The agent was asked to explore one undecided row, \(\mathrm{GL}_2/T\), where (IG) fails because the Steinberg representation is \(T\)-distinguished. It worked in the tree model: \(LX\) is the set of oriented geodesics, and the \(I\)-orbits are two closed orbits \(A_\pm\) (geodesics through the base edge, in either direction) and two infinite families ( (Setup 6.1)). The Hecke action on orbit indicators is explicit.
At function level the Hecke span of \(\varphi_0\) is \(C_K+C_{K'}\), of codimension two, with quotient the two sign characters ( (Proposition 6.6)). At sheaf level every generator \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\) splits into IC sheaves of the closures \(Y_n\) and \(\omega Y_n\), all \(K\)- or \(K'\)-equivariant; so the constant sheaves on \(A_\pm\) are never reached ( (Proposition 6.5)).
The agent then proved a sheaf-level form of the pages' Corollary D ( (Lemma 6.7)). A Hecke functional that kills every generator class and is nonnegative on every IC class excludes the IC sheaves where it is nonzero, because purity rules out cancellation. The Steinberg functional takes both signs and fails this test. A twisted Steinberg functional passes it and gives a second proof ( (Proposition 6.8)). Both levels fail at the same two orbits, so the row is not a divergence. The audit confirmed every step, but noted that the strict level presentation 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). is assumed rather than checked. It also found that the function half of the agent's general closed-orbit test needs an extra condition.
Result as the agent reported it proved
For \(X=\mathrm{GL}_2/T\): over \(\mathbb C\) and over \(\overline{\mathbb F}_q\) (\(p\) odd), \(\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}(\mathrm{For}\,\mathcal C_K\cup\mathrm{For}\,\mathcal C_{K'})\subsetneq\mathcal C_I\) (\(K'=\omega K\omega^{-1}\)); the Iwahori summand set is \(M_I=\{\mathrm{IC}(Y_n),\mathrm{IC}(\omega Y_n)\}\), and the only simple objects not generated are the constant sheaves on the two closed \(I\)-orbits \(A_\pm\) (geodesics through the base edge, either direction). At function level, over any non-archimedean \(F\), \(\mathcal H_I\ast\varphi_0=C_K+C_{K'}\) has codimension \(2\), with complement spanned by \(\mathbf 1_{A_\pm}\) and quotient \(\varepsilon_+\oplus\varepsilon_-\). Both levels fail in the same place: \(\mathrm{GL}_2/T\) is not an Iwahori divergence. By-products: a sheaf-level Corollary D (a Hecke functional that is nonnegative on IC classes excludes summands, using weights; sharp); a second proof of the sheaf half of row 1 (\(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) for the group, via the sign character \(\varepsilon_+\), not the affine Whittaker functor); a closed-orbit sign test for the other symmetric rows.
Gap
None for the GL2/T row beyond cited inputs: Chen–Yi's placid finite-level presentation (hypothesis 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).), the decomposition theorem, and rational smoothness of PGL2 affine Schubert varieties (bypassed by Proposition 6.8). Still open: the C-coefficient case of the group corollary Corollary 6.9 (cited parity argument only), the converse of the closed-orbit test Corollary 6.10, and rows with disconnected I-stabilizers (type IV orbits, non-constant local systems).
Blocked routes
- Deciding sheaves for GL2/T by the Steinberg functional λ of classical.html Proposition F alone: it would need λ ≥ 0 on IC classes, but λ(IC_{A±}) = ±1, so a summand IC_{A+} ⊕ IC_{A−} cannot be excluded. Replaced by the twisted functional ψ (T_ω ↦ +1), which is positive.
- Making the closed-orbit test an iff (mixed signs for every sign-type map ⇒ sheaves generate): needs a sheaf-level converse of Lemma C, i.e. that the sharp certificate of Lemma 6.7(c) can be chosen Hecke-equivariant. No such theorem is available.
- Extending Corollary 6.10 to rows with disconnected I-stabilizers: needs P_s-equivariance and positivity for ICs with non-constant local systems (Chen–Yi types IVa/IVb).
Brief as sent
Target: X = PGL_2/T (equivalently GL_2/T, T the diagonal torus; split, G^θ = T connected). (IG) fails (St_2 is T-distinguished; classical.html F). Decide C_I^{Iwa} = C_I for sheaves, or reduce it to an exact finite computation and carry that computation as far as possible: if sheaves generate, X is an Iwahori divergence; if they fail, both levels agree.
Approach: The affine Lusztig–Vogan module of PGL_2/T in the tree model. Starting points: (1) X(F) = ordered pairs of distinct ends of the Bruhat–Tits tree; list the I-orbits on LX (relative position of the base edge and an oriented geodesic), their closure order, dimensions (relative to L^+X), stabilizers and component groups (local systems), and which are closed; (2) the I-orbits inside L^+X via B-orbits on P^1×P^1 minus the diagonal; (3) the action of C_{s_0}, C_{s_1} and of the length-zero element on standard classes (Chen–Yi's module, formula (29), in chen-yi-v4.pdf; or a direct fibre count: the H_I-module C_I with basis the orbit indicators, which the trace map identifies with the module at v = q^{1/2}); (4) the IC classes (bar involution, KL-type basis of the module) and the expansion of C_w·[IC_0]; (5) decide whether the IC of a closed I-orbit is a summand of some IC_w ∗_I IC_0, using parity/positivity or a sheaf-level Steinberg functional (a linear functional on classes that kills C_w·[IC_0] for all w and is nonnegative on the IC classes); (6) state the verdict for this row with the exact remaining obligations. Python checks are allowed for the module computation.