§1 · Agent 1 · Literature and Table 1 audit · 2026-10-17T03:01:34Z

Lemma 1.5 · Summand criterion: \(\Csat=\mathcal C_K\) if and only if every simple IC sheaf is a summand of a Satake object

proved audit: correct

Second audit: correct.

When Satake objects are semisimple and the perverse heart has finite length, generation is equivalent to every simple IC sheaf occurring as a shifted summand of some \(V_\lambda\ast\IC_0\); a simple that never occurs witnesses \(\Csat\ne\mathcal C_K\). A variant without t-structures detects sign-isotypic parts.

Lemma 1.5.

Work in Chen–Yi's ℓ-adic category over \(\overline{\F}_p\), \(p\ne2\), with \(G^\theta\) connected (or in the Betti category over \(\C\), granted the pages' transfer, open item 1). Assume:

Then $$\Csat=\mathcal C_K\iff\text{every simple }\IC_{\chi,\mu}\text{ is a shifted direct summand of some }V_\lambda\ast\IC_0 .$$ Variant without t-structure: if \(\pi_0\Stab(x_m)=\mu_2\) for an orbit \(\mathcal O_m\), the full subcategory \(\mathcal K_m\) of objects \(M\) such that every cohomology sheaf \(H^i(a_m^{*}M)\) has zero sign-isotypic part is thick.

  1. (\(\Leftarrow\)). A thick span contains shifts and retracts, so it contains every simple \(\IC_{\chi,\mu}\). Every object of \(\mathcal P\) has a finite composition series, so property (T1) of thick spans (closure under extensions in exact triangles) gives \(\mathcal P\subseteq\Csat\). Since the t-structure is bounded, every object of \(\mathcal C_K\) is a finite iterated extension of shifted perverse cohomology objects, so \(\Csat=\mathcal C_K\). This is the remark of Proposition III ("Open"), made two-sided.
  1. (\(\Rightarrow\)). Let \(\Sigma\) be the set of simples occurring as shifted summands of Satake objects, \(\mathcal P_\Sigma\subseteq\mathcal P\) the Serre subcategory of objects all of whose composition factors lie in \(\Sigma\), and
$$\mathcal D_\Sigma=\{M\in\mathcal C_K:{}^pH^i(M)\in\mathcal P_\Sigma\text{ for all }i\}.$$

Hence \(\Csat\subseteq\mathcal D_\Sigma\). A simple \(\IC_{\chi,\mu}\notin\Sigma\) has \({}^pH^0=\IC_{\chi,\mu}\notin\mathcal P_\Sigma\), so it lies in \(\mathcal C_K\) but not in \(\Csat\).

  1. Variant. The cohomology sheaves of \(a_m^{*}M\) are \(L^+G\)-equivariant local systems on \(\mathcal O_m\), i.e. objects of \(\Rep(\mu_2)\otimes\Vect\), so the sign-isotypic part makes sense. \(\mathcal K_m\) is closed under shifts; under retracts, since isotypic parts of a direct sum add; and under cofibres, since \(a_m^{*}\) is exact, the long exact sequence of cohomology sheaves is a sequence of representations of \(\mu_2\), and isotypic projection is exact.

Status: proved given (i) and (ii), which are exactly the inputs Proposition III already imports (its step 6 uses (ii)).

Builds on

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Built on by

  • Theorem 1.6 · Over \(\F_q(\!(t)\!)\), (SG) implies \(\Csat=\mathcal C_K\) when stabilizers are connected
  • Proposition 1.7 · \(\SL_2/\SO_2\): sign local systems never occur in Satake objects (ℓ-adic proof)
  • Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide
  • Theorem (repair of A4-11) X2.4 · Repair of A4-11: \(\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}\) in both \(\mathbb F_q\)-forms, \(q\) odd
Original record, as the agent wrote it at 03:01 UTC

Thick span of semisimple Satake objects: equality iff every simple IC is a summand

Lemma (in Chen–Yi's ℓ-adic category over \(k=\overline{\mathbb F}_p\), \(p\ne2\), \(G^\theta\) connected; in Betti \(\mathcal C\) over \(\mathbb C\) under the pages' open item 1). Inputs: (i) \(\mathcal C\) carries a bounded (perverse) t-structure whose heart \(\mathcal P=\operatorname{Perv}_\delta(L^+G\backslash LX)\) has finite length with simple objects \(\mathrm{IC}_{\chi,\mu}=\mathrm{IC}(\mathcal O_\mu,\mathcal L_\chi)\), \(\mu\in\Lambda_S^+\), \(\chi\in\operatorname{Irr}\pi_0\operatorname{Stab}(x_\mu)\) (Chen–Yi §1.2.3, §8.5.3, Appendix A); (ii) every Satake object is semisimple: \(V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_i\mathrm{IC}_{\chi_i,\nu_i}(\delta/2)[2n_i](n_i)\) (proof of Chen–Yi Theorem 93 = argument of Proposition 70, purity). Then $$\mathcal C^{\mathrm{Sat}}=\mathcal C\iff\text{every simple }\mathrm{IC}_{\chi,\mu}\text{ is a shifted direct summand of some }V_\lambda\ast\mathrm{IC}_0 .$$ Proof. (⟸) Thick spans contain retracts and shifts, so \(\mathcal C^{\mathrm{Sat}}\ni\) every simple; finite length and (T1) give \(\mathcal P\subseteq\mathcal C^{\mathrm{Sat}}\); boundedness gives \(\mathcal C\). (This is the pages' remark in criterion.html "Open", made two-sided.) (⟹) Let \(\Sigma\) be the set of simples occurring as summands of Satake objects, \(\mathcal P_\Sigma\subseteq\mathcal P\) the Serre subcategory of objects all of whose composition factors lie in \(\Sigma\), and \(\mathcal D_\Sigma=\{M\in\mathcal C:\ {}^pH^i(M)\in\mathcal P_\Sigma\ \forall i\}\). \(\mathcal D_\Sigma\) contains \(0\), is closed under shifts, under cofibers (long exact sequence of perverse cohomology; Serre subcategories are closed under extensions), and under retracts (\({}^pH^i\) of a retract is a subobject of \({}^pH^i\) of the sum; Serre subcategories are closed under subobjects); it contains each Satake object by (ii). Hence \(\operatorname{Thick}\{V_\lambda\ast\mathrm{IC}_0\}\subseteq\mathcal D_\Sigma\). A simple \(\mathrm{IC}_{\chi,\mu}\notin\Sigma\) has \({}^pH^0=\) itself \(\notin\mathcal P_\Sigma\), so lies in \(\mathcal C\setminus\mathcal C^{\mathrm{Sat}}\). ∎

Variant without t-structure (used in A1-7): for an orbit \(\mathcal O_m\) with \(\pi_0\operatorname{Stab}=\mu_2\), \(\mathcal K_m:=\{M:\ H^i(a_m^*M)_{\mathrm{sign}}=0\ \forall i\}\) (sign-isotypic parts of the cohomology sheaves, which are \(L^+G\)-equivariant local systems \(=\operatorname{Rep}(\mu_2)\otimes\mathrm{Vect}\)) is thick: closed under shifts, retracts (isotypic parts of a direct sum add) and cofibers (long exact sequence of cohomology sheaves, isotypic projection exact).

Status: proved given (i), (ii), which are exactly the ingredients criterion.html already imports (step 6 uses (ii)).

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