§7 · Agent 7 · Audit: criterion and bridge · 2026-10-18T19:25:20Z

Lemma 7.5 · Hypotheses (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.), Frobenius-stability for $\mathbb A^n$ and the group; $I_{N,M}$ must be connected for Lang

proved

A one-line lemma (stabilizers in the unit group of a finite-dimensional algebra are open in an affine space) proves (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) for $\mathbb A^n$ at every level; orbit counts over $\mathbb F_3,\mathbb F_5$ confirm one rational orbit per piece; for the group the stabilizers are connected, but Lang in A3-1 needs $I_{N,M}$ connected, which fails for the full kernel when $Z(H)$ is disconnected.

Lemma 7.5.

Holds (verification of hypotheses used by A2-1, A3-1 (L1c), (L2), A3-2 and A5-7 "sufficient conditions" for the two rows of (III$_I$)).

(a) Lemma. Let $\bar A$ be a finite-dimensional $k$-algebra, $V$ a finite-dimensional $\bar A$-module, $H=\bar A^\times$. For $v\in V$, $\mathrm{Stab}_H(v)=\bar A^\times\cap(1+\mathrm{Ann}(v))$ is a non-empty open subset of the affine space $1+\mathrm{Ann}(v)$, hence irreducible and connected.

(b) For $X=\mathbb A^n$, $G=\mathrm{GL}n$, $Y{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n$, $I_{N,M}=1+t^{N+M}M_n(\mathcal O)$: $I/I_{N,M}=\bar A^\times$ for the algebra $\bar A=\mathfrak A/t^{N+M}M_n(\mathcal O)$ ($\mathfrak A$ the Iwahori order), so (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) holds at every level; $I_{N,M}$ is pro-unipotent, so Lang applies.

(c) Computation. For $(n,q,N,M)\in\{(1,3,1,1),(2,3,0,2),(2,3,1,1),(2,5,1,1),(3,3,1,1),(3,3,0,2)\}$ the number of $I(\mathbb F_q[[t]])$-orbits on $Y_{N,M}(\mathbb F_q)$ is exactly $n(N+M)+1=|\mathcal P_{N,M}|$ (hypothesis.html, Example A), with orbit sizes $1$ and $(q-1)q^k$, $0\le k<n(N+M)$.

(d) For the group $X=H$ (Example B), the stabilizer of $gI_M$ in $I\times I$ is isomorphic to $(I\cap gIg^{-1})\times I_M$, connected. But if $I_{N,M}$ is taken to be the full kernel, it contains $\Delta Z(H)$; for $H=\mathrm{SL}2$ and $q$ odd, $H^1(\mathbb F_q,\Delta\mu_2)=\mathbb Z/2$, so $I(\mathcal O)\to(I/I{N,M})(\mathbb F_q)$ is not surjective and (D2) of A3-1 fails. With the congruence choice of Example B (pro-unipotent $I_{N,M}$), (L1c) and (D2) hold and $\Delta\mu_2$ lies in the connected stabilizer.

