§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:41:20Z

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

proved

Specialises the criterion (I_I) to the group with every hypothesis taken from the set-up: a simple object IC_z lies in the Iwahori-generated part iff it occurs as a shifted summand (equivalently, perverse constituent) of some IC_x ∗ IC_{w_0} ∗ IC_y. The constituent version needs no equivariant splitting.

Proposition 4.2.

Assume Setup 4.1 (a)–(e). Put $$\mathcal M_I^\flat=\{z\in\widetilde W:\ E_{z,n}(x,y)\ne0\ \text{for some }x,y\in\widetilde W,\ n\in\mathbb Z\},\qquad \mathcal M_I=\{z:\ \mathrm{IC}_z[n]\ \text{is a direct summand of some }M_{x,y}\ \text{in }\mathcal C_I\},$$ and for \(\Sigma\subseteq\widetilde W\), \(\mathcal D_\Sigma=\{\mathcal M\in\mathcal C_I:\ \text{every perverse constituent of every }{}^pH^i(\mathcal M)\ \text{is some }\mathrm{IC}_z,\ z\in\Sigma\}\).

  1. \(\mathcal M_I\subseteq\mathcal M_I^\flat\); equality under the equivariant splitting of Setup 4.1(e).
  2. \(\mathcal D_\Sigma\) is thick, and \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^\flat}\). Hence \(\mathrm{IC}_z\in\mathcal C_I^{\mathrm{Iwa}}\Rightarrow z\in\mathcal M_I^\flat\).
  3. With the equivariant splitting: \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_z:z\in\mathcal M_I\}=\mathcal D_{\mathcal M_I}\); \(\mathrm{IC}_z\in\mathcal C_I^{\mathrm{Iwa}}\iff z\in\mathcal M_I\); and \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\widetilde W\).

Valid over \(\overline{\mathbb F}_q\) and over \(\mathbb C\) (Betti), with the decomposition inputs of Setup 4.1(e) on each side.

  1. Summands are constituents. If \(\mathrm{IC}_z[n]\) is a retract of \(M_{x,y}\) in \(\mathcal C_I\), forgetting equivariance (an additive, t-exact functor) keeps it a retract, so \(\mathrm{IC}_z\) is a summand of \(P_{-n}\) and \(E_{z,-n}\ne0\). Conversely, an equivariant splitting \(M_{x,y}\simeq\bigoplus\mathrm{IC}_z[-n]^{\oplus\dim E_{z,n}}\) exhibits each constituent as a summand.
  2. Thickness. \(\mathcal P_\Sigma\subseteq\mathcal P\) (objects of the heart with constituents in \(\{\mathrm{IC}_z\}_{z\in\Sigma}\)) is a Serre subcategory of a finite-length heart (Setup 4.1(c)); with \(h=q\circ{}^pH^0\), \(q:\mathcal P\to\mathcal P/\mathcal P_\Sigma\), \(\mathcal D_\Sigma=\{\mathcal M:h(\mathcal M[i])=0\ \forall i\}\) is a saturated triangulated subcategory: closed under shifts, cofibres and retracts (Stacks, Lemma 13.6.3, Tag 05RD, as in summand.tex Step 3). The forgetful functor from equivariant to ordinary perverse sheaves is exact, fully faithful and sends \(\mathrm{IC}_z\) to \(\mathrm{IC}_z\) (connected group acting through a finite-dimensional quotient: smooth pullback with connected fibres is fully faithful on perverse sheaves with image closed under subquotients, bbd-perverse.pdf Prop. 4.2.5 — number recalled, the local copy is a scan without text layer), so constituents may be computed non-equivariantly.
  3. Generators lie in \(\mathcal D_{\mathcal M_I^\flat}\). By Setup 4.1(e), \({}^pH^n(M_{x,y})=P_n\simeq\bigoplus_z\mathrm{IC}_z\otimes E_{z,n}\) has constituents in \(\mathcal M_I^\flat\) by definition. By Setup 4.1(d) and thickness, \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^\flat}\). If \(z\notin\mathcal M_I^\flat\), \({}^pH^0(\mathrm{IC}_z)=\mathrm{IC}_z\) is simple and not isomorphic to any \(\mathrm{IC}_{z'}\), \(z'\ne z\) (Setup 4.1(c)), so \(\mathrm{IC}_z\notin\mathcal D_{\mathcal M_I^\flat}\supseteq\mathcal C_I^{\mathrm{Iwa}}\). This is summand.tex Step 4 with orbits replaced by Bruhat cells.
  4. Span. Each \(\mathrm{IC}_z\), \(z\in\mathcal M_I\), is a retract of a generator, so \(\operatorname{Thick}\{\mathrm{IC}_z:z\in\mathcal M_I\}\subseteq\mathcal C_I^{\mathrm{Iwa}}\); conversely an equivariantly split generator is an iterated cofibre of split triangles of its summands. And \(\mathcal D_\Sigma=\operatorname{Thick}\{\mathrm{IC}_z:z\in\Sigma\}\): an object of \(\mathcal D_\Sigma\) is an iterated extension of shifted \({}^pH^i\), each of finite length with constituents in \(\Sigma\).
  5. Everything. If \(\mathcal M_I=\widetilde W\), step 4 and Setup 4.1(c) give \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_z\}=\mathcal C_I\); if \(\mathcal M_I\ne\widetilde W\), step 3 exhibits an \(\mathrm{IC}_z\notin\mathcal C_I^{\mathrm{Iwa}}\).

