§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:23:08Z

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

proved ★ key step

Every simple object S_k of C_I is, up to the shift [k], the single generator δ_{τ^k} ∗ IC_0, so C_I^Iwa = C_I, already by extensions and shifts. The proof is correct over C and F̄_q, uses no weights or decomposition theorem, and does not use IV.1 or 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)..

Audit 8.9.

Holds over \(\mathbb C\) (Betti) and \(\overline{\mathbb F}_q\), for every \(n\ge1\), with \(\Delta_w\) unshifted and \(\mathrm{IC}_w\) perverse: $$\mathrm{IC}_{\tau^k}=\Delta_{\tau^k}=\delta_{\tau^kI},\qquad \delta_{\tau^k}\ast_I\mathcal F\simeq(\tau^k)_!\mathcal F,\qquad \delta_{\tau^k}\ast_I\mathrm{IC}_0=\underline k_{\tau^k\mathcal O^n}=S_k[k],$$ so \(\mathcal M_I=\operatorname{Simple}(\mathcal P_I)\) and \(\mathcal C_I=\operatorname{Tri}\{\delta_{\tau^k}\ast_I\mathrm{IC}_0\}=\mathcal C_I^{\rm Iwa}\). Also correct: \(\mathrm{IC}_{s_i}\ast_IS_k=S_{k-1}\oplus S_{k+1}\) if \(i\equiv-k\pmod n\), else \(S_k[1]\oplus S_k[-1]\); \(\Delta_{t^{\lambda_{m,j}}}\ast_I\mathrm{IC}_0=\underline k_{\Lambda_{m,j}}[-2j(n-j)]\); \(\mathrm{IC}_{\tau^kw_0}\ast_I\mathrm{IC}_0=\bigoplus_iS_k[k+N_0-2i]^{b_{2i}(G/B)}\).

  1. Length-zero convolution. \(I\tau^kI=\tau^kI\) is a point of \(\mathrm{Fl}\); \(\tau^kI\times^ILX\to LX\), \([\tau^ki,y]\mapsto\tau^kiy\), is an isomorphism with inverse \(x\mapsto[\tau^k,\tau^{-k}x]\). Since \(\tau^k\) normalizes \(I\), \((\tau^k)_!\) preserves \(I\)-equivariance. With Audit 8.8, \((\tau^k)_!\underline k_{\mathcal O^n}=\underline k_{\tau^k\mathcal O^n}=S_k[k]\), and \((\tau^k)_!S_l=S_{k+l}[k]\). This is an isomorphism of spaces: no decomposition theorem, no weights, valid over both fields.
  2. Generation. Each simple \(S_k\) is a shift of a generator, a one-term sum. Every object of \(\mathcal C_I\) is an iterated extension of shifted \(S_k\) (Audit 8.8 step 5), so \(\mathcal C_I\subseteq\operatorname{Tri}\{\delta_{\tau^k}\ast\mathrm{IC}_0\}\subseteq\mathcal C_I^{\rm Iwa}\). Only \(\delta_{\tau^k}\in\mathcal H_{c,I}\) is used, an \(I\)-equivariant skyscraper at a point of \(\mathrm{Fl}\). Simple objects are IC's of pieces pulled back from finite levels, and the point \(0\) never appears (checklist 1). The references to A5-4 and A2-4(d) are not needed beyond this direct argument.
  3. Reflections. \(P_{s_i}\) is the stabilizer of all \(\Lambda_{0,j}\), \(j\not\equiv i\), and \(\tau\) shifts the chain by one, so \(\tau s_i\tau^{-1}=s_{i-1}\) and \(s_i\tau^k=\tau^ks_{i+k}\). For \(i=0\), \(P_{s_0}/I\) is the \(\mathbb P^1\) of lattices between \(\tau\mathcal O^n=\langle e_1,\dots,e_{n-1},te_n\rangle\) and \(\tau^{-1}\mathcal O^n=\langle t^{-1}e_1,e_2,\dots,e_n\rangle\) (\(P_{s_0}\cap K=I\)). \(E=\{(L,v):v\in L\}\) is the pullback of \(\mathrm{Bl}_0\mathbb A^2\) along the smooth \(V\to V/Z\). The connecting map lies in \(\operatorname{Hom}(\underline k_0,i^!\underline k[3])=H^{-1}_H(\mathrm{pt})=0\), so \(m_!\underline k_E=\underline k_{\tau^{-1}\mathcal O^n}\oplus\underline k_{\tau\mathcal O^n}[-2]\) and \(\mathrm{IC}_{s_0}\ast\mathrm{IC}_0=S_{-1}\oplus S_1\). For \(i\ne0\), \(P_{s_i}\subseteq K\) gives \(S_0\otimes H^*(\mathbb P^1)[1]\), split since \(H^{\rm odd}_H(V)=0\). The general \(k\) follows by conjugation.
  4. Other translates. \(\ell(t^{\lambda_{m,j}})=\#\{a\le j<b\}=j(n-j)\), \(t^\lambda=\tau^ku\), \(u\in W\), \(\ell(u)=\ell(t^\lambda)\); \(IuI\times^I\mathcal O^n\simeq IuI/I\times\mathcal O^n\) since \(IuI\subseteq K\), so \(\Delta_u\ast\mathrm{IC}_0=\underline k\otimes R\Gamma_c(\mathbb A^{\ell(u)})\). For \(w_{\max}=\tau^kw_0\), the even cohomology of \(G/B\) and \(H^{\rm odd}_T(\mathrm{pt})=0\) split the complex in the equivariant category. Correct as written.
  5. Independence from IV.1 (checklist 9). IV.1 (case-affine.html) runs 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: the standard objects \(\Delta_{O_{m,j}}\) from the triangles \(\Delta_{O_{m,j}}\to\underline k_{\Lambda_{m,j}}\to\underline k_{\Lambda_{m,j-1}}\) with the translates \(\Delta_{t^\lambda}\ast\mathrm{IC}_0\), and Proposition III. The present route identifies the heart of the colimit and its simple objects (Audit 8.8) and uses length-zero translates and the summand criterion. It shares with IV.1 only the elementary fact that each lattice sheaf is a Hecke translate of \(\mathrm{IC}_0\), which it re-proves with different Hecke elements. So the argument is different and not circular.

