§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:49:29Z

Theorem 2.9 · Group: $\Omega\cap\mathcal M_I^+=\varnothing$ by sign character and weights

proved ★ key step

A second proof of the sheaf half of row 1 over F_q-bar: the sign character of the Iwahori-Hecke algebra kills the trace of every generator and of every IC_v with v outside Omega, and Frobenius weights rule out cancellation, so no IC_omega = Delta_omega is a constituent of a generator; the criterion then excludes Delta_e. No affine Whittaker functor is used.

Theorem 2.9.

$k=\overline{\mathbb F}q$, $H$ split over $\mathbb F_q$, $W_H\ne1$, $X=H$ under $H\times H$. For all $w_1,w_2\in\widetilde W$, all $s\in\mathbb Z$ and all $\omega\in\Omega$, $$\mathrm{Hom}\big(\Delta_\omega,\ {}^pH^s(\mathrm{IC}_{w_1}\ast c\ast\mathrm{IC}_{w_2})\big)=0 .$$ Hence $\Omega\cap\mathcal M_I^+=\varnothing$ (with $\mathrm{IC}\omega=\Delta_\omega$, Proposition 2.8(iv)) and, by Theorem 2.4(b), $$\boxed{\ \Delta_\omega\notin\mathcal C_I^{\mathrm{Iwa}}\ \text{for every }\omega\in\Omega;\ \text{in particular }\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\ne\mathcal C_I .\ }$$ Function shadow: $\varepsilon(T_x)=(-1)^{\ell(x)}$ kills $\mathcal H_I\mathbf 1_K\mathcal H_I$ and not $T_e$, so (IG) fails. Over $\mathbb C$ the sheaf statement follows given the transfer input (T$_{\mathbb C}$) of Step 9 (cited, not re-proved).

Put $P=\mathrm{IC}{w_1}\ast\mathrm{IC}{w_0}\ast\mathrm{IC}{w_2}=(\mathrm{IC}{w_1}\ast c\ast\mathrm{IC}{w_2})[\ell(w_0)]$, with Weil structures from $\mathrm{IC}{v,0}:=j_{!*}(\overline{\mathbb Q}_\ell[\ell(v)])$ (trivial Frobenius on the open cell).

