§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:20:03Z

Audit 8.3 · Audit of the sign character (A4-3): $\varepsilon$ kills IC traces off $\Omega$

proved

The sign character, its value on 1_K, the trace-of-convolution identity and the vanishing of ε on IC traces off Ω are all correct; the quadratic relation is the same in every version and matches vol(I) = 1.

Audit 8.3.

Holds for Lemma 4.3 (a)–(e) and Theorem 2.9 steps 3–4. Quadratic relation (all three authors): \(T_s^2=(q-1)T_s+qT_e\), equivalently \((T_s-q)(T_s+1)=0\), which is the \(\operatorname{vol}(I)=1\) normalization since \((T_s\ast T_s)(e)=\#(IsI/I)=q\). Then \(\varepsilon(T_w)=(-1)^{\ell(w)}\) is a ring homomorphism with \(\varepsilon(T_\omega)=1\) (\(\omega\in\Omega\)), \(\varepsilon(\mathbf 1_{K_H})=\sum_{w\in W}(-1)^{\ell(w)}=0\) for \(W\ne1\), and for Weil objects with trivial Frobenius on open cells, \(\varepsilon(\operatorname{tr}_m\mathrm{IC}_z)=0\) for all \(z\notin\Omega\), \(m\ge1\). (a), (b) hold over every non-archimedean \(F\); (c)–(e) are \(\ell\)-adic over \(\overline{\mathbb F}_q\).

  1. Character. \(T_xT_s=qT_{xs}+(q-1)T_x\) for \(xs<x\) (from \(x=x's\) and the quadratic relation; HKP §7.2, display after Lemma 7.2.1, text checked). Then \(\varepsilon\): \(q(-1)^{\ell(x)-1}+(q-1)(-1)^{\ell(x)}=(-1)^{\ell(x)+1}\); the cases \(x<xs\) and \(T_xT_\sigma=T_{x\sigma}\) are immediate. Multiplicativity on the basis plus linearity gives a ring homomorphism. Soergel's form: \(H_s=q^{-1/2}T_s\) satisfies \((H_s+v)(H_s-v^{-1})=0\) at \(v=q^{-1/2}\), and \(\varepsilon(H_s)=-v\) kills \(\underline H_s=H_s+v\).
  2. \(\mathbf 1_K\). \(K_H=\bigsqcup_{w\in W}I_HwI_H\), and the affine length restricted to \(W\) is the Coxeter length of \(W\) (finite simple reflections are affine simple reflections). Right multiplication by a finite simple \(s\) is a fixed-point-free involution changing parity.
  3. Trace of convolution. Fibres of \(m\) over \(x\in\mathrm{Fl}(\mathbb F_{q^m})\) are \(\{gI:g^{-1}x\in\mathrm{Fl}_{\le u'}\}\); proper base change and Grothendieck–Lefschetz give \(\sum_{gI}\operatorname{tr}\mathcal A(g)\operatorname{tr}\mathcal B(g^{-1}x)\) once the \(\mathbb F_{q^m}\)-points of \(LH\times^I\mathrm{Fl}\) are \(H(F_m)\times^{I(\mathcal O_m)}\mathrm{Fl}(\mathbb F_{q^m})\). That needs \(H^1(\mathbb F_{q^m},I)=1\): Lang (milne-iag-v2.pdf Cor. 27.56) for each connected \(I/I_n\), and a limit over \(n\) with surjective transition maps. The authors cite only the finite-dimensional statement; the limit step is routine. This is \(\int_Gf_1(g)f_2(g^{-1}x)\,dg\) with \(\operatorname{vol}(I)=1\).
  4. Values. \(\operatorname{tr}\mathrm{IC}_\omega=T_\omega\) (skyscraper at a rational point; \(H\) split); \(\operatorname{tr}\mathrm{IC}_{w_0}=(-1)^{\ell(w_0)}\mathbf 1_{K_H}\) (smooth \(H/B\)); \(\operatorname{tr}\mathrm{IC}_s=-(T_e+T_s)\); \(\varepsilon\) of the last is \(-(1-1)=0\).
  5. Descent. For \(z=\omega z'\notin\Omega\), a right descent \(s\) of \(z'\) is one of \(z\) (\(\ell(zs)=\ell(z's)\)). \(\overline{IzI}\) is right \(P_s\)-stable, \(\mathrm{IC}_z=\pi_s^*\mathcal G[1]\) with the same Weil structure (no twist), \(\mathrm{IC}_z\ast\underline k_{P_s/I}=\pi_s^*\pi_{s!}\mathrm{IC}_z\), trace \((1+q^m)\operatorname{tr}\pi_s^*\mathcal G=-(1+q^m)\operatorname{tr}\mathrm{IC}_z\). Applying \(\varepsilon\): \(0=-(1+q^m)\varepsilon(\operatorname{tr}\mathrm{IC}_z)\). Theorem 2.9's variant (right \(P_s(\mathcal O_r)\)-invariance, \(\operatorname{tr}\ast T_s=q^r\operatorname{tr}\), so \(-\varepsilon=q^r\varepsilon\)) is equivalent and also correct.
  6. Independent check (computation). Through the dictionary \(\operatorname{tr}\mathrm{IC}_z=(-1)^{\ell(z)}q^{\ell(z)/2}C'_z\), (e) says \(\varepsilon(\underline H_z)=0\) for \(z\ne e\). The Kazhdan–Lusztig bases of \(\widetilde A_1\) (length \(\le10\), 21 elements) and \(\widetilde A_2\) (length \(\le9\), 136 elements) were computed by the standard recursion, and \(\varepsilon(\underline H_z)=0\) held for every \(z\ne e\). The proof does not use this dictionary; the check is independent.

