§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:56:59Z

Theorem 6.12 · Verdict for $\mathrm{GL}2/T$: both levels fail at the closed orbits $A\pm$

proved ★ key step

Sheaves and functions both fail Iwahori generation for GL2/T, and they fail in the same place: the two closed I-orbits A± (geodesics through the base edge, either direction), so this row is not an Iwahori divergence; the Iwahori-generated parts are exactly what is invariant under one of the two maximal parahorics.

Theorem 6.12.

For \(X=\mathrm{GL}_2/T\) (= \(\mathrm{PGL}_2/T\)), \(T\) the diagonal torus: (1) (sheaves, over \(\mathbb C\) or \(\overline{\mathbb F}_q\), \(p\) odd) \(\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}(\mathrm{For}\,\mathcal C_K\cup\mathrm{For}\,\mathcal C_{K'})=\mathrm{Thick}\{\mathrm{IC}_{i,n}\}\subsetneq\mathcal C_I\), \(K'=\omega K\omega^{-1}\); the simple objects not in \(\mathcal C_I^{\mathrm{Iwa}}\) are exactly \(\mathrm{IC}_{A_+}=k_{A_+}\) and \(\mathrm{IC}_{A_-}=k_{A_-}\); (2) (functions, any non-archimedean \(F\)) \(\mathcal H_I\ast\varphi_0=C_K+C_{K'}\), and \(C_I=\mathcal H_I\ast\varphi_0\oplus\mathbb C\mathbf 1_{A_+}\oplus\mathbb C\mathbf 1_{A_-}\), with \(C_I/\mathcal H_I\ast\varphi_0\cong\varepsilon_+\oplus\varepsilon_-\); (3) the trace of \(\mathcal C_I^{\mathrm{Iwa}}\) is \(\mathcal H_I\ast\varphi_0\), and the traces of the two missing simple objects span the complement: the row agrees (both fail), it is not an Iwahori divergence; (4) for this \(X\) every hypothesis of the Iwahori summand criterion (I_I) holds and \(M_I=\{\mathrm{IC}_{i,n}\}\ne\{\text{all simples}\}\); the functions-to-sheaves statement (II_I) is vacuous here because (IG) fails.

  1. (1) is Proposition 6.5 (geometric proof: IC stalks Lemma 6.3, convolution rules Lemma 6.4), reproved over \(\overline{\mathbb F}_q\) by the positive certificate \(2\psi\) (Proposition 6.8, Lemma 6.7).
  2. (2) is Proposition 6.6, independent of Corollary D of classical.html; it refines Proposition F there (failure of (IG)) to an exact codimension-two statement.
  3. (3): Proposition 6.6(e) and Proposition 6.5(c).
  4. (4): the hypotheses — level presentation with \(2N+3\) orbits per level, connected stabilizers, finite-length perverse heart, decomposition of \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\) — are checked in Setup 6.1, Proposition 6.5, Audit 6.11.
  5. Mechanism, in one line: every generator is a sum of objects equivariant for \(K\) or \(K'\) (Lemma 6.4), and the closed orbits \(A_\pm\) are the only orbits whose closure is stable under neither \(P_{s_0}\) nor \(P_{s_1}\) (they are of type IIIa for both reflections).
  6. By-products for the team question: the positive-certificate lemma Lemma 6.7 (a sheaf-level Corollary D), the group case of row 1 by the sign character Corollary 6.9, and the closed-orbit sign test with the exact remaining obligation for the other symmetric rows Corollary 6.10.

Builds on

  • Proposition 6.5 · Sheaves fail for $\mathrm{GL}2/T$: $\mathrm{IC}{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}$
  • Proposition 6.6 · Functions: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two
  • 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

Built on by

Original record, as the agent wrote it at 18:56 UTC

Verdict for $\mathrm{GL}2/T$: both levels fail at the closed orbits $A\pm$

  1. (1) is Proposition 6.5 (geometric proof: IC stalks Lemma 6.3, convolution rules Lemma 6.4), reproved over \(\overline{\mathbb F}_q\) by the positive certificate \(2\psi\) (Proposition 6.8, Lemma 6.7).
  2. (2) is Proposition 6.6, independent of Corollary D of classical.html; it refines Proposition F there (failure of (IG)) to an exact codimension-two statement.
  3. (3): Proposition 6.6(e) and Proposition 6.5(c).
  4. (4): the hypotheses — level presentation with \(2N+3\) orbits per level, connected stabilizers, finite-length perverse heart, decomposition of \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\) — are checked in Setup 6.1, Proposition 6.5, Audit 6.11.
  5. Mechanism, in one line: every generator is a sum of objects equivariant for \(K\) or \(K'\) (Lemma 6.4), and the closed orbits \(A_\pm\) are the only orbits whose closure is stable under neither \(P_{s_0}\) nor \(P_{s_1}\) (they are of type IIIa for both reflections).
  6. By-products for the team question: the positive-certificate lemma Lemma 6.7 (a sheaf-level Corollary D), the group case of row 1 by the sign character Corollary 6.9, and the closed-orbit sign test with the exact remaining obligation for the other symmetric rows Corollary 6.10.

Sources