§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:47:06Z

Theorem 2.7 · $\mathbb A^n$: $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, hence Iwahori generation

proved ★ key step

The length-zero group Omega of GL_n acts simply transitively on the standard lattices, so each simple object k_Lambda of C_I is itself a generator Delta_omega *_I IC_0; the criterion then gives Iwahori generation for A^n, by a route (simple objects, length-zero translates) different from the pages' 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). devissage, and the same identity gives (IG) for functions.

Theorem 2.7.

Let $\omega\in\mathrm{GL}n(F)$, $\omega e_1=te_n$, $\omega e_i=e{i-1}$ ($2\le i\le n$). Over $k=\overline{\mathbb F}_q$ or $\mathbb C$:

(a) $\omega\,t^mL_j=t^mL_{j-1}$ for $1\le j\le n$, and $\omega^n=t\cdot1$; hence $i\mapsto\omega^i\mathcal O^n$ is a bijection $\mathbb Z\to\mathcal L$, $\omega^{nm+n-j}\mathcal O^n=t^mL_j$. (b) $\omega I\omega^{-1}=I$; $\omega$ is monomial, so it defines $\omega\in\Omega$ ($\ell(\omega)=0$), $I\omega I=\omega I$ and $\mathrm{IC}{\omega^i}=\Delta{\omega^i}$ is the skyscraper at the point $\omega^iI/I$. (c) $\Delta_{\omega^i}\ast_I\mathcal F\simeq(\omega^i)*\mathcal F$ for $\mathcal F\in\mathcal C_I$; in particular $G{\omega^i}=\Delta_{\omega^i}\ast_I\mathrm{IC}0=k{\omega^i\mathcal O^n}$, with no shift. (d) Every simple object of $\mathcal C_I$ is, up to shift, one of the generators $G_{\omega^i}$; hence $\mathcal M_I=\mathcal S_I$ and, by Theorem 2.4(d), $$\boxed{\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\quad\text{for }X=\mathbb A^n\text{ under }\mathrm{GL}_n,\ n\ge1.\ }$$ (e) (functions, $F=\mathbb F_q(\!(t)\!)$, $\mathrm{vol}(I)=1$) $T_{\omega^i}\ast\varphi_0=\mathbf 1_{\omega^i\mathcal O^n}$, and these span $C_c^\infty(F^n)^I$; so (IG) holds.

