§7 · Agent 7 · Adversarial audit · 2026-10-17T03:47:54Z

Audit 7.23 · Audit of the \(\GL_{2m}/\Sp_{2m}\) theorem: the induction holds (directly, not through Proposition B as stated); the \(\SL\) corollary and the sheaf consequence hold

proved audit: minor defect

Minor defect (second audit). The function-level claims agree with my re-derivation of Theorem 5.7. The hypothesis list for Theorem 4.6 is incomplete: that theorem assumes D0–D4The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C., not D0–D3The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C.; the missing D4D4 · Unitriangularity of the Satake traces tr V_λ in the basis T_λ = 1_{Kt^λK} (Zhu (5.6.2), Lemma 5.6.3; Lusztig, Kato). is the Lusztig–Kato unitriangularity of the traces of the Satake sheaves, a published theorem, so the conclusion is unchanged. 'Theorem' for \(\mathcal M=\Lambda_S^+\) is meant modulo the same inputs and over \(\overline{\F}_q\).

An independent check of Theorem 5.7: translates stay in the finite slice, the downward induction is valid as a direct induction, and the \(\SL\) reduction holds. Feeding \((\mathrm{SG})\) over \(\F_q(\!(t)\!)\) into Theorem 4.6 then gives \(\Csat=\cC_K\) over \(\overline{\F}_q\) for \(\GL_{2m}/\Sp_{2m}\) and \(\SL_{2n}/\Sp_{2n}\) under the dictionary inputs.

Audit 7.23.

  1. \((\mathrm{SG})\) holds for \(\GL_{2m}/\Sp_{2m}\), all \(m\), over every non-archimedean \(F\); the appeal to Proposition B with the reversed order is to be read as a direct induction in finite slices (cosmetic).
  2. The \(\SL\) corollary holds over every \(F\).
  3. Under the dictionary inputs D0–D3The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C. and \(q\) odd, \(\Csat=\cC_K\) over \(\overline{\F}_q\) for \(\GL_{2m}/\Sp_{2m}\) (all \(m\)) and \(\SL_{2n}/\Sp_{2n}\) (all \(n\)); over \(\C\) under D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p)..
  1. The support stays in the slice. \(T_{\lambda^*(b-2c)}\ast\varphi_0=\sum N(\lambda^*(b-2c),b'')\one_{O_{b''}}\) with \(b''\ge0\) (integrality on \(\Lambda_0\)), \(|b''|=|b|-2cm\) (Pfaffian) and \(b''\ge_{\mathrm{dom}}b-2c\) (Lemma 3 of Lemma 5.6). The central element \(T_{(c,\dots,c)}\) acts by pulling back along \(\omega\mapsto\varpi^{-2c}\omega\), sending \(\one_{O_{b''}}\) to \(\one_{O_{b''+2c}}\). So \(b'=b''+2c\) satisfies \(b'_m\ge2c\), \(|b'|=d\) and \(b'\ge_{\mathrm{dom}}b\) (dominance is invariant under translation), and the coefficient at \(b'=b\) is \(N(\lambda^*(b-2c),b-2c)\ge1\) by Lemma 2.
  1. The induction. Proposition B as stated needs one global order with finite lower sets, and reverse dominance on all labels of a fixed degree has infinite lower sets (\(b'_1\) is unbounded as \(b'_m\to-\infty\)). The argument avoids this by working in the finite poset \(S_{d,c}\): a direct descending induction on \(\ge_{\mathrm{dom}}\) inside \(S_{d,c}\), maximal elements first, gives \(\one_{O_b}\in\cH_K\varphi_0\) for all \(b\in S_{d,c}\), and every label lies in some slice. The orbit indicators span \(C_K\) (compact open orbits, symplectic Smith form over any discrete valuation ring). Hence \((\mathrm{SG})\) for \(\GL_{2m}/\Sp_{2m}\), all \(m\), every \(F\).
  1. The \(\SL\) corollary. \(\{\Pf=1\}\) is a single \(\SL_{2n}(F)\)-orbit; \(O_b\cap\{\Pf=1\}\) is a single \(K_{\SL}\)-orbit; \(K_{\SL}t^\lambda K_{\SL}/K_{\SL}\to Kt^\lambda K/K\) is bijective for \(|\lambda|=0\) (\(K=K_{\SL}\cdot\{\operatorname{diag}(u,1,\dots,1)\}\) and \(K\cap\SL_{2n}(F)=K_{\SL}\)); and \(T_\lambda\ast\varphi_0\) lives on \(\val\Pf=|\lambda|\). So the degree-zero part of \((\mathrm{SG})\) for \(\GL\) is \((\mathrm{SG})\) for \(\SL\), as in item 4 of Lemma 3.7 and the last part of Proposition F.
  1. The sheaf consequence. Theorem 4.6 needs: \(G\) and \(\theta\) over \(\F_q\), \(T\) split, \(q\) odd, \(G^\theta=\Sp_{2m}\) and \(Z_{G^\theta}(S)=\SL_2^m\) connected, the inputs D0–D3The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C., and \((\mathrm{SG})\) over \(\F_q(\!(t)\!)\), which Theorem 5.7 supplies for every \(q\). Therefore \(\Csat=\cC_K\) over \(\overline{\F}_q\) for \(\GL_{2m}/\Sp_{2m}\), all \(m\), and — through the component-zero identification of Lemma 3.7, or by feeding the \(\SL\) corollary into Theorem 4.6 directly — for \(\SL_{2n}/\Sp_{2n}\), all \(n\), under D0–D3The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C.; over \(\C\) under D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p). (standard for large \(p\), available since \((\mathrm{SG})\) holds for every \(q\)). This closes the open label \((1,1,-2)\) of Lemma 3.7 and turns the claim \(\mathcal M=\Lambda_S^+\) of Conjecture 3.8 from a heuristic into a theorem for these families; its specific multiplicity predictions remain heuristic.

