§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:44:34Z

Proposition 3.9 · Group: no $\mathrm{IC}_\omega$ is a summand, so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$

proved ★ key step

A second proof of row 1: sign characters over every $\mathbb F_{q^n}(\!(t)\!)$ kill the traces of all generators and of all $\mathrm{IC}z$ with $\ell(z)>0$, and purity (distinct weights in distinct perverse degrees) plus Vandermonde turns this function-level vanishing into the absence of $\mathrm{IC}\omega$ as a summand; the summand criterion and the bridge's witness finish both levels.

Proposition 3.9.

Let $H$ be split connected reductive over $\mathbb F_q$ with $W_H\neq1$, $X=H$, $G=H\times H$, $(h_1,h_2)x=h_1xh_2^{-1}$. Then $\mathcal C_I=\mathrm{Shv}c(I_H\backslash LH/I_H)$, the simple objects are the $\mathrm{IC}_z$, $z\in\widetilde W_H$, the generators are $\mathrm{IC}_x*c*\mathrm{IC}_y$ with $c=\overline{\mathbb Q}{\ell,K/I}$ (unshifted), and (L1), (L2), (D1)–(D3) hold. Moreover:

(a) For every $\omega\in\Omega$ and all $x,y\in\widetilde W_H$, $\mathrm{IC}\omega=\Delta\omega$ is not a shifted direct summand of $\mathrm{IC}_x*c*\mathrm{IC}_y$ (over $\overline{\mathbb F}_q$).

(b) Hence $\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}$ (by Lemma 3.5(a)), and $T_e=\mathbf 1_I\notin\mathcal H_I*\varphi_0=\mathcal H_{I_H}\mathbf 1_K\mathcal H_{I_H}$ over $\mathbb F_q(\!(t)\!)$ (by Theorem 3.6(c), or directly by step 1).

  1. Inputs. Levels of proposition-III.tex (Example, group): $Y_N=\bigcup_{\ell(w)\le N}\overline{IwI}$, $Y_{N,M}=Y_N/I_M$; transitions are torsors under the split unipotent $I_M/I_{M'}$ (affine-space bundles); $Y_N(\mathbb F_q)$ is a finite union of double cosets $I(\mathcal O)vI(\mathcal O)$ with basis of cosets $xI_M(\mathcal O)$. Pieces are the cells $IvI$, rational, and the stabilizer of $wI_M$ is the image of the connected $(I\cap wIw^{-1})\times I_M$ under $(a,c)\mapsto(a,w^{-1}aw\,c)$. (D1): bounded left and right multiplications. (D3): $\mathrm{Fl}{\le x}\tilde\times K/I\tilde\times\mathrm{Fl}{\le y}\to\mathrm{Fl}$ is proper and the twisted product of pure complexes is pure; Sun, Thm. 1.2. Generators: Lemma 3.5 step 0.
  2. Sign characters. Over $F_n=\mathbb F_{q^n}(\!(t)\!)$ with quadratic relation $(T_s-q^n)(T_s+1)=0$, $\varepsilon_n(T_w)=(-1)^{\ell(w)}$ defines a character of $\mathcal H_{I_n}$ ($T_s\mapsto-1$, $T_\omega\mapsto1$ respects braid, quadratic and $T_\omega T_sT_\omega^{-1}=T_{\omega s\omega^{-1}}$ relations [Iwahori–Matsumoto 1965]). $\varepsilon_n(\mathbf 1_{K_n})=\sum_{w\in W}(-1)^{\ell(w)}=0$ as $W\neq1$.
