§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:22:28Z

Audit 8.8 · Audit of $\mathbb A^n$ orbits, levels and simple objects (A5-1, A5-2, A2-6)

proved

The orbit classification, the length-zero element τ, the connected stabilizers, the shift-normalized t-structure on the colimit and the list of simple objects S_k = k_{τ^k O^n}[−k] are correct; orbit counts recomputed by brute force.

Audit 8.8.

Holds for Lemma 5.1 (a)–(d), Lemma 5.2 (a)–(d) and Proposition 2.6 (ii)–(iv), over \(\mathbb C\) and \(\overline{\mathbb F}_q\) (orbit sets over any field). With \(\tau e_1=te_n\), \(\tau e_i=e_{i-1}\): \(\tau\Lambda_{m,j}=\Lambda_{m,j-1}\), \(\tau^{nm+n-j}\mathcal O^n=\Lambda_{m,j}\), \(\tau I\tau^{-1}=I\), \(\ell(\tau)=0\), \(\tau^n=t\), and \(\langle\tau\rangle=\Omega\) (\(\operatorname{val}\det\tau=1\)). The \(I\)-orbits on \(F^n\) are \(\{0\}\) and \(O_{m,j}=\tau^k\mathcal O^n\setminus\tau^{k+1}\mathcal O^n\). At level \((N,M)\) there are \(n(N+M)+1\) orbits, closures are linear subspaces, and stabilizers are connected. With the t-structure shifted by \(c_{N,M}=nM\), the transitions are t-exact and fully faithful with Serre images, and \(\operatorname{Simple}(\mathcal P_I)=\{S_k=\underline k_{\tau^k\mathcal O^n}[-k]:k\in\mathbb Z\}\), \(S_0=\mathrm{IC}_0\). Every object of \(\mathcal C_I\) is an iterated extension of shifted \(S_k\), and no nonzero object is supported on \(\{0\}\).

  1. Lattices and \(\tau\). The images of the generators of \(\Lambda_{m,j}\) under \(\tau\) are \(t^me_1,\dots,t^me_{j-1},t^{m+1}e_j,\dots,t^{m+1}e_n\) (including \(j=1\): \(t^{m+1}\mathcal O^n=\Lambda_{m,0}\)). \(I\) is the stabilizer of the periodic chain \(\{t^m\Lambda_{0,j}\}\), which \(\tau\) permutes, so \(\tau\) normalizes \(I\), \(I\tau I=\tau I\), and \(\ell(\tau)=0\). \(\tau=P\operatorname{diag}(t,1,\dots,1)\) with \(P\) the cyclic permutation, so \(\tau^k=t^{\mu}\sigma^k\) with \(t^\mu\mathcal O^n=\Lambda_{m,j}\), \(\mu=\lambda_{m,j}\). For \(\mathrm{GL}_n\), \(\Omega\simeq X_*(T)/Q^\vee\simeq\mathbb Z\) via \(\operatorname{val}\det\), and \(\operatorname{val}\det\tau=1\), so \(\tau\) generates \(\Omega\) (the authors do not state this, but it is not needed).
  2. Orbits. \(K_1v=v+t^{m+1}\mathcal O^n\) for \(\operatorname{ord}v=m\), so \(I\)-orbits of order \(m\) are preimages of \(B(k)\)-orbits on \(k^n\setminus0\) (upper triangular, columns): \(B e_j=\{w_j\ne0,\ w_i=0\ (i>j)\}\). Proposition 2.6's invariants and its explicit unipotent \(g\) give the same result.
  3. Brute-force check (computation). The orbits of the full group \(I/I_{N,M}\) on \(t^{-N}\mathcal O^n/t^M\mathcal O^n\) over \(\mathbb F_2\) were counted. The orbits of a subgroup generated by 60 random elements number at least as many as those of the group, and the invariants \((\operatorname{ord},\operatorname{last})\) give at least \(n(N+M)+1\); equality of the subgroup count with \(n(N+M)+1\) therefore proves the count. Results: \(n=2\), \((N,M)=(0,1),(0,2),(1,1),(1,2)\): \(3,5,5,7\); \(n=3\), \((0,1),(1,1)\): \(4,7\). All agree with \(n(N+M)+1\).
  4. Stabilizers. The stabilizer of \(t^me_j\bmod t^M\) is {column \(j\equiv e_j\bmod t^{M-m}\)} intersected with the Iwahori conditions. These are compatible since \(M-m\ge1\), and together they give a nonempty open subset of an affine-linear space, hence connected. Proposition 2.6's units-lemma form (\((1+\operatorname{Ann}\bar v)\cap(\mathfrak A/\mathfrak J)^\times\)) is equivalent. So every equivariant local system on a piece is constant (checklist 4).
  5. t-structure (checklist 1). \(q^*[n]\) is t-exact for the smooth \(\mathbb A^n\)-fibration \(V_{N,M+1}\to V_{N,M}\), and \(i_*\) is t-exact. Compatibility forces \(c_{N,M+1}=c_{N,M}+n\) and \(c_{N+1,M}=c_{N,M}\), so \(c_{N,M}=nM+\)const: the normalization is canonical up to one global shift. \(q^*\) is fully faithful on \(D^b\) (affine-space fibres) and on hearts with image closed under subquotients (BBD 4.2.5–4.2.6); the change of group along \(H_{N,M+1}\to H_{N,M}\) has kernel \(\simeq M_n(k)\), unipotent. Hence the heart of the colimit is the union of the finite-length hearts and simple objects stay simple. \(\mathrm{IC}(V_{\alpha,k})=\underline k[nM-k]\), which is \(\underline k[-k]\) after the shift, giving \(S_k\).
  6. The point \(0\). Objects are pulled back from finite levels, so their supports are unions of cosets of \(t^M\mathcal O^n\); \(\{0\}\) is never a piece, and the finite-level skyscraper at \(0\) is \(S_{nM}\), the constant sheaf on the tail \(t^M\mathcal O^n\).

