§3 · Agent 3 · Symplectic family: fibres · 2026-10-17T02:07:17Z

Proposition 3.4 · Where Proposition I uses 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., and the summand criterion \(\mathcal M=\Lambda_S^+\)

proved audit: minor defect

Minor defect (second audit). Proposition I' and its proof are correct (re-derived; (2) <= uses H1-H3Hypotheses 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. for standard generation on the boundary, which are assumed; (2) => is a valid Jordan-Hoelder/Serre argument). Two imprecisions that change no claim: (i) the side remark that for pure P the restriction a_mu^* P is a sum of shifted constants cites Chen-Yi Prop. 70 (which states semisimplicity of convolutions, not purity) and Proposition III step 6 (which treats only mu = phi(lambda)); for mu strictly below the top orbit the splitting needs pointwise purity of the IC sheaves (Chen-Yi Thm. 81) and connected stabilizers. The statement is true and is not used. (ii) Item (3) is announced as stability of M under mu -> mu + phi(lambda); what is stated and proved is the weaker 'V_lambda * IC_mu lies in Csat and its IC-retracts are labelled in M'.

Shows that the H5-type hypotheses can never hold at an uncovered label, and that a nonzero restriction to an orbit is not enough to produce its standard object. It then proves a modified Proposition I: under 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. and semisimplicity of Satake objects, \(\Csat=\cC_K\) holds exactly when the intersection complex of every orbit is a retract of some Satake object.

Proposition 3.4.

Work in the setting of Proposition I. Write \(\IC_\nu:=\IC(\overline{\cO_\nu},k)\) and let (DT) be the hypothesis: every \(V_\lambda\ast\IC_0\) is a finite direct sum of shifted \(\IC_\nu\) (connected stabilizers; Chen–Yi, Prop. 70 and the proof of their Thm. 93). Put $$\mathcal M:=\{\mu\in\Lambda_S^+:\ \IC_\mu\ \text{is a retract of some}\ V_\lambda\ast\IC_0\}\ \supseteq\ \varphi(\Lambda_T^+).$$