  3. $\varepsilon_n$ on IC traces. If $z\notin\Omega$, pick a simple affine $s$ with $zs<z$. Then $\mathrm{Fl}{\le z}$ is right $P_s$-stable (lifting property), so $\mathrm{IC}_z$ with its canonical Weil structure is pulled back along the $\mathbb P^1$-bundle $\mathrm{Fl}\to LH/P_s$ [BBD, 4.2.6]; $f=\mathrm{tr}_n\mathrm{IC}_z$ is right $P_s$-invariant, so $f*(T_e+T_s)=(q^n+1)f$, i.e. $f*T_s=q^nf$, and $-\varepsilon_n(f)=q^n\varepsilon_n(f)$ forces $\varepsilon_n(f)=0$. If $z=\omega\in\Omega$, $\mathrm{IC}\omega=\Delta_\omega$ and $\varepsilon_n(T_\omega)=1$.
  4. Components. $K$ lies in the neutral component of $LH$ and $\overline{IxI}$ in the component of the $\Omega$-part of $x$, so $\mathrm{IC}x*c*\mathrm{IC}_y$ lives on one component, labelled $\omega_0$; no $\mathrm{IC}\omega$ with $\omega\neq\omega_0$ occurs.
  5. Vanishing coefficient. By Lemma 3.3(a) (two-sided) and Lemma 3.4(ii) over $\mathbb F_{q^n}$ (with $A^n$ in place of $A$, trace.tex footnote 1), $\mathrm{tr}n\mathrm{IC}_x*\mathbf 1{K_n}*\mathrm{tr}n\mathrm{IC}_y^{\vee}=\sum_zm^{(n)}_z\mathrm{tr}_n\mathrm{IC}_z$, $m^{(n)}_z=\sum_i(-1)^i\mathrm{tr}A{z,i}^n$. Apply $\varepsilon_n$: the left side is $0$ by step 1, the right side is $m^{(n)}{\omega_0}$ by steps 2–3. So $m^{(n)}{\omega_0}=0$ for all $n\ge1$.
  6. Weights. The generator is pure of weight $w_{xy}=\ell(x)+\ell(y)$; ${}^p\mathcal H^i$ is pure of weight $w_{xy}+i$ [BBD, 5.4.1]; its isotypic part $\mathrm{IC}{\omega_0}\otimes(E{\omega_0,i},A_{\omega_0,i})$ is a Weil direct summand (Lemma 3.4 step 3), hence pure, and its stalk at the point cell $I\omega_0I/I$ is $E_{\omega_0,i}$ in degree $0$. So every eigenvalue $\alpha$ of $A_{\omega_0,i}$ has $|\iota\alpha|=q^{(w_{xy}+i)/2}$: different $i$ give different $\alpha$.
  7. Vandermonde. $m^{(n)}{\omega_0}=\sum\alpha c_\alpha\alpha^n$ over the distinct eigenvalues, with $c_\alpha=(-1)^{i(\alpha)}\cdot$(multiplicity) $\neq0$ by step 5. Vanishing for $n=1,\dots,r$ ($r$ the number of $\alpha$) and $\det(\alpha_j^n)\neq0$ force $r=0$: all $E_{\omega_0,i}=0$. This is (a).
  8. (b). Lemma 3.5(a) with $\mathrm{IC}e=\Delta_e\notin\mathcal M_I$. For functions, $\kappa{\mathrm{IC}e}$ kills $\mathcal H_I*\varphi_0$ (Theorem 3.6(c)) and $\kappa{\mathrm{IC}_e}(T_e)=1$. $\square$

