The agent was asked whether, for \(\mathrm{GL}_4/\mathrm{Sp}_4\), the IC sheaf of the orbit \((1,1,1,1)\) is a summand of some Satake object even though that orbit has no dominant lift (so the pages' hypothesis H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. fails), and then to push the method to \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\).
Modelling the loop space by alternating forms, it found that the minuscule convolution has one point over the open orbit and the Lagrangian Grassmannian \(LG(2,4)\) over the closed orbit, of codimension 4. The decomposition theorem, the support condition and Verdier duality then give $$V_{\omega_2}\ast\mathrm{IC}_0\simeq\mathrm{IC}_{(2,0)}\oplus\mathrm{IC}_{(1,1)}\otimes\big(k[2]\oplus k\oplus k[-2]\big),$$ proved \(\ell\)-adically, the Betti case resting on the pages' transfer item (Proposition 3.3). It proved a criterion replacing H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.: the Satake span is everything exactly when every IC sheaf is a retract of a Satake object (Proposition 3.4). At function level two Hecke translates combine to the indicator of the uncovered orbit, giving spherical generation for \(\mathrm{GL}_4/\mathrm{Sp}_4\) over every local field without Sakellaridis's hypotheses (Theorem 3.9). It also proved that generation over \(\mathbb F_q(\!(t)\!)\) forces sheaf generation (Proposition 3.10). In higher rank it reached only some uncovered labels.
The audit flagged the closing claim that no disagreement can occur in higher rank: it combined a result over \(\mathbb F_q(\!(t)\!)\) with a \(p\)-adic one (Summary 3.11). The claim became true later, through the lattice-count theorem. A codimension citation assumed a connected loop space; the agent's own dimension count repairs it. The \(\mathrm{GL}_4/\mathrm{Sp}_4\) result is Theorem 2 of the verdict, and its criterion is one source of Proposition 7.
Result as the agent reported it partial
GL_4/Sp_4: IC of the closed uncovered orbit (1,1,1,1) is a retract of V_{omega_2}*IC_0 (multiplicity k[2]+k+k[-2], fibre LG(2,4), codimension 4), all uncovered labels (a,a,a,a) follow by the scalar shift, so M = Lambda_S^+ and C^Sat = C given the decomposition theorem for Satake objects in C (Chen–Yi Prop. 70 / Sun) and its transfer to the Betti category; (SG) for GL_4/Sp_4 holds over every non-archimedean local field unconditionally (explicit T_{omega_2}, T_{(2,0,0,0)}). General criterion: C^Sat = C iff every IC_mu is a retract of some V_lambda*IC_0 (coverage/H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. replaced by this summand condition). For GL_2m/Sp_2m: the labels (1^k 0^{m-k}) + c(1^m) and (2^{k-m} 1^{2m-k}) + c(1^m) lie in M via V_{omega_k} (maximal-defect lemma, defect = number of 1's); SL_2n/Sp_2n is component 0 of GL_2n/Sp_2n. (SG) over F_q((t)) implies C^Sat = C for these X, so the two levels cannot disagree in the direction 'functions hold, sheaves fail'; the other direction is excluded under Sakellaridis's (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies. by Prop. F.
Gap
(i) Fibre-dimension bound 2f_nu - c_nu <= |a| for the Bott–Samelson convolution attached to lambda = (a, 0^m) over all strata nu, with equality at a (equivalently the Hecke-compatibility of Chen–Yi Thm. 87(ii)); this would give M = Lambda_S^+ for all GL_2m/Sp_2m and SL_2n/Sp_2n, in particular for the open SL_6 label (1,1,-2). (ii) Transfer of the ell-adic decomposition theorem to the Betti category C over C (page's open item 1), inherited by every sheaf-level statement. (iii) (SG) for GL_2m/Sp_2m, m >= 3, without (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit., and over F_q((t)).
Blocked routes
Brief as sent
Target: \(\mathrm{GL}_4/\mathrm{Sp}_4\) and the uncovered label \(\mu=(1,1,1,1)\); then \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\). The Summary remarks that \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) would follow if the intersection complex of each uncovered orbit closure were a retract of some \(V_\lambda\ast\mathrm{IC}_0\). Steps: model \(X(F)\) as alternating forms, orbits = elementary-divisor classes; dimension formula and closure relations; choose \(\lambda\) with \(\varphi(\lambda)>\mu\), compute the fibre of the convolution map over \(x_\mu\), its cohomology; apply the decomposition theorem in an explicit finite-type equivariant model and decide whether \(\mathrm{IC}_{\overline{\mathcal O_\mu}}\) is a summand; say which step of Proposition I uses H5 and whether a per-orbit substitute suffices; propagate by central translation; attempt general uncovered labels; compute the function-level shadow as point counts. Shared setting and rules: the problem card of the hub.