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

Proposition 3.10 · Function level implies sheaf level: (SG) over \(\F_q(\!(t)\!)\) forces \(\mathcal M=\Lambda_S^+\)

proved audit: minor defect

Minor defect (second audit). The argument is correct; Audit 7.14 shows it is the same proof as Theorem 4.6 and Theorem 1.6, and the hypothesis 'one K-orbit per label' is not used (only nonemptiness of O_nu(F_q)). Missing hypotheses that hold in the case at hand: q odd (Chen-Yi's orbit geometry, placid formalism and (DT) over F_q-bar are proved in characteristic not 2) and, in the general version, G and theta split over F_q. Both hold for GL_{2m}/Sp_{2m} with q odd, and downstream q can always be chosen odd because (SG) is proved for every q. Wording: 'semisimple perverse sheaf' should be 'geometrically semisimple' (Frobenius need not act semisimply on multiplicity spaces; the trace argument uses Jordan-Hoelder constituents, which is what the node says next), and purity of Satake objects comes from the proof of Chen-Yi Thm. 93 rather than the statement of Prop. 70. Passing to C from a single small p needs p outside the finite exceptional set of BBD 6.1, which the 'transfer item' caveat covers.

Through Frobenius traces, every Hecke translate of the basic function is a combination of trace functions of intersection complexes that occur as summands of Satake objects. So if spherical generation holds over \(\F_q(\!(t)\!)\), every orbit's intersection complex occurs, and the summand criterion gives \(\Csat=\cC_K\) over \(\overline{\F}_q\). The converse is not formal.

Proposition 3.10.

Let \(X=\GL_{2m}/\Sp_{2m}\); more generally assume H1–H4Hypotheses of the pages' Proposition I (H1–H5 imply categorical generation): H1 finite orbit filtration; H2 equivariant recollement; H3 constant coefficients on orbits; H4 the Hecke action preserves the category; H5 every orbit carries a Satake object with nonzero top splitting., connected stabilizers, (DT) of Proposition 3.4, and \(K\backslash X(F)\leftrightarrow\Lambda_S^+\) with \(\cO_\nu(\F_q)\) a single nonempty \(K\)-orbit for every \(\nu\). Work ℓ-adically over \(\F_q\), with \(F=\F_q(\!(t)\!)\). If \(\cH_K\varphi_0=C_K\) for one \(q\), then \(\mathcal M=\Lambda_S^+\); hence \(\Csat=\cC_K\) over \(\overline{\F}_q\), and over \(\C\) modulo the workspace's transfer item.

Remarks. (a) The converse is not formal. \(\mathcal M=\Lambda_S^+\) says that every column of the graded multiplicity matrix \((M^\lambda_\nu(q^{1/2}))_{\lambda,\nu}\) is nonzero, while (SG) says that its rows span all finitely supported vectors; in the relative-Satake model this is the surjectivity of \(r\) at the given \(q\) (Theorem E(2)), which can fail (Sakellaridis's \(\SL_2^3\backslash\Sp_4^2\)) but holds for the symplectic family for every \(q\) (Proposition F). (b) Sakellaridis's (A1)(A1) Field and group: F is a local field of characteristic 0 (p-adic), and G is split over the integers, with connected fibres and simply connected derived group. Its characteristic-p counterpart is (A1ₚ). is \(p\)-adic; Proposition F's computation of \(r\) is field-independent, but Theorem E over \(\F_q(\!(t)\!)\) is not asserted on the pages, so the hypothesis is (SG) over \(\F_q(\!(t)\!)\). For \(\GL_4/\Sp_4\), Theorem 3.9 supplies it unconditionally, which gives a second proof of Theorem 3.6(a).

  1. Dictionary. By the Grothendieck trace formula and proper base change, the trace function of \(V_\lambda\ast\IC_0\) is \(t_{V_\lambda}\ast_K\varphi_0\), and \(\{t_{V_\lambda}\}_\lambda\) is a basis of \(\cH_K\), unitriangular with respect to \(\{T_\lambda\}\) (Lusztig, Kato). Hence \(\cH_K\varphi_0=\operatorname{span}_\lambda t_{V_\lambda\ast\IC_0}\).
  1. Traces of Satake objects. \(K_0:=V_\lambda\ast\IC_0\) is pure (the node cites Chen–Yi, Prop. 70), and \(t_{K_0}=\sum_i(-1)^it_{{}^pH^i(K_0)}\). Each \({}^pH^i(K_0)\) is a semisimple perverse sheaf whose simple constituents over \(\F_q\) are Weil twists \(\IC_\nu\otimes\alpha\) with \(\nu\in\mathcal M\), since its geometric constituents are the \(\IC_\nu\) (connected stabilizers). Hence \(t_{K_0}\in S:=\operatorname{span}_{\C}\{t_{\IC_\nu}:\nu\in\mathcal M\}\).
  1. Extraction. \(t_{\IC_\nu}\) is supported on \(\overline{\cO_\nu}\) and nonzero on \(\cO_\nu(\F_q)\), where \(\IC_\nu\) is a shifted constant sheaf. Suppose \(\mu\notin\mathcal M\) and \(\one_{\cO_\mu}=\sum_{\nu\in\mathcal M}c_\nu t_{\IC_\nu}\) (a finite sum). Choose \(\nu_0\) maximal with \(c_{\nu_0}\ne0\); at a point of \(\cO_{\nu_0}(\F_q)\) only the term \(c_{\nu_0}t_{\IC_{\nu_0}}\ne0\) contributes, so \(\cO_{\nu_0}=\cO_\mu\), i.e. \(\mu=\nu_0\in\mathcal M\), a contradiction. Thus (SG) implies \(\mathcal M=\Lambda_S^+\), and Proposition 3.4(2) concludes. \(\square\)

