§5 · Agent 5 · 𝔸ⁿ: lattices · 2026-10-18T18:44:39Z

Summary 5.9 · Verdict: for $\mathbb A^n$ both Iwahori-level statements hold

proved ★ key step

For $\mathbb A^n$ under $\mathrm{GL}_n$ the two Iwahori-level statements agree (both hold), and each is proved by a route independent of IV.1 and of the dévissage: one length-zero element $\tau$ moves $\mathrm{IC}_0$, resp. $\varphi_0$, onto every simple object, resp. every lattice indicator. The general criterion (I$_I$) and the bridge (II$_I$) are proved with explicit hypotheses, all checked for $\mathbb A^n$.

Summary 5.9.

For $X=\mathbb A^n$ under $\mathrm{GL}_n$, every $n\ge1$:

  1. (sheaves, $k=\mathbb C$ Betti and $k=\overline{\mathbb F}q$ $\ell$-adic) the simple objects of $\mathcal C_I$ are $S_k=\underline k{\tau^k\mathcal O^n}[-k]$, $k\in\mathbb Z$; $\mathrm{IC}{\tau^k}\ast_I\mathrm{IC}_0=S_k[k]$; so $\mathcal M_I=\operatorname{Simple}(\mathcal P_I)$ and $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I=\operatorname{Tri}\{\delta{\tau^k}\ast\mathrm{IC}_0\}$ (Lemma 5.2, Computation 5.3, Theorem 5.5);
  2. (functions, every non-archimedean $F$) $C_I=\mathbb C[\Omega]\ast\varphi_0$, free of rank one, so (IG) holds (Proposition 5.6);
  3. (bridge, $\mathbb F_q((t))$ and $\overline{\mathbb F}_q$) taking traces of 1 gives 2, and (IG) with (II$_I$) gives 1 back (Proposition 5.6(e), Proposition 5.7, Corollary 5.8).

Row 2 of Table 1 at Iwahori level: agree (no divergence).

The logic. $$\{\tau^k\mathcal O^n\}=\{I\text{-stable lattices}\}\ \xrightarrow{\ \ell(\tau)=0\ }\ \mathrm{IC}_{\tau^k}\ast\mathrm{IC}_0=S_k[k]\ \xrightarrow{\ \text{(I}_I)\ }\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\ \xrightarrow{\ \operatorname{tr}\ }\ T_{\tau^k}\ast\varphi_0=b_k\ \xrightarrow{\ \text{(II}_I)\ }\ \mathcal M_I=\text{all}.$$

New compared with IV.1 (case-affine.html).

  1. The perverse t-structure of the colimit (renormalized by $nM$), its finite-length heart and the list of simple objects, with the tails $t^M\mathcal O^n$ as the supports of $S_{nM}$ and no object at $0$ (Lemma 5.2).
  2. Generators that are simple objects of $\mathcal H_{c,I}$: the length-zero skyscrapers $\delta_{\tau^k}$, in place of $\Delta_{t^{\lambda_{m,j}}}$ with its shift $-2j(n-j)$; the latter is recovered as $\delta_{\tau^k}\ast\Delta_u$ (Computation 5.3(b)).
  3. No standard objects of pieces, no 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)., no Proposition III: the summand criterion (I$_I$) decides, and triangulated (not only thick) generation by one $\Omega$-orbit.
  4. The action of every simple reflection on simple objects, $\mathrm{IC}{s_i}\ast S_k\in\{S_k[1]\oplus S_k[-1],\ S{k-1}\oplus S_{k+1}\}$, with splittings by parity (Computation 5.3(c)).

New compared with F3 (classical-affine.html). $T_{\tau^k}\ast\varphi_0=\mathbf 1_{\tau^k\mathcal O^n}$ with no factor $[K:I]$ or $q^{\ell}$; cyclicity already over the commutative subalgebra $\mathbb C[\Omega]$, with $C_I$ free of rank one; the explicit $T_{s_i}$-action; the identification of these identities as traces of 1, and the reverse implication (II$_I$) with its inputs checked.

Decomposition theorem. Not needed for 1 or 2 (isomorphisms of spaces and parity). Needed only for the reverse route 3, where traces alone give constituents, not summands (Corollary 5.8).

What (I$I$), (II$_I$) give for other rows (general, not checked here). Whenever (L), (L$_q$), (W), (Lin), (P), (Tr) hold, a divergence with (IG) true and $\mathcal C_I^{\mathrm{Iwa}}\ne\mathcal C_I$ is impossible over $\mathbb F_q((t))$ / $\overline{\mathbb F}_q$. The remaining kind, sheaves true and (IG) false, needs the converse, whose exact obstruction is a simple $S\in\mathcal M_I$ whose trace is invisible ($m{S,w}=0$ for all $w$) or generation that needs retracts (Proposition 5.7, step 5). Rows with disconnected finite-level stabilizers need (Lin) for traces of non-constant local systems, which is not automatic.

