The agent was asked to find every simple object of \(\mathcal C_I\) for \(\mathbb A^n\) under \(\mathrm{GL}_n\), to decide both levels, and to run the bridge in reverse. The key observation is that the \(I\)-stable lattices form one orbit of the length-zero element \(\tau=t^{\varepsilon_n}\sigma\): \(\Lambda_{m,j}=\tau^{nm+n-j}\mathcal O^n\) ( (Lemma 5.1)). With the perverse t-structure renormalized by \(nM\) at level \((N,M)\), the simple objects are the shifted constant sheaves \(S_k\) on the lattices \(\tau^k\mathcal O^n\), and no object is supported at the origin ( (Lemma 5.2)).
Convolution with the skyscraper at \(\tau^k\) is push-forward along \(\tau^k\), so \(\mathrm{IC}_{\tau^k}\ast_I\mathrm{IC}_0=S_k[k]\) ( (Computation 5.3)). Every simple object is therefore a single Hecke translate of \(\mathrm{IC}_0\), and \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\), generated by extensions and shifts alone, with no decomposition theorem ( (Theorem 5.5)). At function level \(T_{\tau^k}\ast\varphi_0=\mathbf 1_{\tau^k\mathcal O^n}\), and these form a basis of \(C_I\), which is free of rank one over \(\mathbb C[\Omega]\) ( (Proposition 5.6)).
The agent also proved general versions of the criterion and of the bridge, and checked the bridge's inputs for \(\mathbb A^n\), so (IG) gives the sheaf statement a second time ( (Corollary 5.8)). The audit kept every step. It strengthened the level axioms of the general criterion and required the equivariant decomposition theorem in the bridge.
Result as the agent reported it proved
For X = A^n under GL_n (every n >= 1): (1) with the perverse t-structure renormalized by nM at level (N,M), the heart of C_I has finite length and its simple objects are S_k = k_{\tau^k O^n}[-k], k in Z, where \tau = t^{\varepsilon_n}\sigma generates \Omega and \tau^{nm+n-j}O^n = \Lambda_{m,j}; no object of C_I is supported at 0. (2) IC_{\tau^k} *I IC_0 = S_k[k], so M_I = all simple objects and, by the Iwahori summand criterion (I_I) (proved in general under (L), (Hk), (DT)(DT) · Decomposition: every generator IC_w ∗_I IC_0 is, equivariantly, a finite direct sum of shifted simple objects of C_I. Holds over F̄_q from a finite-type proper model of the convolution (purity and the stack decomposition theorem).; the direction used needs only (L)), C_I^{Iwa} = C_I = Tri{\delta{\tau^k} *I IC_0}, over C (Betti) and over \bar F_q, with no decomposition theorem, no standard objects, and no use of IV.1 or Proposition III. Also IC{s_i} * S_k = S_{k-1} + S_{k+1} if i = -k mod n, else S_k[1] + S_k[-1]. (3) For every non-archimedean F, with vol(I)=1: T_{\tau^k} * \varphi_0 = 1_{\tau^k O^n}, and these form a basis of C_I, so (IG) holds and C_I is free of rank one over C[\Omega]; T_{s_i} acts by q, or by b_k -> b_{k-1} + q b_{k+1} - b_k. Over F_q((t)) these are the traces of (2). (4) (II_I), proved in general under (L), (L_q), (W), (P), (Lin), (Tr): (IG) over F_q((t)) implies M_I = all, hence C_I^{Iwa} = C_I over \bar F_q; every input is checked for A^n, so the reverse route re-proves (2) over \bar F_q. Row A^n of Table 1: the two Iwahori-level statements agree (both hold).
Gap
none for A^n beyond cited standard theorems (BBD 4.2.4-4.2.6, 4.3.1 for the sheaf statement; Gabber purity, BBD 5.1.14, 5.3.8, 5.4.5, Sun Thm 1.2 and the Grothendieck-Lefschetz dictionary only for the reverse route). The general (II_I) reaches only \bar F_q; transferring a summand statement to Betti sheaves over C needs a separate comparison (not needed for A^n, proved directly over C). For other rows, (W), (Lin) (traces of non-constant local systems) and (P) must be checked per X, and the converse sheaves => functions needs control of summands whose trace is invisible.
Blocked routes
- Reverse route without purity: traces alone show each simple is a Jordan-Holder constituent of some pH^i(IC_w * IC_0), not a summand; upgrading needs semisimplicity of IC_w * IC_0 (BBD 5.4.5 + 5.3.8 on a proper finite-type model, or the equivariant decomposition theorem, Sun Thm 1.2).
- Sheaves => functions for general X via K_0 of the thick closure: a retract can add classes, so C_I^{Iwa} = C_I gives (IG) only if every simple has trace in the span of traces of generators (no trace-invisible summand, m_{S,w} != 0 for some w) or generation holds without retracts; for A^n this holds because C_I = Tri{delta_{tau^k} * IC_0}.
- (II_I) to Betti sheaves over C: needs a comparison theorem for the summand statement between \bar F_q and C (spreading out, BBD 6.1); not supplied in general.
Brief as sent
Target: X = A^n under G = GL_n. Determine every simple object of C_I (expected: shifted constant sheaves on the I-stable lattices Λ_{m,j} = t^{λ_{m,j}} O^n, the closures of the I-orbits O_{m,j}), show that each is, up to shift, a convolution Δ_{t^λ} ∗_I IC_0 or a summand of some IC_w ∗_I IC_0, so that M_I is everything; prove (IG) for A^n by taking traces; and show the reverse route: (IG) together with a functions ⇒ sheaves bridge gives C_I^{Iwa} = C_I again. Say exactly what is new compared with case-affine.html (IV.1) and classical-affine.html (F3).
Approach: Lattice geometry and the simple objects of the colimit category. Starting points: (1) classify the I/I_{N,M}-orbits on Y_{N,M} = t^{−N}O^n/t^M O^n, their closures and stabilizers, and the pieces; (2) identify the simple perverse objects of C_I and their shifts under raising M; check that the IC of the closure of O_{m,j} is the shifted constant sheaf on the smooth lattice Λ_{m,j}; (3) compute IC_w ∗I IC_0 for w = t^λ and for the simple affine reflections, with its decomposition; (4) (IG): 1{O_{m,j}} = 1_{Λ_{m,j}} − 1_{Λ_{m,j−1}} and 1_{Λ_{m,j}} as a multiple of T_{t^λ} ∗ φ_0 (check the normalization vol(I) = 1); (5) record whether the decomposition theorem is needed here or avoidable; (6) the point 0 and the tails T_M = t^M O^n: check that no step uses an object supported at 0.