§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:49:47Z

Lemma 6.7 · Sheaf-level Corollary D: positive Steinberg certificates exclude summands

proved ★ key step

Gives a sheaf-level analogue of the function-level obstruction principle: a Hecke-equivariant functional to a sign-type character automatically kills the classes of all generators IC_w ∗ IC_0, and if it is nonnegative on all IC classes, weights forbid cancellation, so any IC on which it is positive is not Iwahori-generated; the criterion is sharp.

Lemma 6.7.

Setting over \(k=\overline{\mathbb F}_q\), \(p\) odd: \(X\) with a level presentation having finitely many \(I\)-orbits per level, connected stabilizers, all orbits and \(\mathrm{IC}_Q\) defined over \(\mathbb F_q\); \(\mathcal M_A\) the Grothendieck group of Weil objects of \(\mathcal C_I\), a \(\mathbb Z[A]\)-module (\(A=\overline{\mathbb Q}_\ell^\times/\mu_\infty\)), \(\mathbb Z[v^{\pm1}]\subset\mathbb Z[A]\) via \(v\mapsto-[q^{-1/2}]\); \(\mathcal H\) acts by convolution. Let \(\Lambda:\mathcal M_A\to\mathbb Z[A]\) be \(\mathbb Z[A]\)-linear. (a) If \(\Lambda(\underline H_s m)=0\) for every simple affine reflection \(s\) and \(\Lambda(T_\omega m)=\pm\Lambda(m)\) for \(\omega\in\Omega\), then \(\Lambda([\mathrm{IC}_w\ast_I\mathrm{IC}_0])=0\) for all \(w\in\widetilde W\). (b) If \(\Lambda\) vanishes on all \([\mathrm{IC}_w\ast_I\mathrm{IC}_0]\) (e.g. under (a)) and \(\Lambda([\mathrm{IC}_{Q'}])\in\mathbb Z_{\ge0}[v^{\pm1}]\) for every simple object \(\mathrm{IC}_{Q'}\), then no \(\mathrm{IC}_Q\) with \(\Lambda([\mathrm{IC}_Q])\ne0\) belongs to \(M_I\); if in addition the summand criterion's hypotheses hold (finite-length heart, t-exact transitions), \(\mathrm{IC}_Q\notin\mathcal C_I^{\mathrm{Iwa}}\). (c) (Sharpness.) If \(\mathrm{IC}_Q\notin M_I\), the \(Q\)-coordinate in the IC basis is a functional as in (b) with value \(1\) on \([\mathrm{IC}_Q]\). So: \(\mathrm{IC}_Q\notin M_I\) iff some \(\Lambda\) vanishing on all generator classes is \(\ge0\) on all IC classes and \(\ne0\) on \([\mathrm{IC}_Q]\).