Gaps. None for $\mathbb A^n$ beyond the cited standard theorems (BBD 4.2.4–4.2.6, 4.3.1 for 1; purity, BBD 5.1.14, 5.3.8, 5.4.5, Sun Thm 1.2, and the Grothendieck–Lefschetz dictionary for 3 only). The Betti form of 1 is proved directly; (II$_I$) by itself reaches only $\overline{\mathbb F}_q$.

Builds on

  • Proposition 5.4 · (I$_I$) The Iwahori summand criterion: generation iff $\mathcal M_I$ is all
  • Theorem 5.5 · $\mathbb A^n$: $\mathcal M_I=\{S_k\}$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$
  • Proposition 5.6 · (IG) for $\mathbb A^n$: $T_{\tau^k}\ast\varphi_0=\mathbf 1_{\tau^k\mathcal O^n}$
  • Proposition 5.7 · (II$_I$) (IG) over $\mathbb F_q((t))$ puts every simple object in $\mathcal M_I$
  • Corollary 5.8 · Reverse route for $\mathbb A^n$: (IG) and (II$_I$) give sheaf generation

Built on by

  • —
Original record, as the agent wrote it at 18:44 UTC

Verdict: for $\mathbb A^n$ both Iwahori-level statements hold

The logic. $$\{\tau^k\mathcal O^n\}=\{I\text{-stable lattices}\}\ \xrightarrow{\ \ell(\tau)=0\ }\ \mathrm{IC}_{\tau^k}\ast\mathrm{IC}_0=S_k[k]\ \xrightarrow{\ \text{(I}_I)\ }\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\ \xrightarrow{\ \operatorname{tr}\ }\ T_{\tau^k}\ast\varphi_0=b_k\ \xrightarrow{\ \text{(II}_I)\ }\ \mathcal M_I=\text{all}.$$

New compared with IV.1 (case-affine.html).

  1. The perverse t-structure of the colimit (renormalized by $nM$), its finite-length heart and the list of simple objects, with the tails $t^M\mathcal O^n$ as the supports of $S_{nM}$ and no object at $0$ (Lemma 5.2).
  2. Generators that are simple objects of $\mathcal H_{c,I}$: the length-zero skyscrapers $\delta_{\tau^k}$, in place of $\Delta_{t^{\lambda_{m,j}}}$ with its shift $-2j(n-j)$; the latter is recovered as $\delta_{\tau^k}\ast\Delta_u$ (Computation 5.3(b)).
  3. No standard objects of pieces, no I3, no Proposition III: the summand criterion (I$_I$) decides, and triangulated (not only thick) generation by one $\Omega$-orbit.
  4. The action of every simple reflection on simple objects, $\mathrm{IC}{s_i}\ast S_k\in\{S_k[1]\oplus S_k[-1],\ S{k-1}\oplus S_{k+1}\}$, with splittings by parity (Computation 5.3(c)).

New compared with F3 (classical-affine.html). $T_{\tau^k}\ast\varphi_0=\mathbf 1_{\tau^k\mathcal O^n}$ with no factor $[K:I]$ or $q^{\ell}$; cyclicity already over the commutative subalgebra $\mathbb C[\Omega]$, with $C_I$ free of rank one; the explicit $T_{s_i}$-action; the identification of these identities as traces of 1, and the reverse implication (II$_I$) with its inputs checked.

Decomposition theorem. Not needed for 1 or 2 (isomorphisms of spaces and parity). Needed only for the reverse route 3, where traces alone give constituents, not summands (Corollary 5.8).

What (I$I$), (II$_I$) give for other rows (general, not checked here). Whenever (L), (L$_q$), (W), (Lin), (P), (Tr) hold, a divergence with (IG) true and $\mathcal C_I^{\mathrm{Iwa}}\ne\mathcal C_I$ is impossible over $\mathbb F_q((t))$ / $\overline{\mathbb F}_q$. The remaining kind, sheaves true and (IG) false, needs the converse, whose exact obstruction is a simple $S\in\mathcal M_I$ whose trace is invisible ($m{S,w}=0$ for all $w$) or generation that needs retracts (Proposition 5.7, step 5). Rows with disconnected finite-level stabilizers need (Lin) for traces of non-constant local systems, which is not automatic.

Gaps. None for $\mathbb A^n$ beyond the cited standard theorems (BBD 4.2.4–4.2.6, 4.3.1 for 1; purity, BBD 5.1.14, 5.3.8, 5.4.5, Sun Thm 1.2, and the Grothendieck–Lefschetz dictionary for 3 only). The Betti form of 1 is proved directly; (II$_I$) by itself reaches only $\overline{\mathbb F}_q$.

Sources