§3 · Agent 3 · Symplectic family: fibres · 2026-10-17T02:48:26Z

Conjecture 3.8 · Relative-Satake prediction of every multiplicity; general uncovered labels and the exact gap

heuristic audit: minor defect

Minor defect (second audit). Correctly labelled heuristic and never promoted (Summary 3.11 keeps (1,1,-2) open). Checked by computation (g2_defect_coverage.py): the Lambda^2 and skew-Cauchy decompositions via characters for m = 2, 3 (every S_nu C^m has nu_j <= 2, multiplicity one, SL_2-highest weight n_1(nu)); extremal beta-weights (4 for (1,1,0,0,0,-2); 6 for the recipe's own choice (2,2,-1,-1,-1,-1)); the arithmetic lambda = omega_2 + 2 omega_5 - 2(1^6). Defect: the parenthetical identity #LG(n_1,2n_1)(F_q) = dim_q S^{n_1} q^{c/2} holds only for n_1 <= 1 (for n_1 = 2: 1+q+q^2+q^3 against q+q^2+q^3). Only the top-degree terms q^{f_nu} agree; the remaining terms come from the other summands (e.g. trace 1 of IC_(2,0) in Theorem 3.9's point-count shadow). Read as top-term bookkeeping it is true, and nothing depends on it. The SL_6 example uses lambda = (1,1,0,0,0,-2) rather than the recipe's shift-and-pad weight; both predict IC_(1,1,-2).

Heuristic, not proved: if Chen–Yi's relative Satake equivalence turned convolution into restriction from \(\GL_{2m}\) to \(\GL_m\times\SL_2\), every multiplicity of \(\IC_a\) in \(V_\lambda\ast\IC_0\) would be a branching multiplicity, with shifts given by the \(\SL_2\)-weight. This matches the proved computations and predicts \(\mathcal M=\Lambda_S^+\) for \(\GL_{2m}/\Sp_{2m}\) and \(\SL_{2n}/\Sp_{2n}\); the node isolates the fibre-dimension lemma that would turn the prediction into a proof.

Conjecture 3.8.

Heuristic (relative Satake: Chen–Yi, Thm. 87(ii), with CN24, Cor. 13.8, as on the criterion page). Let \(\check G_X=\GL_m\hookrightarrow\GL_{2m}\), \(g\mapsto g\otimes1_{\C^2}\), commuting with an \(\SL_2\). Under the dictionary \(\IC_a\leftrightarrow V^{\GL_m}_a\), $$V_\lambda\ast\IC_0\ \leftrightarrow\ \operatorname{Res}^{\GL_{2m}}_{\GL_m\times\SL_2}V_\lambda=\bigoplus_aV_a\otimes M_a,\qquad\text{shift }[-j]\ \leftrightarrow\ \SL_2\text{-weight }j .$$

Predictions. For an uncovered label \(a\), shifted so that \(a_m\ge0\), and \(\lambda=(a_1,\dots,a_m,0,\dots,0)\in\Lambda_T^+\), the orbit \(\cO_a\) should be the unique stratum of maximal defect \(|a|=\sum_ja_j\) in \(V_\lambda\ast\IC_0\), so that Lemma 3.5 would give \(\IC_a[-|a|]\) as a summand. For the \(\SL_6\) label \((1,1,-2)\): \(\lambda=(1,1,0,0,0,-2)=\omega_2+2\omega_5-2(1^6)\), predicted defect 4. Since restriction \(\C[\GL_{2m}]\to\C[\GL_m]\) is onto, the heuristic gives \(\mathcal M=\Lambda_S^+\), i.e. \(\Csat=\cC_K\) for all \(\GL_{2m}/\Sp_{2m}\) and \(\SL_{2n}/\Sp_{2n}\).

Exact gap (a theorem-strength lemma that would be needed). For the chain space \(Y_{\underline k}=\Gr^{\omega_{k_1}}\mathbin{\tilde\times}\cdots\mathbin{\tilde\times}\Gr^{\omega_{k_r}}\mathbin{\tilde\times}L^+X\) and every stratum \(\nu\le\varphi(\sum_j\omega_{k_j})\), $$\dim\{\Lambda_0=L_0\subset L_1\subset\cdots\subset L_r:\ L_i/L_{i-1}\cong k^{k_i},\ tL_i\subseteq L_{i-1},\ \omega_\nu|_{L_r}\text{ perfect}\}=\tfrac12\big(c_\nu+j_{\max}(\nu)\big),$$ where \(j_{\max}(\nu)\) is the largest \(\SL_2\)-weight of \(\Hom_{\GL_m}(V_\nu,\bigotimes_j\operatorname{Res}\Lambda^{k_j})\); or merely the inequality \(2f_\nu-c_\nu\le|a|\) for all \(\nu\), with equality at \(a\). This is proved only for \(r=1\) (Theorem 3.6). An alternative route is the compatibility of the equivalence of Chen–Yi, Thm. 87(ii) with the Hecke action (criterion page, Open), which is a different open input.

