§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:43:38Z

Corollary 3.8 · $\mathbb A^n$: inputs checked; (IG) gives sheaf generation over $\overline{\mathbb F}_q$

proved

For the affine space all of (L1), (L2), (D1)-(D3) are verified directly (lattice levels, stabilizers open in affine-linear spaces), and the elementary function theorem of F3 over $\mathbb F_q(\!(t)\!)$ then gives the sheaf statement through the bridge, without 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)., standard objects or Proposition IV.1.

Corollary 3.8.

Let $X=\mathbb A^n$ under $\mathrm{GL}n$ ($n\ge1$), split over $\mathbb F_q$, $p$ odd. With $Y_N=t^{-N}\mathcal O^n$, $Y{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n\simeq\mathbb A^{(N+M)n}$, $I_{N,M}=1+t^{N+M}M_n(\mathcal O)$, the hypotheses (L1), (L2), (D1)–(D3) of Setup 3.1 hold. Since $\mathcal H_I*\varphi_0=C_I$ over every non-archimedean $F$ (classical-affine.html, Theorem 3), Theorem 3.6 gives $$\mathcal M_I=\mathrm{Irr}\,\mathcal P_I\qquad\text{and}\qquad\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\quad\text{over }\overline{\mathbb F}_q .$$ The same holds for $\mathbb G_m=\mathrm{GL}_2/\mathrm{Sp}_2$ (action through $\det$).

  1. (L1). (a) $Y_{N,M'}\to Y_{N,M}$ is a linear projection of rank $n(M'-M)$, equivariant. (b) $Y_N=\pi_{N',M}^{-1}(t^{-N}\mathcal O^n/t^M\mathcal O^n)$, a closed linear subspace. (c) $I/I_{N,M}$ is the image of $I$ in $\mathrm{GL}n(\mathcal O/t^{N+M})$, an extension of $B$ by a unipotent group, connected; $I{N,M}$ is pro-unipotent. (d) $t^{-N}\mathcal O^n$ is compact open in $F^n$, exhausts it, and the cosets of $t^M\mathcal O^n$ are a basis. (e) Holds by the definition of $\mathcal C_I$ in this run's setting (objects pulled back from a finite level).
  2. (L2): orbits. The $I/I_{N,M}$-orbits on $Y_{N,M}$ are the images of $O_{m,j}=\{v:\mathrm{ord}\,v=m,\ \mathrm{last}\,v=j\}$, $-N\le m<M$, and $\{0\}$ (hypothesis.html, Example $\mathbb A^n$): finitely many, with rational representatives $t^me_j$ and $0$, hence $\mathrm{Fr}$-stable.
  3. (L2): stabilizers are connected. Write points as $t^{-N}u$, $u\in(\mathcal O/t^R)^n$, $R=N+M$; $I/I_{N,M}$ is the open subset $\{\det g\not\equiv0\bmod t\}$ of the affine space $L=\{g\in M_n(\mathcal O/t^R):g\bmod t\ \text{upper triangular}\}$. The stabilizer of $u$ is its intersection with the affine-linear subspace $\{g:(g-1)u=0\}$, a non-empty (it contains $1$) open subset of an affine space, hence irreducible, hence connected.
  4. (D1). For $w\in\widetilde W=\mathbb Z^n\rtimes S_n$ there is $a$ with $g,g^{-1}\in t^{-a}M_n(\mathcal O)$ for all $g\in\overline{IwI}$. Then $gY_N\subseteq Y_{N+a}$, and $g^{-1}(x+t^{M+a}\mathcal O^n)\subseteq g^{-1}x+t^M\mathcal O^n$, so $(g,x)\mapsto\pi_{N,M}(g^{-1}x)$ and the condition $g^{-1}x\in Y_N$ factor through $Y_{N+a,M''}$ for $M''\ge M+a$.
  5. (D2). Lang for the connected groups of steps 1 and 3. (D3). $\mathrm{IC}w$ is pure of weight $\ell(w)$, $\mathrm{IC}_0=\overline{\mathbb Q}{\ell,\mathcal O^n}$ is pure of weight $0$, and by step 4 $\mathrm{IC}w*_I\mathrm{IC}_0$ is the proper push-forward along $\mathrm{Fl}{\le w}\times Y_{N',M''}\to Y_{N',M''}$, equivariant for a finite-type quotient of $I$; Sun's theorem on the quotient stack gives the decomposition, with constant-coefficient summands by step 3.
  6. Conclusion. (IG) over $\mathbb F_q(\!(t)\!)$ is Theorem 3 of classical-affine.html, proved there by the lattice identities $\mathbf 1_{O_{m,j}}=[K:I]^{-1}(\mathbf 1_{g_{m,j}K}-\mathbf 1_{g_{m,j-1}K})*\varphi_0$ for every non-archimedean $F$. Apply Theorem 3.6(a).
  7. $\mathbb G_m$. $\det(I)=\mathcal O^\times$; take $Y_N=\bigsqcup_{|m|\le N}t^mL^+\mathbb G_m$, $Y_{N,M}=\bigsqcup_{|m|\le N}t^m(\mathcal O/t^M)^\times$, $I_{N,M}=\ker(\det\bmod t^M)$ (connected: generated by $I\cap\mathrm{SL}2$ and a pro-unipotent torus part). Each $t^m(\mathcal O/t^M)^\times$ is one orbit with trivial stabilizer; $C_I=C_K$ has basis $\mathbf 1{t^m\mathcal O^\times}=c\,T_{t^\lambda}*\varphi_0$ ($|\lambda|=m$, $c\ne0$), so (IG) holds, and (D3) is trivial (sums of shifted constant sheaves on components). $\square$

Independence. Not used: Proposition IV.1, (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).), the standard objects $\Delta_P$, or any sheaf-level dévissage. Used from the pages: only the function theorem of F3 and the level presentation of hypothesis.html. Over $\mathbb C$ the pages' IV.1 is still needed (the bridge is $\ell$-adic, Theorem 3.6 Fields).

