§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:44:21Z

Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$

proved ★ key step

The sign character kills the trace of every generator; in the IC-trace basis only the length-zero IC's survive it, each with value 1; the surviving coefficients are alternating sums of Frobenius traces whose weights pin down the shift, so they cannot cancel over all F_{q^m}. Therefore no Δ_ω occurs, and the summand criterion gives Δ_e ∉ C_I^Iwa without the affine Whittaker functor.

Theorem 4.5.

Let \(k=\overline{\mathbb F}_q\), \(W\ne1\). For all \(x,y\in\widetilde W\), \(\omega\in\Omega\), \(n\in\mathbb Z\): $$E_{\omega,n}(x,y)=\operatorname{Hom}\big(\mathrm{IC}_\omega,{}^pH^n(\mathrm{IC}_x\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_y)\big)=0 .$$ So no \(\Delta_\omega[n]=\mathrm{IC}_\omega[n]\) is a perverse constituent, a fortiori not a direct summand, of any generator \(\mathrm{IC}_x\ast\mathrm{IC}_0\ast\mathrm{IC}_y\); \(\Omega\cap\mathcal M_I^\flat=\emptyset\); and by Proposition 4.2(2) $$\Delta_\omega\notin\mathcal C_I^{\mathrm{Iwa}}\ \ (\omega\in\Omega),\qquad\text{in particular}\quad\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}},\quad \mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\widetilde W\setminus\Omega}\subsetneq\mathcal C_I .$$

  1. The functional kills generators. By Lemma 4.4(i) and Lemma 4.3(a),(b),(d), for every \(m\ge1\): \(\varepsilon(\operatorname{tr}_mM_{x,y})=\varepsilon(\operatorname{tr}_m\mathrm{IC}_x)\cdot(-1)^{\ell(w_0)}\varepsilon(\mathbf 1_{K_H})\cdot\varepsilon(\operatorname{tr}_m\mathrm{IC}_y)=0\). (Over \(\mathbb F_{q^m}\) the objects are the base changes of the same Weil objects, and \(\varepsilon\) is the sign character of \(\mathcal H_{I,m}\).)
  2. Only length zero survives. By Lemma 4.4(ii) and Lemma 4.3(d),(e): \(\varepsilon(\operatorname{tr}_mM_{x,y})=\sum_{\omega\in\Omega}\sum_n(-1)^n\operatorname{tr}(A_{\omega,n}^m)\). So
$$\sum_{\omega\in\Omega}\sum_{n}(-1)^n\operatorname{tr}\big(A_{\omega,n}^m\big)=0\qquad\text{for all }m\ge1 .$$
  1. Separate the eigenvalues. Let \(\Lambda\subset\overline{\mathbb Q}_\ell^\times\) be the finite set of eigenvalues of the finitely many nonzero \(A_{\omega,n}\), and \(c_\alpha=\sum_{\omega,n}(-1)^n\mu_{\omega,n}(\alpha)\) with \(\mu_{\omega,n}(\alpha)\) the algebraic multiplicity of \(\alpha\) in \(A_{\omega,n}\). Step 2 reads \(\sum_{\alpha\in\Lambda}c_\alpha\alpha^m=0\) for \(m=1,\dots,|\Lambda|\); the Vandermonde matrix \((\alpha^m)\) of distinct nonzero numbers is invertible, so \(c_\alpha=0\) for every \(\alpha\).
  2. No cancellation (weights). By Lemma 4.4(iii) with \(\ell(\omega)=0\), every eigenvalue of \(A_{\omega,n}\) has weight \(w+n\), \(w=\ell(x)+\ell(w_0)+\ell(y)\). So \(\alpha\) determines \(n(\alpha)=\operatorname{wt}(\alpha)-w\), and \(c_\alpha=(-1)^{n(\alpha)}\sum_{\omega}\mu_{\omega,n(\alpha)}(\alpha)\) is a signed sum of non-negative integers with a single sign. Hence all \(\mu_{\omega,n}(\alpha)=0\): \(A_{\omega,n}\) has no eigenvalue. It is invertible (Lemma 4.4(iii)), so \(E_{\omega,n}=0\).
  3. Conclusion. \(\Omega\cap\mathcal M_I^\flat=\emptyset\); Proposition 4.2(2) gives \(\mathrm{IC}_\omega\notin\mathcal C_I^{\mathrm{Iwa}}\). \(\mathrm{IC}_e=\Delta_e\) is the skyscraper at the base point, the unit of convolution.