  1. (a). $\omega L_j=\mathrm{span}{\mathcal O}(\omega e_1,\dots,\omega e_j,t\omega e{j+1},\dots,t\omega e_n)=\mathrm{span}(te_n,e_1,\dots,e_{j-1},te_j,\dots,te_{n-1})=L_{j-1}$, and $t$ is central. $\omega$ cycles $e_n\mapsto e_{n-1}\mapsto\dots\mapsto e_1\mapsto te_n$, so $\omega^n=t$; then $\omega^{n-j}L_n=L_j$ and $\omega^{nm+n-j}\mathcal O^n=t^mL_j$; the residue of $i$ mod $n$ fixes $j$ and the quotient fixes $m$.
  2. (b). $g\in\mathrm{GL}n(F)$ stabilizes $L_n=\mathcal O^n$ iff $g\in\mathrm{GL}_n(\mathcal O)$, and then stabilizes every $L_j$ iff $g\bmod t$ preserves each $\mathrm{span}(\bar e_1,\dots,\bar e_j)$, i.e. is upper triangular. So $I$ is the stabilizer of the periodic chain $\{t^mL_j\}$. By (a) $\omega$ permutes this chain, so $\omega I\omega^{-1}=I$. $\omega=P\,\mathrm{diag}(t,1,\dots,1)$ with $P$ the cyclic permutation matrix $e_1\mapsto e_n$, $e_i\mapsto e{i-1}$, so lies in $N_G(T)(F)$; its class in $\widetilde W=N_G(T)(F)/T(\mathcal O)$ has $I\omega I=\omega I$, a single point of $\mathrm{Fl}$, so $\ell(\omega)=\dim I\omega I/I=0$ (Iwahori–Matsumoto, Prop. 3.2: $[I\omega I:I]=q^{\ell(\omega)}$ at function level).
  3. (c). $I\omega I\times^ILX=\omega I\times^ILX\to LX$, $[\omega i,y]\mapsto\omega iy$, is an isomorphism with inverse $x\mapsto[\omega,\omega^{-1}x]$, so $\Delta_\omega\ast_I\mathcal F\simeq\omega_*\mathcal F$, which is $I$-equivariant because $\omega$ normalizes $I$. Iterating ($\Delta_\omega\ast\Delta_{\omega'}=\Delta_{\omega\omega'}$, $\Delta_{\omega^{-1}}\ast\Delta_\omega=\Delta_e$): $\Delta_{\omega^i}\ast_I k_{\mathcal O^n}=k_{\omega^i\mathcal O^n}$ for all $i\in\mathbb Z$.
  4. (d). By Proposition 2.6(iv), $\mathcal S_I=\{k_\Lambda:\Lambda\in\mathcal L\}$ up to shift, and by (a), (c) each $k_\Lambda$ equals $G_{\omega^i}$ for the unique $i$ with $\omega^i\mathcal O^n=\Lambda$. A generator is a summand of itself, so $k_\Lambda\in\mathcal M_I$: $\mathcal M_I=\mathcal S_I$. Theorem 2.4(d) (the direction that needs no (DT$_I$)) gives $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}$. Hypotheses used: (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.), checked in Proposition 2.6; (DT$_I$), also checked there, is not needed for this direction. No weights are used, so the argument is valid over $\overline{\mathbb F}_q$ and over $\mathbb C$ alike (given (L3)).
  5. (e). $(T_\omega\ast\varphi)(x)=\sum_{gI\in G(F)/I}T_\omega(g)\varphi(g^{-1}x)=\varphi(\omega^{-1}x)$, so $T_{\omega^i}\ast\varphi_0=\mathbf 1_{\omega^i\mathcal O^n}$. If $f\in C_c^\infty(F^n)^I$, compact support puts $\operatorname{supp}f\subseteq t^{-N}\mathcal O^n$, local constancy at $0$ makes $f$ constant on some $t^M\mathcal O^n$, and $I$-invariance makes it constant on each $O_{m,j}$ (Proposition 2.6, Step 2). Hence $f=c\,\mathbf 1_{t^ML_n}+\sum_{-N\le m<M,\,j}c_{m,j}\mathbf 1_{O_{m,j}}$ with $\mathbf 1_{O_{m,j}}=\mathbf 1_{t^mL_j}-\mathbf 1_{t^mL_{j-1}}$, a combination of $T_{\omega^i}\ast\varphi_0$. So $\mathcal H_I\ast\varphi_0=C_I$.
  6. Difference from the pages' IV.1 (comparison only). IV.1 reaches the standard objects through translates $\Delta_{t^\lambda}\ast\mathrm{IC}_0$ and one triangle per piece (the 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). dévissage); here the simple objects are identified and each is a length-zero translate of $\mathrm{IC}_0$, i.e. $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, and generation follows from finite-length hearts.

Builds on

  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$
  • Proposition 2.6 · $\mathbb A^n$: all hypotheses hold; simple objects are lattice sheaves $k_\Lambda$

Built on by

  • 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 8.9 · Audit of $\mathcal C_I^{\rm Iwa}=\mathcal C_I$ for $\mathbb A^n$ (A5-3, A5-5, A2-7)
  • 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 18:47 UTC

$\mathbb A^n$: $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, hence Iwahori generation

