§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:44:46Z

Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$

proved ★ key step

The Iwahori-generated part is squeezed between the thick spans of the simple summands and of the simple constituents of the generators IC_w *_I IC_0; under the decomposition hypothesis DT_I both agree, so C_I^Iwa = Thick{S in M_I} and C_I^Iwa = C_I iff every simple object is a shifted summand of some IC_w *_I IC_0.

Theorem 2.4.

Let $k=\overline{\mathbb F}q$ or $\mathbb C$. Assume (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).) and (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) or (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') of Setup 2.1, and that every $G_w=\mathrm{IC}_w\ast_I\mathrm{IC}_0$ lies in $\mathcal C_I$ (for the group: every $\mathrm{IC}{w_1}\ast c\ast\mathrm{IC}_{w_2}$). Then:

(a) $\mathrm{Thick}\{S:S\in\mathcal M_I\}\ \subseteq\ \mathcal C_I^{\mathrm{Iwa}}\ \subseteq\ \mathcal D_{\mathcal M_I^+}=\mathrm{Thick}\{S:S\in\mathcal M_I^+\}.$

(b) For $S\in\mathcal S_I$: $S\in\mathcal M_I\Rightarrow S\in\mathcal C_I^{\mathrm{Iwa}}\Rightarrow S\in\mathcal M_I^+$.

(c) Under (DT$_I$): $\mathcal M_I=\mathcal M_I^+$, hence $$\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{S:S\in\mathcal M_I\}=\mathcal D_{\mathcal M_I},\qquad S\in\mathcal C_I^{\mathrm{Iwa}}\iff S\in\mathcal M_I .$$

(d) $\mathcal M_I=\mathcal S_I\Rightarrow\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ (no (DT$_I$) needed); $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\Rightarrow\mathcal M_I^+=\mathcal S_I$. Under (DT$_I$): $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I$.

(e) Under (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.), $\mathcal S_I=\{\mathrm{IC}(Z):Z\in\mathcal Z\}$ (Lemma 2.2(c)), so with (DT$_I$): $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ iff for every closure $Z$ of a piece of some level, $\mathrm{IC}(Z)[s]$ is a direct summand of some $\mathrm{IC}_w\ast_I\mathrm{IC}_0$. Under (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') the same holds with $\mathcal S_I=\{\mathrm{IC}(Z,\xi)\}$, $\xi$ irreducible equivariant local systems on the open orbit of $Z$ at a fine enough level, all of which must be reached.

  1. Summands are reached. If $S\in\mathcal M_I$, then $S[s]$ is a retract of a generator $G_w\in\mathcal C_I^{\mathrm{Iwa}}$; a thick subcategory contains retracts and shifts, so $S\in\mathcal C_I^{\mathrm{Iwa}}$ and $\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}$.
  2. Only constituents are reached. By Lemma 2.3(b), $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{G_w\}$. By definition of $\mathcal M_I^+$, every $G_w$ lies in $\mathcal D_{\mathcal M_I^+}$ (its perverse cohomologies, at a level containing it, have all constituents in $\mathcal M_I^+$; by Lemma 2.2(c) this does not depend on the level). $\mathcal D_{\mathcal M_I^+}$ is thick and equals $\mathrm{Thick}(\mathcal M_I^+)$ (Lemma 2.2(d)); minimality of $\mathrm{Thick}\{G_w\}$ gives $\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^+}$. This is (a).
  3. (b). The first implication is Step 1. If $S\in\mathcal C_I^{\mathrm{Iwa}}$, take a level at which $S$ is (a shift of) a simple perverse object; ${}^pH^0$ of it is $S$ itself, whose only constituent is $S$; Step 2 forces $S\in\mathcal M_I^+$.
  4. (c). Under (DT$I$), $G_w\simeq\bigoplus_bS_b[s_b]$ with $S_b$ simple perverse at a common level, so ${}^pH^i(G_w)=\bigoplus{s_b=-i}S_b$ is semisimple and its constituents are exactly the $S_b$, which are summands of $G_w$ up to shift: $\mathcal M_I^+=\mathcal M_I$. Then (a) collapses to equalities and (b) to an equivalence.
  5. (d). If $\mathcal M_I=\mathcal S_I$, Step 1 and $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$ (Lemma 2.2(d)) give $\mathcal C_I\subseteq\mathcal C_I^{\mathrm{Iwa}}$. If $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$, every simple object is in $\mathcal C_I^{\mathrm{Iwa}}$, hence in $\mathcal M_I^+$ by (b). With (c), the equivalence.
  6. (e). Rewrite $\mathcal S_I$ by Lemma 2.2(c) (constant coefficients under (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.)) or Lemma 2.2(a),(b) (local systems under (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.'); checklist item 4: an IC with non-constant $\xi$ is a separate simple object and must itself be a summand).
  7. Spherical inputs replaced; fields. summand.tex used H1 and standard objects (its Steps 5–6); here finite length plus boundedness is used level by level, so H1, false for $\mathbb A^n$ at Iwahori level, is never invoked (checklist items 1, 8). Steps 1–6 hold over $\overline{\mathbb F}_q$ and $\mathbb C$; weights enter only through (DT$_I$) (Lemma 2.5).

