§5 · Agent 5 · 𝔸ⁿ: lattices · 2026-10-18T18:44:11Z

Corollary 5.8 · Reverse route for $\mathbb A^n$: (IG) and (II$_I$) give sheaf generation

proved

For $\mathbb A^n$ the inputs (L$_q$), (W), (P), (Lin), (Tr) of the functions-to-sheaves bridge all hold, so (IG) from Proposition 5.6 re-proves sheaf generation over $\overline{\mathbb F}_q$ without the convolution computations of Computation 5.3. This is the one place in the $\mathbb A^n$ story where purity (the decomposition theorem) is used.

Corollary 5.8.

For $X=\mathbb A^n$ under $\mathrm{GL}n$, $F=\mathbb F_q((t))$, sheaves over $\overline{\mathbb F}_q$: the hypotheses (L), (Hk), (L$_q$), (W), (P), (Lin), (Tr) of Proposition 5.7 hold, with $S{k,0}=\overline{\mathbb Q}{\ell,\tau^k\mathcal O^n}[-k]$ and $\operatorname{tr}S{k,0}=(-1)^kb_k$. Since (IG) holds (Proposition 5.6(c)), Proposition 5.7 gives $\mathcal M_I=\{S_k\}_{k\in\mathbb Z}$ and $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ over $\overline{\mathbb F}_q$, using from the sheaf side only Lemma 5.1, Lemma 5.2 and Proposition 5.4.

Check of each input.

  1. (L), (Hk). Theorem 5.5, steps 1–2.
  1. (L$q$). $V{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n$, $H_{N,M}$, the action and the reductions are defined over $\mathbb F_q$; $V_{N,M}(\mathbb F_q)=t^{-N}\mathbb F_q[[t]]^n/t^M\mathbb F_q[[t]]^n$ and $H_{N,M}(\mathbb F_q)=I_F/I_{F,N,M}$.
  1. (W). $\tau\in\mathrm{GL}n(\mathbb F_q((t)))$, so every lattice $\tau^k\mathcal O^n$ is defined over $\mathbb F_q$ and $S_k$ has the Weil structure $S{k,0}$ above (no Tate twist needed for the argument); every simple object is Frobenius-stable (Lemma 5.2(b)).
  1. (Lin). $\operatorname{tr}S_{k,0}=(-1)^kb_k$, and the $b_k$ are linearly independent (Proposition 5.6(c), strictly nested lattices).
  1. (P). Fix $w$. $\mathrm{Fl}{\le w}$ is projective, so there are $N,M$ with $t^M\mathcal O^n\subseteq g\mathcal O^n\subseteq t^{-N}\mathcal O^n$ for all $gI\in\mathrm{Fl}{\le w}$. Put $E_w=\{(x,\bar y)\in\mathrm{Fl}{\le w}\times V{N,M}:\bar y\in g_x\mathcal O^n/t^M\mathcal O^n\}$, a closed $I$-stable vector subbundle of the trivial bundle; $m_w=\mathrm{pr}2:E_w\to V{N,M}$ is proper. The convolution space over $\mathrm{Fl}{\le w}$ is $\{(gI,y):y\in g\mathcal O^n\}=(\mathrm{id}\times\pi{N,M})^{-1}(E_w)$, on which $\mathrm{IC}w\mathbin{\tilde\boxtimes}\mathrm{IC}_0=\mathrm{pr}_1^\mathrm{IC}_w$; smooth base change gives $\mathrm{IC}_w\ast\mathrm{IC}_0=\pi{N,M}^\,m_{w!}\mathrm{pr}1^\mathrm{IC}_w$. $\mathrm{IC}_w$ is pure (Gabber), $\mathrm{pr}_1^$ preserves purity up to shift and twist, and $m{w!}=m_{w*}$ preserves it (BBD 5.1.14 and duality). So (P) holds, and (DT)(DT) · Decomposition: every generator IC_w ∗_I IC_0 is, equivariantly, a finite direct sum of shifted simple objects of C_I. Holds over F̄_q from a finite-type proper model of the convolution (purity and the stack decomposition theorem). follows (BBD 5.4.5, 5.3.8; equivariantly Sun, Thm 1.2).
  1. (Tr). $\operatorname{tr}\mathrm{IC}0=\mathbf 1{\mathcal O^n}=\varphi_0$. For the generators: proper base change along $m_w$ and the Grothendieck–Lefschetz formula on its fibres, with $H^1(\mathbb F_q,I)=1$ ($I$ connected pro-algebraic, Lang), give $\operatorname{tr}(\mathrm{IC}_w\ast\mathrm{IC}_0)=\operatorname{tr}(\mathrm{IC}_w)\ast\varphi_0$ with $\operatorname{vol}I=1$.

