§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:47:02Z

Theorem 3.11 · Sheaf generation as a test on IC-coordinates over all $\mathbb F_{q^n}$

proved ★ key step

With purity, sheaf generation is equivalent to a statement about functions over all $\mathbb F_{q^n}(\!(t)\!)$ (each IC-coordinate form is non-zero on the Hecke translates for some $n$), which pins down exactly how it differs from (IG); and any Hecke-module map to the sign character becomes a sheaf-level obstruction once it isolates one simple object.

Theorem 3.11.

Assume (L1), (L2), (D1)–(D3) of Setup 3.1 and purity of every $\mathrm{IC}w*_I\mathrm{IC}_0$. Over $F_n=\mathbb F{q^n}(\!(t)\!)$ let $\kappa^{(n)}S$ be the coordinate form of $\mathrm{tr}_nS$ in the basis $\{\mathrm{tr}_nS'\}$ of $C{I_n}\otimes\overline{\mathbb Q}_\ell$ (Lemma 3.4(i)).

(a) For a simple object $S$: $S\in\mathcal M_I\iff\exists n\ge1:\ \kappa^{(n)}S(\mathcal H{I_n}\varphi_0)\neq0$. Equivalently, $S\notin\mathcal M_I$ iff for every $n$ some linear form $\lambda_n$ with $\lambda_n(\mathcal H_{I_n}\varphi_0)=0$ satisfies $\lambda_n(\mathrm{tr}nS')=\delta{S'S}$ for all simple $S'$.

(b) Hence: $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ iff no simple object has such a family $(\lambda_n)_n$; while (IG)$_1$ fails iff some non-zero linear form kills $\mathcal H_I*\varphi_0$. A divergence of type "sheaves hold, (IG) fails" is exactly: witnesses exist at $n=1$, but none isolates a single IC-coordinate at every $n$.

(c) (Sign-module exclusion.) Assume $W\neq1$. Let $\lambda_n\colon C_{I_n}\to\overline{\mathbb Q}\ell$ satisfy $\lambda_n(T_s*f)=-\lambda_n(f)$ for all simple affine $s$. Then $\lambda_n$ kills $\mathcal H{I_n}*\varphi_0$ and $\mathrm{tr}_nS$ for every $S$ that is $P_s$-equivariant for some simple affine $s$. If for every $n$ such a $\lambda_n$ vanishes on $\mathrm{tr}_nS$ for all non-equivariant $S\neq S_0$ and not on $\mathrm{tr}_nS_0$, then $S_0\notin\mathcal M_I$ and $\mathcal C_I^{\mathrm{Iwa}}\neq\mathcal C_I$.

  1. (a) $\Leftarrow$. $\mathcal H_{I_n}*\varphi_0\otimes\overline{\mathbb Q}\ell$ is spanned by the $\mathrm{tr}_n(\mathrm{IC}_w*_I\mathrm{IC}_0)$ (Lemma 3.3(c) over $\mathbb F{q^n}$), and $\kappa^{(n)}S$ of that trace is $m^{(n)}{S,w}$ (Lemma 3.4(ii)). If it is non-zero, some $E_{S,i}(w)\neq0$, so $S\in\mathcal M_I$ (extraction, Theorem 3.6 step 2).
  2. (a) $\Rightarrow$. If $S\in\mathcal M_I$, some $E_{S,i}(w)\neq0$, and the weight and Vandermonde argument of Proposition 3.10 step 1 gives $n\le\dim\bigoplus_iE_{S,i}(w)$ with $m^{(n)}_{S,w}\neq0$.
  3. (a), second form. If $S\notin\mathcal M_I$, take $\lambda_n=\kappa^{(n)}S$. Conversely such $\lambda_n$ give $m^{(n)}{S,w}=\lambda_n(\mathrm{tr}n(\mathrm{IC}_w*_I\mathrm{IC}_0))=0$ for all $n,w$, and by weights and Vandermonde (Proposition 3.9 steps 5–6, with the weight $\ell(w)+i-\delta(P)$ of Proposition 3.10 step 1) all $E{S,i}(w)=0$.
  4. (b). Lemma 3.5(b) and (a); the statement about (IG)$_1$ is duality for the subspace $\mathcal H_I*\varphi_0\subseteq C_I$ (restricted to each finite-dimensional $C_I^{(N,M)}$).
  5. (c). For a finite simple $s\in W$, $\varphi_0$ is $P_s$-invariant, so $T_s*\varphi_0=q^n\varphi_0$ and $-\lambda_n(\varphi_0)=q^n\lambda_n(\varphi_0)$, i.e. $\lambda_n(\varphi_0)=0$. For $w=s_1\cdots s_k\omega$ reduced, $\lambda_n(T_w*\varphi_0)=(-1)^k\lambda_n(T_\omega*\varphi_0)$, and $T_\omega*\varphi_0=\mathbf 1_{\omega X(\mathcal O)}$ is left $P_{s'}$-invariant for $s'=\omega s\omega^{-1}$, again simple affine, so it is killed in the same way. Hence $\lambda_n(\mathcal H_{I_n}\varphi_0)=0$. If $S$ is $P_s$-equivariant, $\mathrm{tr}nS$ is left $P_s$-invariant, $T_s\mathrm{tr}_nS=q^n\mathrm{tr}_nS$, and the same computation kills it. Hence $0=\lambda_n(\mathrm{tr}_n(\mathrm{IC}_w*_I\mathrm{IC}_0))=\sum{S\ \text{non-equiv.}}m^{(n)}{S,w}\lambda_n(\mathrm{tr}_nS)=m^{(n)}{S_0,w}\lambda_n(\mathrm{tr}_nS_0)$, and step 3 concludes.