  1. Proof of (a). $g\in\bar A$ fixes $v$ iff $(g-1)v=0$ iff $g\in1+\mathrm{Ann}(v)$, and $\mathrm{Ann}(v)$ is a $k$-subspace. $\bar A^\times$ is the complement of the hypersurface $\det(L_g)=0$ ($L_g$ left multiplication), open in $\bar A$. The intersection contains $1$, so it is a non-empty open subset of an irreducible affine space. $\square$ (Stabilizers are taken reduced; for étale local systems only the reduced group matters.)
  2. Proof of (b). $t^{N+M}M_n(\mathcal O)$ is a two-sided ideal of $\mathfrak A$, and $g\in I$ acts trivially on $t^{-N}\mathcal O^n/t^M\mathcal O^n$ iff $g-1\in t^{N+M}M_n(\mathcal O)$; so $I/I_{N,M}=(\mathfrak A/t^{N+M}M_n(\mathcal O))^\times$ acting $\bar A$-linearly on $Y_{N,M}$. Apply (a). $I_{N,M}$ is a projective limit of unipotent groups, so $H^1(\mathbb F_q,I_{N,M})=1$ (Milne, Cor. 27.56, for each finite quotient, plus Mittag-Leffler).
  3. Computation (c). Breadth-first orbit enumeration on the $q^{n(N+M)}$ points of $Y_{N,M}(\mathbb F_q)$ under generators of $I/I_{N,M}$ (diagonal units $a\in\mathbb F_q^\times$ and $1+t$; upper elementary matrices with entries $t^k$, $k\ge0$; lower ones with $t^k$, $k\ge1$; these generate by the Iwahori decomposition). Result: $3,5,5,5,7,7$ orbits respectively, equal to $n(N+M)+1$, sizes $\{1\}\cup\{(q-1)q^k\}$. Why it matters: with $\mathrm{Fr}$-stable orbits and connected stabilizers, Lang gives exactly one rational orbit per geometric orbit (A3-2(c)); a disconnected stabilizer with trivial Frobenius action on $\pi_0$ would split a rational orbit and raise the count. The sizes are those of $\mathbb G_m\times\mathbb A^k$, matching the pieces $O_{m,j}$ of Example A. This checks, at these levels, the hypotheses (L2) of A3-1 and (W), (Lin) of A5-7 for $\mathbb A^n$; the proof for all levels is (a)–(b) together with the rational representatives $t^me_j$, $0$ of the pieces.
  4. Proof of (d), stabilizers. $(i_1,i_2)$ fixes $gI_M$ iff $i_2=j^{-1}g^{-1}i_1g$ with $j\in I_M$ and $g^{-1}i_1g\in I$; so $(i_1,j)\mapsto(i_1,j^{-1}g^{-1}i_1g)$ identifies $(I\cap gIg^{-1})\times I_M$ with the stabilizer. $I\cap gIg^{-1}$ is conjugate to $I\cap wIw^{-1}$, connected (Chen–Yi, Lemma 71(i)); $I_M$ is pro-unipotent. The image in $I/I_{N,M}$ is connected, and $\Delta Z(H)$ lies in it.
  5. Proof of (d), Lang. $1\to J\to I\to I/J\to1$ gives $I(\mathbb F_q)\to(I/J)(\mathbb F_q)\to H^1(\mathbb F_q,J)\to H^1(\mathbb F_q,I)=1$. For $J\supseteq\Delta\mu_2$ with $J/\Delta\mu_2$ pro-unipotent, $H^1(\mathbb F_q,J)=H^1(\mathbb F_q,\mu_2)=\mathrm{Hom}(\hat{\mathbb Z},\mathbb Z/2)=\mathbb Z/2$ ($q$ odd, $\mu_2$ constant). So the full-kernel choice violates A3-1 (L1c) "$I_{N,M}$ connected" — which A3-1 assumes explicitly, so A3-1 is correct; the point is that the verification for the group must use Example B's congruence subgroup.

Builds on

  • Setup 3.1 · Set-up over $\mathbb F_q$: levels, the perverse heart of $\mathcal C_I$, Weil structures
  • Lemma 3.2 · Functions at a finite level: bases $\{\mathbf 1_{P(\mathbb F_q)}\}$ by Lang
  • Setup 2.1 · Set-up: level hypotheses, simple objects, $\mathcal M_I\subseteq\mathcal M_I^+$
  • Proposition 5.7 · (II$_I$) (IG) over $\mathbb F_q((t))$ puts every simple object in $\mathcal M_I$

Built on by

  • Summary 7.11 · Verdict of the audit: (I$_I$) over $\overline{\mathbb F}_q$ and $\mathbb C$, (II$_I$) over $\overline{\mathbb F}_q$ only, converses as stated
Original record, as the agent wrote it at 19:25 UTC

Hypotheses (I1), (I2), Frobenius-stability for $\mathbb A^n$ and the group; $I_{N,M}$ must be connected for Lang

