§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:19:37Z

Audit 8.2 · Audit of the group summand criterion (A4-2)

proved

The constituent half of the criterion (what the non-membership of Δ_e needs) holds with no decomposition input; the exact description of C_I^Iwa needs the equivariant splitting, which is available.

Audit 8.2.

Holds for Proposition 4.2 (1)–(3), over \(\overline{\mathbb F}_q\) and \(\mathbb C\). Precisely: (i) with no decomposition hypothesis, \(\mathcal C_I^{\rm Iwa}\subseteq\mathcal D_{\mathcal M_I^\flat}\) and \(\mathrm{IC}_z\in\mathcal C_I^{\rm Iwa}\Rightarrow z\in\mathcal M_I^\flat\), where \(\mathcal M_I^\flat\) is computed from the perverse constituents of the generators, whatever their splitting; (ii) given the equivariant splitting of Audit 8.1 step 5, \(\mathcal M_I=\mathcal M_I^\flat\), \(\mathcal C_I^{\rm Iwa}=\operatorname{Thick}\{\mathrm{IC}_z:z\in\mathcal M_I\}\), and \(\mathcal C_I^{\rm Iwa}=\mathcal C_I\iff\mathcal M_I=\widetilde W\).

  1. Thickness of \(\mathcal D_\Sigma\). On each \(\mathrm{Fl}_{\le u}\) the heart has finite length (Audit 8.1 step 3), so objects with all constituents in \(\{\mathrm{IC}_z\}_{z\in\Sigma}\) form a Serre subcategory. For a triangle \(A\to B\to C\to\), the long exact sequence of \({}^pH^i\) puts every constituent of \({}^pH^i(B)\) among those of \({}^pH^i(A)\), \({}^pH^i(C)\); a retract's \({}^pH^i\) is a summand. So \(\mathcal D_\Sigma\) is closed under shifts, cones and retracts (the argument of Stacks Tag 05RD used in summand.tex Step 3). Closed push-forward between \(\mathrm{Fl}_{\le u}\subseteq\mathrm{Fl}_{\le u'}\) is t-exact and preserves \(\mathrm{IC}_z\), so membership is independent of \(u\).
  2. Generators lie in \(\mathcal D_{\mathcal M^\flat_I}\) by definition of \(\mathcal M^\flat_I\) and Audit 8.1 step 4; hence so does \(\mathcal C_I^{\rm Iwa}\). For \(z\notin\mathcal M^\flat_I\), \(\mathrm{IC}_z\) is its own unique constituent, so \(\mathrm{IC}_z\notin\mathcal D_{\mathcal M^\flat_I}\). No splitting, purity or weights enter here; this is the only half used by Audit 8.5.
  3. Equivariant versus plain. The constituents may be computed after forgetting equivariance: for a connected group acting through a finite-dimensional quotient, the equivariant perverse heart is a full subcategory of the plain one, closed under subquotients, and \(\mathrm{IC}_z\mapsto\mathrm{IC}_z\). The authors cite BBD 4.2.5 as “number recalled” (the local copy is a scan); the fact itself is BBD 4.2.5–4.2.6 applied to \(\mathrm{act},\mathrm{pr}:H\times V\to V\) and is standard. Retracts in the equivariant category stay retracts after forgetting, so \(\mathcal M_I\subseteq\mathcal M_I^\flat\).
  4. Span. With the equivariant splitting each constituent is an equivariant summand, so \(\mathcal M^\flat_I\subseteq\mathcal M_I\), and every generator is a finite sum of shifted \(\mathrm{IC}_z\), \(z\in\mathcal M_I\). Every object of \(\mathcal D_\Sigma\) is an iterated extension of shifted constituents, so \(\mathcal D_\Sigma=\operatorname{Thick}\{\mathrm{IC}_z:z\in\Sigma\}\) and \(\mathcal C_I^{\rm Iwa}=\mathcal D_{\mathcal M_I}\).
  5. Relation to (I\(_I\)). This is the Iwahori analogue of satake-two-levels Proposition I for this one \(X\), with Bruhat cells as pieces; its hypotheses (finitely many pieces per level after Audit 8.1(R1), connected stabilizers, finite-length heart, decomposition) are verified in Audit 8.1.

Builds on

  • Proposition 4.2 · Summand criterion for the group: generator constituents decide $\mathcal C_I^{\rm Iwa}$

Built on by

  • Summary 8.13 · Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$
Original record, as the agent wrote it at 19:19 UTC

Audit of the group summand criterion (A4-2)

  1. Thickness of \(\mathcal D_\Sigma\). On each \(\mathrm{Fl}_{\le u}\) the heart has finite length (Audit 8.1 step 3), so objects with all constituents in \(\{\mathrm{IC}_z\}_{z\in\Sigma}\) form a Serre subcategory. For a triangle \(A\to B\to C\to\), the long exact sequence of \({}^pH^i\) puts every constituent of \({}^pH^i(B)\) among those of \({}^pH^i(A)\), \({}^pH^i(C)\); a retract's \({}^pH^i\) is a summand. So \(\mathcal D_\Sigma\) is closed under shifts, cones and retracts (the argument of Stacks Tag 05RD used in summand.tex Step 3). Closed push-forward between \(\mathrm{Fl}_{\le u}\subseteq\mathrm{Fl}_{\le u'}\) is t-exact and preserves \(\mathrm{IC}_z\), so membership is independent of \(u\).
  2. Generators lie in \(\mathcal D_{\mathcal M^\flat_I}\) by definition of \(\mathcal M^\flat_I\) and Audit 8.1 step 4; hence so does \(\mathcal C_I^{\rm Iwa}\). For \(z\notin\mathcal M^\flat_I\), \(\mathrm{IC}_z\) is its own unique constituent, so \(\mathrm{IC}_z\notin\mathcal D_{\mathcal M^\flat_I}\). No splitting, purity or weights enter here; this is the only half used by Audit 8.5.
  3. Equivariant versus plain. The constituents may be computed after forgetting equivariance: for a connected group acting through a finite-dimensional quotient, the equivariant perverse heart is a full subcategory of the plain one, closed under subquotients, and \(\mathrm{IC}_z\mapsto\mathrm{IC}_z\). The authors cite BBD 4.2.5 as “number recalled” (the local copy is a scan); the fact itself is BBD 4.2.5–4.2.6 applied to \(\mathrm{act},\mathrm{pr}:H\times V\to V\) and is standard. Retracts in the equivariant category stay retracts after forgetting, so \(\mathcal M_I\subseteq\mathcal M_I^\flat\).
  4. Span. With the equivariant splitting each constituent is an equivariant summand, so \(\mathcal M^\flat_I\subseteq\mathcal M_I\), and every generator is a finite sum of shifted \(\mathrm{IC}_z\), \(z\in\mathcal M_I\). Every object of \(\mathcal D_\Sigma\) is an iterated extension of shifted constituents, so \(\mathcal D_\Sigma=\operatorname{Thick}\{\mathrm{IC}_z:z\in\Sigma\}\) and \(\mathcal C_I^{\rm Iwa}=\mathcal D_{\mathcal M_I}\).
  5. Relation to (I\(_I\)). This is the Iwahori analogue of satake-two-levels Proposition I for this one \(X\), with Bruhat cells as pieces; its hypotheses (finitely many pieces per level after Audit 8.1(R1), connected stabilizers, finite-length heart, decomposition) are verified in Audit 8.1.

Sources