§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:49:22Z

Summary 3.12 · Verdict: (II$_I$) proved; rows 1–3 agree; one divergence type excluded

proved ★ key step

Collects what the trace bridge proves: functions imply sheaves at Iwahori level under named level-wise inputs, rows 1-3 agree by new arguments, $E_6/F_4$ fails (IG) over $\mathbb F_q(\!(t)\!)$, and for every other row the exact deciding statement is the summand condition $\mathcal M_I=\mathrm{Irr}$, equivalently the function-level test of Theorem 3.11.

Summary 3.12.

Functions over $\mathbb F_q(\!(t)\!)$ ($p$ odd), sheaves $\ell$-adic over $\overline{\mathbb F}_q$.

  1. (II$_I$) Theorem 3.6: under (L1), (L2), (D1)–(D3), (IG) $\Rightarrow\mathcal M_I=\mathrm{Irr}\,\mathcal P_I\Rightarrow\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$; with disconnected stabilizers, (IG)$_n$ for all $n$ suffices Proposition 3.7.
  2. Reverse Proposition 3.10, Theorem 3.11: sheaves give only non-zero rows of $(m^{(n)})n$; (IG) needs spanning columns at $n=1$; (WT) (Weil, cancellation-free (I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages).)) gives both; with purity, $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ is equivalent to "every IC-coordinate is non-zero on $\mathcal H{I_n}*\varphi_0$ for some $n$".
  3. Two general implications: (IG) $\Rightarrow$ (SG) (classical.html, Proposition B) and $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\Rightarrow\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K$ (proposition-I.tex, (a) and the averaging step of Step 5).
Table 2. $X$ and (IG) over $\mathbb F_q(\!(t)\!)$ and $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ over $\overline{\mathbb F}_q$
$X$(IG) over $\mathbb F_q(\!(t)\!)$$\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ over $\overline{\mathbb F}_q$verdict
$H$ under $H\times H$, $W_H\ne1$fails Proposition 3.9fails Proposition 3.9agree
$\mathbb A^n$holds (F3, Thm. 3)holds Corollary 3.8agree
$\mathbb G_m=\mathrm{GL}_2/\mathrm{Sp}_2$holdsholds Corollary 3.8agree
$E_6/F_4$fails for all $n$ (item 4)open; $K$-level holdsagree iff $\mathcal M_I\ne\mathrm{Irr}$
other rowsfails where Table 1 says (fields as there); else openopensee item 5
  1. $E_6/F_4$. converse.tex, Proposition IX(a), gives $C_K=\mathcal H_K*\varphi_0\oplus\mathbb C\mathbf 1_{O_{3\omega_1^\vee}}$ over $\mathbb F_{q^n}(\!(t)\!)$ for every $n$ (from Propositions VI–VII of the satake run); by Proposition B, (IG)$_n$ fails for every $n$. The Table 1 cell "functions: open" is thus settled over $\mathbb F_q(\!(t)\!)$, conditionally on those propositions. Since the $K$-level sheaf statement holds there (Proposition VIII(b)), item 3 gives no obstruction: $E_6/F_4$ is an Iwahori-level divergence iff $\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$.
  1. Other rows ($\mathrm{GL}{2n}/\mathrm{Sp}{2n}$, $\mathrm{GL}n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ incl. $\mathrm{GL}_2/T$, $\mathrm{Spin}{2n}/\mathrm{Spin}{2n-1}$, $\mathrm{SL}{2n}/\mathrm{Sp}_{2n}$). Exact obligations, in order:
    • (O1) the inputs: a level presentation over $\mathbb F_q$ with finitely many orbits per level (for symmetric $X$, from Chen–Yi's finiteness of $I$-orbits in closures), finite-level stabilizers (connected, or use Proposition 3.7), (D1), and (D3) with purity.
    • (O2) where (IG) fails only over $p$-adic $F$ (Speh rows): its status over $\mathbb F_q(\!(t)\!)$; if it held there, (O1) and Theorem 3.6 would give sheaves.
    • (O3) where (IG) fails over $\mathbb F_q(\!(t)\!)$: decide $\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$ (Lemma 3.5), equivalently the test of Theorem 3.11(a). For $\mathrm{St}$-distinguished $X$ (classical.html, Corollary D) the concrete test is Theorem 3.11(c): list the simple objects not $P_s$-equivariant for any simple affine $s$ and check whether the Steinberg functional isolates one at every $n$ (then both fail: agree); if instead every simple object is a summand, the row is a divergence.
  1. What is not proved. Nothing over $\mathbb C$ (the bridge is $\ell$-adic; transfer needs spreading out uniform in $w$, Theorem 3.6 Fields). No Iwahori-level example of "sheaves hold, (IG) fails" is established; $E_6/F_4$ is the candidate. The general implication $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\Rightarrow\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K$ over $\overline{\mathbb F}q$ needs the $\ell$-adic key lemma (unit $M\to p*p^M$ split for the $G/B$-bundle, via the decomposition of $\pi_\overline{\mathbb Q}_\ell$ for $BB\to BG$); used only in item 4's remark, not in the table's proved cells.

