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

Lemma 3.7 · \(\SL_{2n}/\Sp_{2n}\) is the degree-zero component of \(\GL_{2n}/\Sp_{2n}\); its smallest uncovered label stays open

proved audit: correct

Second audit: correct. Verified: the Pfaffian argument gives the stack isomorphism and the double-coset bijection; Satake objects of degree zero land in component 0 and thick spans split along components; (1+c, 1+c, c-2) never lies in {0,1,2}^3 (g2_defect_coverage.py). 'The smallest uncovered label' is one of two equally small ones: (2,-1,-1), the image of (1,1,-2) under the duality omega -> omega^{-1}, is equally unreached; this changes nothing. Agrees with Audit 7.13 and with the SL corollary checked in Audit 7.23.

Identifies the equivariant geometry of \(L(\SL_{2n}/\Sp_{2n})\) with the component of Pfaffian valuation zero of \(L(\GL_{2n}/\Sp_{2n})\), compatibly with Satake objects and with Hecke translates of the basic function. Both statements for \(\SL_{2n}/\Sp_{2n}\) are therefore the degree-zero parts of those for \(\GL_{2n}/\Sp_{2n}\); the \(\SL_6\) label \((1,1,-2)\) is out of reach of the minuscule computation.

Lemma 3.7.

Notation of Lemma 3.1; \(X^{GL}=\GL_{2n}/\Sp_{2n}\), \(LX^{GL}_s=\{\val\Pf=s\}\).

  1. \(X^{SL}=\SL_{2n}/\Sp_{2n}=\{\omega:\Pf\omega=1\}\subset X^{GL}\) and \(X^{SL}(F)=\SL_{2n}(F)/\Sp_{2n}(F)\). The map \([g,\omega]\mapsto g\omega g^T\) is an isomorphism
$$L^+\GL_{2n}\times^{L^+\SL_{2n}}LX^{SL}\xrightarrow{\ \sim\ }LX^{GL}_{0},$$

hence an isomorphism of quotient stacks \(L^+\SL_{2n}\backslash LX^{SL}\simeq L^+\GL_{2n}\backslash LX^{GL}_0\), matching \(\cO^{SL}_a=\cO^{GL}_a\cap\{\Pf=1\}\) (\(\sum a_i=0\)) and the \(\IC_a\).

  1. For \(\lambda\in\Lambda_T(\SL_{2n})^+\) this identification matches \(V^{SL}_\lambda\ast\IC_0\) with \(V^{GL}_\lambda\ast\IC_0\). Consequently \(\Csat(X^{SL})=\cC_K(X^{SL})\) if and only if \(\mathcal M(X^{GL})\supseteq\{a:\sum a_i=0\}\).
  2. The smallest uncovered \(\SL_6\) label \((1,1,-2)\) is not reached by Theorem 3.6; it is open at the sheaf level (see Conjecture 3.8 for a predicted witness).
  3. At the function level, \(C_K(X^{SL})\cong C_K(X^{GL})_{s=0}\) as \(\cH_K(\SL_{2n})\)-modules, so (SG) for \(\SL_{2n}/\Sp_{2n}\) is the degree-zero part of (SG) for \(\GL_{2n}/\Sp_{2n}\), restricted to \(\cH_K(\SL_{2n})\subset\cH_K(\GL_{2n})\).
  1. Item 1. \(\Pf(g\omega g^T)=\det g\cdot\Pf\omega\). If \(g\in\GL_{2n}(\mathcal O)\) carries \(\omega\) to \(\omega'\) with \(\Pf\omega=\Pf\omega'\), then \(\det g=1\); this is injectivity. For surjectivity, \(\Pf(LX^{GL}_0)=\mathcal O^\times=\det L^+\GL_{2n}\): if \(\Pf\omega=u\), then \(h=\operatorname{diag}(u,1,\dots,1)\) moves a form of Pfaffian 1 to \(\omega\). On \(F\)-points, \(H^1(F,\Sp_{2n})=0\).
  1. Item 2. \(L^+\GL=L^+\SL\cdot\{\operatorname{diag}(u,1,\dots,1)\}\) and diagonal elements commute with \(t^\lambda\), so \(L^+\GL\,t^\lambda L^+\GL=L^+\GL\cdot L^+\SL\,t^\lambda L^+\SL\); moreover \(\Gr^0_{\GL}=\Gr_{\SL}\). The convolution spaces therefore correspond under item 1. The Satake objects of \(\GL_{2n}\) with \(\sum\lambda_i=0\) are exactly those supported on component 0, and thick spans respect the decomposition of \(\cC_K(X^{GL})\) into components; combine with Proposition 3.4.
  1. Item 3. \((1+c,1+c,-2+c)\in\{0,1,2\}^3\) would require \(c\ge2\) and \(c\le1\).
  1. Item 4. The same double-coset bijection \(K_{\SL}t^\lambda K_{\SL}/K_{\SL}\to Kt^\lambda K/K\) holds for \(\sum\lambda_i=0\), and \(T_\lambda\ast\varphi_0\) lives on \(\val\Pf=\sum\lambda_i\).

