§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:45:12Z

Corollary 4.6 · (IG) fails by the same character; (II$_I$) holds for the group

proved

Derives the failure of (IG) from the same sign character, identifies it as the trace shadow of the sheaf argument, and verifies the dictionary inputs of (II_I) for the group, so that the sheaf theorem also re-derives the function failure over F_q((t)). The function statement alone does not give the sheaf statement; the gap is exactly the weight step of the key theorem.

Corollary 4.6.

(a) For every non-archimedean local field \(F\) and \(W\ne1\): with \(\mathcal H_I=\mathcal H_{I_H}\otimes\mathcal H_{I_H}^{\rm op}\) acting on \(C_I=\mathcal H_{I_H}\) by \((a\otimes b)\cdot\varphi=a\ast\varphi\ast b\), $$\mathcal H_I\ast\varphi_0=\mathcal H_{I_H}\,\mathbf 1_{K_H}\,\mathcal H_{I_H}\subseteq\ker\varepsilon,\qquad\varepsilon(T_e)=1,$$ so (IG) fails, and \(C_I/\mathcal H_I\ast\varphi_0\twoheadrightarrow\overline{\mathbb Q}_\ell\) (the sign line). (b) For \(F=\mathbb F_{q^m}(\!(t)\!)\): \(\operatorname{span}_{\overline{\mathbb Q}_\ell}\{\operatorname{tr}_mM_{x,y}\}=\mathcal H_{I_H}\mathbf 1_{K_H}\mathcal H_{I_H}\otimes\overline{\mathbb Q}_\ell\); step 1 of Theorem 4.5 is (a) evaluated on these traces. (c) (II\(_I\)) for \(X=H\). The inputs D1 (points), D2 (trace of convolution), D3 (IC-trace basis), D4 (Weil decomposition) hold, and: if (IG) holds over \(\mathbb F_q(\!(t)\!)\) then \(\mathcal M_I^\flat=\widetilde W\), hence (with the equivariant splitting) \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) over \(\overline{\mathbb F}_q\). With Theorem 4.5 (\(e\notin\mathcal M_I^\flat\)) this re-derives the failure of (IG) over \(\mathbb F_q(\!(t)\!)\). (d) (a) does not imply Theorem 4.5 by linear algebra; the missing input is the weight bound of Lemma 4.4(iii).

  1. (a). \(\varphi_0=\mathbf 1_{X(\mathcal O)}=\mathbf 1_{K_H}\), so \(\mathcal H_I\ast\varphi_0=\operatorname{span}\{a\mathbf 1_{K_H}b\}\) (classical.html §3). \(\varepsilon\) is a ring homomorphism (Lemma 4.3(a)), so \(\varepsilon(a\mathbf 1_{K_H}b)=\varepsilon(a)\varepsilon(\mathbf 1_{K_H})\varepsilon(b)=0\) by Lemma 4.3(b), and \(\ker\varepsilon\) is a proper two-sided ideal. This is classical.html Proposition E, rederived from the Iwahori–Matsumoto relations only.
  2. D1. \(H(F)=\bigsqcup_{z\in\widetilde W}I_HzI_H\); \(\mathrm{Fl}(\mathbb F_q)=H(F)/I_H\) and \(\mathrm{Fl}_z(\mathbb F_q)=I_HzI_H/I_H\) by Lang's theorem for the connected stabilizers of Setup 4.1(b) (milne-iag-v2.pdf Cor. 27.56): each \(I\)-orbit on \(LX\) is \(\mathrm{Fr}\)-stable and its \(\mathbb F_q\)-points form one \(I(\mathbb F_q[[t]])\)-orbit (checklist 5). D2 is Lemma 4.3(c). D4 is Lemma 4.4(ii).
  3. D3. \(\operatorname{tr}_m\mathrm{IC}_z\) is bi-\(I_H\)-invariant, supported on \(\overline{\mathrm{Fl}_z}(\mathbb F_{q^m})=\bigsqcup_{y\le z}I_HyI_H/I_H\), with value \((-1)^{\ell(z)}\) on the open cell; so \(\operatorname{tr}_m\mathrm{IC}_z=(-1)^{\ell(z)}T_z+\sum_{y<z}c_{yz}T_y\). Bruhat lower sets are finite, so the transition matrix to \(\{T_z\}\) is unitriangular up to sign on every finite lower set and \(\{\operatorname{tr}_m\mathrm{IC}_z\}\) is a basis of \(\mathcal H_{I,m}\otimes\overline{\mathbb Q}_\ell\) (as trace.tex, Lemma icbasis). Then (b) follows from Lemma 4.4(i).
  4. (c). Assume (IG) and let \(\mu\in\widetilde W\). Then \(T_\mu\), hence (D3) \(\operatorname{tr}\mathrm{IC}_\mu\), lies in \(\operatorname{span}_{x,y}\operatorname{tr}M_{x,y}\). By Lemma 4.4(ii) at \(m=1\), \(\operatorname{tr}M_{x,y}=\sum_zm_z^{(x,y)}\operatorname{tr}\mathrm{IC}_z\) with \(m_z^{(x,y)}=\sum_n(-1)^n\operatorname{tr}A_{z,n}\), which is \(0\) unless \(z\in\mathcal M_I^\flat\). So \(\operatorname{tr}\mathrm{IC}_\mu\in\operatorname{span}\{\operatorname{tr}\mathrm{IC}_z:z\in\mathcal M_I^\flat\}\) and independence (D3) gives \(\mu\in\mathcal M_I^\flat\). Then Proposition 4.2(1),(3). Only the trivial direction “nonzero coefficient \(\Rightarrow\) constituent” is used (trace.tex, Lemma exp), so no weights enter here.
  5. (d). Linear algebra gives: \(\varepsilon\circ\operatorname{tr}\) vanishes on traces of generators. A retract \(R\) of a generator need not have \(\varepsilon(\operatorname{tr}R)=0\) a priori, and an object with \(\varepsilon(\operatorname{tr})=0\) can contain \(\mathrm{IC}_e\) (e.g. \(\mathrm{IC}_e\oplus\mathrm{IC}_e[1]\)). Theorem 4.5 closes this with purity: every potential \(\Delta_\omega\)-constituent carries a weight that fixes its shift. Field: (a) over any \(F\); (b), (c) over \(\mathbb F_{q^m}(\!(t)\!)\) with \(\ell\)-adic sheaves over \(\overline{\mathbb F}_q\).