Builds on

  • Setup 2.1 · Set-up: level hypotheses, simple objects, $\mathcal M_I\subseteq\mathcal M_I^+$
  • Lemma 2.2 · Lemma (PH$_I$): perverse hearts glue; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$
  • Lemma 2.3 · Lemma: generators $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; spherical summands persist

Built on by

  • 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
  • Summary 2.12 · Verdict: the Iwahori summand criterion and the two test cases
  • Audit 7.2 · Audit of A2-3, A2-4: generators $\mathrm{IC}w\ast_I\mathrm{IC}_0$ and $\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D{\mathcal M_I^+}$
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
  • 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 18:44 UTC

Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$

  1. Summands are reached. If $S\in\mathcal M_I$, then $S[s]$ is a retract of a generator $G_w\in\mathcal C_I^{\mathrm{Iwa}}$; a thick subcategory contains retracts and shifts, so $S\in\mathcal C_I^{\mathrm{Iwa}}$ and $\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}$.
  2. Only constituents are reached. By Lemma 2.3(b), $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{G_w\}$. By definition of $\mathcal M_I^+$, every $G_w$ lies in $\mathcal D_{\mathcal M_I^+}$ (its perverse cohomologies, at a level containing it, have all constituents in $\mathcal M_I^+$; by Lemma 2.2(c) this does not depend on the level). $\mathcal D_{\mathcal M_I^+}$ is thick and equals $\mathrm{Thick}(\mathcal M_I^+)$ (Lemma 2.2(d)); minimality of $\mathrm{Thick}\{G_w\}$ gives $\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^+}$. This is (a).
  3. (b). The first implication is Step 1. If $S\in\mathcal C_I^{\mathrm{Iwa}}$, take a level at which $S$ is (a shift of) a simple perverse object; ${}^pH^0$ of it is $S$ itself, whose only constituent is $S$; Step 2 forces $S\in\mathcal M_I^+$.
  4. (c). Under (DT$I$), $G_w\simeq\bigoplus_bS_b[s_b]$ with $S_b$ simple perverse at a common level, so ${}^pH^i(G_w)=\bigoplus{s_b=-i}S_b$ is semisimple and its constituents are exactly the $S_b$, which are summands of $G_w$ up to shift: $\mathcal M_I^+=\mathcal M_I$. Then (a) collapses to equalities and (b) to an equivalence.
  5. (d). If $\mathcal M_I=\mathcal S_I$, Step 1 and $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$ (Lemma 2.2(d)) give $\mathcal C_I\subseteq\mathcal C_I^{\mathrm{Iwa}}$. If $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$, every simple object is in $\mathcal C_I^{\mathrm{Iwa}}$, hence in $\mathcal M_I^+$ by (b). With (c), the equivalence.
  6. (e). Rewrite $\mathcal S_I$ by Lemma 2.2(c) (constant coefficients under (I2)) or Lemma 2.2(a),(b) (local systems under (I2'); checklist item 4: an IC with non-constant $\xi$ is a separate simple object and must itself be a summand).
  7. Spherical inputs replaced; fields. summand.tex used H1 and standard objects (its Steps 5–6); here finite length plus boundedness is used level by level, so H1, false for $\mathbb A^n$ at Iwahori level, is never invoked (checklist items 1, 8). Steps 1–6 hold over $\overline{\mathbb F}_q$ and $\mathbb C$; weights enter only through (DT$_I$) (Lemma 2.5).

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