Builds on

  • Computation 5.3 · $\mathrm{IC}_{\tau^k}\ast_I\mathrm{IC}_0=S_k[k]$ and the simple reflections
  • Theorem 5.5 · $\mathbb A^n$: $\mathcal M_I=\{S_k\}$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$
  • Theorem 2.7 · $\mathbb A^n$: $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, hence Iwahori generation

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 19:23 UTC

Audit of $\mathcal C_I^{\rm Iwa}=\mathcal C_I$ for $\mathbb A^n$ (A5-3, A5-5, A2-7)

  1. Length-zero convolution. \(I\tau^kI=\tau^kI\) is a point of \(\mathrm{Fl}\); \(\tau^kI\times^ILX\to LX\), \([\tau^ki,y]\mapsto\tau^kiy\), is an isomorphism with inverse \(x\mapsto[\tau^k,\tau^{-k}x]\). Since \(\tau^k\) normalizes \(I\), \((\tau^k)_!\) preserves \(I\)-equivariance. With Audit 8.8, \((\tau^k)_!\underline k_{\mathcal O^n}=\underline k_{\tau^k\mathcal O^n}=S_k[k]\), and \((\tau^k)_!S_l=S_{k+l}[k]\). This is an isomorphism of spaces: no decomposition theorem, no weights, valid over both fields.
  2. Generation. Each simple \(S_k\) is a shift of a generator, a one-term sum. Every object of \(\mathcal C_I\) is an iterated extension of shifted \(S_k\) (Audit 8.8 step 5), so \(\mathcal C_I\subseteq\operatorname{Tri}\{\delta_{\tau^k}\ast\mathrm{IC}_0\}\subseteq\mathcal C_I^{\rm Iwa}\). Only \(\delta_{\tau^k}\in\mathcal H_{c,I}\) is used, an \(I\)-equivariant skyscraper at a point of \(\mathrm{Fl}\). Simple objects are IC's of pieces pulled back from finite levels, and the point \(0\) never appears (checklist 1). The references to A5-4 and A2-4(d) are not needed beyond this direct argument.
  3. Reflections. \(P_{s_i}\) is the stabilizer of all \(\Lambda_{0,j}\), \(j\not\equiv i\), and \(\tau\) shifts the chain by one, so \(\tau s_i\tau^{-1}=s_{i-1}\) and \(s_i\tau^k=\tau^ks_{i+k}\). For \(i=0\), \(P_{s_0}/I\) is the \(\mathbb P^1\) of lattices between \(\tau\mathcal O^n=\langle e_1,\dots,e_{n-1},te_n\rangle\) and \(\tau^{-1}\mathcal O^n=\langle t^{-1}e_1,e_2,\dots,e_n\rangle\) (\(P_{s_0}\cap K=I\)). \(E=\{(L,v):v\in L\}\) is the pullback of \(\mathrm{Bl}_0\mathbb A^2\) along the smooth \(V\to V/Z\). The connecting map lies in \(\operatorname{Hom}(\underline k_0,i^!\underline k[3])=H^{-1}_H(\mathrm{pt})=0\), so \(m_!\underline k_E=\underline k_{\tau^{-1}\mathcal O^n}\oplus\underline k_{\tau\mathcal O^n}[-2]\) and \(\mathrm{IC}_{s_0}\ast\mathrm{IC}_0=S_{-1}\oplus S_1\). For \(i\ne0\), \(P_{s_i}\subseteq K\) gives \(S_0\otimes H^*(\mathbb P^1)[1]\), split since \(H^{\rm odd}_H(V)=0\). The general \(k\) follows by conjugation.
  4. Other translates. \(\ell(t^{\lambda_{m,j}})=\#\{a\le j<b\}=j(n-j)\), \(t^\lambda=\tau^ku\), \(u\in W\), \(\ell(u)=\ell(t^\lambda)\); \(IuI\times^I\mathcal O^n\simeq IuI/I\times\mathcal O^n\) since \(IuI\subseteq K\), so \(\Delta_u\ast\mathrm{IC}_0=\underline k\otimes R\Gamma_c(\mathbb A^{\ell(u)})\). For \(w_{\max}=\tau^kw_0\), the even cohomology of \(G/B\) and \(H^{\rm odd}_T(\mathrm{pt})=0\) split the complex in the equivariant category. Correct as written.
  5. Independence from IV.1 (checklist 9). IV.1 (case-affine.html) runs the I3 dévissage: the standard objects \(\Delta_{O_{m,j}}\) from the triangles \(\Delta_{O_{m,j}}\to\underline k_{\Lambda_{m,j}}\to\underline k_{\Lambda_{m,j-1}}\) with the translates \(\Delta_{t^\lambda}\ast\mathrm{IC}_0\), and Proposition III. The present route identifies the heart of the colimit and its simple objects (Audit 8.8) and uses length-zero translates and the summand criterion. It shares with IV.1 only the elementary fact that each lattice sheaf is a Hecke translate of \(\mathrm{IC}_0\), which it re-proves with different Hecke elements. So the argument is different and not circular.

Sources