Builds on

  • Lemma 4.3 · Sign character on IC traces: $\varepsilon(\operatorname{tr}\mathrm{IC}_z)=[z\in\Omega]$
  • Lemma 4.4 · Trace expansion of $M_{x,y}$ with pure multiplicity Frobenii
  • Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$

Built on by

  • Proposition 4.10 · Pure sign-type functionals force sheaf failure: obligations for other rows
  • Summary 4.11 · Verdict for $X=H$: both levels fail, no divergence
  • Audit 8.6 · Audit of (IG) and (II$_I$) for the group (A4-6, A2-9 step 8)
  • 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 18:45 UTC

(IG) fails by the same character; (II$_I$) holds for the group

  1. (a). \(\varphi_0=\mathbf 1_{X(\mathcal O)}=\mathbf 1_{K_H}\), so \(\mathcal H_I\ast\varphi_0=\operatorname{span}\{a\mathbf 1_{K_H}b\}\) (classical.html §3). \(\varepsilon\) is a ring homomorphism (Lemma 4.3(a)), so \(\varepsilon(a\mathbf 1_{K_H}b)=\varepsilon(a)\varepsilon(\mathbf 1_{K_H})\varepsilon(b)=0\) by Lemma 4.3(b), and \(\ker\varepsilon\) is a proper two-sided ideal. This is classical.html Proposition E, rederived from the Iwahori–Matsumoto relations only.
  2. D1. \(H(F)=\bigsqcup_{z\in\widetilde W}I_HzI_H\); \(\mathrm{Fl}(\mathbb F_q)=H(F)/I_H\) and \(\mathrm{Fl}_z(\mathbb F_q)=I_HzI_H/I_H\) by Lang's theorem for the connected stabilizers of Setup 4.1(b) (milne-iag-v2.pdf Cor. 27.56): each \(I\)-orbit on \(LX\) is \(\mathrm{Fr}\)-stable and its \(\mathbb F_q\)-points form one \(I(\mathbb F_q[[t]])\)-orbit (checklist 5). D2 is Lemma 4.3(c). D4 is Lemma 4.4(ii).
  3. D3. \(\operatorname{tr}_m\mathrm{IC}_z\) is bi-\(I_H\)-invariant, supported on \(\overline{\mathrm{Fl}_z}(\mathbb F_{q^m})=\bigsqcup_{y\le z}I_HyI_H/I_H\), with value \((-1)^{\ell(z)}\) on the open cell; so \(\operatorname{tr}_m\mathrm{IC}_z=(-1)^{\ell(z)}T_z+\sum_{y<z}c_{yz}T_y\). Bruhat lower sets are finite, so the transition matrix to \(\{T_z\}\) is unitriangular up to sign on every finite lower set and \(\{\operatorname{tr}_m\mathrm{IC}_z\}\) is a basis of \(\mathcal H_{I,m}\otimes\overline{\mathbb Q}_\ell\) (as trace.tex, Lemma icbasis). Then (b) follows from Lemma 4.4(i).
  4. (c). Assume (IG) and let \(\mu\in\widetilde W\). Then \(T_\mu\), hence (D3) \(\operatorname{tr}\mathrm{IC}_\mu\), lies in \(\operatorname{span}_{x,y}\operatorname{tr}M_{x,y}\). By Lemma 4.4(ii) at \(m=1\), \(\operatorname{tr}M_{x,y}=\sum_zm_z^{(x,y)}\operatorname{tr}\mathrm{IC}_z\) with \(m_z^{(x,y)}=\sum_n(-1)^n\operatorname{tr}A_{z,n}\), which is \(0\) unless \(z\in\mathcal M_I^\flat\). So \(\operatorname{tr}\mathrm{IC}_\mu\in\operatorname{span}\{\operatorname{tr}\mathrm{IC}_z:z\in\mathcal M_I^\flat\}\) and independence (D3) gives \(\mu\in\mathcal M_I^\flat\). Then Proposition 4.2(1),(3). Only the trivial direction “nonzero coefficient \(\Rightarrow\) constituent” is used (trace.tex, Lemma exp), so no weights enter here.
  5. (d). Linear algebra gives: \(\varepsilon\circ\operatorname{tr}\) vanishes on traces of generators. A retract \(R\) of a generator need not have \(\varepsilon(\operatorname{tr}R)=0\) a priori, and an object with \(\varepsilon(\operatorname{tr})=0\) can contain \(\mathrm{IC}_e\) (e.g. \(\mathrm{IC}_e\oplus\mathrm{IC}_e[1]\)). Theorem 4.5 closes this with purity: every potential \(\Delta_\omega\)-constituent carries a weight that fixes its shift. Field: (a) over any \(F\); (b), (c) over \(\mathbb F_{q^m}(\!(t)\!)\) with \(\ell\)-adic sheaves over \(\overline{\mathbb F}_q\).

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