§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:54:50Z

Corollary 6.10 · Closed-orbit test: sign-type Hecke maps nonnegative on closed orbits

proved

For symmetric X in Chen–Yi's setting with connected I-stabilizers, any Hecke-module map to a sign-type character vanishes on every IC of a non-closed orbit, so its positivity need only be checked on the finitely described closed orbits; at function level any nonzero such map already breaks (IG). The two levels can disagree only if every such map has mixed signs on the closed orbits.

Corollary 6.10.

Let \(X=G/G^\theta\) be symmetric over \(\overline{\mathbb F}_q\), \(p\ne2\), \(G^\theta\) connected, with all \(I\)-stabilizers on \(LX\) connected (Chen–Yi §8 setting, \(T_0\) split over \(\mathbb F_q\)). Let \(\psi:\mathcal M_A\to\mathbb Z[A]\) be \(\mathbb Z[A]\)-linear with \(\psi\circ\underline H_s=0\) for all simple affine \(s\) and \(\psi\circ T_\omega=\pm\psi\) for \(\omega\in\Omega\). (a) \(\psi([\mathrm{IC}_O])=0\) for every non-closed \(I\)-orbit \(O\), and \(\psi\) kills every \([\mathrm{IC}_w\ast_I\mathrm{IC}_0]\). (b) If \(\psi([\mathrm{IC}_O])\in\mathbb Z_{\ge0}[v^{\pm1}]\) for every closed orbit \(O\) (there \(\mathrm{IC}_O=\Delta_O\)) and \(\psi([\mathrm{IC}_{O_1}])\ne0\) for one closed \(O_1\), then \(\mathrm{IC}_{O_1}\notin\mathcal C_I^{\mathrm{Iwa}}\): sheaves fail. (c) If the specialization of \(\psi\) at \(v=-q^{-1/2}\) is nonzero, it is a nonzero \(\mathcal H_I\)-map \(C_I\to\mathbb C_{\varepsilon}\) killing \(\varphi_0\), so (IG) fails (classical.html, Lemma C(1)); \(\psi\) is determined by its values on closed-orbit classes, which generate \(\mathcal M\) over \(\mathcal H\). Consequently a row of this kind is an Iwahori divergence (sheaves generate, functions fail) only if every nonzero such \(\psi\) takes values of both signs on closed orbits, as \(\lambda\) does for \(\mathrm{GL}_2/T\) (Proposition 6.8); for \(\mathrm{GL}_2/T\) the second map \(\psi\) is positive, so the levels agree.

  1. Non-closed orbits. By Chen–Yi, Lemma 31 ((i)\(\Leftrightarrow\)(iv)), a non-closed orbit \(O\) is of type IIb, IIIb or IVb for some simple affine \(s\); by Lemma 28 it is then open in \(P_sO\), so \(\overline O\) is \(P_s\)-stable. With connected stabilizers \(\mathrm{IC}_O=\mathrm{IC}(\overline O,k)\) is the intersection complex of a \(P_s\)-stable irreducible closed subset with constant coefficients, hence \(P_s\)-equivariant (\(P_s\) acts at a finite level through a connected group). So \(\underline H_s[\mathrm{IC}_O]=(v+v^{-1})[\mathrm{IC}_O]\) and \(\psi([\mathrm{IC}_O])=0\) (\(\mathbb Z[A]\) a domain). The generator classes: Lemma 6.7(a).
  2. (b). Lemma 6.7(b) with \(\Lambda=\psi\): positivity holds on non-closed ICs (value \(0\)) and on closed ones by hypothesis. Hypotheses of Lemma 6.7: finitely many \(I\)-orbits in each \(L^+G\)-orbit closure (Chen–Yi, as used on hypothesis.html), placid finite levels (Chen–Yi §5, Appendix A), purity of \(\mathrm{IC}_w\ast\mathrm{IC}_0\) (BBD 5.1.14).
  3. (c). \(\psi(\varphi_0)=\psi([\mathrm{IC}_0])=0\) by step 1 (\(L^+X\) is \(K\)-stable). Generation: by induction on \(\delta\), a non-closed \(O_v\) of type IIb (resp. IIIb) for \(s\) has \(\mathbf 1_v=T_s\mathbf 1_{v'}\) (resp. \(T_s\mathbf 1_{v'}-\mathbf 1_{v''}\)) with \(v',v''\) lower (formula (29), types IIa, IIIa); type IV involves a disconnected stabilizer and is excluded by hypothesis.
  4. Scope and gap. Disconnected stabilizers (types IV, non-constant local systems) are not covered: one must check \(P_s\)-equivariance of the non-constant ICs and positivity on all of them. Only a sufficient condition for sheaf failure is proved; whether mixed signs on closed orbits for all \(\psi\) forces \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) is open (the sharp certificate of Lemma 6.7(c) need not be a module map).
  5. Exact obligation for each symmetric row (\(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(E_6/F_4\)): list the closed \(I\)-orbits (Chen–Yi, Lemma 31: \(w_v\in\Omega\)), compute the space of sign-type maps \(\psi\) on their classes from (29), and decide whether one is \(\ge0\) and nonzero. If yes: sheaves fail, and functions fail too, so the row agrees. If no: the row is undecided by this test; a divergence needs \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\), i.e. every closed-orbit IC in \(M_I\).

