§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:54:01Z

Corollary 6.9 · Group case: the sign character $\varepsilon_+$ excludes every $\delta_\omega$

proved

Applies the positive-certificate lemma to row 1 of Table 1: the two-sided sign character (T_s ↦ −1, T_ω ↦ +1) kills the class of every generator and is 0 or 1 on Kazhdan–Lusztig classes, so the unit Δ_e (and every δ_ω) is not Iwahori-generated; its trace is the function-level sign obstruction. This is a proof of the sheaf half of row 1 different from the affine Whittaker functor of Proposition II.

Corollary 6.9.

Let \(H\) be split connected reductive with finite Weyl group \(W\ne1\), \(X=H\), \(G=H\times H\), \((h_1,h_2)\cdot x=h_1xh_2^{-1}\), over \(\overline{\mathbb F}_q\), \(p\) odd. Then \(\mathcal C_I=\mathcal D_c(I_H\backslash LH/I_H)\), \(\mathcal M_A=\mathcal H\otimes\mathbb Z[A]\) (\(\mathcal H\) the extended affine Hecke algebra of \(H\)) with \((a\otimes b)\cdot m=a\,m\,\iota(b)\), \(\iota(T_w)=T_{w^{-1}}\), and \([\mathrm{IC}_0]=[\underline k_{K_H}]=\underline H_{w_0}\). Let \(\varepsilon_+:\mathcal H\to\mathbb Z[v^{\pm1}]\) be the character \(T_s\mapsto-1\) (\(s\) simple affine), \(T_\omega\mapsto1\) (\(\omega\in\Omega\)). Then (a) \(\varepsilon_+\) kills \([(\mathrm{IC}_a\boxtimes\mathrm{IC}_b)\ast\mathrm{IC}_0]\) for all \(a,b\in\widetilde W\); (b) \(\varepsilon_+([\mathrm{IC}_w])=0\) for \(\ell(w)\ge1\) and \(=1\) for \(w\in\Omega\); (c) hence no \(\delta_\omega=\mathrm{IC}_\omega\) (\(\omega\in\Omega\)) is a shifted summand of any generator, and \(\Delta_e=\delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\); at \(v=-q^{-1/2}\), \(\varepsilon_+\) is the sign character of classical.html, Proposition E, killing \(\mathcal H_I\ast\varphi_0\) but not \(T_e\).

  1. Character. \(\varepsilon_+\) respects \((T_s-q)(T_s+1)=0\), the braid relations (both sides of a braid relation have the same length) and \(T_\omega T_sT_\omega^{-1}=T_{\omega s\omega^{-1}}\); \(\varepsilon_+\circ\iota=\varepsilon_+\) since \(\ell(w^{-1})=\ell(w)\). So \(\Lambda(m):=\varepsilon_+(m)\) satisfies \(\Lambda((a\otimes b)\cdot m)=\varepsilon_+(a)\Lambda(m)\varepsilon_+(b)\) and \(\Lambda\circ\underline H_s=0\) on either side (\(\underline H_s=v(T_s+1)\mapsto0\)).
  2. (a). \(\underline H_s\underline H_{w_0}=(v+v^{-1})\underline H_{w_0}\) for a finite simple \(s\) (\(\underline k_{K_H}\) is left \(P_s\)-equivariant), so \(\varepsilon_+(\underline H_{w_0})=0\) (here \(W\ne1\)); the generator class is \([\mathrm{IC}_a]\,\underline H_{w_0}\,\iota[\mathrm{IC}_b]\).
  3. (b). If \(\ell(w)\ge1\) pick a simple affine \(s\) with \(sw<w\); \(\overline{IwI}\) is left \(P_s\)-stable, so \(\underline H_s[\mathrm{IC}_w]=(v+v^{-1})[\mathrm{IC}_w]\) and \(\varepsilon_+([\mathrm{IC}_w])=0\). For \(\omega\in\Omega\), \(I\omega I=\omega I\), \([\mathrm{IC}_\omega]=T_\omega\mapsto1\).
  4. (c). Lemma 6.7(b): \(\Lambda\) vanishes on all generator classes, is \(\ge0\) on all IC classes (the simple objects are the \(\mathrm{IC}_w\): Bruhat cells, connected stabilizers \(I_H\cap wI_Hw^{-1}\)), and is \(1\) on \(\delta_\omega\); weights exclude cancellation. The Jordan–Hölder argument of Proposition 6.5 step 5 (perverse t-structure on each finite union of Bruhat cells) gives \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\). Function level: \(\varepsilon_+(\mathbf 1_{K_H})=\sum_W(-1)^{\ell(w)}=0\), \(\varepsilon_+(T_e)=1\).
  5. Scope. Inputs: decomposition theorem with weights (BBD 5.1.14, 5.4.5), the Bruhat stratification, and the fact that an \(\mathrm{IC}\) of a left-\(P_s\)-stable closure is \(P_s\)-equivariant. Not used: the affine Whittaker category (Proposition II, group.html) or the failure 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). (IV.2). Over \(\mathbb C\) the same conclusion follows from the parity of Schubert ICs (stalks in degrees of one parity, so the decomposition multiplicities of a convolution have no sign cancellation) — cited, not re-verified here. Remark (cited, not used): the summands that do occur are the \(\mathrm{IC}_w\) with \(w\) in Lusztig's lowest two-sided cell, the ideal generated by \(\underline H_{w_0}\).