Why step 4 is needed (checklist 6). \(\varepsilon\circ\operatorname{tr}\) is additive on triangles, so step 1 makes it vanish on everything built from generators by shifts and cofibres, but not on retracts: a generator of the shape \(\mathrm{IC}_e[n]\oplus\mathrm{IC}_e[n+1]\oplus(\text{terms killed by }\varepsilon)\) with equal Frobenius scalars on the two copies would have \(\varepsilon\)-trace \(0\) at every \(m\) and still have \(\Delta_e\) as a retract. Purity forbids it: the two copies would carry weights \(w+n\) and \(w+n+1\). This is where the sheaf statement is stronger than the function statement of Corollary 4.6.

Second route at one \(q\) (not needed). Bezrukavnikov–Yun, bezrukavnikov-yun-v2.pdf Prop. 3.2.5 (Kac–Moody flag varieties): \(\mathrm{IC}_{w_1}\ast\mathrm{IC}_{w_2}\) is, as a mixed complex, a direct sum of \(\mathrm{IC}_w[n](n/2)\) with \(n\equiv\ell(w_1)+\ell(w_2)-\ell(w)\pmod 2\) (normalised weight 0). With weight-0 normalisations, \(\operatorname{tr}(\mathrm{IC}_\omega[n](n/2))=(-1)^nq^{-n/2}T_\omega\), so at \(m=1\) the coefficient of \(\operatorname{tr}\mathrm{IC}_\omega\) in \(\operatorname{tr}M_{x,y}\) is \(\pm\sum_n\dim E_{\omega,n}\,q^{-n/2}\) with one sign (parity); \(\varepsilon(\operatorname{tr}M_{x,y})=0\) forces it to vanish, hence all \(E_{\omega,n}=0\). Applying it to \(\mathrm{Fl}_H\) needs the identification of each component of \(\mathrm{Fl}_H\) with a Kac–Moody flag variety (reduced structure); the weight argument above avoids this.

Comparison (not used). The pages prove \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) over \(\mathbb C\) by \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\ker\Phi\) for the affine Whittaker averaging \(\Phi\) (group.html, Proposition II) and locate it at 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). on closed pieces (case-group.html, IV.2). The present proof uses neither \(\Phi\) nor 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).; its only inputs are Setup 4.1–Lemma 4.4.

Builds on

  • Proposition 4.2 · Summand criterion for the group: generator constituents decide $\mathcal C_I^{\rm Iwa}$
  • 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

Built on by

  • 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
  • Summary 4.11 · Verdict for $X=H$: both levels fail, no divergence
  • Audit 8.5 · Audit of the main group theorem (A4-5, A2-9): no $\Delta_\omega$ constituent
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
  • 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 18:44 UTC