  1. Use of 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.. In the proof of Proposition I, 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.(a) (support in \(\overline{\cO_\mu}\)) is used in step 2 (top part) to obtain the triangle \(a_{\mu!}a_\mu^*P\to P\to i_{\mu*}i_\mu^*P\) with \(P\in\Csat\); 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.(b) (equivariant splitting into shifted constants) is used in steps 3–4 (split, extract): \(\Delta_\mu[d_1]\) is a retract of \(a_{\mu!}a_\mu^*P\).
  2. The weaker input fails. For \(P=V_\lambda\ast\IC_0\), the conditions \(\supp P\subseteq\overline{\cO_\mu}\) and \(a_\mu^*P\ne0\) together force \(\varphi(\lambda)=\mu\); so at an uncovered \(\mu\) no Satake object satisfies them. A nonzero restriction \(a_\mu^*P\) with constant underlying cohomology does not by itself give \(\Delta_\mu\in\Thick(a_{\mu!}a_\mu^*P)\). For pure \(P\) the restriction \(a_\mu^*P\) is a sum of shifted constants, so purity, not nonvanishing, is what makes 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.(b) automatic.
  3. Proposition I′. 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. and (DT). Then
    • (1) \(\Csat=\Thick_{\cC_K}\{\IC_\nu:\nu\in\mathcal M\}\);
    • (2) \(\Csat=\cC_K\) if and only if \(\mathcal M=\Lambda_S^+\);
    • (3) \(\mathcal M\) is stable under \(\mu\mapsto\mu+\varphi(\lambda)\) in the sense that \(V_\lambda\ast\IC_\mu\in\Csat\) for \(\mu\in\mathcal M\), and every retract of such an object has its label in \(\mathcal M\).
  1. Item 1 is read off the proof of Proposition I.
  1. Item 2. By the fibre proposition of the generation pages (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. pins \(\lambda\) to the fibre of \(\varphi\)), parts (a) and (b): \(\supp P\subseteq\overline{\cO_\mu}\iff\varphi(\lambda)\le\mu\), and \(a_\mu^*P\ne0\Rightarrow\mu\le\varphi(\lambda)\); together they give \(\varphi(\lambda)=\mu\). For the second statement take the equivariant category of a point with stabilizer \(\mathbb G_m\) and \(E=\operatorname{Cone}(u:k\to k[2])\), \(u\) the generator of \(H^2(B\mathbb G_m)\): its cohomology is nonzero and non-equivariantly constant, yet \(k\notin\Thick(E)\). Indeed, for connected stabilizers the orbit category \(\cC_\mu\) is equivalent to perfect modules over \(H^*(B\Stab)\), with \(k_{\cO_\mu}\) the free module, and the modules with finite-dimensional cohomology form a proper thick subcategory. For the last statement the node cites Chen–Yi, Prop. 70 (purity of Satake objects) and Proposition III, step 6 (splitting of the restriction).
  1. (1). Retracts of objects of \(\Csat\) lie in \(\Csat\), which gives \(\supseteq\). Conversely each generator \(V_\lambda\ast\IC_0\) is, by (DT), a finite sum of shifted \(\IC_\nu\) with \(\nu\in\mathcal M\).
  1. (2), \(\Leftarrow\). Replace steps 3–4 of the proof of Proposition I, keeping the induction on \(h(\mu)=\#\{\nu\le\mu\}\) and steps 1–2 with \(P=\IC_\mu\). Since \(\mu\in\mathcal M\), \(\IC_\mu\in\Csat\); it is supported on \(\overline{\cO_\mu}\) and \(a_\mu^*\IC_\mu=k[d_\mu]\). The triangle \(a_{\mu!}a_\mu^*\IC_\mu\to\IC_\mu\to i_{\mu*}i_\mu^*\IC_\mu\), with the boundary term in \(\Thick\{\Delta_\nu:\nu<\mu\}\) by standard generation, gives \(\Delta_\mu[d_\mu]\in\Thick(\IC_\mu,\Delta_\nu\ (\nu<\mu))\subseteq\Csat\). The lemma "Standard generation" finishes as before.
  1. (2), \(\Rightarrow\). If \(\mu\notin\mathcal M\), consider the full subcategory of objects all of whose perverse cohomology sheaves have Jordan–Hölder constituents in \(\{\IC_\nu:\nu\in\mathcal M\}\). It is thick (long exact sequences of perverse cohomology; the heart has finite length on each finite-type model), contains every Satake object by (DT), and does not contain \(\IC_\mu\). So \(\Csat\ne\cC_K\).
  1. (3). \(\Csat\) is stable under \(V_\lambda\ast(-)\): the action is associative and \(V_\lambda\ast V_{\lambda'}\) is a finite sum of shifted \(V_{\lambda''}\) (semisimplicity of geometric Satake). If \(\IC_\mu\) is a retract of \(V_{\lambda'}\ast\IC_0\), then \(V_\lambda\ast\IC_\mu\) is a retract of \(V_\lambda\ast V_{\lambda'}\ast\IC_0\in\Csat\). An \(\IC_\nu\) lying in \(\Csat\) has \(\nu\in\mathcal M\) by the subcategory of step 5.

Coverage is thus replaced by \(\mathcal M=\Lambda_S^+\): the summand condition of the criterion page's Open paragraph, now with its converse.

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 \(k[2]\oplus k\oplus k[-2]\)

Built on by

  • Proposition 3.10 · Function level implies sheaf level: (SG) over \(\F_q(\!(t)\!)\) forces \(\mathcal M=\Lambda_S^+\)
  • Lemma 3.5 · Maximal-defect lemma: at a stratum of maximal defect the top stalk forces an IC summand
  • 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 02:07 UTC

Where Proposition I uses H5; the sharp summand criterion replacing coverage

Use of H5 in pdf/proposition-I.tex. H5(a) (support in \(\overline{\mathcal O_\mu}\)) is used in step 2 (Top part) to get the triangle \(a_{\mu!}a_\mu^*P\to P\to i_{\mu*}i_\mu^*P\) with \(P\in\mathcal C^{\mathrm{Sat}}\); H5(b) (equivariant splitting into shifted constants) is used in step 3 (Split) and step 4 (Extract): \(\Delta_\mu[d_1]\) is a retract of \(a_{\mu!}a_\mu^*P\).

