§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:53:08Z

Summary 2.12 · Verdict: the Iwahori summand criterion and the two test cases

proved ★ key step

The Iwahori summand criterion holds under (L1)-(L3), (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).), (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.)/(I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') and DT_I, all verified for A^n and the group (with the Omega-truncation repair); it gives second proofs that C_I^Iwa = C_I for A^n (Omega-translates) and that Delta_e is not Iwahori-generated for the group (weights), plus the G_m row; the other rows reduce to one explicit summand statement each.

Summary 2.12.

  1. (I$_I$) (Theorem 2.4): under (L1)–(L3), (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).), (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.)/(I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.'), $\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathrm{Thick}(\mathcal M_I^+)$; with (DT$_I$), $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{S\in\mathcal M_I\}$ and $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I$. PH$_I$ is a consequence (Lemma 2.2); (DT$_I$) follows from a finite-type proper model over $\overline{\mathbb F}_q$ (Lemma 2.5).
  2. $\mathbb A^n$ (Proposition 2.6, Theorem 2.7): all hypotheses hold; $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$; (IG) holds.
  3. Group (Proposition 2.8, Theorem 2.9): all hypotheses hold ($\Omega$-truncated levels); over $\overline{\mathbb F}q$, $\Omega\cap\mathcal M_I^+=\varnothing$, so $\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}$; (IG) fails. Over $\mathbb C$ given (T${\mathbb C}$).
  4. Other rows (Proposition 2.10): $\mathbb G_m$ agrees (both hold). Symmetric rows: given (L$_X$), sheaves hold iff $(\ast_X)$; where (IG) fails, the row differs iff $(\ast_X)$ holds, and agrees once the sign-obstruction lemma applies to one simple object.
  1. Divergences decided here. Rows 1 (group), 2 ($\mathbb A^n$) and $\mathbb G_m$: no divergence (both levels fail, both hold, both hold respectively), consistent with Table 1 and proved by routes different from the pages' Propositions II and IV.1.
  2. Remaining obligations. (L3) as a theorem (the pages assume it too); the equivariant decomposition theorem over $\mathbb C$ and the transfer (T$_{\mathbb C}$) (cited); (L$_X$) and $(\ast_X)$ for the six open symmetric rows.

Builds on

  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$
  • Theorem 2.7 · $\mathbb A^n$: $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, hence Iwahori generation
  • Theorem 2.9 · Group: $\Omega\cap\mathcal M_I^+=\varnothing$ by sign character and weights
  • Proposition 2.10 · Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction
  • Audit 2.11 · Audit: the nine failure modes checked against the criterion and the two cases

Built on by

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Original record, as the agent wrote it at 18:53 UTC

Verdict: the Iwahori summand criterion and the two test cases

  1. Divergences decided here. Rows 1 (group), 2 ($\mathbb A^n$) and $\mathbb G_m$: no divergence (both levels fail, both hold, both hold respectively), consistent with Table 1 and proved by routes different from the pages' Propositions II and IV.1.
  2. Remaining obligations. (L3) as a theorem (the pages assume it too); the equivariant decomposition theorem over $\mathbb C$ and the transfer (T$_{\mathbb C}$) (cited); (L$_X$) and $(\ast_X)$ for the six open symmetric rows.

Sources