§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:07:17Z

Setup 4.1 · Set-up of the sheaf–function dictionary over $\F_q(\!(t)\!)$: Weil objects, trace functions, obligations D0–D5

open audit: minor defect

Minor defect (second audit). Two attributions are imprecise; no claim used later is affected. (1) The standing assumption of Chen–Yi §8.2 is that T0 is split for a θ-stable Borel pair (B0,T0), not that the θ-stable torus T ⊇ S of a θ-split pair is split; what the track uses is that S (hence every t^μ, μ∈Λ_S) is defined over F_q, which the node's own hypothesis 'T split' gives. (2) Chen–Yi §8.2.1–8.2.2 define Weil structures as compatible systems Φ_n for sufficiently divisible n, on abelian categories, with Frobenius eigenvalues recorded modulo roots of unity; this is not 'exactly' the category of pairs (M, Φ: Fr*M ≅ M) of complexes, and §8.5.3 does not define the L+G Weil category (it is implicit, cf. §9.1). Checked and correct: rationality of x_μ (τ injective on k((t))-points, t^μ ∈ S(F)); the trace properties; the description of Chen–Yi Prop. 17/Thm. 18/Remark 19; BBD 5.4.1 (K pure of weight w iff each pH^i K pure of weight w+i — read in the scanned source); BF Prop. 5 (comparison for all residue fields of a suitable localization).

Puts the symmetric space, its loop space and the category $\cC_K$ over $\F_q$, defines Frobenius trace functions of Weil objects, and lists the six inputs D0–D5 on which every result of the dictionary track rests. Only $\ell$-adic sheaves over $\overline{\F}_q$ and functions over $\F_q(\!(t)\!)$ are compared; Betti sheaves over $\C$ and $p$-adic functions are not.

Setup 4.1.

Setting. $G,\theta,X=G/G^\theta$ are defined over $\F_q$, $q$ odd, with $G^\theta$ connected and $T\supseteq S$ split over $\F_q$. Put $k=\overline{\F}_q$, $F=\F_q(\!(t)\!)$, $\cO=\F_q[\![t]\!]$, $K=G(\cO)$, and for $n\ge1$ let $F_n=\F_{q^n}(\!(t)\!)$, $K_n=G(\F_{q^n}[\![t]\!])$. Then $$LX(\F_q)=X(F),\qquad L^+G(\F_q)=K,\qquad \Gr(\F_q)=G(F)/K,$$ and the Frobenius $\mathrm{Fr}$ preserves every orbit $\cO_\mu=L^+G\cdot x_\mu$, $\mu\in\Lambda_S^+$.

Weil objects and traces. $\cC_{K,k}=\Shv_{\mathrm{cons}}(L^+G\backslash LX)$ is Chen–Yi's $\ell$-adic category over $k$ (Chen–Yi, App. A.5). A Weil object is a pair $(M,\Phi)$ with $M\in\cC_{K,k}$ and $\Phi:\mathrm{Fr}^*M\xrightarrow{\sim}M$. For $x\in X(F_n)$, $$\tr_n(M,\Phi)(x)=\sum_i(-1)^i\tr\big(\Phi^n_x,\ \mathcal H^i(M)_x\big),\qquad \tr:=\tr_1 .$$ It is $K_n$-invariant, additive on exact triangles, $\tr(M[1])=-\tr M$, and $\tr(M\otimes\alpha)=\alpha\,\tr M$ for a rank-one twist by $\alpha\in\overline{\Q}_\ell^\times$. An isomorphism $\iota:\overline{\Q}_\ell\simeq\C$ and a square root $q^{1/2}$ are fixed.

Objects compared. Sheaves: $\Csat_k=\Thick_{\cC_{K,k}}\{V_\lambda\ast\IC_0\}\overset?=\cC_{K,k}$. Functions: (SG), $\cH_K\cdot\varphi_0=C_K$, for $F=\F_q(\!(t)\!)$. Not compared directly: $\cC_K$ over $\C$, and (SG) over a $p$-adic field.

Obligations. The results of this track are proved modulo:

