§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:41:17Z

Lemma 3.5 · Iwahori summand criterion: $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\,\mathcal M_I$

proved

With a level-wise finite-length heart and the decomposition of the generators, the thick span of the generators is exactly the thick span of the simple objects occurring as their shifted summands, and a simple object outside that set is not reached; this is the form of (I_I) the bridge and the group argument use.

Lemma 3.5.

Assume (L1) of Setup 3.1, finitely many orbits in each $Y_{N,M}$ (local systems allowed), and the decomposition (D3) with simple objects of $\mathcal P_I$ (possibly with local systems). Put $$\mathcal M_I:=\{S\in\mathrm{Irr}\,\mathcal P_I:\ S[s]\ \text{is a direct summand of}\ \mathrm{IC}_w*_I\mathrm{IC}_0\ \text{for some}\ w\in\widetilde W,\ s\in\mathbb Z\}.$$ Then (a) $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}_{\mathcal C_I}\{S:S\in\mathcal M_I\}$, and for $S\in\mathrm{Irr}\,\mathcal P_I$: $S\in\mathcal C_I^{\mathrm{Iwa}}\iff S\in\mathcal M_I$; (b) $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$. Proved over $k=\overline{\mathbb F}_q$; the same proof works over $\mathbb C$ wherever the t-structure of Setup 3.1 and (D3) are available there.

  1. Generators. $\mathcal H_{c,I}=\mathrm{Thick}\{\Delta_w\}$ by dévissage along the finitely many affine-space cells in each $\mathrm{Fl}{\le w}$ (connected stabilizers; proposition-I.tex, Step 4), and $\mathrm{Thick}\{\Delta_v:v\le w\}=\mathrm{Thick}\{\mathrm{IC}_v:v\le w\}$ by induction on the finite Bruhat interval using $\Delta_w[\ell(w)]\to\mathrm{IC}_w\to ii^\mathrm{IC}w\xrightarrow{+1}$. Since $-*_I\mathrm{IC}_0$ is exact, (T2) gives $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{\mathrm{IC}_w*_I\mathrm{IC}_0:w\in\widetilde W\}$. In the group case $G=H\times H$, $\widetilde W_G=\widetilde W_H\times\widetilde W_H$ and $\mathrm{IC}{(x,y)}=\mathrm{IC}_x\boxtimes\mathrm{IC}_y$, so the generators are the two-sided products $\mathrm{IC}_x*c*\mathrm{IC}_y$.
  2. Span. For $S\in\mathcal M_I$, $S[s]$ is a retract of a generator, so $S\in\mathcal C_I^{\mathrm{Iwa}}$. Conversely each generator is a finite sum $\bigoplus_bS_b[s_b]$ with $S_b\in\mathcal M_I$ by definition, an iterated cofibre of split triangles. Minimality gives the first claim of (a).
  3. Constituent subcategories are thick. Every object of $\mathcal P_I$ lies in some finite-length $\mathcal P^{(N,M)}$, which is closed under subquotients in $\mathcal P_I$ (Setup 3.1(i)–(ii)), so Jordan–Hölder constituents are defined. For $\Sigma\subseteq\mathrm{Irr}\,\mathcal P_I$ let $\mathcal P_\Sigma$ be the Serre subcategory of objects with constituents in $\Sigma$ and $\mathcal D_\Sigma=\{K:{}^p\mathcal H^i(K)\in\mathcal P_\Sigma\ \forall i\}$, the common kernel of the cohomological functors $K\mapsto q\,{}^p\mathcal H^0(K[i])$, $q\colon\mathcal P_I\to\mathcal P_I/\mathcal P_\Sigma$. It is thick [Stacks, Tag 05RD] (summand.tex, Step 3, verbatim).
  4. Only summands. By (D3) every generator lies in $\mathcal D_{\mathcal M_I}$, so $\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I}$. If $S\notin\mathcal M_I$, ${}^p\mathcal H^0(S)=S$ is a simple object not in $\mathcal M_I$, so $S\notin\mathcal D_{\mathcal M_I}$. This finishes (a) and gives ($\Rightarrow$) of (b).
  5. Everything from simple objects. If $\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$, every simple object lies in $\mathcal C_I^{\mathrm{Iwa}}$. Any $K\in\mathcal C_I$ lies at one level, has finitely many non-zero ${}^p\mathcal H^i$ (bounded t-structure), each of finite length; truncation triangles and Jordan–Hölder filtrations with (T1) put $K$ in $\mathrm{Thick}\{\text{simple objects}\}\subseteq\mathcal C_I^{\mathrm{Iwa}}$. This is ($\Leftarrow$). $\square$

Checks. No hypothesis H1 (finitely many orbits in a closure) is used: finiteness is only level-wise, which is all that holds for $\mathbb A^n$. Thick, not localizing: step 4 uses only finite filtrations. The standard objects $\Delta_P$ never enter.