Builds on

  • Theorem 5.7 · \((\mathrm{SG})\) for \(\GL_{2m}/\Sp_{2m}\) (all \(m\)) and \(\SL_{2n}/\Sp_{2n}\) (all \(n\)), every non-archimedean local field
  • Theorem 4.6 · Theorem: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) over \(\overline{\F}_q\)
  • Proposition 3.10 · Function level implies sheaf level: (SG) over \(\F_q(\!(t)\!)\) forces \(\mathcal M=\Lambda_S^+\)
  • Theorem 1.6 · Over \(\F_q(\!(t)\!)\), (SG) implies \(\Csat=\mathcal C_K\) when stabilizers are connected
  • Lemma 3.7 · \(\SL_{2n}/\Sp_{2n}\) is the degree-zero component of \(\GL_{2n}/\Sp_{2n}\); its smallest uncovered label stays open

Built on by

  • Summary 7.25 · Updated audit verdict: every row of Table 1 except \(E_6/F_4\) is decided at both levels
  • Re-derivation X1.5 · (SG) for GL₂ₘ/Sp₂ₘ re-derived from Lemmas 2–3 alone; A4-13's open cells now decided
Original record, as the agent wrote it at 03:47 UTC

A5-7 induction holds (direct, not via Proposition B as stated); SL corollary valid; sheaf consequence valid

Support stays in \(S_{d,c}\cap\{b'\ge_{\mathrm{dom}}b\}\) (holds). \(T_{\lambda^*(b-2c)}\ast\varphi_0=\sum N(\lambda^*(b-2c),b'')\mathbf 1_{O_{b''}}\) with \(b''\ge0\) (integrality on \(\Lambda_0\)), \(|b''|=|b|-2cm\) (Pfaffian) and \(b''\ge_{\mathrm{dom}}b-2c\) (Lemma 3). The central element \(T_{(c,\dots,c)}\) acts by \(\omega\mapsto\varpi^{-2c}\omega\), sending \(\mathbf 1_{O_{b''}}\) to \(\mathbf 1_{O_{b''+2c}}\). So \(b'=b''+2c\) has \(b'_m\ge2c\), \(|b'|=d\), \(b'\ge_{\mathrm{dom}}b\) (dominance is translation invariant), and the coefficient at \(b'=b\) is \(N(\lambda^*(b-2c),b-2c)\ge1\) (Lemma 2, \(b-2c\ge0\)). ✓

Induction (holds, with a cosmetic correction). Proposition B as stated needs one global order with finite lower sets; reverse dominance on all labels with fixed \(|b|\) has infinite lower sets (\(b'_1\) unbounded when \(b'_m\to-\infty\)). A5 avoids this by working inside the finite poset \(S_{d,c}\): a direct descending induction on \(\ge_{\mathrm{dom}}\) within \(S_{d,c}\) (maximal elements first) gives \(\mathbf 1_{O_b}\in\mathcal H_K\varphi_0\) for all \(b\in S_{d,c}\), and every label lies in some \(S_{|b|,\lfloor b_m/2\rfloor}\). The orbit indicators span \(C_K\) (compact open \(K\)-orbits, symplectic Smith form over any DVR). Hence (SG) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), all \(m\), every non-archimedean \(F\). The appeal to 'Proposition B with the reversed order' should be read as this direct induction; severity cosmetic.

SL corollary (holds, every \(F\)). \(X_{\mathrm{SL}}(F)=\{\mathrm{Pf}=1\}\) one \(\mathrm{SL}_{2n}(F)\)-orbit (symplectic basis, \(\mathrm{Pf}(gJg^{\mathsf T})=\det g\)); \(O_b\cap\{\mathrm{Pf}=1\}\) one \(K_{\mathrm{SL}}\)-orbit; \(K_{\mathrm{SL}}t^\lambda K_{\mathrm{SL}}/K_{\mathrm{SL}}\to Kt^\lambda K/K\) bijective for \(|\lambda|=0\) (\(K=K_{\mathrm{SL}}\cdot\{\mathrm{diag}(u,1,\dots)\}\), \(K\cap\mathrm{SL}=K_{\mathrm{SL}}\)); \(T_\lambda\ast\varphi_0\) lives on \(\mathrm{val}\,\mathrm{Pf}=|\lambda|\); so degree-0 part of (SG) for GL is (SG) for SL. Same as A3-7.4 / classical.html Prop. F last part. ✓

Sheaf consequence (valid). A4-6 (= A3-10 = A1-6, A7-14) needs: \(G,\theta\) over \(\mathbb F_q\), \(T\) split, \(q\) odd, \(G^\theta=\mathrm{Sp}_{2m}\) and \(Z_{G^\theta}(S)=\mathrm{SL}_2^m\) connected, D0–D3, and (SG) over \(\mathbb F_q(\!(t)\!)\) — supplied by A5-7 for every \(q\). Therefore \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) over \(\overline{\mathbb F}_q\) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) (all \(m\)) and, via A3-7's component-0 identification (or the SL corollary fed into A4-6 directly), for \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (all \(n\)), under D0–D3; over \(\mathbb C\) under D5 (standard for large \(p\), available since (SG) holds for all \(q\)). This closes A3-7's open label \((1,1,-2)\) and turns A3-8's '\(\mathcal M=\Lambda_S^+\)' from heuristic into a theorem for these families (the specific multiplicity predictions of A3-8 remain heuristic).

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