Instances. The group: $\lambda_n=\varepsilon_n$ on $C_I=\mathcal H_{I_H}$, non-equivariant simple objects $=\{\mathrm{IC}_\omega\}$, isolated by components (Proposition 3.9). For $X$ with $\mathrm{St}$ (or another $\pi$ with $\pi^I$ the sign character, $\pi^K=0$) $X$-distinguished, classical.html Corollary D supplies $\lambda_1$; the sheaf-level obligation is to compute the non-equivariant simple objects and isolate one of them for all $n$.

Builds on

  • Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$
  • Proposition 3.9 · Group: no $\mathrm{IC}_\omega$ is a summand, so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Proposition 3.10 · Sheaves to functions: non-zero rows only; (WT) gives (IG)

Built on by

  • Summary 3.12 · Verdict: (II$_I$) proved; rows 1–3 agree; one divergence type excluded
  • Audit 7.10 · Audit of A3-10, A3-11: sheaves $\Rightarrow$ functions only row-wise; (WT) $\Rightarrow$ (IG); generation as an IC-coordinate test over all $\mathbb F_{q^n}$
Original record, as the agent wrote it at 18:47 UTC

Sheaf generation as a test on IC-coordinates over all $\mathbb F_{q^n}$

  1. (a) $\Leftarrow$. $\mathcal H_{I_n}*\varphi_0\otimes\overline{\mathbb Q}\ell$ is spanned by the $\mathrm{tr}_n(\mathrm{IC}_w*_I\mathrm{IC}_0)$ (Lemma 3.3(c) over $\mathbb F{q^n}$), and $\kappa^{(n)}S$ of that trace is $m^{(n)}{S,w}$ (Lemma 3.4(ii)). If it is non-zero, some $E_{S,i}(w)\neq0$, so $S\in\mathcal M_I$ (extraction, Theorem 3.6 step 2).
  2. (a) $\Rightarrow$. If $S\in\mathcal M_I$, some $E_{S,i}(w)\neq0$, and the weight and Vandermonde argument of Proposition 3.10 step 1 gives $n\le\dim\bigoplus_iE_{S,i}(w)$ with $m^{(n)}_{S,w}\neq0$.
  3. (a), second form. If $S\notin\mathcal M_I$, take $\lambda_n=\kappa^{(n)}S$. Conversely such $\lambda_n$ give $m^{(n)}{S,w}=\lambda_n(\mathrm{tr}n(\mathrm{IC}_w*_I\mathrm{IC}_0))=0$ for all $n,w$, and by weights and Vandermonde (Proposition 3.9 steps 5–6, with the weight $\ell(w)+i-\delta(P)$ of Proposition 3.10 step 1) all $E{S,i}(w)=0$.
  4. (b). Lemma 3.5(b) and (a); the statement about (IG)$_1$ is duality for the subspace $\mathcal H_I*\varphi_0\subseteq C_I$ (restricted to each finite-dimensional $C_I^{(N,M)}$).
  5. (c). For a finite simple $s\in W$, $\varphi_0$ is $P_s$-invariant, so $T_s*\varphi_0=q^n\varphi_0$ and $-\lambda_n(\varphi_0)=q^n\lambda_n(\varphi_0)$, i.e. $\lambda_n(\varphi_0)=0$. For $w=s_1\cdots s_k\omega$ reduced, $\lambda_n(T_w*\varphi_0)=(-1)^k\lambda_n(T_\omega*\varphi_0)$, and $T_\omega*\varphi_0=\mathbf 1_{\omega X(\mathcal O)}$ is left $P_{s'}$-invariant for $s'=\omega s\omega^{-1}$, again simple affine, so it is killed in the same way. Hence $\lambda_n(\mathcal H_{I_n}\varphi_0)=0$. If $S$ is $P_s$-equivariant, $\mathrm{tr}nS$ is left $P_s$-invariant, $T_s\mathrm{tr}_nS=q^n\mathrm{tr}_nS$, and the same computation kills it. Hence $0=\lambda_n(\mathrm{tr}_n(\mathrm{IC}_w*_I\mathrm{IC}_0))=\sum{S\ \text{non-equiv.}}m^{(n)}{S,w}\lambda_n(\mathrm{tr}_nS)=m^{(n)}{S_0,w}\lambda_n(\mathrm{tr}_nS_0)$, and step 3 concludes.

Instances. The group: $\lambda_n=\varepsilon_n$ on $C_I=\mathcal H_{I_H}$, non-equivariant simple objects $=\{\mathrm{IC}_\omega\}$, isolated by components (Proposition 3.9). For $X$ with $\mathrm{St}$ (or another $\pi$ with $\pi^I$ the sign character, $\pi^K=0$) $X$-distinguished, classical.html Corollary D supplies $\lambda_1$; the sheaf-level obligation is to compute the non-equivariant simple objects and isolate one of them for all $n$.

Sources