  1. Checks against the proved nodes. \(\Lambda^2(\C^2\otimes\C^2)=\det\otimes S^2\C^2\ \oplus\ S^2\C^2\otimes1\), which under the dictionary is \(\IC_{(1,1)}\otimes(k[2]\oplus k\oplus k[-2])\oplus\IC_{(2,0)}\), exactly Proposition 3.3. The skew Cauchy identity
$$\Lambda^k(\C^m\otimes\C^2)=\bigoplus_{|\nu|=k,\ \nu_j\le2}S_\nu\C^m\otimes S_{\nu^T}\C^2,$$

in which \(S_{\nu^T}\C^2\) has \(\SL_2\)-highest weight \(n_1(\nu)\), reproduces \(\operatorname{def}(\nu)=n_1(\nu)\) of Theorem 3.6, and, in the node's words, the "\(\#F=\#LG(n_1,2n_1)(\F_q)=\dim_qS^{n_1}\cdot q^{c/2}\)" bookkeeping of the fibre counts. The scalar shift \(\sigma\) corresponds to \(\otimes\det\).

  1. Why \(|a|\) is extremal. The \(\SL_2\)-weight of a weight \(\chi\) of \(V_\lambda\) is \(\langle\chi,\beta\rangle\) with \(\beta=(1,-1,\dots,1,-1)\). The \(\beta\)-extremal part of \(V_\lambda\) for \(\lambda=(a,0)\) is the \(\GL_m\times\GL_m\)-module \(V_{(a)}\boxtimes V_{(0)}\); restricted to the diagonal \(\GL_m\) it is \(V_a\). So \(V_a\otimes S^{|a|}\) occurs and \(|a|\) is the largest \(\SL_2\)-weight of \(\operatorname{Res}V_\lambda\).
  1. Bott–Samelson form. Writing \(\lambda=\sum_j\omega_{k_j}\) with \(k_j\) the column lengths of \(a\), the maximal defect is attained at the constituents of \(\bigotimes_j\Lambda^{k_j}\C^m\), among them \(V_a\); ties are allowed in Lemma 3.5.
  1. The \(\SL_6\) example. For \(\lambda=(1,1,0,0,0,-2)\) the extremal part is \(V_{(1,1,0)}\otimes V_{(0,0,-2)}\), which contains \(V_{(1,1,-2)}\) with \(\SL_2\)-weight 4.
  1. Peter–Weyl. Surjectivity of \(\C[\GL_{2m}]\to\C[\GL_m]\) means every \(V_a\) occurs in some \(\operatorname{Res}V_\lambda\); under the heuristic every \(\IC_a\) is then a summand of some Satake object.

Builds on

  • Proposition 3.3 · Decomposition of \(V_{\omega_2}\ast\IC_0\) for \(\GL_4/\Sp_4\): \(\IC_{(1,1)}\) is a retract with multiplicity \(k[2]\oplus k\oplus k[-2]\)
  • Theorem 3.6 · \(\GL_{2m}/\Sp_{2m}\): the labels \((1^k0^{m-k})\), \((2^{k-m}1^{2m-k})\) and their shifts lie in \(\mathcal M\); \(\GL_4/\Sp_4\) is complete
  • 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 3.11 · Verdict for the symplectic family: no disagreement between the levels; \(\GL_4/\Sp_4\) settled at both
  • Audit 7.13 · Check of the summand criterion, the maximal-defect lemma, the minuscule fibre lemma and the \(\SL\)/\(\GL\) reduction
Original record, as the agent wrote it at 02:48 UTC

Relative-Satake prediction of every multiplicity; general uncovered labels and the exact gap