Remark. Only steps 2–3 (the constituent version) are needed for \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) in Theorem 4.5; the equivariant splitting is needed only for the exact description of \(\mathcal C_I^{\mathrm{Iwa}}\) in Proposition 4.7.

Builds on

  • Setup 4.1 · Group case: cell pieces, IC heart, two-sided generators $M_{x,y}$

Built on by

  • Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Proposition 4.7 · $\mathcal M_I$ for the group: top cells in, $\Omega$ out, exact in rank one
  • Audit 8.2 · Audit of the group summand criterion (A4-2)
Original record, as the agent wrote it at 18:41 UTC

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

  1. Summands are constituents. If \(\mathrm{IC}_z[n]\) is a retract of \(M_{x,y}\) in \(\mathcal C_I\), forgetting equivariance (an additive, t-exact functor) keeps it a retract, so \(\mathrm{IC}_z\) is a summand of \(P_{-n}\) and \(E_{z,-n}\ne0\). Conversely, an equivariant splitting \(M_{x,y}\simeq\bigoplus\mathrm{IC}_z[-n]^{\oplus\dim E_{z,n}}\) exhibits each constituent as a summand.
  2. Thickness. \(\mathcal P_\Sigma\subseteq\mathcal P\) (objects of the heart with constituents in \(\{\mathrm{IC}_z\}_{z\in\Sigma}\)) is a Serre subcategory of a finite-length heart (Setup 4.1(c)); with \(h=q\circ{}^pH^0\), \(q:\mathcal P\to\mathcal P/\mathcal P_\Sigma\), \(\mathcal D_\Sigma=\{\mathcal M:h(\mathcal M[i])=0\ \forall i\}\) is a saturated triangulated subcategory: closed under shifts, cofibres and retracts (Stacks, Lemma 13.6.3, Tag 05RD, as in summand.tex Step 3). The forgetful functor from equivariant to ordinary perverse sheaves is exact, fully faithful and sends \(\mathrm{IC}_z\) to \(\mathrm{IC}_z\) (connected group acting through a finite-dimensional quotient: smooth pullback with connected fibres is fully faithful on perverse sheaves with image closed under subquotients, bbd-perverse.pdf Prop. 4.2.5 — number recalled, the local copy is a scan without text layer), so constituents may be computed non-equivariantly.
  3. Generators lie in \(\mathcal D_{\mathcal M_I^\flat}\). By Setup 4.1(e), \({}^pH^n(M_{x,y})=P_n\simeq\bigoplus_z\mathrm{IC}_z\otimes E_{z,n}\) has constituents in \(\mathcal M_I^\flat\) by definition. By Setup 4.1(d) and thickness, \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^\flat}\). If \(z\notin\mathcal M_I^\flat\), \({}^pH^0(\mathrm{IC}_z)=\mathrm{IC}_z\) is simple and not isomorphic to any \(\mathrm{IC}_{z'}\), \(z'\ne z\) (Setup 4.1(c)), so \(\mathrm{IC}_z\notin\mathcal D_{\mathcal M_I^\flat}\supseteq\mathcal C_I^{\mathrm{Iwa}}\). This is summand.tex Step 4 with orbits replaced by Bruhat cells.
  4. Span. Each \(\mathrm{IC}_z\), \(z\in\mathcal M_I\), is a retract of a generator, so \(\operatorname{Thick}\{\mathrm{IC}_z:z\in\mathcal M_I\}\subseteq\mathcal C_I^{\mathrm{Iwa}}\); conversely an equivariantly split generator is an iterated cofibre of split triangles of its summands. And \(\mathcal D_\Sigma=\operatorname{Thick}\{\mathrm{IC}_z:z\in\Sigma\}\): an object of \(\mathcal D_\Sigma\) is an iterated extension of shifted \({}^pH^i\), each of finite length with constituents in \(\Sigma\).
  5. Everything. If \(\mathcal M_I=\widetilde W\), step 4 and Setup 4.1(c) give \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_z\}=\mathcal C_I\); if \(\mathcal M_I\ne\widetilde W\), step 3 exhibits an \(\mathrm{IC}_z\notin\mathcal C_I^{\mathrm{Iwa}}\).

Remark. Only steps 2–3 (the constituent version) are needed for \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\) in Theorem 4.5; the equivariant splitting is needed only for the exact description of \(\mathcal C_I^{\mathrm{Iwa}}\) in Proposition 4.7.

Sources