  1. Multiplicity spaces. $P_0$ is pure of weight $w_P=\ell(w_1)+\ell(w_0)+\ell(w_2)$ (Proposition 2.8, Step 7), so ${}^pH^s(P_0)$ is pure of weight $w_P+s$ (Sun, Thm. 3.5 = stack form of BBD 5.4.1, 5.4.4) and geometrically semisimple (BBD 5.3.8; Sun, Thm. 3.11). Let $V_{v,s}=\mathrm{Hom}(\mathrm{IC}v,{}^pH^s(P))$ with its Frobenius; the evaluation map $\bigoplus_v\mathrm{IC}{v,0}\otimes V_{v,s}\to{}^pH^s(P_0)$ is an isomorphism of Weil perverse sheaves (isotypic decomposition, $\mathrm{End}\,\mathrm{IC}v=\overline{\mathbb Q}\ell$; the same construction as $E_{\nu,i}(\lambda)$, $A_{\nu,i}(\lambda)$ in trace.tex, (D2) and Def. A). A summand of a pure perverse sheaf is pure of the same weight; restricting $\mathrm{IC}{v,0}\otimes V{v,s}$ to the open cell, where $\mathrm{IC}{v,0}=\overline{\mathbb Q}\ell[\ell(v)]$ has weight $\ell(v)$, shows $V_{v,s}$ is pure of weight $w_P+s-\ell(v)$. For $v=\omega\in\Omega$: weight $w_P+s$.
  2. Support. Write $\sigma(y)\in\Omega$ for the $\Omega$-part of $y=y'\sigma(y)$. From $IsI\cdot IyI\subseteq IsyI\cup IyI$ ($s$ simple affine) and $I\tau I\cdot IyI=I\tau yI$ ($\tau\in\Omega$), $IuI\cdot IyI$ is a union of cells with $\Omega$-part $\sigma(u)\sigma(y)$, and Schubert closures keep the $\Omega$-part. So $\operatorname{supp}P$ only meets cells with $\Omega$-part $\sigma:=\sigma(w_1)\sigma(w_2)$ ($\sigma(w_0)=1$), and $V_{\omega,s}=0$ for $\omega\ne\sigma$. It remains to treat $\omega=\sigma$.
  3. Traces. For $r\ge1$ let $F_r=\mathbb F_{q^r}(\!(t)\!)$, $\mathcal H_r=C_c(I_r\backslash H(F_r)/I_r)$ with $\mathrm{vol}(I_r)=1$, and $t^{(r)}A$ the trace function of $F^r$. (i) $t^{(r)}{A\ast B}=t^{(r)}A\ast t^{(r)}_B$: proper base change and the Grothendieck–Lefschetz formula on the fibres of the convolution map (Laumon (1.1.1.3)), the $\mathbb F{q^r}$-points of $LH\times^ILH/I$ being $LH(F_r)\times^{I_r}(LH/I)(F_r)$ by Lang's theorem for the connected groups $I/I_M$ (Milne, Cor. 27.56), and Künneth for the twisted product (Laumon (1.1.1.2)). (ii) $t^{(r)}{\mathrm{IC}{w_0}}=(-1)^{\ell(w_0)}\mathbf 1_{K_r}$ and $t^{(r)}{\mathrm{IC}\omega}=T_\omega$. (iii) Additivity along the Frobenius-stable perverse truncation triangles and the sign of shifts (Laumon (1.1.1.1)), with Step 1: $t^{(r)}P=\sum{s,v}(-1)^s\operatorname{Tr}(F^r\,|\,V_{v,s})\,t^{(r)}_{\mathrm{IC}_v}$.
  4. The sign character. $\varepsilon_r(T_x)=(-1)^{\ell(x)}$ is an algebra homomorphism $\mathcal H_r\to\overline{\mathbb Q}\ell$. Quadratic relation used: $T_s^2=q^r+(q^r-1)T_s$ (Iwahori–Matsumoto, Prop. 3.4(i)). Check on the Iwahori–Matsumoto relations (as displayed in haines-kottwitz-prasad-v3.pdf, end of §3): $T_xT_s=T{xs}$ if $x<xs$; if $xs<x$, $\varepsilon_r(q^rT_{xs}+(q^r-1)T_x)=(-1)^{\ell(x)}(-q^r+q^r-1)=\varepsilon_r(T_x)\varepsilon_r(T_s)$; $T_xT_\tau=T_{x\tau}$, $\ell(x\tau)=\ell(x)$. Values: $\varepsilon_r(\mathbf 1_{K_r})=\sum_{w\in W}(-1)^{\ell(w)}=0$ ($K=\bigsqcup_{w\in W}IwI$; $w\mapsto ws$ reverses signs, $W\ne1$); $\varepsilon_r(T_\omega)=1$. For $v\notin\Omega$: $\ell(v)\ge1$ gives a simple affine $s$ with $vs<v$ (a right descent of $v'$, conjugated by $\sigma(v)$); then $\overline{IvI}$ is right $P_s$-stable and $\mathrm{IC}{v,0}$ is the pullback, shifted by $1$, of the IC of $\overline{IvI}/P_s$ along the $\mathbb P^1$-bundle (BBD 4.2.6.1 and Lemma 4.3.2, compatibly with the trivial Weil structures), so $t=t^{(r)}{\mathrm{IC}v}$ is right $P_s(\mathcal O_r)$-invariant and $(t\ast T_s)(x)=\sum{gI\subseteq xIsI}t(g)=q^rt(x)$. Applying $\varepsilon_r$: $-\varepsilon_r(t)=q^r\varepsilon_r(t)$, so $\varepsilon_r(t^{(r)}_{\mathrm{IC}_v})=0$.
  5. The generators are killed. By Step 3(i),(ii), $\varepsilon_r(t^{(r)}P)=\pm\varepsilon_r(t^{(r)}{\mathrm{IC}{w_1}})\,\varepsilon_r(\mathbf 1{K_r})\,\varepsilon_r(t^{(r)}{\mathrm{IC}{w_2}})=0$. By Step 3(iii), Step 4 and Step 2, $\varepsilon_r(t^{(r)}P)=\sum_s(-1)^s\operatorname{Tr}(F^r\,|\,V{\sigma,s})$. Hence $\sum_s(-1)^s\operatorname{Tr}(F^r\,|\,V_{\sigma,s})=0$ for all $r\ge1$.
  6. Weights forbid cancellation (checklist items 5–6). Write $m_{\alpha,s}$ for the multiplicity of the eigenvalue $\alpha$ of $F$ on $V_{\sigma,s}$. Then $\sum_{r\ge1}\big(\sum_s(-1)^s\operatorname{Tr}(F^r|V_{\sigma,s})\big)T^r=\sum_\alpha\big(\sum_s(-1)^sm_{\alpha,s}\big)\frac{\alpha T}{1-\alpha T}=0$; the rational functions $\alpha T/(1-\alpha T)$, $\alpha\ne0$ distinct, are linearly independent (distinct poles), so $\sum_s(-1)^sm_{\alpha,s}=0$ for each $\alpha$. By Step 1, $|\iota\alpha|=q^{(w_P+s)/2}$ for every eigenvalue on $V_{\sigma,s}$, so each $\alpha$ occurs for at most one $s$: all $m_{\alpha,s}=0$ and $V_{\sigma,s}=0$.
  7. Conclusion. Steps 2 and 6: $\mathrm{Hom}(\Delta_\omega,{}^pH^s(P))=0$ for all $\omega,s$, i.e. $\Omega\cap\mathcal M_I^+=\varnothing$. Theorem 2.4(b) (which needs no (DT$I$)) gives $\Delta\omega\notin\mathcal C_I^{\mathrm{Iwa}}$; with (DT$_I$) from Proposition 2.8(v) and Theorem 2.4(c), moreover $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}(\mathcal M_I)$ with $\mathcal M_I\subseteq\{\mathrm{IC}_v:v\notin\Omega\}$. (Two-sidedness, checklist 2, enters through Lemma 2.3(b) and Step 3(i).)
  8. Functions. With $r=1$, $\varepsilon_1(T_{w_1}\ast\mathbf 1_K\ast T_{w_2})=0$ by Step 4, while $\varepsilon_1(T_e)=1$; for the group $C_I=\mathcal H_I$ with the two-sided action, so $\mathcal H_I\ast\varphi_0\ast\mathcal H_I\subseteq\ker\varepsilon_1\subsetneq C_I$ and (IG) fails. The sheaf statement needed Step 6: $\Delta_e\oplus\Delta_e[1]$ has trace $0$, so $\varepsilon$ alone cannot exclude a summand.
  9. Over $\mathbb C$ (T${\mathbb C}$, cited). $Z$, $m$ and the $\mathrm{IC}_v$ are defined over $\mathbb Z[1/N]$; by the good-model comparison (BBD 6.1.9–6.1.10; Bezrukavnikov–Finkelberg, Prop. 5, input D5 of trace.tex) $\dim\mathrm{Hom}(\Delta\omega,{}^pH^s(P))$ agrees over $\mathbb C$ and $\overline{\mathbb F}_p$ for almost all $p$, so the conclusion holds for Betti sheaves, conditionally on this citation. No step uses the affine Whittaker functor or the pages' Propositions II, IV.2.