  1. (a). $\omega L_j=\mathrm{span}{\mathcal O}(\omega e_1,\dots,\omega e_j,t\omega e{j+1},\dots,t\omega e_n)=\mathrm{span}(te_n,e_1,\dots,e_{j-1},te_j,\dots,te_{n-1})=L_{j-1}$, and $t$ is central. $\omega$ cycles $e_n\mapsto e_{n-1}\mapsto\dots\mapsto e_1\mapsto te_n$, so $\omega^n=t$; then $\omega^{n-j}L_n=L_j$ and $\omega^{nm+n-j}\mathcal O^n=t^mL_j$; the residue of $i$ mod $n$ fixes $j$ and the quotient fixes $m$.
  2. (b). $g\in\mathrm{GL}n(F)$ stabilizes $L_n=\mathcal O^n$ iff $g\in\mathrm{GL}_n(\mathcal O)$, and then stabilizes every $L_j$ iff $g\bmod t$ preserves each $\mathrm{span}(\bar e_1,\dots,\bar e_j)$, i.e. is upper triangular. So $I$ is the stabilizer of the periodic chain $\{t^mL_j\}$. By (a) $\omega$ permutes this chain, so $\omega I\omega^{-1}=I$. $\omega=P\,\mathrm{diag}(t,1,\dots,1)$ with $P$ the cyclic permutation matrix $e_1\mapsto e_n$, $e_i\mapsto e{i-1}$, so lies in $N_G(T)(F)$; its class in $\widetilde W=N_G(T)(F)/T(\mathcal O)$ has $I\omega I=\omega I$, a single point of $\mathrm{Fl}$, so $\ell(\omega)=\dim I\omega I/I=0$ (Iwahori–Matsumoto, Prop. 3.2: $[I\omega I:I]=q^{\ell(\omega)}$ at function level).
  3. (c). $I\omega I\times^ILX=\omega I\times^ILX\to LX$, $[\omega i,y]\mapsto\omega iy$, is an isomorphism with inverse $x\mapsto[\omega,\omega^{-1}x]$, so $\Delta_\omega\ast_I\mathcal F\simeq\omega_*\mathcal F$, which is $I$-equivariant because $\omega$ normalizes $I$. Iterating ($\Delta_\omega\ast\Delta_{\omega'}=\Delta_{\omega\omega'}$, $\Delta_{\omega^{-1}}\ast\Delta_\omega=\Delta_e$): $\Delta_{\omega^i}\ast_I k_{\mathcal O^n}=k_{\omega^i\mathcal O^n}$ for all $i\in\mathbb Z$.
  4. (d). By Proposition 2.6(iv), $\mathcal S_I=\{k_\Lambda:\Lambda\in\mathcal L\}$ up to shift, and by (a), (c) each $k_\Lambda$ equals $G_{\omega^i}$ for the unique $i$ with $\omega^i\mathcal O^n=\Lambda$. A generator is a summand of itself, so $k_\Lambda\in\mathcal M_I$: $\mathcal M_I=\mathcal S_I$. Theorem 2.4(d) (the direction that needs no (DT$_I$)) gives $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}$. Hypotheses used: (L1)–(L3), (I1), (I2), checked in Proposition 2.6; (DT$_I$), also checked there, is not needed for this direction. No weights are used, so the argument is valid over $\overline{\mathbb F}_q$ and over $\mathbb C$ alike (given (L3)).
  5. (e). $(T_\omega\ast\varphi)(x)=\sum_{gI\in G(F)/I}T_\omega(g)\varphi(g^{-1}x)=\varphi(\omega^{-1}x)$, so $T_{\omega^i}\ast\varphi_0=\mathbf 1_{\omega^i\mathcal O^n}$. If $f\in C_c^\infty(F^n)^I$, compact support puts $\operatorname{supp}f\subseteq t^{-N}\mathcal O^n$, local constancy at $0$ makes $f$ constant on some $t^M\mathcal O^n$, and $I$-invariance makes it constant on each $O_{m,j}$ (Proposition 2.6, Step 2). Hence $f=c\,\mathbf 1_{t^ML_n}+\sum_{-N\le m<M,\,j}c_{m,j}\mathbf 1_{O_{m,j}}$ with $\mathbf 1_{O_{m,j}}=\mathbf 1_{t^mL_j}-\mathbf 1_{t^mL_{j-1}}$, a combination of $T_{\omega^i}\ast\varphi_0$. So $\mathcal H_I\ast\varphi_0=C_I$.
  6. Difference from the pages' IV.1 (comparison only). IV.1 reaches the standard objects through translates $\Delta_{t^\lambda}\ast\mathrm{IC}_0$ and one triangle per piece (the I3 dévissage); here the simple objects are identified and each is a length-zero translate of $\mathrm{IC}_0$, i.e. $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, and generation follows from finite-length hearts.

Sources