§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:52:31Z

Summary 4.11 · Verdict for $X=H$: both levels fail, no divergence

proved ★ key step

Second proof of the first row of Table 1, independent of the affine Whittaker functor and of 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).: the sign character kills every generator's trace, only length-zero IC's survive it, and purity prevents cancellation; the summand criterion converts this into Δ_e ∉ C_I^Iwa, and the same character gives the failure of (IG). The row is not a divergence.

Summary 4.11.

For \(H\) split connected reductive with \(W\ne1\), \(X=H\) under \(H\times H\):

  1. (proved over \(\overline{\mathbb F}_q\); over \(\mathbb C\) modulo the standard comparison, Audit 4.9) No \(\Delta_\omega\), \(\omega\in\Omega\), is a perverse constituent of any \(\mathrm{IC}_x\ast\mathrm{IC}_0\ast\mathrm{IC}_y\); hence \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) (Theorem 4.5, with the criterion Proposition 4.2 whose hypotheses are Setup 4.1).
  2. (proved, any \(F\)) (IG) fails: \(\mathcal H_I\ast\varphi_0\subseteq\ker\varepsilon\not\ni T_e\) (Corollary 4.6); over \(\mathbb F_q(\!(t)\!)\) this is the trace of the sheaf argument, (II\(_I\)) holds for \(X=H\), and its contrapositive re-derives (IG)'s failure from 1.
  3. (proved) \(\{xw_0y\ \text{reduced}\}\subseteq\mathcal M_I\subseteq\widetilde W\setminus\Omega\); \(\mathcal M_I=\widetilde W\setminus\Omega\) in semisimple rank one; (cited) \(\mathcal M_I=c_0\Omega\), the lowest two-sided cell, in general (Proposition 4.7, Computation 4.8).
  4. The group row is not a divergence: both levels fail. For the other rows only “functions fail, sheaves hold” can diverge, and the exact obligation is a pure cohomological lift (\(\Lambda\)) of the function-level obstruction (Proposition 4.10).
  1. Chain. Setup 4.1 (pieces = Bruhat cells, finite closures, connected stabilizers, heart, two-sided generators, decomposition) \(\to\) Proposition 4.2 (constituents of generators control the thick span) \(\to\) Lemma 4.3 (\(\varepsilon\) is a character; \(\varepsilon(\mathbf 1_K)=0\); \(\varepsilon\operatorname{tr}\mathrm{IC}_z=[z\in\Omega]\)) \(\to\) Lemma 4.4 (trace expansion, pure multiplicity Frobenii) \(\to\) Theorem 4.5 (no cancellation; \(\Delta_\omega\notin\mathcal C_I^{\mathrm{Iwa}}\)).
  2. Difference from the pages. group.html Proposition II separates \(\Delta_e\) by an exact functor \(\Phi\) (affine Whittaker averaging) with \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\ker\Phi\); here the separation is by a trace functional plus weights, and it identifies more: every \(\mathrm{IC}_z\) with \(z\notin\mathcal M_I\) is missing, e.g. every \(\mathrm{IC}_s\) for \(\mathrm{SL}_3\) (cited cell theory), although \(\Phi(\mathrm{IC}_s)=0\) (group.html, step 4: \(\Phi(c_s)=0\)). So, granting Proposition 4.7(4), \(\mathcal C_I^{\mathrm{Iwa}}\subsetneq\ker\Phi\) for simple \(H\) of rank \(\ge2\), while in semisimple rank one \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal D_{\widetilde W\setminus\Omega}\) (proved).
  3. Fields. Every step of 1 is over \(\overline{\mathbb F}_q\) with \(\overline{\mathbb Q}_\ell\)-coefficients; the function statement holds over every non-archimedean \(F\); the \(\mathbb C\)-statement uses the comparison of Audit 4.9.
  4. What is not proved here. The exact \(\mathcal M_I\) beyond semisimple rank one (cell theory cited, numbers unverified locally); the \(\mathbb A^n\) row and (I\(_I\)), (II\(_I\)) in general (other targets); the obligations (\(\Lambda\)) for the symmetric rows.

Builds on

  • Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Corollary 4.6 · (IG) fails by the same character; (II$_I$) holds for the group
  • Proposition 4.7 · $\mathcal M_I$ for the group: top cells in, $\Omega$ out, exact in rank one
  • Audit 4.9 · Transfer to $\mathbb C$ and checklist audit for the group
  • Proposition 4.10 · Pure sign-type functionals force sheaf failure: obligations for other rows

Built on by

  • —
Original record, as the agent wrote it at 18:52 UTC

Verdict for $X=H$: both levels fail, no divergence

  1. Chain. Setup 4.1 (pieces = Bruhat cells, finite closures, connected stabilizers, heart, two-sided generators, decomposition) \(\to\) Proposition 4.2 (constituents of generators control the thick span) \(\to\) Lemma 4.3 (\(\varepsilon\) is a character; \(\varepsilon(\mathbf 1_K)=0\); \(\varepsilon\operatorname{tr}\mathrm{IC}_z=[z\in\Omega]\)) \(\to\) Lemma 4.4 (trace expansion, pure multiplicity Frobenii) \(\to\) Theorem 4.5 (no cancellation; \(\Delta_\omega\notin\mathcal C_I^{\mathrm{Iwa}}\)).
  2. Difference from the pages. group.html Proposition II separates \(\Delta_e\) by an exact functor \(\Phi\) (affine Whittaker averaging) with \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\ker\Phi\); here the separation is by a trace functional plus weights, and it identifies more: every \(\mathrm{IC}_z\) with \(z\notin\mathcal M_I\) is missing, e.g. every \(\mathrm{IC}_s\) for \(\mathrm{SL}_3\) (cited cell theory), although \(\Phi(\mathrm{IC}_s)=0\) (group.html, step 4: \(\Phi(c_s)=0\)). So, granting Proposition 4.7(4), \(\mathcal C_I^{\mathrm{Iwa}}\subsetneq\ker\Phi\) for simple \(H\) of rank \(\ge2\), while in semisimple rank one \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal D_{\widetilde W\setminus\Omega}\) (proved).
  3. Fields. Every step of 1 is over \(\overline{\mathbb F}_q\) with \(\overline{\mathbb Q}_\ell\)-coefficients; the function statement holds over every non-archimedean \(F\); the \(\mathbb C\)-statement uses the comparison of Audit 4.9.
  4. What is not proved here. The exact \(\mathcal M_I\) beyond semisimple rank one (cell theory cited, numbers unverified locally); the \(\mathbb A^n\) row and (I\(_I\)), (II\(_I\)) in general (other targets); the obligations (\(\Lambda\)) for the symmetric rows.

Sources