Builds on

  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$
  • Proposition 2.8 · Group: all hypotheses hold ($\Omega$-truncated levels); $\mathcal S_I=\{\mathrm{IC}_v\}$

Built on by

  • Audit 2.11 · Audit: the nine failure modes checked against the criterion and the two cases
  • Summary 2.12 · Verdict: the Iwahori summand criterion and the two test cases
  • Audit 8.5 · Audit of the main group theorem (A4-5, A2-9): no $\Delta_\omega$ constituent
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
  • 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

Group: $\Omega\cap\mathcal M_I^+=\varnothing$ by sign character and weights

Put $P=\mathrm{IC}{w_1}\ast\mathrm{IC}{w_0}\ast\mathrm{IC}{w_2}=(\mathrm{IC}{w_1}\ast c\ast\mathrm{IC}{w_2})[\ell(w_0)]$, with Weil structures from $\mathrm{IC}{v,0}:=j_{!*}(\overline{\mathbb Q}_\ell[\ell(v)])$ (trivial Frobenius on the open cell).

  1. Multiplicity spaces. $P_0$ is pure of weight $w_P=\ell(w_1)+\ell(w_0)+\ell(w_2)$ (Proposition 2.8, Step 7), so ${}^pH^s(P_0)$ is pure of weight $w_P+s$ (Sun, Thm. 3.5 = stack form of BBD 5.4.1, 5.4.4) and geometrically semisimple (BBD 5.3.8; Sun, Thm. 3.11). Let $V_{v,s}=\mathrm{Hom}(\mathrm{IC}v,{}^pH^s(P))$ with its Frobenius; the evaluation map $\bigoplus_v\mathrm{IC}{v,0}\otimes V_{v,s}\to{}^pH^s(P_0)$ is an isomorphism of Weil perverse sheaves (isotypic decomposition, $\mathrm{End}\,\mathrm{IC}v=\overline{\mathbb Q}\ell$; the same construction as $E_{\nu,i}(\lambda)$, $A_{\nu,i}(\lambda)$ in trace.tex, (D2) and Def. A). A summand of a pure perverse sheaf is pure of the same weight; restricting $\mathrm{IC}{v,0}\otimes V{v,s}$ to the open cell, where $\mathrm{IC}{v,0}=\overline{\mathbb Q}\ell[\ell(v)]$ has weight $\ell(v)$, shows $V_{v,s}$ is pure of weight $w_P+s-\ell(v)$. For $v=\omega\in\Omega$: weight $w_P+s$.
  2. Support. Write $\sigma(y)\in\Omega$ for the $\Omega$-part of $y=y'\sigma(y)$. From $IsI\cdot IyI\subseteq IsyI\cup IyI$ ($s$ simple affine) and $I\tau I\cdot IyI=I\tau yI$ ($\tau\in\Omega$), $IuI\cdot IyI$ is a union of cells with $\Omega$-part $\sigma(u)\sigma(y)$, and Schubert closures keep the $\Omega$-part. So $\operatorname{supp}P$ only meets cells with $\Omega$-part $\sigma:=\sigma(w_1)\sigma(w_2)$ ($\sigma(w_0)=1$), and $V_{\omega,s}=0$ for $\omega\ne\sigma$. It remains to treat $\omega=\sigma$.
  3. Traces. For $r\ge1$ let $F_r=\mathbb F_{q^r}(\!(t)\!)$, $\mathcal H_r=C_c(I_r\backslash H(F_r)/I_r)$ with $\mathrm{vol}(I_r)=1$, and $t^{(r)}A$ the trace function of $F^r$. (i) $t^{(r)}{A\ast B}=t^{(r)}A\ast t^{(r)}_B$: proper base change and the Grothendieck–Lefschetz formula on the fibres of the convolution map (Laumon (1.1.1.3)), the $\mathbb F{q^r}$-points of $LH\times^ILH/I$ being $LH(F_r)\times^{I_r}(LH/I)(F_r)$ by Lang's theorem for the connected groups $I/I_M$ (Milne, Cor. 27.56), and Künneth for the twisted product (Laumon (1.1.1.2)). (ii) $t^{(r)}{\mathrm{IC}{w_0}}=(-1)^{\ell(w_0)}\mathbf 1_{K_r}$ and $t^{(r)}{\mathrm{IC}\omega}=T_\omega$. (iii) Additivity along the Frobenius-stable perverse truncation triangles and the sign of shifts (Laumon (1.1.1.1)), with Step 1: $t^{(r)}P=\sum{s,v}(-1)^s\operatorname{Tr}(F^r\,|\,V_{v,s})\,t^{(r)}_{\mathrm{IC}_v}$.