Differs from the pages. No Whittaker category, no averaging functor, no shift trick: the categorical sign character of Proposition II is replaced by the classical one over all $\mathbb F_{q^n}$ plus weights. The cancellation obstruction (checklist 6) is handled by purity: within one perverse degree all signs agree, across degrees absolute values differ. Fields: sheaves over $\overline{\mathbb F}q$, functions over $\mathbb F{q^n}(\!(t)\!)$.

Builds on

  • Lemma 3.4 · Unitriangular IC-trace basis of $C_I$ and the decomposition matrix $m_{S,w}$
  • Lemma 3.5 · Iwahori summand criterion: $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\,\mathcal M_I$
  • Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$

Built on by

  • Proposition 3.10 · Sheaves to functions: non-zero rows only; (WT) gives (IG)
  • Theorem 3.11 · Sheaf generation as a test on IC-coordinates over all $\mathbb F_{q^n}$
  • Summary 3.12 · Verdict: (II$_I$) proved; rows 1–3 agree; one divergence type excluded
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
Original record, as the agent wrote it at 18:44 UTC

Group: no $\mathrm{IC}_\omega$ is a summand, so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$

  1. Inputs. Levels of proposition-III.tex (Example, group): $Y_N=\bigcup_{\ell(w)\le N}\overline{IwI}$, $Y_{N,M}=Y_N/I_M$; transitions are torsors under the split unipotent $I_M/I_{M'}$ (affine-space bundles); $Y_N(\mathbb F_q)$ is a finite union of double cosets $I(\mathcal O)vI(\mathcal O)$ with basis of cosets $xI_M(\mathcal O)$. Pieces are the cells $IvI$, rational, and the stabilizer of $wI_M$ is the image of the connected $(I\cap wIw^{-1})\times I_M$ under $(a,c)\mapsto(a,w^{-1}aw\,c)$. (D1): bounded left and right multiplications. (D3): $\mathrm{Fl}{\le x}\tilde\times K/I\tilde\times\mathrm{Fl}{\le y}\to\mathrm{Fl}$ is proper and the twisted product of pure complexes is pure; Sun, Thm. 1.2. Generators: Lemma 3.5 step 0.
  2. Sign characters. Over $F_n=\mathbb F_{q^n}(\!(t)\!)$ with quadratic relation $(T_s-q^n)(T_s+1)=0$, $\varepsilon_n(T_w)=(-1)^{\ell(w)}$ defines a character of $\mathcal H_{I_n}$ ($T_s\mapsto-1$, $T_\omega\mapsto1$ respects braid, quadratic and $T_\omega T_sT_\omega^{-1}=T_{\omega s\omega^{-1}}$ relations [Iwahori–Matsumoto 1965]). $\varepsilon_n(\mathbf 1_{K_n})=\sum_{w\in W}(-1)^{\ell(w)}=0$ as $W\neq1$.
  3. $\varepsilon_n$ on IC traces. If $z\notin\Omega$, pick a simple affine $s$ with $zs<z$. Then $\mathrm{Fl}{\le z}$ is right $P_s$-stable (lifting property), so $\mathrm{IC}_z$ with its canonical Weil structure is pulled back along the $\mathbb P^1$-bundle $\mathrm{Fl}\to LH/P_s$ [BBD, 4.2.6]; $f=\mathrm{tr}_n\mathrm{IC}_z$ is right $P_s$-invariant, so $f*(T_e+T_s)=(q^n+1)f$, i.e. $f*T_s=q^nf$, and $-\varepsilon_n(f)=q^n\varepsilon_n(f)$ forces $\varepsilon_n(f)=0$. If $z=\omega\in\Omega$, $\mathrm{IC}\omega=\Delta_\omega$ and $\varepsilon_n(T_\omega)=1$.
  4. Components. $K$ lies in the neutral component of $LH$ and $\overline{IxI}$ in the component of the $\Omega$-part of $x$, so $\mathrm{IC}x*c*\mathrm{IC}_y$ lives on one component, labelled $\omega_0$; no $\mathrm{IC}\omega$ with $\omega\neq\omega_0$ occurs.
  5. Vanishing coefficient. By Lemma 3.3(a) (two-sided) and Lemma 3.4(ii) over $\mathbb F_{q^n}$ (with $A^n$ in place of $A$, trace.tex footnote 1), $\mathrm{tr}n\mathrm{IC}_x*\mathbf 1{K_n}*\mathrm{tr}n\mathrm{IC}_y^{\vee}=\sum_zm^{(n)}_z\mathrm{tr}_n\mathrm{IC}_z$, $m^{(n)}_z=\sum_i(-1)^i\mathrm{tr}A{z,i}^n$. Apply $\varepsilon_n$: the left side is $0$ by step 1, the right side is $m^{(n)}{\omega_0}$ by steps 2–3. So $m^{(n)}{\omega_0}=0$ for all $n\ge1$.
  6. Weights. The generator is pure of weight $w_{xy}=\ell(x)+\ell(y)$; ${}^p\mathcal H^i$ is pure of weight $w_{xy}+i$ [BBD, 5.4.1]; its isotypic part $\mathrm{IC}{\omega_0}\otimes(E{\omega_0,i},A_{\omega_0,i})$ is a Weil direct summand (Lemma 3.4 step 3), hence pure, and its stalk at the point cell $I\omega_0I/I$ is $E_{\omega_0,i}$ in degree $0$. So every eigenvalue $\alpha$ of $A_{\omega_0,i}$ has $|\iota\alpha|=q^{(w_{xy}+i)/2}$: different $i$ give different $\alpha$.
  7. Vandermonde. $m^{(n)}{\omega_0}=\sum\alpha c_\alpha\alpha^n$ over the distinct eigenvalues, with $c_\alpha=(-1)^{i(\alpha)}\cdot$(multiplicity) $\neq0$ by step 5. Vanishing for $n=1,\dots,r$ ($r$ the number of $\alpha$) and $\det(\alpha_j^n)\neq0$ force $r=0$: all $E_{\omega_0,i}=0$. This is (a).
  8. (b). Lemma 3.5(a) with $\mathrm{IC}e=\Delta_e\notin\mathcal M_I$. For functions, $\kappa{\mathrm{IC}e}$ kills $\mathcal H_I*\varphi_0$ (Theorem 3.6(c)) and $\kappa{\mathrm{IC}_e}(T_e)=1$. $\square$

Differs from the pages. No Whittaker category, no averaging functor, no shift trick: the categorical sign character of Proposition II is replaced by the classical one over all $\mathbb F_{q^n}$ plus weights. The cancellation obstruction (checklist 6) is handled by purity: within one perverse degree all signs agree, across degrees absolute values differ. Fields: sheaves over $\overline{\mathbb F}q$, functions over $\mathbb F{q^n}(\!(t)\!)$.

Sources