§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:49:04Z

Proposition 6.6 · Functions: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two

proved ★ key step

Computes the Iwahori-generated part at function level exactly: it is the sum of the functions invariant under the two maximal parahorics, of codimension two, with quotient spanned by the indicators of the same two closed orbits A± that sheaves miss; this reproves the failure of (IG) without Corollary D and matches the sheaf answer trace by trace.

Proposition 6.6.

Let \(F\) be a non-archimedean local field with residue field \(\mathbb F_q\), \(X=\mathrm{GL}_2/T\), \(\mathrm{vol}(I)=1\), \(f_n(\gamma)=\max(0,n+1-d(v_0,\gamma))\), \(f^{(1)}_n=T_\omega f_n\). Then (a) \(\mathcal H_I\ast\varphi_0=\mathrm{span}\{f_n,f^{(1)}_n:n\ge0\}=C_K+C_{K'}\), \(C_K=C_c(X(F))^K\), \(C_{K'}=C_c(X(F))^{\omega K\omega^{-1}}\); (b) \(C_I=(C_K+C_{K'})\oplus\mathbb C\mathbf 1_{A_+}\oplus\mathbb C\mathbf 1_{A_-}\); (c) \(C_I/\mathcal H_I\ast\varphi_0\cong\varepsilon_-\oplus\varepsilon_+\) as \(\mathcal H_I\)-modules, \(\varepsilon_\pm(T_{s_0})=\varepsilon_\pm(T_{s_1})=-1\), \(\varepsilon_\pm(T_\omega)=\pm1\), via $$\lambda(f)=f(A_+)-f(A_-),\qquad \psi(f)=\tfrac12\big(f(A_+)+f(A_-)\big)+\sum_{i\in\{0,1\},\,m\ge0}(-1)^{m+1}f(O_{i,m});$$ (d) \(\lambda\) is, up to the factor \(\mathrm{vol}(A_+)\), the restriction to \(C_I\) of the Steinberg map \(\mathcal S(X)\to\mathrm{St}\) defined by the \(T\)-invariant functional \(h\mapsto h([1:0])-h([0:1])\) of classical.html, Proposition F; (e) the trace of every object of \(\mathcal C_I^{\mathrm{Iwa}}\) (over \(\mathbb F_q\), \(p\) odd) lies in \(\mathcal H_I\ast\varphi_0\), and the traces of \(\mathrm{IC}_{A_\pm}\) are \(\mathbf 1_{A_\pm}\).