Builds on

  • Lemma 5.1 · Orbits of $I$ on $L\mathbb A^n$: closures $\tau^k\mathcal O^n$, $\tau\in\Omega$
  • Lemma 5.2 · Simple objects of $\mathcal C_I$: $S_k=\underline k_{\tau^k\mathcal O^n}[-k]$
  • Proposition 2.6 · $\mathbb A^n$: all hypotheses hold; simple objects are lattice sheaves $k_\Lambda$

Built on by

  • Summary 8.13 · Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$
Original record, as the agent wrote it at 19:22 UTC

Audit of $\mathbb A^n$ orbits, levels and simple objects (A5-1, A5-2, A2-6)

  1. Lattices and \(\tau\). The images of the generators of \(\Lambda_{m,j}\) under \(\tau\) are \(t^me_1,\dots,t^me_{j-1},t^{m+1}e_j,\dots,t^{m+1}e_n\) (including \(j=1\): \(t^{m+1}\mathcal O^n=\Lambda_{m,0}\)). \(I\) is the stabilizer of the periodic chain \(\{t^m\Lambda_{0,j}\}\), which \(\tau\) permutes, so \(\tau\) normalizes \(I\), \(I\tau I=\tau I\), and \(\ell(\tau)=0\). \(\tau=P\operatorname{diag}(t,1,\dots,1)\) with \(P\) the cyclic permutation, so \(\tau^k=t^{\mu}\sigma^k\) with \(t^\mu\mathcal O^n=\Lambda_{m,j}\), \(\mu=\lambda_{m,j}\). For \(\mathrm{GL}_n\), \(\Omega\simeq X_*(T)/Q^\vee\simeq\mathbb Z\) via \(\operatorname{val}\det\), and \(\operatorname{val}\det\tau=1\), so \(\tau\) generates \(\Omega\) (the authors do not state this, but it is not needed).
  2. Orbits. \(K_1v=v+t^{m+1}\mathcal O^n\) for \(\operatorname{ord}v=m\), so \(I\)-orbits of order \(m\) are preimages of \(B(k)\)-orbits on \(k^n\setminus0\) (upper triangular, columns): \(B e_j=\{w_j\ne0,\ w_i=0\ (i>j)\}\). Proposition 2.6's invariants and its explicit unipotent \(g\) give the same result.
  3. Brute-force check (computation). The orbits of the full group \(I/I_{N,M}\) on \(t^{-N}\mathcal O^n/t^M\mathcal O^n\) over \(\mathbb F_2\) were counted. The orbits of a subgroup generated by 60 random elements number at least as many as those of the group, and the invariants \((\operatorname{ord},\operatorname{last})\) give at least \(n(N+M)+1\); equality of the subgroup count with \(n(N+M)+1\) therefore proves the count. Results: \(n=2\), \((N,M)=(0,1),(0,2),(1,1),(1,2)\): \(3,5,5,7\); \(n=3\), \((0,1),(1,1)\): \(4,7\). All agree with \(n(N+M)+1\).
  4. Stabilizers. The stabilizer of \(t^me_j\bmod t^M\) is {column \(j\equiv e_j\bmod t^{M-m}\)} intersected with the Iwahori conditions. These are compatible since \(M-m\ge1\), and together they give a nonempty open subset of an affine-linear space, hence connected. Proposition 2.6's units-lemma form (\((1+\operatorname{Ann}\bar v)\cap(\mathfrak A/\mathfrak J)^\times\)) is equivalent. So every equivariant local system on a piece is constant (checklist 4).
  5. t-structure (checklist 1). \(q^*[n]\) is t-exact for the smooth \(\mathbb A^n\)-fibration \(V_{N,M+1}\to V_{N,M}\), and \(i_*\) is t-exact. Compatibility forces \(c_{N,M+1}=c_{N,M}+n\) and \(c_{N+1,M}=c_{N,M}\), so \(c_{N,M}=nM+\)const: the normalization is canonical up to one global shift. \(q^*\) is fully faithful on \(D^b\) (affine-space fibres) and on hearts with image closed under subquotients (BBD 4.2.5–4.2.6); the change of group along \(H_{N,M+1}\to H_{N,M}\) has kernel \(\simeq M_n(k)\), unipotent. Hence the heart of the colimit is the union of the finite-length hearts and simple objects stay simple. \(\mathrm{IC}(V_{\alpha,k})=\underline k[nM-k]\), which is \(\underline k[-k]\) after the shift, giving \(S_k\).
  6. The point \(0\). Objects are pulled back from finite levels, so their supports are unions of cosets of \(t^M\mathcal O^n\); \(\{0\}\) is never a piece, and the finite-level skyscraper at \(0\) is \(S_{nM}\), the constant sheaf on the tail \(t^M\mathcal O^n\).

Sources