The agent was asked to state and prove the Iwahori analogue of the spherical summand criterion, with every hypothesis named, and to check those hypotheses for the group and \(\mathbb A^n\). It first fixed level axioms: affine-space transitions in \(M\), Cartesian closed embeddings in \(N\), connected finite quotients of \(I\), and full faithfulness ( (Setup 2.1)). Then it glued the perverse hearts of the finite levels into one heart on \(\mathcal C_I\) whose simple objects are the IC sheaves of pieces, two pieces with the same closure giving the same object ( (Lemma 2.2)).
The criterion ( (Theorem 2.4)) holds without any decomposition theorem as a sandwich: the summands of the generators \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\) are reached, and only their constituents can be. Given the decomposition, which follows from a finite-type proper model by purity ( (Lemma 2.5)), the two coincide, and \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) exactly when every simple object is a summand.
For \(\mathbb A^n\) every simple object is a translate of \(\mathrm{IC}_0\) by a length-zero element ( (Theorem 2.7)). For the group, the sign character and Frobenius weights show that no \(\Delta_\omega\) is reached ( (Theorem 2.9)); \(\mathbb G_m\) works like \(\mathbb A^n\). The audit kept every step. It repaired two points: the level groups must be pro-unipotent, and the local systems must be counted at every level.
Result as the agent reported it proved
(I_I), over F_q-bar or C: under (L1)-(L3) [compatible level presentation: affine-space transitions in M, Cartesian closed embeddings in N, connected I/I_{N,M} with unipotent kernels, pi^* fully faithful], (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).) [finitely many pieces per level] and (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) [connected stabilizers] or (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') [local systems], the perverse hearts glue along the levels (PH_I is a consequence), C_I = Thick(S_I), and Thick(M_I) <= C_I^Iwa <= Thick(M_I^+) (M_I = simple shifted summands, M_I^+ = simple constituents of the IC_w *_I IC_0, two-sided for the group); under DT_I, C_I^Iwa = Thick{S in M_I} and C_I^Iwa = C_I iff M_I = S_I. DT_I follows from a finite-type proper model over F_q-bar (purity + Sun Thm 1.2). For A^n under GL_n all hypotheses hold, the simple objects are the constant sheaves on the lattices t^m L_j, and they form the single orbit Omega*IC_0 (omega = cyclic shift times diag(t,1,...,1)), so C_I^Iwa = C_I and (IG) holds: a second proof, without the 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). devissage. For the group H under H x H (W_H != 1) all hypotheses hold once the levels are truncated in Omega; the simple objects are the IC_v; over F_q-bar the sign character T_x -> (-1)^l(x), applied to traces, plus Frobenius weights, shows no Delta_omega (omega in Omega) is a constituent of any IC_w1 * c * IC_w2, so Delta_e is not in C_I^Iwa and (IG) fails: a second proof, without the affine Whittaker functor. GL_2/Sp_2 = G_m: both hold by the same Omega-argument. Symmetric rows: given (L_X), C_I^Iwa = C_I iff every IC(closure O_v, L_xi) is a shifted summand of some IC_w * IC_0 (Chen-Yi supply 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)., I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.', DT_I); a sign-obstruction lemma gives a sufficient test for sheaf failure.
Gap
Cited, not re-proved: (i) the full-faithfulness clause of (L3) (homotopy invariance and inflation along unipotent kernels, Bernstein-Lunts), assumed by the pages as well; (ii) the equivariant decomposition theorem over C (Bernstein-Lunts 5.3; Sun Thm 3.15 stated only), so DT_I over C is cited; (iii) the F_q-bar to C transfer (BBD 6.1.9-6.1.10, Bezrukavnikov-Finkelberg Prop 5) for the group statement over C. Open: (L_X) for Chen-Yi's presentation and the summand statement (*_X) for GL_2n/Sp_2n (n>=2), GL_2/T, GL_n/(GL_p x GL_q), Spin_2n/Spin_2n-1, SL_2n/Sp_2n, E6/F4.
Blocked routes
- Global perverse t-structure on C_I from a dimension theory (Chen-Yi Prop 63): needs a dimension function compatible with every transition map; bypassed because all conditions used are shift-invariant.
- Group case over C without transfer: ruling out cancellation needs a weight theory over C (mixed Hodge modules) or parity sheaves (Juteau-Mautner-Williamson: convolution of parity complexes is parity, IC_w parity); not attempted, replaced by the cited good-model transfer.
- Omega-argument beyond A^n and G_m: needs S_I = Omega*IC_0; heuristically false for the other symmetric rows (I-orbits of L^+X over non-open B-orbits, non-constant L_xi).
- Sheaves => functions via traces: M_I = S_I does not put each trace t_S in the span of the traces of generators; would need an inverse unitriangularity statement between summand multiplicities and traces.
- (L3) from the paper's definition of coherent D-modules: needs the colimit description D(Y_N) = colim_M D(Y_{N,M}) for equivariant constructible objects (the pages' unproved reading of 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).).
Brief as sent
Target: State and prove the Iwahori summand criterion (I_I) for a general X with a level presentation, with every hypothesis named (suggested names: I1 finite levels, I2 connected stabilizers or its local-system variant, PH_I a finite-length perverse heart on each level compatible with the colimit, DT_I the decomposition of IC_w ∗_I IC_0 into shifted simple objects), and verify those hypotheses for the group H under H×H and for A^n under GL_n.
Approach: Constituent subcategories and the perverse heart over a colimit of finite levels. Starting points: (1) define the perverse t-structure on C_I^{(N,M)} (pull back from Y_{N,M} with the shift by relative dimension, or normalize by the codimension of pieces) and show that the transition functors (raising M; the closed embedding Y_N ⊂ Y_{N+1}) are t-exact and send simple objects to simple objects; identify the simple objects of the union (IC of the closure of a piece with an irreducible equivariant local system; two pieces with the same closure give the same simple); (2) show H_{c,I} = Thick{IC_w : w ∈ W~} and hence C_I^{Iwa} = Thick{IC_w ∗_I IC_0 : w ∈ W~} (two-sided for the group); compare with the K-level description Thick{F ∗_K IC_0 : F ∈ Shv_c(I\Gr)} of the pages' Proposition I(a); (3) prove DT_I from a finite-type model of the convolution (properness on bounded supports; equivariant decomposition theorem over C, or purity over F_q), citing exact results; (4) prove the criterion with the constituent subcategory D_Σ (objects whose perverse cohomologies have all constituents in Σ), as in satake-two-levels props/summand.html; check that D_Σ is thick (Serre subcategory of the heart, long exact sequence, retracts); (5) verify each hypothesis for the group and for A^n.