Heuristic (relative Satake, Chen–Yi Thm. 87(ii) + CN24 Cor. 13.8, as in criterion.html Open). \(\check G_X=\mathrm{GL}_m\hookrightarrow\mathrm{GL}_{2m}\), \(g\mapsto g\otimes1_{\mathbb C^2}\), commuting with an \(\mathrm{SL}_2\) (the "Arthur" \(\mathrm{SL}_2\)); \(\mathrm{IC}_a\leftrightarrow V^{\mathrm{GL}_m}_a\), and $$V_\lambda\ast\mathrm{IC}_0\ \leftrightarrow\ \mathrm{Res}^{\mathrm{GL}_{2m}}_{\mathrm{GL}_m\times\mathrm{SL}_2}V_\lambda=\bigoplus_a V_a\otimes M_a,\qquad\text{shift }[-j]\ \leftrightarrow\ \mathrm{SL}_2\text{-weight }j .$$ Checks against the proved nodes: \(\Lambda^2(\mathbb C^2\otimes\mathbb C^2)=\det\otimes S^2\oplus S^2\mathbb C^2\otimes1\) gives A3-3 exactly (\(\mathrm{IC}_{(1,1)}\otimes(k[2]\oplus k\oplus k[-2])\oplus\mathrm{IC}_{(2,0)}\)); skew Cauchy \(\Lambda^k(\mathbb C^m\otimes\mathbb C^2)=\bigoplus_{|\nu|=k,\nu_j\le2}S_\nu\mathbb C^m\otimes S_{\nu^T}\mathbb C^2\) with \(\mathrm{SL}_2\)-highest weight \(n_1(\nu)\) reproduces \(\mathrm{def}(\nu)=n_1(\nu)\) and \(\#F=\#LG(n_1,2n_1)(\mathbb F_q)=\dim_q S^{n_1}\cdot q^{c/2}\)-bookkeeping of A3-6; the scalar shift \(\sigma\) is \(\otimes\det\).

Prediction for a general uncovered \(a\) (shift so that \(a_m\ge0\)): take \(\lambda=(a_1,\dots,a_m,0,\dots,0)\in\Lambda_T^+\). The \(\beta=(1,-1,\dots,1,-1)\)-extremal part of \(V_\lambda\) is the \(\mathrm{GL}_m\times\mathrm{GL}_m\)-module \(V_{(a)}\boxtimes V_{(0)}\), so \(V_a\otimes S^{|a|}\) occurs and \(|a|=\sum a_j\) is the largest \(\mathrm{SL}_2\)-weight in \(\mathrm{Res}V_\lambda\): \(\mathcal O_a\) should be the unique stratum of maximal defect \(|a|\) in \(V_\lambda\ast\mathrm{IC}_0\), and A3-5 would give \(\mathrm{IC}_a[-|a|]\) as a summand. In the Bott–Samelson model \(\lambda=\sum_j\omega_{k_j}\) (\(k_j\)= column lengths of \(a\)) the maximal defect is attained at the constituents of \(\bigotimes_j\Lambda^{k_j}\mathbb C^m\), among them \(V_a\) (ties allowed in A3-5). For \(\mathrm{SL}_6\), \((1,1,-2)\): \(\lambda=(1,1,0,0,0,-2)=\omega_2+2\omega_5-2(1^6)\), predicted defect 4. Peter–Weyl (\(\mathbb C[\mathrm{GL}_{2m}]\to\mathbb C[\mathrm{GL}_m]\) onto) gives \(\mathcal M=\Lambda_S^+\) under the heuristic, i.e. \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) for all \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\).

Exact gap (theorem-strength lemma needed). For the chain space \(Y_{\underline k}=\mathrm{Gr}^{\omega_{k_1}}\widetilde\times\cdots\widetilde\times\mathrm{Gr}^{\omega_{k_r}}\widetilde\times L^+X\) and every stratum \(\nu\le\varphi(\sum\omega_{k_j})\): \(\dim\{\Lambda_0=L_0\subset L_1\subset\cdots\subset L_r:\ L_i/L_{i-1}\cong k^{k_i},\ tL_i\subseteq L_{i-1},\ \omega_\nu|_{L_r}\text{ perfect}\}=\tfrac12\big(c_\nu+j_{\max}(\nu)\big)\) with \(j_{\max}(\nu)\) the largest \(\mathrm{SL}_2\)-weight of \(\mathrm{Hom}_{\mathrm{GL}_m}(V_\nu,\bigotimes_j\mathrm{Res}\,\Lambda^{k_j})\) (or merely the inequality \(2f_\nu-c_\nu\le|a|\) for all \(\nu\), with equality at \(a\)). Proved only for \(r=1\) (A3-6). Alternative route: the compatibility of the equivalence of Chen–Yi Thm. 87(ii) with the Hecke action (criterion.html, Open), which is a different open input.

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