§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:20:56Z

Audit 8.5 · Audit of the main group theorem (A4-5, A2-9): no $\Delta_\omega$ constituent

proved ★ key step

Both versions correctly prove that no Δ_ω (ω ∈ Ω) is a perverse constituent of any generator, hence Δ_e ∉ C_I^Iwa, over F̄_q; the weight argument genuinely excludes cancellation. Over C the conclusion needs the good-model comparison (T_C), cited but not proved.

Audit 8.5.

Holds over \(k=\overline{\mathbb F}_q\) (every \(p\); \(H\) split, \(W\ne1\)): for all \(x,y\in\widetilde W\), \(\omega\in\Omega\), \(n\in\mathbb Z\), \(\operatorname{Hom}(\mathrm{IC}_\omega,{}^pH^n(\mathrm{IC}_x\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_y))=0\); hence \(\Omega\cap\mathcal M_I^\flat=\emptyset\) and, by Audit 8.2(i), \(\Delta_\omega\notin\mathcal C_I^{\rm Iwa}\), in particular \(\Delta_e\notin\mathcal C_I^{\rm Iwa}\subseteq\mathcal D_{\widetilde W\setminus\Omega}\subsetneq\mathcal C_I\).

Needs hypothesis (T\(_{\mathbb C}\)) over \(\mathbb C\): for each fixed \((x,y)\), \(\dim\operatorname{Hom}(\mathrm{IC}_\omega,{}^pH^n M_{x,y})\) is the same for Betti sheaves over \(\mathbb C\) and \(\ell\)-adic sheaves over \(\overline{\mathbb F}_p\) for almost all \(p\). It is used only in Theorem 2.9 step 9; Theorem 4.5 makes no claim over \(\mathbb C\).

  1. Vanishing on generators. By Audit 8.3, \(\varepsilon\) is a ring homomorphism of \(\mathcal H_{I,m}\), \(\operatorname{tr}_m\) is multiplicative, and \(\varepsilon(\operatorname{tr}_m\mathrm{IC}_{w_0})=(-1)^{\ell(w_0)}\varepsilon(\mathbf 1_{K_H})=0\); so \(\varepsilon(\operatorname{tr}_mM_{x,y})=0\) for every \(m\ge1\). The two-sided generator \(\mathrm{IC}_x\ast\mathrm{IC}_0\ast\mathrm{IC}_{y^{-1}}\) is an element of the Hecke algebra itself, so the ring-homomorphism property is exactly what is needed.
  2. Only \(\Omega\) survives. By Audit 8.4 and Audit 8.3(e), \(\varepsilon(\operatorname{tr}_mM_{x,y})=\sum_{\omega,n}(-1)^n\operatorname{tr}(A^m_{\omega,n})\). Theorem 2.9 step 2 sharpens this to the single \(\omega=\sigma(x)\sigma(y)\) (cells in different \(\Omega\)-cosets are disjoint and \(IuI\cdot IvI\) keeps the \(\Omega\)-part); correct, not needed.
  3. Separation. \(\sum_\alpha c_\alpha\alpha^m=0\) for all \(m\ge1\) with finitely many distinct nonzero \(\alpha\) forces every \(c_\alpha=0\) (Vandermonde on \(m=1,\dots,|\Lambda|\); equivalently the independence of \(\alpha T/(1-\alpha T)\) used in Theorem 2.9 step 6).
  4. No cancellation (checklist 6). By Audit 8.4, every eigenvalue of \(A_{\omega,n}\) has weight \(w+n\) (\(\ell(\omega)=0\)). So \(n\) is a function of \(\alpha\), and \(c_\alpha=(-1)^{n(\alpha)}\sum_\omega\mu_{\omega,n(\alpha)}(\alpha)\) is a sum of non-negative integers with one sign. Hence all \(\mu=0\), and since \(A_{\omega,n}\) is an endomorphism of a finite-dimensional space over the algebraically closed \(\overline{\mathbb Q}_\ell\), \(E_{\omega,n}=0\). The authors' counterexample \(\mathrm{IC}_e[n]\oplus\mathrm{IC}_e[n+1]\) (equal scalars) correctly shows that \(\varepsilon\circ\operatorname{tr}\) alone, or one value of \(m\), cannot conclude: \(\varepsilon\circ\operatorname{tr}\) is additive on triangles, not on retracts. All \(m\) are needed only to separate eigenvalues; weights fix the sign.
  5. What is cited versus proved. Proved from Audit 8.1–Audit 8.4: everything above. The optional second route through Bezrukavnikov–Yun Prop. 3.2.5 is correctly flagged as needing the Kac–Moody identification and is not used.
  6. Independence from the pages (checklist 9). No step uses the affine Whittaker functor \(\Phi\) of group.html Proposition II 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). of case-group.html IV.2. Comment: \(\varepsilon\circ\operatorname{tr}\) is the decategorified shadow of \(\Phi\) (which kills every \(\mathrm{IC}_z\), \(z\notin\Omega\)); \(\Phi\), being an exact functor, kills retracts automatically, and the present proof replaces that exactness by the weight argument of step 4. So the route is genuinely different, and nothing being re-proved is assumed.
  7. (T\(_{\mathbb C}\)). For fixed \((x,y)\) all spaces and maps have models over \(\mathbb Z[1/N]\) with the affine-space Schubert stratification, and \(M_{x,y}\) is built by \(j_{!*}\), \(\widetilde\boxtimes\) and proper push-forward. The good-model comparison (BBD 6.1.9–6.1.10, as in Bezrukavnikov–Finkelberg Prop. 5, extract checked) identifies the Hom dimensions over \(\mathbb C\) and \(\overline{\mathbb F}_p\), \(p\gg0\), granted its compatibility with \(m_!\) and perverse truncation. Since step 4 holds for every \(p\), the \(\mathbb C\) statement follows. Standard, but a citation, not a proof in the file.