Builds on

  • Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$
  • Proposition 3.7 · Disconnected stabilizers: (IG)$_n$ for all $n$ gives $\mathcal M_I=\mathrm{Irr}$
  • Corollary 3.8 · $\mathbb A^n$: inputs checked; (IG) gives sheaf generation over $\overline{\mathbb F}_q$
  • 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)
  • Theorem 3.11 · Sheaf generation as a test on IC-coordinates over all $\mathbb F_{q^n}$

Built on by

  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 18:49 UTC

Verdict: (II$_I$) proved; rows 1–3 agree; one divergence type excluded

  1. $E_6/F_4$. converse.tex, Proposition IX(a), gives $C_K=\mathcal H_K*\varphi_0\oplus\mathbb C\mathbf 1_{O_{3\omega_1^\vee}}$ over $\mathbb F_{q^n}(\!(t)\!)$ for every $n$ (from Propositions VI–VII of the satake run); by Proposition B, (IG)$_n$ fails for every $n$. The Table 1 cell "functions: open" is thus settled over $\mathbb F_q(\!(t)\!)$, conditionally on those propositions. Since the $K$-level sheaf statement holds there (Proposition VIII(b)), item 3 gives no obstruction: $E_6/F_4$ is an Iwahori-level divergence iff $\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$.
  1. Other rows ($\mathrm{GL}{2n}/\mathrm{Sp}{2n}$, $\mathrm{GL}n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ incl. $\mathrm{GL}_2/T$, $\mathrm{Spin}{2n}/\mathrm{Spin}{2n-1}$, $\mathrm{SL}{2n}/\mathrm{Sp}_{2n}$). Exact obligations, in order:
    • (O1) the inputs: a level presentation over $\mathbb F_q$ with finitely many orbits per level (for symmetric $X$, from Chen–Yi's finiteness of $I$-orbits in closures), finite-level stabilizers (connected, or use Proposition 3.7), (D1), and (D3) with purity.
    • (O2) where (IG) fails only over $p$-adic $F$ (Speh rows): its status over $\mathbb F_q(\!(t)\!)$; if it held there, (O1) and Theorem 3.6 would give sheaves.
    • (O3) where (IG) fails over $\mathbb F_q(\!(t)\!)$: decide $\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$ (Lemma 3.5), equivalently the test of Theorem 3.11(a). For $\mathrm{St}$-distinguished $X$ (classical.html, Corollary D) the concrete test is Theorem 3.11(c): list the simple objects not $P_s$-equivariant for any simple affine $s$ and check whether the Steinberg functional isolates one at every $n$ (then both fail: agree); if instead every simple object is a summand, the row is a divergence.
  1. What is not proved. Nothing over $\mathbb C$ (the bridge is $\ell$-adic; transfer needs spreading out uniform in $w$, Theorem 3.6 Fields). No Iwahori-level example of "sheaves hold, (IG) fails" is established; $E_6/F_4$ is the candidate. The general implication $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\Rightarrow\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K$ over $\overline{\mathbb F}q$ needs the $\ell$-adic key lemma (unit $M\to p*p^M$ split for the $G/B$-bundle, via the decomposition of $\pi_\overline{\mathbb Q}_\ell$ for $BB\to BG$); used only in item 4's remark, not in the table's proved cells.

Sources