Builds on

  • 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

Built on by

  • Summary 3.11 · Verdict for the symplectic family: no disagreement between the levels; \(\GL_4/\Sp_4\) settled at both
  • Conjecture 3.8 · Relative-Satake prediction of every multiplicity; general uncovered labels and the exact gap
  • Audit 7.13 · Check of the summand criterion, the maximal-defect lemma, the minuscule fibre lemma and the \(\SL\)/\(\GL\) reduction
  • 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
  • Correction 10.2 · Codimensions of orbits in \(L(\GL_{2m}/\Sp_{2m})\) in every component, without connectedness of \(LX\)
  • Proposition X1.4 · L(GL₂ₘ/Sp₂ₘ): codim = ⟨ρ,μ−ν⟩ in every component, char ≠ 2; σ reaches only m | s
Original record, as the agent wrote it at 02:48 UTC

SL_2n/Sp_2n is the degree-zero component of GL_2n/Sp_2n; its smallest uncovered label stays open

  1. \(X^{SL}=\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}=\{\omega:\mathrm{Pf}\,\omega=1\}\subset X^{GL}\), \(X^{SL}(F)=\mathrm{SL}_{2n}(F)/\mathrm{Sp}_{2n}(F)\) (\(H^1(F,\mathrm{Sp})=0\)). Any \(g\in\mathrm{GL}_{2n}(\mathcal O)\) with \(g\omega g^T=\omega'\), \(\mathrm{Pf}\,\omega=\mathrm{Pf}\,\omega'\), has \(\det g=1\). Hence \([g,\omega]\mapsto g\omega g^T\) is an isomorphism \(L^+\mathrm{GL}_{2n}\times^{L^+\mathrm{SL}_{2n}}LX^{SL}\xrightarrow{\sim}LX^{GL}_{s=0}\) (surjective since \(\mathrm{Pf}(LX^{GL}_{0})=\mathcal O^\times=\det L^+\mathrm{GL}_{2n}\)), i.e. an isomorphism of quotient stacks \(L^+\mathrm{SL}_{2n}\backslash LX^{SL}\simeq L^+\mathrm{GL}_{2n}\backslash LX^{GL}_0\) matching orbits \(\mathcal O^{SL}_a=\mathcal O^{GL}_a\cap\{\mathrm{Pf}=1\}\), \(\sum a_i=0\), and the \(\mathrm{IC}_a\).
  2. For \(\lambda\in\Lambda_T(\mathrm{SL}_{2n})^+\): \(L^+\mathrm{GL}\,t^\lambda L^+\mathrm{GL}=L^+\mathrm{GL}\cdot L^+\mathrm{SL}\,t^\lambda L^+\mathrm{SL}\) (\(L^+\mathrm{GL}=L^+\mathrm{SL}\cdot\{\mathrm{diag}(u,1,\dots)\}\), diagonal elements commute with \(t^\lambda\)), and \(\mathrm{Gr}^0_{\mathrm{GL}}=\mathrm{Gr}_{\mathrm{SL}}\); the convolution spaces correspond under 1. So \(V^{SL}_\lambda\ast\mathrm{IC}_0\leftrightarrow V^{GL}_\lambda\ast\mathrm{IC}_0\) and \(\mathcal C^{\mathrm{Sat}}(X^{SL})=\mathcal C(X^{SL})\iff\mathcal M(X^{GL})\supseteq\{a:\sum a_i=0\}\) (Satake objects of \(\mathrm{GL}_{2n}\) with \(\sum\lambda_i=0\) are exactly those landing in component 0; thick spans respect the component decomposition).
  3. The smallest uncovered \(\mathrm{SL}_6\) label \((1,1,-2)\): its shifts \((1+c,1+c,-2+c)\) never lie in \(\{0,1,2\}^3\), so A3-6 does not reach it. Open at the sheaf level (see A3-8 for the predicted witness).
  4. At the function level the same identification gives \(C_K(X^{SL})\cong C_K(X^{GL})_{s=0}\) as \(\mathcal H_K(\mathrm{SL}_{2n})\)-modules (same double-coset bijection), so (SG) for \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) is the degree-0 part of (SG) for \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\) restricted to \(\mathcal H_K(\mathrm{SL}_{2n})\subset\mathcal H_K(\mathrm{GL}_{2n})\).

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