The agent was asked to prove, for \(X=H\) under \(H\times H\), that the unit \(\Delta_e\) is not Iwahori-generated, without the affine Whittaker functor. It first checked every hypothesis of a summand criterion for this \(X\): the pieces are the Bruhat cells, stabilizers are connected, the generators \(\mathrm{IC}_x\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_y\) are two-sided, and they decompose ( (Setup 4.1)).
The argument has three parts. The sign character \(\varepsilon(T_w)=(-1)^{\ell(w)}\) of the Iwahori–Hecke algebra kills the trace of every \(\mathrm{IC}_z\) with \(z\) of positive length, since such a \(z\) has a descent, and it kills \(\mathbf 1_{K_H}\) because \(W\ne1\) ( (Lemma 4.3)). Purity fixes the Frobenius weight of each multiplicity space of the generators ( (Lemma 4.4)). So the vanishing of \(\varepsilon\) on the traces over every \(\mathbb F_{q^m}\) leaves no room for cancellation, and no \(\Delta_\omega\), \(\omega\in\Omega\), occurs in any generator ( (Theorem 4.5)). The function failure of (IG) is the same computation ( (Corollary 4.6)).
The agent then determined the summand set: exactly the elements of positive length in type \(A_1\), and the lowest two-sided cell in general, by cited cell theory ( (Proposition 4.7)). The audit confirmed the argument over \(\overline{\mathbb F}_q\). It noted that the statement over \(\mathbb C\) rests on a cited comparison and that the cell description beyond rank one is cited.
Result as the agent reported it partial
For X = H under H×H with W ≠ 1 (H split connected reductive), over k = F̄q with Q̄_ℓ-coefficients: no Δ_ω (ω ∈ Ω, in particular Δ_e) is a perverse constituent, hence not a shifted direct summand, of any generator IC_x ∗ IC_0 ∗ IC_y; by the group summand criterion, whose hypotheses (finite Bruhat intervals, connected stabilizers, finite-length heart with simples IC_z, two-sided generators, decomposition) are checked, Δ_e ∉ C_I^Iwa. Proof: the sign character ε(T_w) = (−1)^{ℓ(w)} kills tr(IC_x ∗ IC_0 ∗ IC_y) because ε(1_K) = 0, satisfies ε(tr IC_z) = 0 for z ∉ Ω (descent) and = 1 for z ∈ Ω, and purity fixes the shift of each Δ_ω-multiplicity by its Frobenius weight, so the vanishing over all F{q^m} admits no cancellation. No affine Whittaker functor, no I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages).. At the same time (IG) fails over every non-archimedean F (H_I 1_K H_I ⊆ ker ε ∌ T_e), this is the trace of the sheaf argument, and (II_I) holds for X = H with inputs D1–D4 checked, re-deriving the function failure from the sheaf one over F_q((t)). M_I: {reduced x·w_0·y} ⊆ M_I ⊆ W~ ∖ Ω, z missing a whole simple factor excluded (mixed sign/trivial character); M_I = W~ ∖ Ω, i.e. C_I^Iwa = Thick{IC_z : ℓ(z) > 0}, in semisimple rank one (SL_2, PGL_2, GL_2); M_I = c_0·Ω (lowest two-sided cell {x·w_J·y reduced, ℓ(w_J) = ℓ(w_0)}) in general, given cited cell theory, which gives IC_s ∉ C_I^Iwa for simple H of rank ≥ 2, so C_I^Iwa ⊊ ker Φ there. Over C (Betti / D-modules) by comparison of stalk dimensions. The group row is not a divergence. For the other rows only 'functions fail, sheaves hold' can diverge, and a pure cohomological lift (Λ) of the function-level Steinberg/Speh functional rules it out.