Builds on

  • Lemma 6.7 · Sheaf-level Corollary D: positive Steinberg certificates exclude summands

Built on by

  • Audit 6.11 · Audit of the $\mathrm{GL}_2/T$ argument against the nine failure modes
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
Original record, as the agent wrote it at 18:54 UTC

Group case: the sign character $\varepsilon_+$ excludes every $\delta_\omega$

  1. Character. \(\varepsilon_+\) respects \((T_s-q)(T_s+1)=0\), the braid relations (both sides of a braid relation have the same length) and \(T_\omega T_sT_\omega^{-1}=T_{\omega s\omega^{-1}}\); \(\varepsilon_+\circ\iota=\varepsilon_+\) since \(\ell(w^{-1})=\ell(w)\). So \(\Lambda(m):=\varepsilon_+(m)\) satisfies \(\Lambda((a\otimes b)\cdot m)=\varepsilon_+(a)\Lambda(m)\varepsilon_+(b)\) and \(\Lambda\circ\underline H_s=0\) on either side (\(\underline H_s=v(T_s+1)\mapsto0\)).
  2. (a). \(\underline H_s\underline H_{w_0}=(v+v^{-1})\underline H_{w_0}\) for a finite simple \(s\) (\(\underline k_{K_H}\) is left \(P_s\)-equivariant), so \(\varepsilon_+(\underline H_{w_0})=0\) (here \(W\ne1\)); the generator class is \([\mathrm{IC}_a]\,\underline H_{w_0}\,\iota[\mathrm{IC}_b]\).
  3. (b). If \(\ell(w)\ge1\) pick a simple affine \(s\) with \(sw<w\); \(\overline{IwI}\) is left \(P_s\)-stable, so \(\underline H_s[\mathrm{IC}_w]=(v+v^{-1})[\mathrm{IC}_w]\) and \(\varepsilon_+([\mathrm{IC}_w])=0\). For \(\omega\in\Omega\), \(I\omega I=\omega I\), \([\mathrm{IC}_\omega]=T_\omega\mapsto1\).
  4. (c). Lemma 6.7(b): \(\Lambda\) vanishes on all generator classes, is \(\ge0\) on all IC classes (the simple objects are the \(\mathrm{IC}_w\): Bruhat cells, connected stabilizers \(I_H\cap wI_Hw^{-1}\)), and is \(1\) on \(\delta_\omega\); weights exclude cancellation. The Jordan–Hölder argument of Proposition 6.5 step 5 (perverse t-structure on each finite union of Bruhat cells) gives \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\). Function level: \(\varepsilon_+(\mathbf 1_{K_H})=\sum_W(-1)^{\ell(w)}=0\), \(\varepsilon_+(T_e)=1\).
  5. Scope. Inputs: decomposition theorem with weights (BBD 5.1.14, 5.4.5), the Bruhat stratification, and the fact that an \(\mathrm{IC}\) of a left-\(P_s\)-stable closure is \(P_s\)-equivariant. Not used: the affine Whittaker category (Proposition II, group.html) or the failure of I3 (IV.2). Over \(\mathbb C\) the same conclusion follows from the parity of Schubert ICs (stalks in degrees of one parity, so the decomposition multiplicities of a convolution have no sign cancellation) — cited, not re-verified here. Remark (cited, not used): the summands that do occur are the \(\mathrm{IC}_w\) with \(w\) in Lusztig's lowest two-sided cell, the ideal generated by \(\underline H_{w_0}\).

Sources