  4. The sign character. $\varepsilon_r(T_x)=(-1)^{\ell(x)}$ is an algebra homomorphism $\mathcal H_r\to\overline{\mathbb Q}\ell$. Quadratic relation used: $T_s^2=q^r+(q^r-1)T_s$ (Iwahori–Matsumoto, Prop. 3.4(i)). Check on the Iwahori–Matsumoto relations (as displayed in haines-kottwitz-prasad-v3.pdf, end of §3): $T_xT_s=T{xs}$ if $x<xs$; if $xs<x$, $\varepsilon_r(q^rT_{xs}+(q^r-1)T_x)=(-1)^{\ell(x)}(-q^r+q^r-1)=\varepsilon_r(T_x)\varepsilon_r(T_s)$; $T_xT_\tau=T_{x\tau}$, $\ell(x\tau)=\ell(x)$. Values: $\varepsilon_r(\mathbf 1_{K_r})=\sum_{w\in W}(-1)^{\ell(w)}=0$ ($K=\bigsqcup_{w\in W}IwI$; $w\mapsto ws$ reverses signs, $W\ne1$); $\varepsilon_r(T_\omega)=1$. For $v\notin\Omega$: $\ell(v)\ge1$ gives a simple affine $s$ with $vs<v$ (a right descent of $v'$, conjugated by $\sigma(v)$); then $\overline{IvI}$ is right $P_s$-stable and $\mathrm{IC}{v,0}$ is the pullback, shifted by $1$, of the IC of $\overline{IvI}/P_s$ along the $\mathbb P^1$-bundle (BBD 4.2.6.1 and Lemma 4.3.2, compatibly with the trivial Weil structures), so $t=t^{(r)}{\mathrm{IC}v}$ is right $P_s(\mathcal O_r)$-invariant and $(t\ast T_s)(x)=\sum{gI\subseteq xIsI}t(g)=q^rt(x)$. Applying $\varepsilon_r$: $-\varepsilon_r(t)=q^r\varepsilon_r(t)$, so $\varepsilon_r(t^{(r)}_{\mathrm{IC}_v})=0$.
  5. The generators are killed. By Step 3(i),(ii), $\varepsilon_r(t^{(r)}P)=\pm\varepsilon_r(t^{(r)}{\mathrm{IC}{w_1}})\,\varepsilon_r(\mathbf 1{K_r})\,\varepsilon_r(t^{(r)}{\mathrm{IC}{w_2}})=0$. By Step 3(iii), Step 4 and Step 2, $\varepsilon_r(t^{(r)}P)=\sum_s(-1)^s\operatorname{Tr}(F^r\,|\,V{\sigma,s})$. Hence $\sum_s(-1)^s\operatorname{Tr}(F^r\,|\,V_{\sigma,s})=0$ for all $r\ge1$.
  6. Weights forbid cancellation (checklist items 5–6). Write $m_{\alpha,s}$ for the multiplicity of the eigenvalue $\alpha$ of $F$ on $V_{\sigma,s}$. Then $\sum_{r\ge1}\big(\sum_s(-1)^s\operatorname{Tr}(F^r|V_{\sigma,s})\big)T^r=\sum_\alpha\big(\sum_s(-1)^sm_{\alpha,s}\big)\frac{\alpha T}{1-\alpha T}=0$; the rational functions $\alpha T/(1-\alpha T)$, $\alpha\ne0$ distinct, are linearly independent (distinct poles), so $\sum_s(-1)^sm_{\alpha,s}=0$ for each $\alpha$. By Step 1, $|\iota\alpha|=q^{(w_P+s)/2}$ for every eigenvalue on $V_{\sigma,s}$, so each $\alpha$ occurs for at most one $s$: all $m_{\alpha,s}=0$ and $V_{\sigma,s}=0$.
  7. Conclusion. Steps 2 and 6: $\mathrm{Hom}(\Delta_\omega,{}^pH^s(P))=0$ for all $\omega,s$, i.e. $\Omega\cap\mathcal M_I^+=\varnothing$. Theorem 2.4(b) (which needs no (DT$I$)) gives $\Delta\omega\notin\mathcal C_I^{\mathrm{Iwa}}$; with (DT$_I$) from Proposition 2.8(v) and Theorem 2.4(c), moreover $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}(\mathcal M_I)$ with $\mathcal M_I\subseteq\{\mathrm{IC}_v:v\notin\Omega\}$. (Two-sidedness, checklist 2, enters through Lemma 2.3(b) and Step 3(i).)
  8. Functions. With $r=1$, $\varepsilon_1(T_{w_1}\ast\mathbf 1_K\ast T_{w_2})=0$ by Step 4, while $\varepsilon_1(T_e)=1$; for the group $C_I=\mathcal H_I$ with the two-sided action, so $\mathcal H_I\ast\varphi_0\ast\mathcal H_I\subseteq\ker\varepsilon_1\subsetneq C_I$ and (IG) fails. The sheaf statement needed Step 6: $\Delta_e\oplus\Delta_e[1]$ has trace $0$, so $\varepsilon$ alone cannot exclude a summand.
  9. Over $\mathbb C$ (T${\mathbb C}$, cited). $Z$, $m$ and the $\mathrm{IC}_v$ are defined over $\mathbb Z[1/N]$; by the good-model comparison (BBD 6.1.9–6.1.10; Bezrukavnikov–Finkelberg, Prop. 5, input D5 of trace.tex) $\dim\mathrm{Hom}(\Delta\omega,{}^pH^s(P))$ agrees over $\mathbb C$ and $\overline{\mathbb F}_p$ for almost all $p$, so the conclusion holds for Betti sheaves, conditionally on this citation. No step uses the affine Whittaker functor or the pages' Propositions II, IV.2.

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