  1. (a). Let \(s\) be a finite simple reflection. \(\mathrm{IC}_0\) is \(K\)-equivariant, so \(\mathrm{IC}_s\ast\mathrm{IC}_0\cong\mathrm{IC}_0[1]\oplus\mathrm{IC}_0[-1]\) (Lemma 6.4 step 5), i.e. \(\underline H_s[\mathrm{IC}_0]=(v+v^{-1})[\mathrm{IC}_0]\). Applying \(\Lambda\): \(0=(v+v^{-1})\Lambda([\mathrm{IC}_0])\), and \(\mathbb Z[A]\) is an integral domain (\(A\) is torsion-free abelian, so every finitely generated subgroup is free and its group ring is a Laurent polynomial ring), so \(\Lambda([\mathrm{IC}_0])=0\). Write \([\mathrm{IC}_w]\in\mathcal H\) as a \(\mathbb Z[v^{\pm1}]\)-combination of monomials \(\underline H_{s_{i_1}}\cdots\underline H_{s_{i_r}}T_{\omega}\) (possible since the \(\underline H_s,T_\omega\) generate \(\mathcal H\) over \(\mathbb Z[v^{\pm1}]\)). Each monomial with \(r\ge1\) is killed by the hypothesis applied to \(m=\underline H_{s_{i_2}}\cdots[\mathrm{IC}_0]\), and \(r=0\) gives \(\pm\Lambda([\mathrm{IC}_0])=0\). As \([\mathrm{IC}_w\ast\mathrm{IC}_0]=[\mathrm{IC}_w]\cdot[\mathrm{IC}_0]\) (Computation 6.2), (a) follows.
  2. Decomposition with weights. \(\mathcal F=\mathrm{IC}_w\ast\mathrm{IC}_0\) (normalized Tate twists) is pure of weight \(0\) (proper pushforward of a pure complex, BBD 5.1.14), so over \(\overline{\mathbb F}_q\), \(\mathcal F\cong\bigoplus_n{}^p\mathcal H^n(\mathcal F)[-n]\) and \({}^p\mathcal H^n(\mathcal F)\cong\bigoplus_Q\mathrm{IC}_Q\otimes V^n_Q\) with \(V^n_Q\) a Frobenius module pure of weight \(n\) (BBD 5.4.5, 5.4.6, 5.3.8). Hence \([\mathcal F]=\sum_Q m_Q[\mathrm{IC}_Q]\), \(m_Q=\sum_n(-1)^n[V^n_Q]\in\mathbb Z[A]\).
  3. No cancellation. \(0=\Lambda([\mathcal F])=\sum_{Q,n}(-1)^n[V_Q^n]\,\Lambda([\mathrm{IC}_Q])\). Write \(\Lambda([\mathrm{IC}_Q])=\sum_kc_{Q,k}v^k\), \(c_{Q,k}\ge0\); \(v^k=(-1)^k[q^{-k/2}]\) has weight \(-k\). A term of \([V^n_Q]v^k\) is \((-1)^{n+k}c\,[\alpha]\) with \(c\ge0\) and \([\alpha]\) of weight \(n-k\equiv n+k\bmod2\): the sign is determined by the weight. Elements of \(A\) of different weights are distinct, so for each weight \(w\) the weight-\(w\) part is \((-1)^w\) times a sum of nonnegative multiples of basis elements of \(\mathbb Z[A]\), hence each term vanishes: \(c_{Q,k}[V^n_Q]=0\) for all \(Q,n,k\). If \(\Lambda([\mathrm{IC}_Q])\ne0\), some \(c_{Q,k}>0\), so \(V^n_Q=0\) for all \(n\): \(\mathrm{IC}_Q\) is not a shifted summand of \(\mathcal F\), for every \(w\): \(\mathrm{IC}_Q\notin M_I\).
  4. Membership. If the summand criterion holds — \(\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}(M_I)\subseteq\{\)objects with all perverse Jordan–Hölder factors in \(M_I\}\) (the argument of Proposition 6.5 steps 1, 2, 5) — then \(\mathrm{IC}_Q\notin\mathcal C_I^{\mathrm{Iwa}}\).
  5. (c). The \([\mathrm{IC}_Q]\) form a \(\mathbb Z[A]\)-basis (unitriangular against the standard classes); by step 2 every \([\mathrm{IC}_w\ast\mathrm{IC}_0]\) lies in the \(\mathbb Z[A]\)-span of \(M_I\), so the \(Q\)-coordinate kills it, and it takes the values \(0,1\) on IC classes.

Compared with the function level (classical.html, Lemma C, Corollary D): there a nonzero module map to a character killed by \(e_K\) suffices; here one also needs positivity on every IC class (checklist 6), and weights are what make positivity decisive.

Builds on

  • Computation 6.2 · Affine Lusztig–Vogan module of $\mathrm{GL}_2/T$: Hecke action on orbit indicators

Built on by

  • Proposition 6.8 · Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not
  • Corollary 6.9 · Group case: the sign character $\varepsilon_+$ excludes every $\delta_\omega$
  • Corollary 6.10 · Closed-orbit test: sign-type Hecke maps nonnegative on closed orbits
  • Audit 9.8 · Audit of A6-7: positive certificates exclude summands
  • 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 18:49 UTC