  1. (a). \(T_w=T_{s_{i_1}}\cdots T_{s_{i_r}}T_\omega^j\) and \(T_s=(T_s+1)-1\); by Lemma 6.4 (shadow) the span of the \(f_n,f^{(1)}_n\) is \(\mathcal H_I\)-stable and contains \(f_0=\varphi_0\), so it contains \(\mathcal H_I\ast\varphi_0\); conversely \(f^{(1)}_0=T_\omega\varphi_0\), \(f^{(1)}_{n+1}=(T_{s_0}+1)f_n-qf^{(1)}_{n-1}\), \(f_{n+1}=T_\omega f^{(1)}_{n+1}\). Since \(f_n-f_{n-1}=\mathbf 1_{Y_n}\) and \(C_K\) has basis \(\mathbf 1_{\{d(v_0,\cdot)=n\}}=\mathbf 1_{Y_n}-\mathbf 1_{Y_{n-1}}\), \(\mathrm{span}\{f_n\}=C_K\); apply \(\omega\). The orbit list and Hecke formulas of Setup 6.1, Computation 6.2 hold over any such \(F\) (only the tree is used).
  2. (b). \(\mathbf 1_{O_{0,0}}=\mathbf 1_{Y_0}-\mathbf 1_{A_+}-\mathbf 1_{A_-}\), \(\mathbf 1_{O_{0,m}}=\mathbf 1_{\{d(v_0,\cdot)=m\}}-\mathbf 1_{O_{1,m-1}}\), and symmetrically, so \(C_I=C_K+C_{K'}+\mathbb C\mathbf 1_{A_+}+\mathbb C\mathbf 1_{A_-}\). A functional \(\mu\) kills \(C_K+C_{K'}\) iff \(\mu(O_{0,0})=\mu(O_{1,0})=-\mu(A_+)-\mu(A_-)\) and \(\mu(O_{0,m})=-\mu(O_{1,m-1})\), \(\mu(O_{1,m})=-\mu(O_{0,m-1})\): a \(2\)-dimensional space, free in \((\mu(A_+),\mu(A_-))\), spanned by \(\lambda,\psi\). The pairing of \(\{\lambda,\psi\}\) with \(\{\mathbf 1_{A_+},\mathbf 1_{A_-}\}\) has determinant \(1\cdot\tfrac12-(-1)\cdot\tfrac12=1\), so the sum is direct.
  3. (c). \(\lambda(T_sx)=-\lambda(x)\), \(\psi(T_sx)=-\psi(x)\) for \(s=s_0,s_1\), \(\lambda\circ T_\omega=-\lambda\), \(\psi\circ T_\omega=\psi\): check on the basis with Computation 6.2, e.g. \(\psi(T_{s_1}\mathbf 1_{O_{0,0}})=(q-1)-(q-2)=1=-\psi(\mathbf 1_{O_{0,0}})\), \(\psi(T_{s_1}\mathbf 1_{O_{0,m}})=(-1)^m(q-(q-1))=-\psi(\mathbf 1_{O_{0,m}})\). So \((\lambda,\psi):C_I\to\varepsilon_-\oplus\varepsilon_+\) is \(\mathcal H_I\)-linear with kernel \(C_K+C_{K'}\) (step 2) and onto.
  4. (d). The \(G\)-map \(\mathcal S(X)\to\mathrm{St}^\vee\), \(f\mapsto\int_{X(F)}f(a,b)(\delta_a-\delta_b)\), paired with the \(I\)-fixed class of \(\mathbf 1_{\mathcal O}\in C^\infty(\mathbb P^1)\) (\(\mathcal O\subset\mathbb P^1(F)\) one of the two \(I\)-orbits of ends), gives \(\int f\,(\mathbf 1_{\mathcal O}(a)-\mathbf 1_{\mathcal O}(b))\); the integrand vanishes unless \(\gamma\) crosses the base edge, i.e. on \(A_+\cup A_-\), where it is \(\pm f\). \(\mathrm{vol}(A_+)=\mathrm{vol}(A_-)\) as \(\omega A_+=A_-\). \(\psi\) is the analogous map for the twist \(\mathrm{St}\otimes\chi_0\), \(\chi_0=(-1)^{\mathrm{val}\det}\) (heuristic identification; not used).
  5. (e). By Proposition 6.5(c) the class of any \(\mathcal F\in\mathcal C_I^{\mathrm{Iwa}}\) is \(\sum_i(-1)^i[{}^p\mathcal H^i\mathcal F]\in\mathrm{span}[\mathrm{IC}_{i,n}](\text{twists})\), whose traces are multiples of \(f_n,f_n^{(1)}\) (Lemma 6.3(b)); trace of a Weil object is additive (Laumon 1987, 1.1).

Builds on

  • Computation 6.2 · Affine Lusztig–Vogan module of $\mathrm{GL}_2/T$: Hecke action on orbit indicators
  • Lemma 6.4 · Convolution rules for $\mathrm{IC}s\ast_I\mathrm{IC}{i,n}$ and $\mathrm{IC}_\omega$
  • Proposition 6.5 · Sheaves fail for $\mathrm{GL}2/T$: $\mathrm{IC}{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}$

Built on by

  • Proposition 6.8 · Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not
  • 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 9.7 · Audit of A6-6: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two
  • 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 18:49 UTC