D0 Chen–Yi's orbit geometry over $k$, $\operatorname{char}k\ne2$: orbits $\leftrightarrow\Lambda_S^+$ with representatives $x_\mu$ (Prop. 17), closures (Thm. 18), the support bound $\supp(V_\lambda\ast\IC_0)\subseteq\overline{\cO_{\varphi(\lambda)}}$ (Proposition III, step 2), and the stabilizer lemma $\pi_0\Stab_{L^+G}(x_\mu)\cong\pi_0Z_{G^\theta}(\mu)$, $Z_{G^\theta}(\mu)=Z_{G^\theta}(\mu)^\circ Z_{G^\theta}(S)$ (Proposition III, step 4).
D1 Proper base change and the Grothendieck–Lefschetz trace formula on the fibres of $m:\Gr_{\le\lambda}\widetilde\times\overline{\cO_\nu}\to LX$ over rational points (closed subschemes of the finite-type $\Gr_{\le\lambda}$); a Frobenius-compatible perverse t-structure on $\cC_{K,k}$ (Chen–Yi A.7) with finite-length heart whose simple objects are the $\IC_{\chi,\mu}$, $\chi\in\operatorname{Irr}\pi_0\Stab(x_\mu)$, with $\End(\IC_{\chi,\mu})=\overline{\Q}_\ell$.
D2 $V_\lambda\ast\IC_0$ is pure and $\simeq\bigoplus_b\IC(\overline{\cO_{\nu_b}},\mathcal L_b)[s_b]$ over $k$ (decomposition theorem, BBD 5.4.5–5.4.6, as in the proof of Chen–Yi Thm. 93; Deligne's purity of proper pushforward), and ${}^pH^i$ of a pure Weil complex of weight $w$ is pure of weight $w+i$ (BBD 5.4.1) in Chen–Yi's sense (Def. 109, via finite-type smooth covers).
D3 Lang's theorem for the pro-algebraic groups $L^+G=\varprojlim G(\cO/t^n)$ and $\Stab_{L^+G}(x_\mu)$.
D4 Unitriangularity of $\tr V_\lambda$ in the basis $T_\lambda=\one_{Kt^\lambda K}$ (Zhu (5.6.2) and Lemma 5.6.3; Lusztig, Kato).
D5 The comparison $\overline{\F}_q\leftrightarrow\C$ of the statement "$\IC_\mu$ is a summand of $V_\lambda\ast\IC_0$", a statement on a finite-type equivariant model (BBD 6.1.9; Bezrukavnikov–Finkelberg, Prop. 5); the workspace already assumes it (open item 1 of the criterion page).

Rationality of the orbits. The Cartan map $\tau(gG^\theta)=g\theta(g)^{-1}$ is injective on points over an algebraically closed field, hence on $k(\!(t)\!)$-points. Since $S$ is split, $t^\mu\in S(F)$ for $\mu\in\Lambda_S$, so $\tau(\mathrm{Fr}(x_\mu))=\mathrm{Fr}(t^\mu)=t^\mu=\tau(x_\mu)$: the point $x_\mu$ is the unique point with $\tau(x_\mu)=t^\mu$, and it is rational. Therefore $\mathrm{Fr}(\cO_\mu)=\mathrm{Fr}(L^+G)\cdot x_\mu=\cO_\mu$.

Weil objects. The pairs $(M,\Phi)$ are the Weil objects of Chen–Yi §8.2.1–8.2.2 (defined there for the Iwahori subgroup $I_0$) and §8.5.3 (for $L^+G$); the splitting of $T$ over $\F_q$ is the standing assumption of Chen–Yi §8.2. The listed properties of $\tr_n$ are the usual ones: equivariance gives $K_n$-invariance, and the trace is additive along exact triangles.

What is compared. The two statements are compared only in one pair of settings: $\ell$-adic sheaves over $\overline{\F}_q$ and functions over $\F_q(\!(t)\!)$. Passing to $\cC_K$ over $\C$ needs D5; passing to a $p$-adic field needs a separate transfer that is not available (see Objection 4.12).

Why D0 is listed. Chen–Yi state Theorem 18 over any algebraically closed field of characteristic $\ne2$ (Remark 19) and work over $\F_q$ in §8.2, but their Proposition 17 is proved by citing Gaitsgory–Nadler and Nadler, over $\C$. D5 is the transfer the criterion page lists as its open item 1.

Builds on

Built on by

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
Objection 4.12 · Obstacles: $\C$ versus $\overline{\F}_q$, characteristic $0$ in (A1) versus $\F_q(\!(t)\!)$, Weil versus geometric, eigenvalue cancellation
Audit 7.3 · Audit of the orbit count, the trace dictionary and the IC expansion: they hold under D0–D3, and D4 is elementary
Lemma X2.2 · Weil inertia lemma: $\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}$

Original record, as the agent wrote it at 02:07 UTC

Dictionary set-up: Weil category over F_q, trace functions, and the obligations D0-D5

Setting for the dictionary. Take $G,\theta,X=G/G^\theta$ over $\mathbb F_q$ ($q$ odd), $T$ split over $\mathbb F_q$ (the standing assumption of Chen–Yi §8.2), $k=\overline{\mathbb F}_q$, $F=\mathbb F_q(\!(t)\!)$, $\mathcal O=\mathbb F_q[[t]]$. Then $LX(\mathbb F_q)=X(F)$, $L^+G(\mathbb F_q)=K=G(\mathcal O)$, $\mathrm{Gr}(\mathbb F_q)=G(F)/K$, and Frobenius $\mathrm{Fr}$ acts on $LX$, $LG$, on the orbits $\mathcal O_\mu$ (each $\mathrm{Fr}$-stable: $x_\mu$ is the unique point with $\tau(x_\mu)=t^\mu\in S(F)$, so it is rational), and on $\mathcal C_k:=\mathrm{Shv}_{\mathrm{cons}}(L^+G\backslash LX)$ in Chen–Yi's $\ell$-adic theory (App. A.5).

