§1 · Agent 1 · Literature and Table 1 audit · 2026-10-17T02:07:02Z

Setup 1.1 · Table 1 of the Summary, re-derived row by row

proved audit: minor defect

Minor defect (second audit). All cells agree with the workspace pages (lift and centralizer columns checked against the family computations; function cells against the classical page). Imprecisions: the SL₄/Sp₄ function cell is called 'unconditional by Proposition 1.3', but Proposition 1.3 covers only odd residue characteristic (unconditional over every F follows only later, from Theorem 3.9 or Proposition 1.4 plus the degree-0 reduction); the node, timestamped 02:07, cites results recorded at 03:01–03:06 (Proposition 1.3, Proposition 1.4, Theorem 1.6, Proposition 1.7) — forward references in the record, not mathematical defects.

For every symmetric space on the pages, records what is proved at the sheaf level (\(\Csat=\mathcal C_K\)) and at the function level ((SG)), and under which hypotheses. This is the baseline that the rest of the search updates.

Setup 1.1.

For each symmetric space on the pages, the status proved there is:

Table 2. \(X\) and sheaves and functions
\(X\)sheavesfunctionsclass
\((H\times H)/H\)holds: \(Z_{G^\theta}(S)=T_H\), lift \((a,0)\)holds for every \(F\) (Proposition D)both hold
\(\Spin_{2n}/\Spin_{2n-1}\), \(n\ge2\)holds: \(Z_{G^\theta}(S)=\Spin_{2n-2}\), lift \(m(\varepsilon_1+\varepsilon_2)\)\(n=2\) holds (group case of \(\SL_2\)); \(n=3\) under (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. (\(=\SL_4/\Sp_4\)); \(n\ge4\) open on the pagesboth hold for \(n\le3\); \(n\ge4\) decided in Proposition 1.3
\(\GL_n/(\GL_p\times\GL_q)\)holds: \(Z_{G^\theta}(S)=\mathbb G_m^p\times\GL_{q-p}\), lift \((a_1,\dots,a_p,0,\dots,0)\)under (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. (Proposition F); \(p=q=1\) unconditional (Proposition J)both hold; functions under (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. except \(\GL_2/T\)
\(\GL_2/\Sp_2\simeq\mathbb G_m\)holds (one pair, every label lifts)holds (Proposition J)both hold
\(\GL_{2m}/\Sp_{2m}\), \(m\ge2\)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 at \((1,1,1,1,0,\dots,0)\); undecidedunder (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. (Proposition F)sheaf open; \(m=2\) decided in Proposition 1.4, Theorem 1.6
\(\SL_4/\Sp_4\)holds (every \((k,k,-k,-k)\) lifts)under (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., through \(\GL_4/\Sp_4\)both hold; functions unconditional by Proposition 1.3
\(\SL_{2n}/\Sp_{2n}\), \(n\ge3\)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 at \((1,1,1,1,0,\dots,0,-2,-2)\); undecidedunder (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., through \(\GL_{2n}/\Sp_{2n}\)sheaf open
\(E_6/F_4\)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 at \(3k\omega_1^\vee\), \(3k\omega_6^\vee\), \(k\) odd; undecidedopen (not in Sakellaridis's table, \(r\) not computed)both open
\(\SL_2/\SO_2=\SL_2/T\)both conditions of Proposition III fail; undecided on the pagesfails for \(q\) odd (Proposition J)sheaf open on the pages; decided in Proposition 1.7
\(\GL_n/\mathrm O_n\), \(\PGL_2/\mathrm{PO}_2\)\(G^\theta\) disconnected; not equal, by supportsfails (several \(G(F)\)-orbits)both fail

The non-symmetric varieties of the pages (\(\mathbb A^n\), \(M_n\), \(\SL_2^3\backslash\Sp_4^2\)) are outside the question. Three entries of the pages' table are imprecise: (a) for \(\SL_{2n}/\Sp_{2n}\) the reduction to \(\GL_{2n}/\Sp_{2n}\) in Proposition F needs no hypothesis in any characteristic, so the only hypothesis is (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. for \(\GL_{2n}/\Sp_{2n}\); (b) the \(\SL_2/T\) function entry is proved for odd \(q\) only, while Proposition 1.2 proves failure for every \(q\); (c) the spheres' function entry for \(n\ge4\) is open on the pages but decidable (Proposition 1.3).

Conventions. Write SH for the sheaf statement \(\Csat=\mathcal C_K\) and FN for (SG). "SH holds" means that H1–H5Hypotheses 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. are verified and Proposition I applies, with the standing caveat of Proposition III: its steps 3 and 6 import Chen–Yi's ℓ-adic theory over \(\overline{\F}_p\), \(p\ne2\), and carrying it to Betti sheaves over \(\C\) is the pages' unproved open item 1. "Under (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." means under Sakellaridis's hypotheses (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. of Theorem E, which are never verified on the pages; (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). forces \(\operatorname{char}F=0\).

