§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:20:31Z

Audit 8.4 · Audit of the trace expansion (A4-4): $E_{z,n}$ has weight $w+n-\ell(z)$

proved

The trace of each generator expands in IC traces with coefficients Σ(−1)^n tr(A^m_{z,n}), and every Frobenius eigenvalue on E_{z,n} has weight exactly w+n−ℓ(z); both proofs of the weight are correct.

Audit 8.4.

Holds for Lemma 4.4 (i)–(iii) and Theorem 2.9 steps 1, 3, over \(\overline{\mathbb F}_q\). For \(M=\mathrm{IC}_x\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_y\) with the Weil structures trivial on open cells, \(w=\ell(x)+\ell(w_0)+\ell(y)\), \(P_n={}^pH^n(M)\simeq\bigoplus_z\mathrm{IC}_z\otimes E_{z,n}\) (Frobenius \(\Phi_z\otimes A_{z,n}\)): $$\operatorname{tr}_mM=\sum_{z,n}(-1)^n\operatorname{tr}(A_{z,n}^m)\operatorname{tr}_m\mathrm{IC}_z,\qquad |\iota\alpha|=q^{(w+n-\ell(z))/2}\ \text{for every eigenvalue }\alpha\text{ of }A_{z,n}.$$ Inputs used: purity of \(M\) (Audit 8.1 step 5), geometric semisimplicity of \(P_n\) (BBD 5.4.5–5.3.8), Frobenius-compatibility of the evaluation map, Gabber purity of \(\mathrm{IC}_z\).

  1. Expansion. \(P_n\) carries the Weil structure \(\Phi_n\) induced by the \(\mathbb F_q\)-structure of \(M\) (the perverse t-structure is Frobenius-stable). Since \(P_n\) is geometrically semisimple, \(\operatorname{End}\mathrm{IC}_z=\overline{\mathbb Q}_\ell\) and \(\operatorname{Hom}(\mathrm{IC}_z,\mathrm{IC}_{z'})=0\) for \(z\ne z'\), the evaluation map \(\bigoplus_z\mathrm{IC}_z\otimes\operatorname{Hom}(\mathrm{IC}_z,P_n)\to P_n\) is an isomorphism, and it intertwines \(\Phi_z\otimes A_{z,n}\) with \(\Phi_n\) for \(A_{z,n}(f)=\Phi_n\circ\mathrm{Fr}^*f\circ\Phi_z^{-1}\) (trace.tex, Def. A, Lemma frob). Additivity on perverse truncation triangles and \(\operatorname{tr}(K[-n])=(-1)^n\operatorname{tr}K\) give the expansion; over \(\mathbb F_{q^m}\) the \(m\)-th iterate gives \(A^m_{z,n}\). Every \(\mathrm{IC}_z\) is Frobenius-stable (cells are defined over \(\mathbb F_q\)), so no orbit of non-stable constituents occurs here (checklist 5).
  2. Upper bound. \(P_n\) is pure of weight \(w+n\) (BBD 5.4.1–5.4.4). At the rational point \(\dot zI\), \(\mathcal H^{-\ell(z)}(P_n)_{\dot z}=\bigoplus_{z'}E_{z',n}\otimes\mathcal H^{-\ell(z)}(\mathrm{IC}_{z'})_{\dot z}\). For \(z'\ne z\) with \(\mathrm{Fl}_z\subseteq\overline{\mathrm{Fl}_{z'}}\) the support condition gives \(\mathcal H^j(\mathrm{IC}_{z'})|_{\mathrm{Fl}_z}\ne0\Rightarrow\ell(z)<-j\), so degree \(j=-\ell(z)\) vanishes; the term \(z'=z\) is \((E_{z,n},A_{z,n})\). Mixed of weight \(\le w+n\) bounds the weights of this stalk by \(w+n-\ell(z)\).
  3. Lower bound. \(\mathbb D\mathrm{IC}_z\simeq\mathrm{IC}_z(\ell(z))\) (on the open cell \(\mathbb D(\overline{\mathbb Q}_\ell[d])=\overline{\mathbb Q}_\ell[d](d)\)), so \(\mathbb DP_n\), pure of weight \(-(w+n)\), has multiplicity Frobenius with eigenvalues \(\alpha^{-1}\) and stalk eigenvalues \(q^{-\ell(z)}\alpha^{-1}\); step 2 for \(\mathbb DP_n\) gives \(-2\ell(z)-\operatorname{wt}\alpha\le-(w+n)-\ell(z)\), i.e. \(\operatorname{wt}\alpha\ge w+n-\ell(z)\). Theorem 2.9's route (a Weil summand of a pure perverse sheaf is pure, and \(\mathrm{IC}_{z}\otimes(E,A)\) restricted to the open cell has weights \(\ell(z)+\operatorname{wt}A\)) gives the same conclusion; Jordan blocks of \(A_{z,n}\) do not matter for either.
  4. Invertibility of \(A_{z,n}\) is clear (composite of bijections).
  5. Scope. Only purity of \(M\) is used, not Tate-ness or parity of the \(E_{z,n}\) (Bezrukavnikov–Yun Prop. 3.2.5, text checked: \(\mathrm{IC}_{w_1}\ast\mathrm{IC}_{w_2}=\bigoplus\mathrm{IC}_w[n](n/2)\) with parity, for Kac–Moody flag varieties). That stronger input is not needed, which avoids identifying components of \(\mathrm{Fl}_H\) with Kac–Moody flag varieties.

