrm{id}$ (split $F=F^+\oplus F^-$ by A2-2(c); $F^-$ has all composition factors with $\varepsilon=-1$, i.e. of the form $\mathrm{IC}(\cdot,\mathcal L^-)$); and $\varepsilon=\mathrm{id}$ on a simple iff its coefficient system is constant (A2-2.5, A2-1).
Strictness: $\mathrm{IC}(\overline{\mathcal O_1},\underline k)\in\mathcal C^+\setminus\mathcal C^{\mathrm{Sat}}$; $\mathrm{IC}(\overline{\mathcal O_1},\mathcal L_1^-)\in\mathcal C\setminus\mathcal C^+$.
Moreover $\mathcal C^+=\operatorname{Thick}\{V^{\mathrm{PGL}_2}_{m\omega^\vee}\bar\ast\mathrm{IC}_0\}$ by the $\mathrm{PGL}_2$ half of A2-6 ($t^{m\omega^\vee}\cdot I=\operatorname{diag}(t^m,t^{-m})=x_m$: the leading-term map of $(\mathrm{PGL}_2,X)$ is $m\omega^\vee\mapsto m\alpha^\vee$, onto). The odd $\mathrm{PGL}_2$-Satake sheaves, living on the non-neutral component of $\mathrm{Gr}_{\mathrm{PGL}_2}$, are exactly what the $\mathrm{SL}_2$-Hecke category lacks.

Reading. Both obstructions of Proposition III are realised by concrete non-generated objects: the H5/coverage failure ($\varphi(\Lambda_T^+)=2\Lambda_S^+$) is the gap $\mathcal C^+\setminus\mathcal C^{\mathrm{Sat}}$ (odd orbits, constant coefficients), the H3 failure ($Z_{G^\theta}(S)=\{\pm I\}$, central) is the gap $\mathcal C\setminus\mathcal C^+$. This matches Chen–Yi Remark 13 / [CN24, Cor. 13.8]: $\mathrm{Rep}(\check G_X)\subset\mathrm{Perv}$ consists of the $\mathrm{IC}$ with trivial coefficients on orbits $-\theta\mu+\mu$, here the even labels, and nothing else is reached.

Sources

sources/bbd-perverse.pdf Thm 4.3.1, §1.4
criterion.html (Open: summand condition)
sources/chen-yi-v4.pdf Remark 13, Thm 87(ii) proof, Lemma 90

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