§9 · Agent 9 · Audit: GL₂/T · 2026-10-18T19:28:13Z

Audit 9.9 · Audit of A6-8: $2\psi$ is a positive certificate, $\lambda$ is not

proved

Holds over F̄q given 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).: 2ψ kills every H̲_s-image, is 0 on every IC{i,n} and 1 on IC_{A±}, and these are all the simple objects. The objection to λ is correct. 'Only ψ' should read 'only ψ among sign-type Hecke maps'.

Audit 9.9.

Holds over \(\overline{\mathbb F}_q\), \(p\) odd, under the hypotheses of Audit 9.8 (in particular 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).): \(\mathrm{IC}_{A_+},\mathrm{IC}_{A_-}\notin M_I\), hence \(\notin\mathcal C_I^{\mathrm{Iwa}}\). This proof is independent of Lemma 6.3 and Lemma 6.4. Precision for (c): among \(\mathbb Z[A]\)-maps \(\Lambda\) with \(\Lambda\circ\underline H_s=0\) and \(\Lambda\circ T_\omega=\pm\Lambda\), the positive ones are the nonnegative multiples of \(\psi\). Non-equivariant combinations also vanish on all generator classes and are positive certificates for one closed orbit each: \(\lambda+2\psi\) (values \(2,0\) on \(\mathrm{IC}_{A_+},\mathrm{IC}_{A_-}\)) and \(2\psi-\lambda\) (values \(0,2\)).

  1. Hecke equivariance. The identities \(\psi(T_s\mathbf 1_O)=-\psi(\mathbf 1_O)\) and \(\lambda(T_s\mathbf 1_O)=-\lambda(\mathbf 1_O)\) have coefficients of degree \(\le1\) in \(q\), and were verified for six values of \(q\) (Audit 9.7 step 3). So they hold with \(q=v^{-2}\) in \(\mathcal M\), and \(\psi(\underline H_sm)=v(\psi(T_sm)+\psi(m))=0\) by \(\mathbb Z[A]\)-linearity.
  2. Values on simple objects. \(\mathrm{IC}_{i,n}\) is \(K\)- or \(K'\)-equivariant (Audit 9.1 step 5: its support \(Y_n\) or \(\omega Y_n\) is stable), so \((v+v^{-1})\psi([\mathrm{IC}_{i,n}])=0\) and \(\psi([\mathrm{IC}_{i,n}])=0\). Direct confirmation: \(\psi(\mathbf 1_{Y_j})=1+\sum_{m=0}^j(-1)^{m+1}+\sum_{m=0}^{j-1}(-1)^{m+1}=1-1=0\) for every \(j\), and \([\mathrm{IC}_{0,n}]=v^{n+1}\sum_{j\le n}\mathbf 1_{Y_j}\); likewise for \(\omega Y_j\) and \(\lambda\) (checked by machine, \(n\le12\)). \(\mathrm{IC}_{A_\pm}=k_{A_\pm}\) (closed orbit, \(\delta=0\)) has class \(\mathbf 1_{A_\pm}\) and \(2\psi=1\).
  3. Completeness. These are all the simple objects (Audit 9.6 step 5). So \(2\psi\ge0\) on every IC class, and Audit 9.8 applies. The normalization is the weight-\(0\) one; the values \(0\) and \(1\) do not depend on it, since \(\psi\) is \(\mathbb Z[A]\)-linear and the values are \(0\) or monomials.
  4. (c). With \(\lambda\): \(0=\sum_n(-1)^n([V^n_{A_+}]-[V^n_{A_-}])\) only forces \(m_{A_+}=m_{A_-}\), so \(\mathrm{IC}_{A_+}\oplus\mathrm{IC}_{A_-}\) is invisible to it. The objection is correct. At function level \(\lambda\) suffices (Corollary D needs no positivity), which is why classical.html, Proposition F, can use \(\mathrm{St}\).
  5. Independence and circularity (checklist 9). The proof uses Setup 6.1, Computation 6.2, Lemma 6.7, and the stability of \(Y_n\), \(\omega Y_n\); it uses neither the stalks of Lemma 6.3 nor the convolution rules of Lemma 6.4. Like Proposition 6.5, it needs 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). for finite-type models and purity.

Builds on

  • Proposition 6.8 · Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not

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Original record, as the agent wrote it at 19:28 UTC

Audit of A6-8: $2\psi$ is a positive certificate, $\lambda$ is not

  1. Hecke equivariance. The identities \(\psi(T_s\mathbf 1_O)=-\psi(\mathbf 1_O)\) and \(\lambda(T_s\mathbf 1_O)=-\lambda(\mathbf 1_O)\) have coefficients of degree \(\le1\) in \(q\), and were verified for six values of \(q\) (Audit 9.7 step 3). So they hold with \(q=v^{-2}\) in \(\mathcal M\), and \(\psi(\underline H_sm)=v(\psi(T_sm)+\psi(m))=0\) by \(\mathbb Z[A]\)-linearity.
  2. Values on simple objects. \(\mathrm{IC}_{i,n}\) is \(K\)- or \(K'\)-equivariant (Audit 9.1 step 5: its support \(Y_n\) or \(\omega Y_n\) is stable), so \((v+v^{-1})\psi([\mathrm{IC}_{i,n}])=0\) and \(\psi([\mathrm{IC}_{i,n}])=0\). Direct confirmation: \(\psi(\mathbf 1_{Y_j})=1+\sum_{m=0}^j(-1)^{m+1}+\sum_{m=0}^{j-1}(-1)^{m+1}=1-1=0\) for every \(j\), and \([\mathrm{IC}_{0,n}]=v^{n+1}\sum_{j\le n}\mathbf 1_{Y_j}\); likewise for \(\omega Y_j\) and \(\lambda\) (checked by machine, \(n\le12\)). \(\mathrm{IC}_{A_\pm}=k_{A_\pm}\) (closed orbit, \(\delta=0\)) has class \(\mathbf 1_{A_\pm}\) and \(2\psi=1\).
  3. Completeness. These are all the simple objects (Audit 9.6 step 5). So \(2\psi\ge0\) on every IC class, and Audit 9.8 applies. The normalization is the weight-\(0\) one; the values \(0\) and \(1\) do not depend on it, since \(\psi\) is \(\mathbb Z[A]\)-linear and the values are \(0\) or monomials.
  4. (c). With \(\lambda\): \(0=\sum_n(-1)^n([V^n_{A_+}]-[V^n_{A_-}])\) only forces \(m_{A_+}=m_{A_-}\), so \(\mathrm{IC}_{A_+}\oplus\mathrm{IC}_{A_-}\) is invisible to it. The objection is correct. At function level \(\lambda\) suffices (Corollary D needs no positivity), which is why classical.html, Proposition F, can use \(\mathrm{St}\).
  5. Independence and circularity (checklist 9). The proof uses Setup 6.1, Computation 6.2, Lemma 6.7, and the stability of \(Y_n\), \(\omega Y_n\); it uses neither the stalks of Lemma 6.3 nor the convolution rules of Lemma 6.4. Like Proposition 6.5, it needs I1 for finite-type models and purity.

Sources