Functions: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two

  1. (a). \(T_w=T_{s_{i_1}}\cdots T_{s_{i_r}}T_\omega^j\) and \(T_s=(T_s+1)-1\); by Lemma 6.4 (shadow) the span of the \(f_n,f^{(1)}_n\) is \(\mathcal H_I\)-stable and contains \(f_0=\varphi_0\), so it contains \(\mathcal H_I\ast\varphi_0\); conversely \(f^{(1)}_0=T_\omega\varphi_0\), \(f^{(1)}_{n+1}=(T_{s_0}+1)f_n-qf^{(1)}_{n-1}\), \(f_{n+1}=T_\omega f^{(1)}_{n+1}\). Since \(f_n-f_{n-1}=\mathbf 1_{Y_n}\) and \(C_K\) has basis \(\mathbf 1_{\{d(v_0,\cdot)=n\}}=\mathbf 1_{Y_n}-\mathbf 1_{Y_{n-1}}\), \(\mathrm{span}\{f_n\}=C_K\); apply \(\omega\). The orbit list and Hecke formulas of Setup 6.1, Computation 6.2 hold over any such \(F\) (only the tree is used).
  2. (b). \(\mathbf 1_{O_{0,0}}=\mathbf 1_{Y_0}-\mathbf 1_{A_+}-\mathbf 1_{A_-}\), \(\mathbf 1_{O_{0,m}}=\mathbf 1_{\{d(v_0,\cdot)=m\}}-\mathbf 1_{O_{1,m-1}}\), and symmetrically, so \(C_I=C_K+C_{K'}+\mathbb C\mathbf 1_{A_+}+\mathbb C\mathbf 1_{A_-}\). A functional \(\mu\) kills \(C_K+C_{K'}\) iff \(\mu(O_{0,0})=\mu(O_{1,0})=-\mu(A_+)-\mu(A_-)\) and \(\mu(O_{0,m})=-\mu(O_{1,m-1})\), \(\mu(O_{1,m})=-\mu(O_{0,m-1})\): a \(2\)-dimensional space, free in \((\mu(A_+),\mu(A_-))\), spanned by \(\lambda,\psi\). The pairing of \(\{\lambda,\psi\}\) with \(\{\mathbf 1_{A_+},\mathbf 1_{A_-}\}\) has determinant \(1\cdot\tfrac12-(-1)\cdot\tfrac12=1\), so the sum is direct.
  3. (c). \(\lambda(T_sx)=-\lambda(x)\), \(\psi(T_sx)=-\psi(x)\) for \(s=s_0,s_1\), \(\lambda\circ T_\omega=-\lambda\), \(\psi\circ T_\omega=\psi\): check on the basis with Computation 6.2, e.g. \(\psi(T_{s_1}\mathbf 1_{O_{0,0}})=(q-1)-(q-2)=1=-\psi(\mathbf 1_{O_{0,0}})\), \(\psi(T_{s_1}\mathbf 1_{O_{0,m}})=(-1)^m(q-(q-1))=-\psi(\mathbf 1_{O_{0,m}})\). So \((\lambda,\psi):C_I\to\varepsilon_-\oplus\varepsilon_+\) is \(\mathcal H_I\)-linear with kernel \(C_K+C_{K'}\) (step 2) and onto.
  4. (d). The \(G\)-map \(\mathcal S(X)\to\mathrm{St}^\vee\), \(f\mapsto\int_{X(F)}f(a,b)(\delta_a-\delta_b)\), paired with the \(I\)-fixed class of \(\mathbf 1_{\mathcal O}\in C^\infty(\mathbb P^1)\) (\(\mathcal O\subset\mathbb P^1(F)\) one of the two \(I\)-orbits of ends), gives \(\int f\,(\mathbf 1_{\mathcal O}(a)-\mathbf 1_{\mathcal O}(b))\); the integrand vanishes unless \(\gamma\) crosses the base edge, i.e. on \(A_+\cup A_-\), where it is \(\pm f\). \(\mathrm{vol}(A_+)=\mathrm{vol}(A_-)\) as \(\omega A_+=A_-\). \(\psi\) is the analogous map for the twist \(\mathrm{St}\otimes\chi_0\), \(\chi_0=(-1)^{\mathrm{val}\det}\) (heuristic identification; not used).
  5. (e). By Proposition 6.5(c) the class of any \(\mathcal F\in\mathcal C_I^{\mathrm{Iwa}}\) is \(\sum_i(-1)^i[{}^p\mathcal H^i\mathcal F]\in\mathrm{span}[\mathrm{IC}_{i,n}](\text{twists})\), whose traces are multiples of \(f_n,f_n^{(1)}\) (Lemma 6.3(b)); trace of a Weil object is additive (Laumon 1987, 1.1).

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