§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:56:21Z

Audit 6.11 · Audit of the $\mathrm{GL}_2/T$ argument against the nine failure modes

proved

Checks each of the nine standard failure modes against the proofs for GL2/T and the group case; records the three inputs taken from the literature rather than re-proved.

Audit 6.11.

Each item of the adversarial checklist is either avoided or discharged in Setup 6.1–Corollary 6.10; the inputs used without proof are: (i) the finite-level placid presentation of \(LX\) with orbits as pieces (Chen–Yi §5, Appendix A; hypothesis.html I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).); (ii) rational smoothness of \(\mathrm{Gr}_{\le n}\) for \(\mathrm{PGL}_2\) (dihedral Kazhdan–Lusztig polynomials \(=1\)), used only in the first proof of Lemma 6.3 and bypassed by Proposition 6.8; (iii) the decomposition theorem (BBD 5.1.14, 5.4.5, 5.4.6, 6.2.5).

  1. Infinite orbit closures. Not present: \(Y_N\) is a union of \(2N+3\) orbits (Setup 6.1(d)), every orbit is a piece at large level, every simple object is an orbit IC pulled back with shift \(\delta\); the perverse t-structure is that of each level, with t-exact transitions (Proposition 6.5 step 5).
  2. Generators. \(\mathcal H_{c,I}=\mathrm{Thick}\{\mathrm{IC}_w\}\) by cell dévissage (Proposition 6.5 step 1); the action is one-sided for \(\mathrm{GL}_2/T\), two-sided (with \(\iota\)) for the group (Corollary 6.9).
  3. Decomposition. Finite-type models: \(\nu_n\), \(\nu'\) and Bott–Samelson maps are proper at a finite level; semisimplicity from BBD over \(\overline{\mathbb F}_q\) (purity) and over \(\mathbb C\) (6.2.5).
  4. Local systems. All \(I\)-stabilizers are connected (Setup 6.1(c); Chen–Yi Lemma 71(ii) gives the component group as \(\pi_0\) of a torus fixed-point group, trivial here), so no non-constant local systems; Corollary 6.10 flags the disconnected case as not covered.
  5. Traces. Orbits are defined over \(\mathbb F_q\) and have connected stabilizers, so Lang's theorem gives the same orbit list over \(\mathbb F_q((t))\) (Setup 6.1(e)); \(\mathrm{vol}(I)=1\); trace of a convolution is the convolution of traces (proper base change and Grothendieck–Lefschetz, Computation 6.2 step 4); IC classes are unitriangular against standards (Lemma 6.3(c)).
  6. Cancellation. Proposition 6.5 uses no Grothendieck group: stalks are peeled degree by degree and Krull–Schmidt identifies summands. Lemma 6.7 uses weights to exclude cancellation; Proposition 6.8(c) shows the Steinberg functional \(\lambda\) alone cannot exclude \(\mathrm{IC}_{A_+}\oplus\mathrm{IC}_{A_-}\).
  7. Fields. Proposition 6.5: \(\mathbb C\) (Betti) and \(\overline{\mathbb F}_q\), \(p\) odd. Proposition 6.6: every non-archimedean local field. Lemma 6.7, Proposition 6.8, Corollary 6.9: \(\overline{\mathbb F}_q\); \(\mathbb C\) for Corollary 6.9 only via the cited parity argument. \(p=2\) is excluded where the involution model of Setup 6.1 is used.
  8. Hypothesis transfer. No spherical-level hypothesis (H1–H3, (R), (C)) is used at Iwahori level; the finiteness used is checked directly on the tree. The spherical statement \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) (Lemma 6.3(d)) is a by-product, not an input.
  9. Circularity. The group-case proof Corollary 6.9 cites neither group.html Proposition II nor IV.2; it uses Lemma 6.7, the Bruhat stratification and BBD. \(\mathrm{GL}_2\) versus \(\mathrm{PGL}_2\): \(\widetilde W=\mathbb Z^2\rtimes S_2\), \(\omega^2=t\cdot\mathrm{Id}\) acts trivially on \(LX\), so the \(\mathrm{PGL}_2\) tree computation is the \(\mathrm{GL}_2\) one.

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

Built on by

  • Theorem 6.12 · Verdict for $\mathrm{GL}2/T$: both levels fail at the closed orbits $A\pm$
Original record, as the agent wrote it at 18:56 UTC

Audit of the $\mathrm{GL}_2/T$ argument against the nine failure modes

  1. Infinite orbit closures. Not present: \(Y_N\) is a union of \(2N+3\) orbits (Setup 6.1(d)), every orbit is a piece at large level, every simple object is an orbit IC pulled back with shift \(\delta\); the perverse t-structure is that of each level, with t-exact transitions (Proposition 6.5 step 5).
  2. Generators. \(\mathcal H_{c,I}=\mathrm{Thick}\{\mathrm{IC}_w\}\) by cell dévissage (Proposition 6.5 step 1); the action is one-sided for \(\mathrm{GL}_2/T\), two-sided (with \(\iota\)) for the group (Corollary 6.9).
  3. Decomposition. Finite-type models: \(\nu_n\), \(\nu'\) and Bott–Samelson maps are proper at a finite level; semisimplicity from BBD over \(\overline{\mathbb F}_q\) (purity) and over \(\mathbb C\) (6.2.5).
  4. Local systems. All \(I\)-stabilizers are connected (Setup 6.1(c); Chen–Yi Lemma 71(ii) gives the component group as \(\pi_0\) of a torus fixed-point group, trivial here), so no non-constant local systems; Corollary 6.10 flags the disconnected case as not covered.
  5. Traces. Orbits are defined over \(\mathbb F_q\) and have connected stabilizers, so Lang's theorem gives the same orbit list over \(\mathbb F_q((t))\) (Setup 6.1(e)); \(\mathrm{vol}(I)=1\); trace of a convolution is the convolution of traces (proper base change and Grothendieck–Lefschetz, Computation 6.2 step 4); IC classes are unitriangular against standards (Lemma 6.3(c)).
  6. Cancellation. Proposition 6.5 uses no Grothendieck group: stalks are peeled degree by degree and Krull–Schmidt identifies summands. Lemma 6.7 uses weights to exclude cancellation; Proposition 6.8(c) shows the Steinberg functional \(\lambda\) alone cannot exclude \(\mathrm{IC}_{A_+}\oplus\mathrm{IC}_{A_-}\).
  7. Fields. Proposition 6.5: \(\mathbb C\) (Betti) and \(\overline{\mathbb F}_q\), \(p\) odd. Proposition 6.6: every non-archimedean local field. Lemma 6.7, Proposition 6.8, Corollary 6.9: \(\overline{\mathbb F}_q\); \(\mathbb C\) for Corollary 6.9 only via the cited parity argument. \(p=2\) is excluded where the involution model of Setup 6.1 is used.
  8. Hypothesis transfer. No spherical-level hypothesis (H1–H3, (R), (C)) is used at Iwahori level; the finiteness used is checked directly on the tree. The spherical statement \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) (Lemma 6.3(d)) is a by-product, not an input.
  9. Circularity. The group-case proof Corollary 6.9 cites neither group.html Proposition II nor IV.2; it uses Lemma 6.7, the Bruhat stratification and BBD. \(\mathrm{GL}_2\) versus \(\mathrm{PGL}_2\): \(\widetilde W=\mathbb Z^2\rtimes S_2\), \(\omega^2=t\cdot\mathrm{Id}\) acts trivially on \(LX\), so the \(\mathrm{PGL}_2\) tree computation is the \(\mathrm{GL}_2\) one.

Sources