Conclusion and its scope. Proposition 5.7 applies: $\mathcal M_I=\{S_k\}$, and Proposition 5.4(c) gives $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ over $\overline{\mathbb F}_q$. This route needs (P), i.e. purity and the decomposition theorem, because traces alone only show that each $S_k$ is a constituent of some $^pH^i(\mathrm{IC}_w\ast\mathrm{IC}_0)$; the summand property needs semisimplicity. The direct route Theorem 5.5 avoids this, holds over $\mathbb C$ as well, and also names the summands ($w=\tau^k$). Consistency: both routes give the same $\mathcal M_I$, and the traces of Computation 5.3 are the identities of Proposition 5.6(e).

Builds on

  • Lemma 5.2 · Simple objects of $\mathcal C_I$: $S_k=\underline k_{\tau^k\mathcal O^n}[-k]$
  • 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 $\mathcal M_I$

Built on by

  • Summary 5.9 · Verdict: for $\mathbb A^n$ both Iwahori-level statements hold
  • Audit 8.10 · Audit of (IG) for $\mathbb A^n$ and the reverse route (A5-6, A5-8)
Original record, as the agent wrote it at 18:44 UTC

Reverse route for $\mathbb A^n$: (IG) and (II$_I$) give sheaf generation

Check of each input.

  1. (L), (Hk). Theorem 5.5, steps 1–2.
  1. (L$q$). $V{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n$, $H_{N,M}$, the action and the reductions are defined over $\mathbb F_q$; $V_{N,M}(\mathbb F_q)=t^{-N}\mathbb F_q[[t]]^n/t^M\mathbb F_q[[t]]^n$ and $H_{N,M}(\mathbb F_q)=I_F/I_{F,N,M}$.
  1. (W). $\tau\in\mathrm{GL}n(\mathbb F_q((t)))$, so every lattice $\tau^k\mathcal O^n$ is defined over $\mathbb F_q$ and $S_k$ has the Weil structure $S{k,0}$ above (no Tate twist needed for the argument); every simple object is Frobenius-stable (Lemma 5.2(b)).
  1. (Lin). $\operatorname{tr}S_{k,0}=(-1)^kb_k$, and the $b_k$ are linearly independent (Proposition 5.6(c), strictly nested lattices).
  1. (P). Fix $w$. $\mathrm{Fl}{\le w}$ is projective, so there are $N,M$ with $t^M\mathcal O^n\subseteq g\mathcal O^n\subseteq t^{-N}\mathcal O^n$ for all $gI\in\mathrm{Fl}{\le w}$. Put $E_w=\{(x,\bar y)\in\mathrm{Fl}{\le w}\times V{N,M}:\bar y\in g_x\mathcal O^n/t^M\mathcal O^n\}$, a closed $I$-stable vector subbundle of the trivial bundle; $m_w=\mathrm{pr}2:E_w\to V{N,M}$ is proper. The convolution space over $\mathrm{Fl}{\le w}$ is $\{(gI,y):y\in g\mathcal O^n\}=(\mathrm{id}\times\pi{N,M})^{-1}(E_w)$, on which $\mathrm{IC}w\mathbin{\tilde\boxtimes}\mathrm{IC}_0=\mathrm{pr}_1^\mathrm{IC}_w$; smooth base change gives $\mathrm{IC}_w\ast\mathrm{IC}_0=\pi{N,M}^\,m_{w!}\mathrm{pr}1^\mathrm{IC}_w$. $\mathrm{IC}_w$ is pure (Gabber), $\mathrm{pr}_1^$ preserves purity up to shift and twist, and $m{w!}=m_{w*}$ preserves it (BBD 5.1.14 and duality). So (P) holds, and (DT) follows (BBD 5.4.5, 5.3.8; equivariantly Sun, Thm 1.2).
  1. (Tr). $\operatorname{tr}\mathrm{IC}0=\mathbf 1{\mathcal O^n}=\varphi_0$. For the generators: proper base change along $m_w$ and the Grothendieck–Lefschetz formula on its fibres, with $H^1(\mathbb F_q,I)=1$ ($I$ connected pro-algebraic, Lang), give $\operatorname{tr}(\mathrm{IC}_w\ast\mathrm{IC}_0)=\operatorname{tr}(\mathrm{IC}_w)\ast\varphi_0$ with $\operatorname{vol}I=1$.

Conclusion and its scope. Proposition 5.7 applies: $\mathcal M_I=\{S_k\}$, and Proposition 5.4(c) gives $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ over $\overline{\mathbb F}_q$. This route needs (P), i.e. purity and the decomposition theorem, because traces alone only show that each $S_k$ is a constituent of some $^pH^i(\mathrm{IC}_w\ast\mathrm{IC}_0)$; the summand property needs semisimplicity. The direct route Theorem 5.5 avoids this, holds over $\mathbb C$ as well, and also names the summands ($w=\tau^k$). Consistency: both routes give the same $\mathcal M_I$, and the traces of Computation 5.3 are the identities of Proposition 5.6(e).

Sources