Builds on

  • Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Theorem 2.9 · Group: $\Omega\cap\mathcal M_I^+=\varnothing$ by sign character and weights

Built on by

  • Summary 8.13 · Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$
  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 19:20 UTC

Audit of the main group theorem (A4-5, A2-9): no $\Delta_\omega$ constituent

  1. Vanishing on generators. By Audit 8.3, \(\varepsilon\) is a ring homomorphism of \(\mathcal H_{I,m}\), \(\operatorname{tr}_m\) is multiplicative, and \(\varepsilon(\operatorname{tr}_m\mathrm{IC}_{w_0})=(-1)^{\ell(w_0)}\varepsilon(\mathbf 1_{K_H})=0\); so \(\varepsilon(\operatorname{tr}_mM_{x,y})=0\) for every \(m\ge1\). The two-sided generator \(\mathrm{IC}_x\ast\mathrm{IC}_0\ast\mathrm{IC}_{y^{-1}}\) is an element of the Hecke algebra itself, so the ring-homomorphism property is exactly what is needed.
  2. Only \(\Omega\) survives. By Audit 8.4 and Audit 8.3(e), \(\varepsilon(\operatorname{tr}_mM_{x,y})=\sum_{\omega,n}(-1)^n\operatorname{tr}(A^m_{\omega,n})\). Theorem 2.9 step 2 sharpens this to the single \(\omega=\sigma(x)\sigma(y)\) (cells in different \(\Omega\)-cosets are disjoint and \(IuI\cdot IvI\) keeps the \(\Omega\)-part); correct, not needed.
  3. Separation. \(\sum_\alpha c_\alpha\alpha^m=0\) for all \(m\ge1\) with finitely many distinct nonzero \(\alpha\) forces every \(c_\alpha=0\) (Vandermonde on \(m=1,\dots,|\Lambda|\); equivalently the independence of \(\alpha T/(1-\alpha T)\) used in Theorem 2.9 step 6).
  4. No cancellation (checklist 6). By Audit 8.4, every eigenvalue of \(A_{\omega,n}\) has weight \(w+n\) (\(\ell(\omega)=0\)). So \(n\) is a function of \(\alpha\), and \(c_\alpha=(-1)^{n(\alpha)}\sum_\omega\mu_{\omega,n(\alpha)}(\alpha)\) is a sum of non-negative integers with one sign. Hence all \(\mu=0\), and since \(A_{\omega,n}\) is an endomorphism of a finite-dimensional space over the algebraically closed \(\overline{\mathbb Q}_\ell\), \(E_{\omega,n}=0\). The authors' counterexample \(\mathrm{IC}_e[n]\oplus\mathrm{IC}_e[n+1]\) (equal scalars) correctly shows that \(\varepsilon\circ\operatorname{tr}\) alone, or one value of \(m\), cannot conclude: \(\varepsilon\circ\operatorname{tr}\) is additive on triangles, not on retracts. All \(m\) are needed only to separate eigenvalues; weights fix the sign.
  5. What is cited versus proved. Proved from Audit 8.1–Audit 8.4: everything above. The optional second route through Bezrukavnikov–Yun Prop. 3.2.5 is correctly flagged as needing the Kac–Moody identification and is not used.
  6. Independence from the pages (checklist 9). No step uses the affine Whittaker functor \(\Phi\) of group.html Proposition II or I3 of case-group.html IV.2. Comment: \(\varepsilon\circ\operatorname{tr}\) is the decategorified shadow of \(\Phi\) (which kills every \(\mathrm{IC}_z\), \(z\notin\Omega\)); \(\Phi\), being an exact functor, kills retracts automatically, and the present proof replaces that exactness by the weight argument of step 4. So the route is genuinely different, and nothing being re-proved is assumed.
  7. (T\(_{\mathbb C}\)). For fixed \((x,y)\) all spaces and maps have models over \(\mathbb Z[1/N]\) with the affine-space Schubert stratification, and \(M_{x,y}\) is built by \(j_{!*}\), \(\widetilde\boxtimes\) and proper push-forward. The good-model comparison (BBD 6.1.9–6.1.10, as in Bezrukavnikov–Finkelberg Prop. 5, extract checked) identifies the Hom dimensions over \(\mathbb C\) and \(\overline{\mathbb F}_p\), \(p\gg0\), granted its compatibility with \(m_!\) and perverse truncation. Since step 4 holds for every \(p\), the \(\mathbb C\) statement follows. Standard, but a citation, not a proof in the file.

Sources