Builds on

  • Setup 3.1 · Set-up over $\mathbb F_q$: levels, the perverse heart of $\mathcal C_I$, Weil structures

Built on by

  • 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)
  • Audit 7.7 · Audit of A3-4, A3-5, A3-6: unitriangular IC-trace basis, $m_{S,w}=\sum_i(-1)^i\mathrm{tr}\,A_{S,i}(w)$, and (IG) $\Rightarrow\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
Original record, as the agent wrote it at 18:41 UTC

Iwahori summand criterion: $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\,\mathcal M_I$

  1. Generators. $\mathcal H_{c,I}=\mathrm{Thick}\{\Delta_w\}$ by dévissage along the finitely many affine-space cells in each $\mathrm{Fl}{\le w}$ (connected stabilizers; proposition-I.tex, Step 4), and $\mathrm{Thick}\{\Delta_v:v\le w\}=\mathrm{Thick}\{\mathrm{IC}_v:v\le w\}$ by induction on the finite Bruhat interval using $\Delta_w[\ell(w)]\to\mathrm{IC}_w\to ii^\mathrm{IC}w\xrightarrow{+1}$. Since $-*_I\mathrm{IC}_0$ is exact, (T2) gives $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{\mathrm{IC}_w*_I\mathrm{IC}_0:w\in\widetilde W\}$. In the group case $G=H\times H$, $\widetilde W_G=\widetilde W_H\times\widetilde W_H$ and $\mathrm{IC}{(x,y)}=\mathrm{IC}_x\boxtimes\mathrm{IC}_y$, so the generators are the two-sided products $\mathrm{IC}_x*c*\mathrm{IC}_y$.
  2. Span. For $S\in\mathcal M_I$, $S[s]$ is a retract of a generator, so $S\in\mathcal C_I^{\mathrm{Iwa}}$. Conversely each generator is a finite sum $\bigoplus_bS_b[s_b]$ with $S_b\in\mathcal M_I$ by definition, an iterated cofibre of split triangles. Minimality gives the first claim of (a).
  3. Constituent subcategories are thick. Every object of $\mathcal P_I$ lies in some finite-length $\mathcal P^{(N,M)}$, which is closed under subquotients in $\mathcal P_I$ (Setup 3.1(i)–(ii)), so Jordan–Hölder constituents are defined. For $\Sigma\subseteq\mathrm{Irr}\,\mathcal P_I$ let $\mathcal P_\Sigma$ be the Serre subcategory of objects with constituents in $\Sigma$ and $\mathcal D_\Sigma=\{K:{}^p\mathcal H^i(K)\in\mathcal P_\Sigma\ \forall i\}$, the common kernel of the cohomological functors $K\mapsto q\,{}^p\mathcal H^0(K[i])$, $q\colon\mathcal P_I\to\mathcal P_I/\mathcal P_\Sigma$. It is thick [Stacks, Tag 05RD] (summand.tex, Step 3, verbatim).
  4. Only summands. By (D3) every generator lies in $\mathcal D_{\mathcal M_I}$, so $\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I}$. If $S\notin\mathcal M_I$, ${}^p\mathcal H^0(S)=S$ is a simple object not in $\mathcal M_I$, so $S\notin\mathcal D_{\mathcal M_I}$. This finishes (a) and gives ($\Rightarrow$) of (b).
  5. Everything from simple objects. If $\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$, every simple object lies in $\mathcal C_I^{\mathrm{Iwa}}$. Any $K\in\mathcal C_I$ lies at one level, has finitely many non-zero ${}^p\mathcal H^i$ (bounded t-structure), each of finite length; truncation triangles and Jordan–Hölder filtrations with (T1) put $K$ in $\mathrm{Thick}\{\text{simple objects}\}\subseteq\mathcal C_I^{\mathrm{Iwa}}$. This is ($\Leftarrow$). $\square$

Checks. No hypothesis H1 (finitely many orbits in a closure) is used: finiteness is only level-wise, which is all that holds for $\mathbb A^n$. Thick, not localizing: step 4 uses only finite filtrations. The standard objects $\Delta_P$ never enter.

Sources