Sheaf-level Corollary D: positive Steinberg certificates exclude summands

  1. (a). Let \(s\) be a finite simple reflection. \(\mathrm{IC}_0\) is \(K\)-equivariant, so \(\mathrm{IC}_s\ast\mathrm{IC}_0\cong\mathrm{IC}_0[1]\oplus\mathrm{IC}_0[-1]\) (Lemma 6.4 step 5), i.e. \(\underline H_s[\mathrm{IC}_0]=(v+v^{-1})[\mathrm{IC}_0]\). Applying \(\Lambda\): \(0=(v+v^{-1})\Lambda([\mathrm{IC}_0])\), and \(\mathbb Z[A]\) is an integral domain (\(A\) is torsion-free abelian, so every finitely generated subgroup is free and its group ring is a Laurent polynomial ring), so \(\Lambda([\mathrm{IC}_0])=0\). Write \([\mathrm{IC}_w]\in\mathcal H\) as a \(\mathbb Z[v^{\pm1}]\)-combination of monomials \(\underline H_{s_{i_1}}\cdots\underline H_{s_{i_r}}T_{\omega}\) (possible since the \(\underline H_s,T_\omega\) generate \(\mathcal H\) over \(\mathbb Z[v^{\pm1}]\)). Each monomial with \(r\ge1\) is killed by the hypothesis applied to \(m=\underline H_{s_{i_2}}\cdots[\mathrm{IC}_0]\), and \(r=0\) gives \(\pm\Lambda([\mathrm{IC}_0])=0\). As \([\mathrm{IC}_w\ast\mathrm{IC}_0]=[\mathrm{IC}_w]\cdot[\mathrm{IC}_0]\) (Computation 6.2), (a) follows.
  2. Decomposition with weights. \(\mathcal F=\mathrm{IC}_w\ast\mathrm{IC}_0\) (normalized Tate twists) is pure of weight \(0\) (proper pushforward of a pure complex, BBD 5.1.14), so over \(\overline{\mathbb F}_q\), \(\mathcal F\cong\bigoplus_n{}^p\mathcal H^n(\mathcal F)[-n]\) and \({}^p\mathcal H^n(\mathcal F)\cong\bigoplus_Q\mathrm{IC}_Q\otimes V^n_Q\) with \(V^n_Q\) a Frobenius module pure of weight \(n\) (BBD 5.4.5, 5.4.6, 5.3.8). Hence \([\mathcal F]=\sum_Q m_Q[\mathrm{IC}_Q]\), \(m_Q=\sum_n(-1)^n[V^n_Q]\in\mathbb Z[A]\).
  3. No cancellation. \(0=\Lambda([\mathcal F])=\sum_{Q,n}(-1)^n[V_Q^n]\,\Lambda([\mathrm{IC}_Q])\). Write \(\Lambda([\mathrm{IC}_Q])=\sum_kc_{Q,k}v^k\), \(c_{Q,k}\ge0\); \(v^k=(-1)^k[q^{-k/2}]\) has weight \(-k\). A term of \([V^n_Q]v^k\) is \((-1)^{n+k}c\,[\alpha]\) with \(c\ge0\) and \([\alpha]\) of weight \(n-k\equiv n+k\bmod2\): the sign is determined by the weight. Elements of \(A\) of different weights are distinct, so for each weight \(w\) the weight-\(w\) part is \((-1)^w\) times a sum of nonnegative multiples of basis elements of \(\mathbb Z[A]\), hence each term vanishes: \(c_{Q,k}[V^n_Q]=0\) for all \(Q,n,k\). If \(\Lambda([\mathrm{IC}_Q])\ne0\), some \(c_{Q,k}>0\), so \(V^n_Q=0\) for all \(n\): \(\mathrm{IC}_Q\) is not a shifted summand of \(\mathcal F\), for every \(w\): \(\mathrm{IC}_Q\notin M_I\).
  4. Membership. If the summand criterion holds — \(\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}(M_I)\subseteq\{\)objects with all perverse Jordan–Hölder factors in \(M_I\}\) (the argument of Proposition 6.5 steps 1, 2, 5) — then \(\mathrm{IC}_Q\notin\mathcal C_I^{\mathrm{Iwa}}\).
  5. (c). The \([\mathrm{IC}_Q]\) form a \(\mathbb Z[A]\)-basis (unitriangular against the standard classes); by step 2 every \([\mathrm{IC}_w\ast\mathrm{IC}_0]\) lies in the \(\mathbb Z[A]\)-span of \(M_I\), so the \(Q\)-coordinate kills it, and it takes the values \(0,1\) on IC classes.

Compared with the function level (classical.html, Lemma C, Corollary D): there a nonzero module map to a character killed by \(e_K\) suffices; here one also needs positivity on every IC class (checklist 6), and weights are what make positivity decisive.

Sources