  1. Sheaf column. For each row, the dominant-lift column of Proposition III, step 7, was compared with the coverage computation of the corresponding family (the lemmas for the symplectic and \(E_6\) families are proved there and confirmed by finite searches), and the centralizer column with the centralizer computation. They agree in every row.
  2. Function column. Each entry was compared with the classical page: Proposition D for the group, Proposition J in rank one, Theorem E and Proposition F for the \(\GL\) families. For the rows with disconnected \(G^\theta\), \(X(F)\) has several \(G(F)\)-orbits and every Hecke translate of \(\varphi_0\) stays in \(G(F)X(\cO)\); at the sheaf level every Satake object lives on \(LG\cdot x_0\), a proper union of components of \(LX\) (Chen–Yi, §11).
  3. Identifications. \(\Spin_4/\Spin_3\) is the group case of \(\SL_2\), and \(\Spin_6/\Spin_5=\SL_4/\Sp_4\), through the exceptional isomorphisms; so the rows of the table may be compared.
  4. Imprecision (a). The passage from \(\GL_{2n}/\Sp_{2n}\) to \(\SL_{2n}/\Sp_{2n}\) (last paragraph of Proposition F) uses only symplectic bases over \(\cO\), the grading of the Hecke algebra by \(\val\det\), and the fact that forms of Pfaffian \(1\) form one \(\SL_{2n}(F)\)-orbit; all of this holds in every characteristic and needs no hypothesis. Imprecisions (b) and (c) are settled in Proposition 1.2 and Proposition 1.3.

Builds on

  • —

Built on by

  • Proposition 1.2 · \(\SL_2/T\) versus \(\GL_2/T\): one variety, two Hecke algebras
  • Proposition 1.3 · Spheres \(\Spin_{2n}/\Spin_{2n-1}\): (SG) for odd residue characteristic
  • Proposition 1.4 · \(\GL_4/\Sp_4\): (SG) holds, and the uncovered label is reached by a cancellation
  • Objection 1.8 · Comparability: the same pair \((G,X)\) in every row, but three different fields
  • Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide
Original record, as the agent wrote it at 02:07 UTC

Table 1 reconstructed and verified row by row against the proving pages

Row-by-row audit of summary.html Table 1 against criterion.html (step 7 table), the six family pages and classical.html §2–§7. Notation: SH = sheaf statement \(\mathcal C^{\mathrm{Sat}}=\mathcal C\), FN = (SG). "SH holds" everywhere means: H1–H5 verified, Proposition I applied, with the standing caveat of criterion.html (steps 3 and 6 are ℓ-adic, Chen–Yi, carried to Betti \(\mathcal C\) over \(\mathbb C\) by the unproved "open item 1"). "under (A)" = Sakellaridis's (A1)–(A5), never verified on the pages; (A1) forces \(\operatorname{char}F=0\).

Table 3. \(X\) (group \(G\)) and SH, as proved and FN, as proved
\(X\) (group \(G\))SH, as provedFN, as provedclass
\((H\times H)/H\)holds: \(Z=T_H\), lift \((a,0)\) (families/group.html §4–5; Prop II)holds unconditionally, any \(F\) (Prop D)both-hold
\(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(n\ge2\)holds: \(Z=\mathrm{Spin}_{2n-2}\), lift \(m(\varepsilon_1+\varepsilon_2)\) (families/spin.html)\(n=2\) holds (= group \(\mathrm{SL}_2\)); \(n=3\) under (A) (= \(\mathrm{SL}_4/\mathrm{Sp}_4\), degree-0 part of Prop F); \(n\ge4\) open on the pageboth-hold for \(n\le3\) (n=3 under (A)); function-open for \(n\ge4\) on the page — decided in A1-3
\(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\)holds: \(Z=\mathbb G_m^p\times\mathrm{GL}_{q-p}\), lift \((a_1,\dots,a_p,0,\dots)\) (families/gl-pq.html)under (A) (Prop F(2)); \(p=q=1\) unconditional (Prop J(1))both-hold, FN under (A) except \(\mathrm{GL}_2/T\)
\(\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\)holds (one pair, every label lifts)holds (Prop J(3))both-hold
\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge2\)H5 fails at \((1,1,1,1,0,\dots)\) (families/symplectic.html Lemma §5); SH undecidedunder (A) (Prop F(1))sheaf-open — \(m=2\) decided in A1-4/A1-6
\(\mathrm{SL}_4/\mathrm{Sp}_4\)holds (every \((k,k,-k,-k)\) lifts)under (A) via GL (Prop F, last paragraph, char-free reduction)both-hold, FN under (A) — unconditional by A1-3
\(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\)H5 fails at \((1,1,1,1,0,\dots,-2,-2)\); SH undecidedunder (A) via GLsheaf-open
\(E_6/F_4\)H5 fails at \(3k\omega_1^\vee,3k\omega_6^\vee\), \(k\) odd (families/e6-f4.html Lemma §5); SH undecidedopen (not in Sakellaridis's table, \(r\) not computed)both-open
\(\mathrm{SL}_2/\mathrm{SO}_2=\mathrm{SL}_2/T\)both conditions fail (\(Z=\{\pm I\}\), \(\varphi=2\)); SH undecided on the pagesfails, \(q\) odd (Prop J(2))sheaf-open on the pages — decided in A1-7
\(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\)\(G^\theta\) disconnected, outside III; not equal by support (\(LG\cdot x_0\) a proper union of components; Chen–Yi §11)fails (several \(G(F)\)-orbits)both-fail
\(\mathbb A^n\), \(M_n\), \(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\)not symmetric (IV / not on pages)G, H, E(3)outside the question

Verified facts behind the table (all checked against the cited pages): the lift column of criterion.html step 7 agrees with each family page's §5 (lemmas for the symplectic and \(E_6\) families are proved there and confirmed by the pages' finite searches); the centralizer column agrees with each §4. Three places where Table 1 is imprecise: (a) for \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) the degree-0 reduction in Prop F is char-free and hypothesis-free, so the only hypothesis is (A) for \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\); (b) the \(\mathrm{SL}_2/T\) function row is proved for \(q\) odd only, but A1-2 shows failure for every \(q\); (c) the \(\mathrm{Spin}\) row's \(n\ge4\) function cell is open on the page but decidable (A1-3).

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