Weil objects and trace functions. $\mathcal C^{\mathrm W}$: pairs $(M,\Phi)$, $M\in\mathcal C_k$, $\Phi:\mathrm{Fr}^*M\xrightarrow{\sim}M$ (Chen–Yi §8.2.1–8.2.2 define exactly this for $I_0$ and, in §8.5.3, for $L^+G$). For $x\in X(F)$ and $n\ge1$ with $x\in LX(\mathbb F_{q^n})$: $$\operatorname{tr}_n(M,\Phi)(x):=\sum_i(-1)^i\operatorname{tr}\big(\Phi^n_x,\ \mathcal H^i(M)_x\big),\qquad \operatorname{tr}:=\operatorname{tr}_1 .$$ It is $G(\mathcal O_n)$-invariant (equivariance), additive on exact triangles, and $\operatorname{tr}(M[1])=-\operatorname{tr}M$, $\operatorname{tr}(M\otimes\alpha)=\alpha\operatorname{tr}M$ for a rank-one twist by $\alpha\in\overline{\mathbb Q}_\ell^\times$. Fix $\iota:\overline{\mathbb Q}_\ell\simeq\mathbb C$ and $q^{1/2}$.

Objects compared. Sheaf side: $\mathcal C^{\mathrm{Sat}}_k=\mathrm{Thick}_{\mathcal C_k}\{V_\lambda\ast\mathrm{IC}_0\}$ over $k=\overline{\mathbb F}_q$. Function side: (SG) for $F=\mathbb F_q(\!(t)\!)$: $\mathcal H_K\cdot\varphi_0=C_K$. Not compared directly: $\mathcal C$ over $\mathbb C$ and (SG) over $p$-adic $F$; these need D5 and a separate (unavailable) transfer, see A4-12.

Obligations (the gap). Everything below is proved modulo:

D0 Chen–Yi's geometry over $k=\overline{\mathbb F}_q$, $\operatorname{char}\neq2$: Prop. 17 (orbits $\leftrightarrow\Lambda_S^+$, $x_\mu$), Thm 18 (closures), Lemma Step 2 of criterion.html (support bound $\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_{\varphi(\lambda)}}$, geometric), Lemma Step 4 ($\pi_0\mathrm{Stab}_{L^+G}(x_\mu)\cong\pi_0Z_K(\mu)$, $Z_K(\mu)=Z_K(\mu)^\circ Z_K(S)$). Chen–Yi state Thm 18 for any algebraically closed field of characteristic $\ne2$ (Remark 19) and use the $\mathbb F_q$-setting in §8.2, but Prop. 17 cites [GN10],[Nad04] (over $\mathbb C$).
D1 Proper base change and the Grothendieck–Lefschetz trace formula for $m:\mathrm{Gr}_{\le\lambda}\tilde\times\overline{\mathcal O_\nu}\to LX$ restricted to the fibre over a rational point (a closed subscheme of the finite-type $\mathrm{Gr}_{\le\lambda}$); and the perverse t-structure $\mathrm{Perv}_\delta$ of Chen–Yi A.7 on $\mathcal C_k$, compatible with $\mathrm{Fr}^*$, with heart of finite length whose simple objects are the $\mathrm{IC}_{\chi,\mu}$ ($\chi\in\mathrm{Irr}\,\pi_0\mathrm{Stab}(x_\mu)$), $\operatorname{End}(\mathrm{IC}_{\chi,\mu})=\overline{\mathbb Q}_\ell$.
D2 Purity/decomposition: $V_\lambda\ast\mathrm{IC}_0$ is pure and $\simeq\bigoplus_b\mathrm{IC}(\overline{\mathcal O_{\nu_b}},\mathcal L_b)[s_b]$ over $k$ (criterion.html Step 5 Lemma (b), Chen–Yi Thm 93 proof, BBD 5.4.5–5.4.6, Deligne's purity of proper pushforward), and $\mathcal{}^p\mathcal H^i$ of a pure weight-$w$ Weil complex is pure of weight $w+i$ (BBD 5.4.1) in Chen–Yi's sense (Def. 109, via finite-type smooth covers).
D3 Lang's theorem for the pro-algebraic groups $L^+G=\varprojlim G(\mathcal O/t^n)$ and $\mathrm{Stab}_{L^+G}(x_\mu)$ (A4-2).
D4 The Lusztig–Kato unitriangularity for $\operatorname{tr}V_\lambda$ (Zhu's notes (5.6.2), Lemma 5.6.3; Lusztig 1983, Kato 1982).
D5 Comparison $\overline{\mathbb F}_q\leftrightarrow\mathbb C$ of the statement "$\mathrm{IC}_\mu$ is a summand of $V_\lambda\ast\mathrm{IC}_0$" (a statement on a finite-type equivariant model: BBD 6.1.9, Bezrukavnikov–Finkelberg Prop. 5; already assumed by criterion.html, "open item 1").

Sources

/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/criterion.html (Steps 2,4,5)
sources/chen-yi-v4.pdf §8.2, §8.5.3, App. A.5, A.7, A.8, Thm 93
sources/zhu-satake-v2.pdf §5.6
sources/bbd-perverse.pdf 5.4.1, 5.4.5, 5.4.6, 6.1.9

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