Gap
(1) The exact M_I = c_0·Ω beyond semisimple rank one rests on cited results (Kazhdan–Lusztig 1980 pointwise purity incl. the affine case, Bezrukavnikov–Yun Prop. 3.2.5 positivity, Lusztig 1985 and Shi 1987–88 on the lowest two-sided cell), whose theorem numbers were not checked against local copies. (2) The C-version uses the standard F̄p↔C comparison (generic base change, BBD §6.1), cited, not re-proved. (3) BBD 4.2.5 and 5.4.1–5.4.4 are cited from memory; the local BBD file is a scan without a text layer. (4) The other rows: the lift (Λ) of the Steinberg-type functionals (GL_2/T, GL_n/(GL_p×GL_q), Spin{2n}/Spin_{2n−1}) and of the Speh functional (GL_{2n}/Sp_{2n}, SL_{2n}/Sp_{2n}, where the function failure is also only stated for char F = 0) is open; E_6/F_4 is open at function level; A^n is not treated here.
Blocked routes
- Ungraded Euler characteristic at v = −1 alone: cancellation between IC_e[n] and IC_e[n+1] cannot be excluded without a parity theorem for IC_x ∗ IC_0 ∗ IC_y (Bezrukavnikov–Yun Prop. 3.2.5, or parity sheaves); replaced by Frobenius weights over all F_{q^m}.
- Trace at a single q without Tate/parity input: eigenvalues of equal absolute value and opposite sign on multiplicity spaces could cancel; needs either the Tate-and-parity decomposition IC_w[n](n/2) (BY 3.2.5, plus identifying components of Fl_H with Kac–Moody flag varieties) or all q^m (used).
- Elementary proof of M_I ⊆ c_0 in rank ≥ 2 (e.g. IC_s ∉ M_I): needs Lusztig's a-function inequality a(z) ≥ a(w_0) for z ≤_LR w_0, or a pure functional (Λ) separating IC_s. Mixed sign/trivial characters only exclude elements missing a whole simple factor. Characters with sign on one root-length class (non-simply-laced) would need (−1)^{ℓ(z)}κ(tr IC_z) ∈ N[q] for all z with descents in the trivial class, which is unproved.
- Direct proof over C without F_q: needs a weight theory (mixed Hodge modules) together with a substitute for Frobenius traces over all q^m; replaced by the comparison of Audit 4.9.
Brief as sent
Target: X = H under G = H×H with W_H ≠ 1, so C_I = H_{c,I_H} = Shv_c(I_H\LH/I_H) and IC_0 is the constant sheaf on L^+H = K_H, i.e. IC_{w_0}[−ℓ(w_0)] up to the normalization (w_0 the longest element of W_H). Prove, without the affine Whittaker functor of the pages' Proposition II, that IC_e = Δ_e is not a shifted direct summand of any IC_x ∗ IC_{w_0} ∗ IC_y, hence (by a summand criterion) Δ_e ∉ C_I^{Iwa}; determine M_I exactly; and derive the failure of (IG) at the same time.
Approach: Kazhdan–Lusztig basis, the sign character and parity. Starting points: (1) the character map from semisimple objects of H_{c,I} to the affine Hecke algebra over Z[v,v^{-1}], IC_w ↦ C_w (state the normalization, e.g. (H_s+v)(H_s−v^{-1}) = 0, C_s = H_s + v), multiplicative for convolution — cite exactly; or the ungraded Euler characteristic at v = −1 together with parity vanishing of the IC stalks; (2) the sign character ε with ε(C_s) = 0 kills every C_w with w ≠ e (a right descent s gives C_w C_s = (v+v^{-1}) C_w) and ε(C_e) = 1; (3) ε(C_x C_{w_0} C_y) = 0 forces the graded multiplicity of IC_e in IC_x ∗ IC_{w_0} ∗ IC_y to vanish — prove there is no cancellation (multiplicities in N[v,v^{-1}], or parity at v = −1); (4) identify M_I with {z : z ≤LR w_0} by positivity and cite Shi/Lusztig for the lowest two-sided cell {x·w_0·y reduced}; (5) at function level, the sign character of H_I kills H_I 1{K_H} H_I and not 1_I — relate it to the trace of the sheaf argument and to classical.html E; (6) check the hypotheses of the summand criterion for this X (pieces are Bruhat cells, connected stabilizers, the decomposition theorem on Fl).