Builds on

  • Lemma 6.7 · Sheaf-level Corollary D: positive Steinberg certificates exclude summands
  • Proposition 6.8 · Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not

Built on by

  • Objection 9.10 · Objection to A6-10: positivity does not give a nonzero specialization
  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 18:54 UTC

Closed-orbit test: sign-type Hecke maps nonnegative on closed orbits

  1. Non-closed orbits. By Chen–Yi, Lemma 31 ((i)\(\Leftrightarrow\)(iv)), a non-closed orbit \(O\) is of type IIb, IIIb or IVb for some simple affine \(s\); by Lemma 28 it is then open in \(P_sO\), so \(\overline O\) is \(P_s\)-stable. With connected stabilizers \(\mathrm{IC}_O=\mathrm{IC}(\overline O,k)\) is the intersection complex of a \(P_s\)-stable irreducible closed subset with constant coefficients, hence \(P_s\)-equivariant (\(P_s\) acts at a finite level through a connected group). So \(\underline H_s[\mathrm{IC}_O]=(v+v^{-1})[\mathrm{IC}_O]\) and \(\psi([\mathrm{IC}_O])=0\) (\(\mathbb Z[A]\) a domain). The generator classes: Lemma 6.7(a).
  2. (b). Lemma 6.7(b) with \(\Lambda=\psi\): positivity holds on non-closed ICs (value \(0\)) and on closed ones by hypothesis. Hypotheses of Lemma 6.7: finitely many \(I\)-orbits in each \(L^+G\)-orbit closure (Chen–Yi, as used on hypothesis.html), placid finite levels (Chen–Yi §5, Appendix A), purity of \(\mathrm{IC}_w\ast\mathrm{IC}_0\) (BBD 5.1.14).
  3. (c). \(\psi(\varphi_0)=\psi([\mathrm{IC}_0])=0\) by step 1 (\(L^+X\) is \(K\)-stable). Generation: by induction on \(\delta\), a non-closed \(O_v\) of type IIb (resp. IIIb) for \(s\) has \(\mathbf 1_v=T_s\mathbf 1_{v'}\) (resp. \(T_s\mathbf 1_{v'}-\mathbf 1_{v''}\)) with \(v',v''\) lower (formula (29), types IIa, IIIa); type IV involves a disconnected stabilizer and is excluded by hypothesis.
  4. Scope and gap. Disconnected stabilizers (types IV, non-constant local systems) are not covered: one must check \(P_s\)-equivariance of the non-constant ICs and positivity on all of them. Only a sufficient condition for sheaf failure is proved; whether mixed signs on closed orbits for all \(\psi\) forces \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) is open (the sharp certificate of Lemma 6.7(c) need not be a module map).
  5. Exact obligation for each symmetric row (\(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(E_6/F_4\)): list the closed \(I\)-orbits (Chen–Yi, Lemma 31: \(w_v\in\Omega\)), compute the space of sign-type maps \(\psi\) on their classes from (29), and decide whether one is \(\ge0\) and nonzero. If yes: sheaves fail, and functions fail too, so the row agrees. If no: the row is undecided by this test; a divergence needs \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\), i.e. every closed-orbit IC in \(M_I\).

Sources