§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:52:27Z

Proposition 6.8 · Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not

proved

Re-derives that IC_{A+} and IC_{A−} are not Iwahori-generated from the positive-certificate lemma alone, without the IC stalks or convolution rules: the functional ψ (Hecke character T_s ↦ −1, T_ω ↦ +1) kills every K- or K'-equivariant class and equals 1/2 on both closed orbits; the plain Steinberg functional λ fails positivity, so it cannot decide sheaves.

Proposition 6.8.

Over \(\overline{\mathbb F}_q\), \(p\) odd, for \(X=\mathrm{GL}_2/T\), define \(\mathbb Z[A]\)-linear maps on \(\mathcal M_A\) by their values on the classes \(\mathbf 1_O=[j_{O!}k_O]\) (independent of \(v\)): $$\lambda(\mathbf 1_{A_\pm})=\pm1,\quad\lambda(\mathbf 1_{O_{i,m}})=0;\qquad 2\psi(\mathbf 1_{A_\pm})=1,\quad 2\psi(\mathbf 1_{O_{i,m}})=2(-1)^{m+1}.$$ (a) \(\lambda\circ T_s=-\lambda\), \(\psi\circ T_s=-\psi\) for \(s\in\{s_0,s_1\}\) with \(q\) formal; \(\lambda\circ T_\omega=-\lambda\), \(\psi\circ T_\omega=\psi\). Hence \(\lambda\circ\underline H_s=\psi\circ\underline H_s=0\). (b) \(2\psi([\mathrm{IC}_{A_\pm}])=1\) and \(2\psi([\mathcal F])=0\) for every \(K\)- or \(K'\)-equivariant \(\mathcal F\), in particular for all \(\mathrm{IC}_{i,n}\). By Lemma 6.7, \(\mathrm{IC}_{A_+},\mathrm{IC}_{A_-}\notin M_I\), hence \(\notin\mathcal C_I^{\mathrm{Iwa}}\). (c) (Objection to the naive route.) \(\lambda([\mathrm{IC}_{A_\pm}])=\pm1\): \(\lambda\) is not of one sign on IC classes, and from \(\lambda\) alone one only gets \([V^n_{A_+}]=[V^n_{A_-}]\) for the multiplicity spaces; a summand \(\mathrm{IC}_{A_+}\oplus\mathrm{IC}_{A_-}\) would be invisible to it.

  1. (a), from Computation 6.2. \(\lambda\): \(\lambda(T_{s_1}\mathbf 1_{A_+})=\lambda(\mathbf 1_{A_-}+\mathbf 1_{O_{0,0}})=-1\); \(\lambda(T_{s_1}\mathbf 1_{O_{0,0}})=(q-1)(1-1)=0\); \(\lambda\) vanishes on both sides for the blocks \(m\ge1\). \(\psi\): \(\psi(T_{s_1}\mathbf 1_{A_\pm})=\tfrac12-1=-\tfrac12\); \(\psi(T_{s_1}\mathbf 1_{O_{0,0}})=(q-1)-(q-2)=1\); \(\psi(T_{s_1}\mathbf 1_{O_{1,m-1}})=\psi(\mathbf 1_{O_{0,m}})=(-1)^{m+1}=-\psi(\mathbf 1_{O_{1,m-1}})\); \(\psi(T_{s_1}\mathbf 1_{O_{0,m}})=(-1)^m(q-(q-1))=-\psi(\mathbf 1_{O_{0,m}})\). All identities are polynomial in \(q\), so they hold in \(\mathcal M\) with \(q=v^{-2}\); \(s_0\) by the symmetry \(0\leftrightarrow1\); \(\omega\) by inspection. Then \(\psi(\underline H_sm)=v(\psi(T_sm)+\psi(m))=0\).
  2. (b). If \(\mathcal F\) is \(K\)-equivariant, \(\underline H_{s_1}[\mathcal F]=(v+v^{-1})[\mathcal F]\) (Lemma 6.4 step 5), so \((v+v^{-1})\psi([\mathcal F])=0\) and \(\psi([\mathcal F])=0\) (\(\mathbb Z[A]\) is a domain). Same with \(s_0\subset K'\). By Setup 6.1(d) every orbit other than \(A_\pm\) has closure \(Y_n\) (\(K\)-stable) or \(\omega Y_n\) (\(K'\)-stable) whose open \(K\)- (resp. \(K'\)-) orbit is smooth and contains the \(I\)-orbit as an open subset; so \(\mathrm{IC}_{i,n}\) is \(K\)- or \(K'\)-equivariant. \(\mathrm{IC}_{A_\pm}=j_{A_\pm!}k\) (closed orbit, \(\delta=0\)) has class \(\mathbf 1_{A_\pm}\). So \(2\psi\) is \(\ge0\) on all IC classes and equals \(1\) on \(\mathrm{IC}_{A_\pm}\); Lemma 6.7(a),(b) apply.
  3. Independence. This proof uses only Setup 6.1, Computation 6.2 and Lemma 6.7; not the stalks of Lemma 6.3 nor the convolution rules of Lemma 6.4. It agrees with Proposition 6.5.
  4. (c). \(\lambda([\mathrm{IC}_{A_\pm}])=\pm1\), \(\lambda([\mathrm{IC}_{i,n}])=0\) (same argument). Step 3 of Lemma 6.7 then gives only \([V^n_{A_+}]=[V^n_{A_-}]\) for all \(n\). At function level either of \(\lambda,\psi\) witnesses the failure of (IG) (Proposition 6.6); at sheaf level only the twisted one, \(\psi\) (\(T_\omega\mapsto+1\), the \(\mathcal H_I\)-module \((\mathrm{St}\otimes\chi_0)^I\) heuristically), is positive. The Steinberg distinction used on classical.html (Proposition F) is \(\lambda\).