Builds on

  • Lemma 4.4 · Trace expansion of $M_{x,y}$ with pure multiplicity Frobenii

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 19:20 UTC

Audit of the trace expansion (A4-4): $E_{z,n}$ has weight $w+n-\ell(z)$

  1. Expansion. \(P_n\) carries the Weil structure \(\Phi_n\) induced by the \(\mathbb F_q\)-structure of \(M\) (the perverse t-structure is Frobenius-stable). Since \(P_n\) is geometrically semisimple, \(\operatorname{End}\mathrm{IC}_z=\overline{\mathbb Q}_\ell\) and \(\operatorname{Hom}(\mathrm{IC}_z,\mathrm{IC}_{z'})=0\) for \(z\ne z'\), the evaluation map \(\bigoplus_z\mathrm{IC}_z\otimes\operatorname{Hom}(\mathrm{IC}_z,P_n)\to P_n\) is an isomorphism, and it intertwines \(\Phi_z\otimes A_{z,n}\) with \(\Phi_n\) for \(A_{z,n}(f)=\Phi_n\circ\mathrm{Fr}^*f\circ\Phi_z^{-1}\) (trace.tex, Def. A, Lemma frob). Additivity on perverse truncation triangles and \(\operatorname{tr}(K[-n])=(-1)^n\operatorname{tr}K\) give the expansion; over \(\mathbb F_{q^m}\) the \(m\)-th iterate gives \(A^m_{z,n}\). Every \(\mathrm{IC}_z\) is Frobenius-stable (cells are defined over \(\mathbb F_q\)), so no orbit of non-stable constituents occurs here (checklist 5).
  2. Upper bound. \(P_n\) is pure of weight \(w+n\) (BBD 5.4.1–5.4.4). At the rational point \(\dot zI\), \(\mathcal H^{-\ell(z)}(P_n)_{\dot z}=\bigoplus_{z'}E_{z',n}\otimes\mathcal H^{-\ell(z)}(\mathrm{IC}_{z'})_{\dot z}\). For \(z'\ne z\) with \(\mathrm{Fl}_z\subseteq\overline{\mathrm{Fl}_{z'}}\) the support condition gives \(\mathcal H^j(\mathrm{IC}_{z'})|_{\mathrm{Fl}_z}\ne0\Rightarrow\ell(z)<-j\), so degree \(j=-\ell(z)\) vanishes; the term \(z'=z\) is \((E_{z,n},A_{z,n})\). Mixed of weight \(\le w+n\) bounds the weights of this stalk by \(w+n-\ell(z)\).
  3. Lower bound. \(\mathbb D\mathrm{IC}_z\simeq\mathrm{IC}_z(\ell(z))\) (on the open cell \(\mathbb D(\overline{\mathbb Q}_\ell[d])=\overline{\mathbb Q}_\ell[d](d)\)), so \(\mathbb DP_n\), pure of weight \(-(w+n)\), has multiplicity Frobenius with eigenvalues \(\alpha^{-1}\) and stalk eigenvalues \(q^{-\ell(z)}\alpha^{-1}\); step 2 for \(\mathbb DP_n\) gives \(-2\ell(z)-\operatorname{wt}\alpha\le-(w+n)-\ell(z)\), i.e. \(\operatorname{wt}\alpha\ge w+n-\ell(z)\). Theorem 2.9's route (a Weil summand of a pure perverse sheaf is pure, and \(\mathrm{IC}_{z}\otimes(E,A)\) restricted to the open cell has weights \(\ell(z)+\operatorname{wt}A\)) gives the same conclusion; Jordan blocks of \(A_{z,n}\) do not matter for either.
  4. Invertibility of \(A_{z,n}\) is clear (composite of bijections).
  5. Scope. Only purity of \(M\) is used, not Tate-ness or parity of the \(E_{z,n}\) (Bezrukavnikov–Yun Prop. 3.2.5, text checked: \(\mathrm{IC}_{w_1}\ast\mathrm{IC}_{w_2}=\bigoplus\mathrm{IC}_w[n](n/2)\) with parity, for Kac–Moody flag varieties). That stronger input is not needed, which avoids identifying components of \(\mathrm{Fl}_H\) with Kac–Moody flag varieties.

Sources