No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$

  1. The functional kills generators. By Lemma 4.4(i) and Lemma 4.3(a),(b),(d), for every \(m\ge1\): \(\varepsilon(\operatorname{tr}_mM_{x,y})=\varepsilon(\operatorname{tr}_m\mathrm{IC}_x)\cdot(-1)^{\ell(w_0)}\varepsilon(\mathbf 1_{K_H})\cdot\varepsilon(\operatorname{tr}_m\mathrm{IC}_y)=0\). (Over \(\mathbb F_{q^m}\) the objects are the base changes of the same Weil objects, and \(\varepsilon\) is the sign character of \(\mathcal H_{I,m}\).)
  2. Only length zero survives. By Lemma 4.4(ii) and Lemma 4.3(d),(e): \(\varepsilon(\operatorname{tr}_mM_{x,y})=\sum_{\omega\in\Omega}\sum_n(-1)^n\operatorname{tr}(A_{\omega,n}^m)\). So
$$\sum_{\omega\in\Omega}\sum_{n}(-1)^n\operatorname{tr}\big(A_{\omega,n}^m\big)=0\qquad\text{for all }m\ge1 .$$
  1. Separate the eigenvalues. Let \(\Lambda\subset\overline{\mathbb Q}_\ell^\times\) be the finite set of eigenvalues of the finitely many nonzero \(A_{\omega,n}\), and \(c_\alpha=\sum_{\omega,n}(-1)^n\mu_{\omega,n}(\alpha)\) with \(\mu_{\omega,n}(\alpha)\) the algebraic multiplicity of \(\alpha\) in \(A_{\omega,n}\). Step 2 reads \(\sum_{\alpha\in\Lambda}c_\alpha\alpha^m=0\) for \(m=1,\dots,|\Lambda|\); the Vandermonde matrix \((\alpha^m)\) of distinct nonzero numbers is invertible, so \(c_\alpha=0\) for every \(\alpha\).
  2. No cancellation (weights). By Lemma 4.4(iii) with \(\ell(\omega)=0\), every eigenvalue of \(A_{\omega,n}\) has weight \(w+n\), \(w=\ell(x)+\ell(w_0)+\ell(y)\). So \(\alpha\) determines \(n(\alpha)=\operatorname{wt}(\alpha)-w\), and \(c_\alpha=(-1)^{n(\alpha)}\sum_{\omega}\mu_{\omega,n(\alpha)}(\alpha)\) is a signed sum of non-negative integers with a single sign. Hence all \(\mu_{\omega,n}(\alpha)=0\): \(A_{\omega,n}\) has no eigenvalue. It is invertible (Lemma 4.4(iii)), so \(E_{\omega,n}=0\).
  3. Conclusion. \(\Omega\cap\mathcal M_I^\flat=\emptyset\); Proposition 4.2(2) gives \(\mathrm{IC}_\omega\notin\mathcal C_I^{\mathrm{Iwa}}\). \(\mathrm{IC}_e=\Delta_e\) is the skyscraper at the base point, the unit of convolution.

Why step 4 is needed (checklist 6). \(\varepsilon\circ\operatorname{tr}\) is additive on triangles, so step 1 makes it vanish on everything built from generators by shifts and cofibres, but not on retracts: a generator of the shape \(\mathrm{IC}_e[n]\oplus\mathrm{IC}_e[n+1]\oplus(\text{terms killed by }\varepsilon)\) with equal Frobenius scalars on the two copies would have \(\varepsilon\)-trace \(0\) at every \(m\) and still have \(\Delta_e\) as a retract. Purity forbids it: the two copies would carry weights \(w+n\) and \(w+n+1\). This is where the sheaf statement is stronger than the function statement of Corollary 4.6.

Second route at one \(q\) (not needed). Bezrukavnikov–Yun, bezrukavnikov-yun-v2.pdf Prop. 3.2.5 (Kac–Moody flag varieties): \(\mathrm{IC}_{w_1}\ast\mathrm{IC}_{w_2}\) is, as a mixed complex, a direct sum of \(\mathrm{IC}_w[n](n/2)\) with \(n\equiv\ell(w_1)+\ell(w_2)-\ell(w)\pmod 2\) (normalised weight 0). With weight-0 normalisations, \(\operatorname{tr}(\mathrm{IC}_\omega[n](n/2))=(-1)^nq^{-n/2}T_\omega\), so at \(m=1\) the coefficient of \(\operatorname{tr}\mathrm{IC}_\omega\) in \(\operatorname{tr}M_{x,y}\) is \(\pm\sum_n\dim E_{\omega,n}\,q^{-n/2}\) with one sign (parity); \(\varepsilon(\operatorname{tr}M_{x,y})=0\) forces it to vanish, hence all \(E_{\omega,n}=0\). Applying it to \(\mathrm{Fl}_H\) needs the identification of each component of \(\mathrm{Fl}_H\) with a Kac–Moody flag variety (reduced structure); the weight argument above avoids this.

Comparison (not used). The pages prove \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) over \(\mathbb C\) by \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\ker\Phi\) for the affine Whittaker averaging \(\Phi\) (group.html, Proposition II) and locate it at I3 on closed pieces (case-group.html, IV.2). The present proof uses neither \(\Phi\) nor I3; its only inputs are Setup 4.1–Lemma 4.4.

Sources