Builds on

  • Setup 6.1 · The $I$-orbits on $L(\mathrm{GL}_2/T)$ as relative positions of an edge and a geodesic
  • Computation 6.2 · Affine Lusztig–Vogan module of $\mathrm{GL}_2/T$: Hecke action on orbit indicators
  • Lemma 6.7 · Sheaf-level Corollary D: positive Steinberg certificates exclude summands
  • Proposition 6.6 · Functions: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two

Built on by

  • Corollary 6.10 · Closed-orbit test: sign-type Hecke maps nonnegative on closed orbits
  • Audit 6.11 · Audit of the $\mathrm{GL}_2/T$ argument against the nine failure modes
  • Theorem 6.12 · Verdict for $\mathrm{GL}2/T$: both levels fail at the closed orbits $A\pm$
  • Audit 9.9 · Audit of A6-8: $2\psi$ is a positive certificate, $\lambda$ is not
  • 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 18:52 UTC

Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not

  1. (a), from Computation 6.2. \(\lambda\): \(\lambda(T_{s_1}\mathbf 1_{A_+})=\lambda(\mathbf 1_{A_-}+\mathbf 1_{O_{0,0}})=-1\); \(\lambda(T_{s_1}\mathbf 1_{O_{0,0}})=(q-1)(1-1)=0\); \(\lambda\) vanishes on both sides for the blocks \(m\ge1\). \(\psi\): \(\psi(T_{s_1}\mathbf 1_{A_\pm})=\tfrac12-1=-\tfrac12\); \(\psi(T_{s_1}\mathbf 1_{O_{0,0}})=(q-1)-(q-2)=1\); \(\psi(T_{s_1}\mathbf 1_{O_{1,m-1}})=\psi(\mathbf 1_{O_{0,m}})=(-1)^{m+1}=-\psi(\mathbf 1_{O_{1,m-1}})\); \(\psi(T_{s_1}\mathbf 1_{O_{0,m}})=(-1)^m(q-(q-1))=-\psi(\mathbf 1_{O_{0,m}})\). All identities are polynomial in \(q\), so they hold in \(\mathcal M\) with \(q=v^{-2}\); \(s_0\) by the symmetry \(0\leftrightarrow1\); \(\omega\) by inspection. Then \(\psi(\underline H_sm)=v(\psi(T_sm)+\psi(m))=0\).
  2. (b). If \(\mathcal F\) is \(K\)-equivariant, \(\underline H_{s_1}[\mathcal F]=(v+v^{-1})[\mathcal F]\) (Lemma 6.4 step 5), so \((v+v^{-1})\psi([\mathcal F])=0\) and \(\psi([\mathcal F])=0\) (\(\mathbb Z[A]\) is a domain). Same with \(s_0\subset K'\). By Setup 6.1(d) every orbit other than \(A_\pm\) has closure \(Y_n\) (\(K\)-stable) or \(\omega Y_n\) (\(K'\)-stable) whose open \(K\)- (resp. \(K'\)-) orbit is smooth and contains the \(I\)-orbit as an open subset; so \(\mathrm{IC}_{i,n}\) is \(K\)- or \(K'\)-equivariant. \(\mathrm{IC}_{A_\pm}=j_{A_\pm!}k\) (closed orbit, \(\delta=0\)) has class \(\mathbf 1_{A_\pm}\). So \(2\psi\) is \(\ge0\) on all IC classes and equals \(1\) on \(\mathrm{IC}_{A_\pm}\); Lemma 6.7(a),(b) apply.
  3. Independence. This proof uses only Setup 6.1, Computation 6.2 and Lemma 6.7; not the stalks of Lemma 6.3 nor the convolution rules of Lemma 6.4. It agrees with Proposition 6.5.
  4. (c). \(\lambda([\mathrm{IC}_{A_\pm}])=\pm1\), \(\lambda([\mathrm{IC}_{i,n}])=0\) (same argument). Step 3 of Lemma 6.7 then gives only \([V^n_{A_+}]=[V^n_{A_-}]\) for all \(n\). At function level either of \(\lambda,\psi\) witnesses the failure of (IG) (Proposition 6.6); at sheaf level only the twisted one, \(\psi\) (\(T_\omega\mapsto+1\), the \(\mathcal H_I\)-module \((\mathrm{St}\otimes\chi_0)^I\) heuristically), is positive. The Steinberg distinction used on classical.html (Proposition F) is \(\lambda\).

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