The weaker input fails for uncovered labels. For \(P=V_\lambda\ast\mathrm{IC}_0\), nodes/fibre.html (a),(b): \(\operatorname{supp}P\subseteq\overline{\mathcal O_\mu}\iff\varphi(\lambda)\le\mu\), and \(a_\mu^*P\ne0\Rightarrow\mu\le\varphi(\lambda)\). So "\(\operatorname{supp}P\subseteq\overline{\mathcal O_\mu}\) and \(a_\mu^*P\ne0\)" already forces \(\varphi(\lambda)=\mu\); for uncovered \(\mu\) (e.g. \((1,1)\), a closed orbit) no Satake object satisfies it. Also, by itself \(a_\mu^*P\ne0\) (non-equivariantly constant cohomology) does not give \(\Delta_\mu\in\operatorname{Thick}(a_{\mu!}a_\mu^*P)\): sources/notes-obstruction.txt, \(E=\mathrm{Cone}(u)\) on \(B\mathbb G_m\), \(\mathbf 1\notin\operatorname{Thick}(E)\) (\(\mathcal C_\mu\simeq D^{fg}(H^*(B\,\mathrm{Stab}))\) with \(\underline k_{\mathcal O_\mu}\leftrightarrow\) the free module, and \(D^{fd}\) is a proper thick subcategory). For \(P\) pure (every Satake object is, Chen–Yi Prop. 70) the restriction \(a_\mu^*P\) is a sum of shifted constants (Prop. III step 6), so purity, not merely nonvanishing, is what makes H5(b) automatic.

Modified Proposition I′. Assume H1–H4 and (DT): every \(V_\lambda\ast\mathrm{IC}_0\) is a finite direct sum of shifted \(\mathrm{IC}_\nu:=\mathrm{IC}(\overline{\mathcal O_\nu},\underline k)\) (connected stabilizers; Chen–Yi Prop. 70 / Thm. 93 proof). Put $$\mathcal M:=\{\mu\in\Lambda_S^+:\ \mathrm{IC}_\mu\text{ is a retract of some }V_\lambda\ast\mathrm{IC}_0\}\ \supseteq\ \varphi(\Lambda_T^+).$$ Then (1) \(\mathcal C^{\mathrm{Sat}}=\operatorname{Thick}_{\mathcal C}\{\mathrm{IC}_\nu:\nu\in\mathcal M\}\); (2) \(\mathcal C^{\mathrm{Sat}}=\mathcal C\iff\mathcal M=\Lambda_S^+\); (3) \(\mathcal M\) is stable under \(\mu\mapsto\mu+\varphi(\lambda)\) in the sense that \(V_\lambda\ast\mathrm{IC}_\mu\in\mathcal C^{\mathrm{Sat}}\) for \(\mu\in\mathcal M\) (\(\mathcal C^{\mathrm{Sat}}\) is \(\mathcal H\)-stable: \(V_\lambda\ast V_{\lambda'}\) is a sum of \(V_{\lambda''}\)'s by Satake semisimplicity), and every retract of such an object has its label in \(\mathcal M\).

Proof. (1) \(\supseteq\): retracts. \(\subseteq\): each generator is a sum of \(\mathrm{IC}_\nu\), \(\nu\in\mathcal M\). (2) \(\Leftarrow\): replace steps 3–4 of Prop. I by: \(\mathrm{IC}_\mu\in\mathcal C^{\mathrm{Sat}}\); the triangle \(a_{\mu!}a_\mu^*\mathrm{IC}_\mu\to\mathrm{IC}_\mu\to i_{\mu*}i_\mu^*\mathrm{IC}_\mu\) with \(a_\mu^*\mathrm{IC}_\mu=\underline k[d_\mu]\) gives \(\Delta_\mu[d_\mu]\in\operatorname{Thick}(\mathrm{IC}_\mu,\ \Delta_\nu\ (\nu<\mu))\), and the induction on \(h(\mu)\) and Lemma "Standard generation" finish as before (steps 1–2 unchanged, with \(P=\mathrm{IC}_\mu\)). \(\Rightarrow\): if \(\mu\notin\mathcal M\), the full subcategory of complexes all of whose perverse cohomology sheaves have Jordan–Hölder constituents in \(\{\mathrm{IC}_\nu:\nu\in\mathcal M\}\) is thick (long exact sequences, artinian heart on each finite-type model), contains every Satake object by (DT), and misses \(\mathrm{IC}_\mu\). (3) as stated. \(\square\)

Coverage is thus replaced by \(\mathcal M=\Lambda_S^+\), exactly the "summand condition" of criterion.html, Open, now with the converse.

Sources