Builds on

Built on by

  • Summary 3.11 · Verdict for the symplectic family: no disagreement between the levels; \(\GL_4/\Sp_4\) settled at both
  • Audit 7.14 · Audit: the three proofs that (SG) implies \(\Csat=\cC_K\) are one proof in three notations; inputs listed and compared
  • 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
Original record, as the agent wrote it at 02:48 UTC

Function level implies sheaf level: (SG) over F_q((t)) forces M = Lambda_S^+

Proposition. Let \(X=\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) (more generally: H1–H4, connected stabilizers, (DT), and \(K\backslash X(F)\leftrightarrow\Lambda_S^+\) with \(\mathcal O_\nu(\mathbb F_q)\) a single nonempty \(K\)-orbit for every \(\nu\)). Work \(\ell\)-adically over \(\mathbb F_q\), \(F=\mathbb F_q(\!(t)\!)\). If \(\mathcal H_K\varphi_0=C_K\) for one \(q\), then \(\mathcal M=\Lambda_S^+\), hence \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) (over \(\overline{\mathbb F}_q\); over \(\mathbb C\) modulo the page's transfer item).

Proof. 1. Sheaf–function dictionary (Grothendieck trace formula, proper base change): \(t_{V_\lambda\ast\mathrm{IC}_0}=t_{V_\lambda}\ast_K\varphi_0\), and \(\{t_{V_\lambda}\}_\lambda\) is a basis of \(\mathcal H_K\) (unitriangular in \(\{T_\lambda\}\), Lusztig/Kato). So \(\mathcal H_K\varphi_0=\operatorname{span}_\lambda t_{V_\lambda\ast\mathrm{IC}_0}\).

  1. \(K_0:=V_\lambda\ast\mathrm{IC}_0\) is pure (Chen–Yi Prop. 70); \(t_{K_0}=\sum_i(-1)^it_{{}^pH^i(K_0)}\), and each \({}^pH^i(K_0)\) is a semisimple perverse sheaf whose simple constituents over \(\mathbb F_q\) are Weil twists \(\mathrm{IC}_\nu\otimes\alpha\) with \(\nu\in\mathcal M\) (geometric constituents are \(\mathrm{IC}_\nu\), connected stabilizers). Hence \(t_{K_0}\in S:=\operatorname{span}_{\mathbb C}\{t_{\mathrm{IC}_\nu}:\nu\in\mathcal M\}\).
  2. \(t_{\mathrm{IC}_\nu}\) is supported on \(\overline{\mathcal O_\nu}\) and nonzero on \(\mathcal O_\nu(\mathbb F_q)\) (\(\mathrm{IC}_\nu|_{\mathcal O_\nu}=\underline k[d]\)). If \(\mu\notin\mathcal M\) and \(\mathbf 1_{\mathcal O_\mu}=\sum_{\nu\in\mathcal M}c_\nu t_{\mathrm{IC}_\nu}\) (finite sum), a maximal \(\nu_0\) with \(c_{\nu_0}\ne0\) gives coefficient \(c_{\nu_0}t_{\mathrm{IC}_{\nu_0}}\ne0\) on \(\mathcal O_{\nu_0}\), so \(\nu_0=\mu\in\mathcal M\): contradiction. Thus (SG) \(\Rightarrow\mathcal M=\Lambda_S^+\), and A3-4(2) concludes. \(\square\)

Remarks. (a) The converse is not formal: \(\mathcal M=\Lambda_S^+\) says every column of the graded multiplicity matrix \((M^\lambda_\nu(q^{1/2}))_{\lambda,\nu}\) is nonzero, while (SG) says its rows span all finitely supported vectors; in the relative-Satake model this is the surjectivity of \(r\) at the given \(q\) (Theorem E(2)), which can fail (Sakellaridis's \(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\)) but holds for the symplectic family for every \(q\) (Prop. F). (b) Sakellaridis's (A1) is \(p\)-adic; Prop. F's computation of \(r\) is field-independent, but Theorem E over \(\mathbb F_q(\!(t)\!)\) is not asserted on the pages, so the hypothesis of the Proposition is "(SG) over \(\mathbb F_q(\!(t)\!)\)". For \(\mathrm{GL}_4/\mathrm{Sp}_4\) A3-9 supplies it unconditionally, giving a second proof of A3-6(a).

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