Builds on

  • Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$

Built on by

  • Summary 3.12 · Verdict: (II$_I$) proved; rows 1–3 agree; one divergence type excluded
Original record, as the agent wrote it at 18:43 UTC

$\mathbb A^n$: inputs checked; (IG) gives sheaf generation over $\overline{\mathbb F}_q$

  1. (L1). (a) $Y_{N,M'}\to Y_{N,M}$ is a linear projection of rank $n(M'-M)$, equivariant. (b) $Y_N=\pi_{N',M}^{-1}(t^{-N}\mathcal O^n/t^M\mathcal O^n)$, a closed linear subspace. (c) $I/I_{N,M}$ is the image of $I$ in $\mathrm{GL}n(\mathcal O/t^{N+M})$, an extension of $B$ by a unipotent group, connected; $I{N,M}$ is pro-unipotent. (d) $t^{-N}\mathcal O^n$ is compact open in $F^n$, exhausts it, and the cosets of $t^M\mathcal O^n$ are a basis. (e) Holds by the definition of $\mathcal C_I$ in this run's setting (objects pulled back from a finite level).
  2. (L2): orbits. The $I/I_{N,M}$-orbits on $Y_{N,M}$ are the images of $O_{m,j}=\{v:\mathrm{ord}\,v=m,\ \mathrm{last}\,v=j\}$, $-N\le m<M$, and $\{0\}$ (hypothesis.html, Example $\mathbb A^n$): finitely many, with rational representatives $t^me_j$ and $0$, hence $\mathrm{Fr}$-stable.
  3. (L2): stabilizers are connected. Write points as $t^{-N}u$, $u\in(\mathcal O/t^R)^n$, $R=N+M$; $I/I_{N,M}$ is the open subset $\{\det g\not\equiv0\bmod t\}$ of the affine space $L=\{g\in M_n(\mathcal O/t^R):g\bmod t\ \text{upper triangular}\}$. The stabilizer of $u$ is its intersection with the affine-linear subspace $\{g:(g-1)u=0\}$, a non-empty (it contains $1$) open subset of an affine space, hence irreducible, hence connected.
  4. (D1). For $w\in\widetilde W=\mathbb Z^n\rtimes S_n$ there is $a$ with $g,g^{-1}\in t^{-a}M_n(\mathcal O)$ for all $g\in\overline{IwI}$. Then $gY_N\subseteq Y_{N+a}$, and $g^{-1}(x+t^{M+a}\mathcal O^n)\subseteq g^{-1}x+t^M\mathcal O^n$, so $(g,x)\mapsto\pi_{N,M}(g^{-1}x)$ and the condition $g^{-1}x\in Y_N$ factor through $Y_{N+a,M''}$ for $M''\ge M+a$.
  5. (D2). Lang for the connected groups of steps 1 and 3. (D3). $\mathrm{IC}w$ is pure of weight $\ell(w)$, $\mathrm{IC}_0=\overline{\mathbb Q}{\ell,\mathcal O^n}$ is pure of weight $0$, and by step 4 $\mathrm{IC}w*_I\mathrm{IC}_0$ is the proper push-forward along $\mathrm{Fl}{\le w}\times Y_{N',M''}\to Y_{N',M''}$, equivariant for a finite-type quotient of $I$; Sun's theorem on the quotient stack gives the decomposition, with constant-coefficient summands by step 3.
  6. Conclusion. (IG) over $\mathbb F_q(\!(t)\!)$ is Theorem 3 of classical-affine.html, proved there by the lattice identities $\mathbf 1_{O_{m,j}}=[K:I]^{-1}(\mathbf 1_{g_{m,j}K}-\mathbf 1_{g_{m,j-1}K})*\varphi_0$ for every non-archimedean $F$. Apply Theorem 3.6(a).
  7. $\mathbb G_m$. $\det(I)=\mathcal O^\times$; take $Y_N=\bigsqcup_{|m|\le N}t^mL^+\mathbb G_m$, $Y_{N,M}=\bigsqcup_{|m|\le N}t^m(\mathcal O/t^M)^\times$, $I_{N,M}=\ker(\det\bmod t^M)$ (connected: generated by $I\cap\mathrm{SL}2$ and a pro-unipotent torus part). Each $t^m(\mathcal O/t^M)^\times$ is one orbit with trivial stabilizer; $C_I=C_K$ has basis $\mathbf 1{t^m\mathcal O^\times}=c\,T_{t^\lambda}*\varphi_0$ ($|\lambda|=m$, $c\ne0$), so (IG) holds, and (D3) is trivial (sums of shifted constant sheaves on components). $\square$

Independence. Not used: Proposition IV.1, (I3), the standard objects $\Delta_P$, or any sheaf-level dévissage. Used from the pages: only the function theorem of F3 and the level presentation of hypothesis.html. Over $\mathbb C$ the pages' IV.1 is still needed (the bridge is $\ell$-adic, Theorem 3.6 Fields).

Sources