  1. Proof of (a). $g\in\bar A$ fixes $v$ iff $(g-1)v=0$ iff $g\in1+\mathrm{Ann}(v)$, and $\mathrm{Ann}(v)$ is a $k$-subspace. $\bar A^\times$ is the complement of the hypersurface $\det(L_g)=0$ ($L_g$ left multiplication), open in $\bar A$. The intersection contains $1$, so it is a non-empty open subset of an irreducible affine space. $\square$ (Stabilizers are taken reduced; for étale local systems only the reduced group matters.)
  2. Proof of (b). $t^{N+M}M_n(\mathcal O)$ is a two-sided ideal of $\mathfrak A$, and $g\in I$ acts trivially on $t^{-N}\mathcal O^n/t^M\mathcal O^n$ iff $g-1\in t^{N+M}M_n(\mathcal O)$; so $I/I_{N,M}=(\mathfrak A/t^{N+M}M_n(\mathcal O))^\times$ acting $\bar A$-linearly on $Y_{N,M}$. Apply (a). $I_{N,M}$ is a projective limit of unipotent groups, so $H^1(\mathbb F_q,I_{N,M})=1$ (Milne, Cor. 27.56, for each finite quotient, plus Mittag-Leffler).
  3. Computation (c). Breadth-first orbit enumeration on the $q^{n(N+M)}$ points of $Y_{N,M}(\mathbb F_q)$ under generators of $I/I_{N,M}$ (diagonal units $a\in\mathbb F_q^\times$ and $1+t$; upper elementary matrices with entries $t^k$, $k\ge0$; lower ones with $t^k$, $k\ge1$; these generate by the Iwahori decomposition). Result: $3,5,5,5,7,7$ orbits respectively, equal to $n(N+M)+1$, sizes $\{1\}\cup\{(q-1)q^k\}$. Why it matters: with $\mathrm{Fr}$-stable orbits and connected stabilizers, Lang gives exactly one rational orbit per geometric orbit (A3-2(c)); a disconnected stabilizer with trivial Frobenius action on $\pi_0$ would split a rational orbit and raise the count. The sizes are those of $\mathbb G_m\times\mathbb A^k$, matching the pieces $O_{m,j}$ of Example A. This checks, at these levels, the hypotheses (L2) of A3-1 and (W), (Lin) of A5-7 for $\mathbb A^n$; the proof for all levels is (a)–(b) together with the rational representatives $t^me_j$, $0$ of the pieces.
  4. Proof of (d), stabilizers. $(i_1,i_2)$ fixes $gI_M$ iff $i_2=j^{-1}g^{-1}i_1g$ with $j\in I_M$ and $g^{-1}i_1g\in I$; so $(i_1,j)\mapsto(i_1,j^{-1}g^{-1}i_1g)$ identifies $(I\cap gIg^{-1})\times I_M$ with the stabilizer. $I\cap gIg^{-1}$ is conjugate to $I\cap wIw^{-1}$, connected (Chen–Yi, Lemma 71(i)); $I_M$ is pro-unipotent. The image in $I/I_{N,M}$ is connected, and $\Delta Z(H)$ lies in it.
  5. Proof of (d), Lang. $1\to J\to I\to I/J\to1$ gives $I(\mathbb F_q)\to(I/J)(\mathbb F_q)\to H^1(\mathbb F_q,J)\to H^1(\mathbb F_q,I)=1$. For $J\supseteq\Delta\mu_2$ with $J/\Delta\mu_2$ pro-unipotent, $H^1(\mathbb F_q,J)=H^1(\mathbb F_q,\mu_2)=\mathrm{Hom}(\hat{\mathbb Z},\mathbb Z/2)=\mathbb Z/2$ ($q$ odd, $\mu_2$ constant). So the full-kernel choice violates A3-1 (L1c) "$I_{N,M}$ connected" — which A3-1 assumes explicitly, so A3-1 is correct; the point is that the verification for the group must use Example B's congruence subgroup.

Sources