Builds on

  • Lemma 4.3 · Sign character on IC traces: $\varepsilon(\operatorname{tr}\mathrm{IC}_z)=[z\in\Omega]$

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 sign character (A4-3): $\varepsilon$ kills IC traces off $\Omega$

  1. Character. \(T_xT_s=qT_{xs}+(q-1)T_x\) for \(xs<x\) (from \(x=x's\) and the quadratic relation; HKP §7.2, display after Lemma 7.2.1, text checked). Then \(\varepsilon\): \(q(-1)^{\ell(x)-1}+(q-1)(-1)^{\ell(x)}=(-1)^{\ell(x)+1}\); the cases \(x<xs\) and \(T_xT_\sigma=T_{x\sigma}\) are immediate. Multiplicativity on the basis plus linearity gives a ring homomorphism. Soergel's form: \(H_s=q^{-1/2}T_s\) satisfies \((H_s+v)(H_s-v^{-1})=0\) at \(v=q^{-1/2}\), and \(\varepsilon(H_s)=-v\) kills \(\underline H_s=H_s+v\).
  2. \(\mathbf 1_K\). \(K_H=\bigsqcup_{w\in W}I_HwI_H\), and the affine length restricted to \(W\) is the Coxeter length of \(W\) (finite simple reflections are affine simple reflections). Right multiplication by a finite simple \(s\) is a fixed-point-free involution changing parity.
  3. Trace of convolution. Fibres of \(m\) over \(x\in\mathrm{Fl}(\mathbb F_{q^m})\) are \(\{gI:g^{-1}x\in\mathrm{Fl}_{\le u'}\}\); proper base change and Grothendieck–Lefschetz give \(\sum_{gI}\operatorname{tr}\mathcal A(g)\operatorname{tr}\mathcal B(g^{-1}x)\) once the \(\mathbb F_{q^m}\)-points of \(LH\times^I\mathrm{Fl}\) are \(H(F_m)\times^{I(\mathcal O_m)}\mathrm{Fl}(\mathbb F_{q^m})\). That needs \(H^1(\mathbb F_{q^m},I)=1\): Lang (milne-iag-v2.pdf Cor. 27.56) for each connected \(I/I_n\), and a limit over \(n\) with surjective transition maps. The authors cite only the finite-dimensional statement; the limit step is routine. This is \(\int_Gf_1(g)f_2(g^{-1}x)\,dg\) with \(\operatorname{vol}(I)=1\).
  4. Values. \(\operatorname{tr}\mathrm{IC}_\omega=T_\omega\) (skyscraper at a rational point; \(H\) split); \(\operatorname{tr}\mathrm{IC}_{w_0}=(-1)^{\ell(w_0)}\mathbf 1_{K_H}\) (smooth \(H/B\)); \(\operatorname{tr}\mathrm{IC}_s=-(T_e+T_s)\); \(\varepsilon\) of the last is \(-(1-1)=0\).
  5. Descent. For \(z=\omega z'\notin\Omega\), a right descent \(s\) of \(z'\) is one of \(z\) (\(\ell(zs)=\ell(z's)\)). \(\overline{IzI}\) is right \(P_s\)-stable, \(\mathrm{IC}_z=\pi_s^*\mathcal G[1]\) with the same Weil structure (no twist), \(\mathrm{IC}_z\ast\underline k_{P_s/I}=\pi_s^*\pi_{s!}\mathrm{IC}_z\), trace \((1+q^m)\operatorname{tr}\pi_s^*\mathcal G=-(1+q^m)\operatorname{tr}\mathrm{IC}_z\). Applying \(\varepsilon\): \(0=-(1+q^m)\varepsilon(\operatorname{tr}\mathrm{IC}_z)\). Theorem 2.9's variant (right \(P_s(\mathcal O_r)\)-invariance, \(\operatorname{tr}\ast T_s=q^r\operatorname{tr}\), so \(-\varepsilon=q^r\varepsilon\)) is equivalent and also correct.
  6. Independent check (computation). Through the dictionary \(\operatorname{tr}\mathrm{IC}_z=(-1)^{\ell(z)}q^{\ell(z)/2}C'_z\), (e) says \(\varepsilon(\underline H_z)=0\) for \(z\ne e\). The Kazhdan–Lusztig bases of \(\widetilde A_1\) (length \(\le10\), 21 elements) and \(\widetilde A_2\) (length \(\le9\), 136 elements) were computed by the standard recursion, and \(\varepsilon(\underline H_z)=0\) held for every \(z\ne e\). The proof does not use this dictionary; the check is independent.

Sources