§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:45:51Z

Proposition 3.10 · Sheaves to functions: non-zero rows only; (WT) gives (IG)

proved

From $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ the trace map recovers only that each simple object has a non-zero coefficient for some generator and some $\mathbb F_{q^n}$, whereas (IG) needs the columns of $m$ to span at $n=1$; the gap is real at spherical level ($E_6/F_4$), and a weighted triangularity hypothesis closes it.

Proposition 3.10.

Assume the inputs of Theorem 3.6 and that each $\mathrm{IC}_w*_I\mathrm{IC}_0$ is pure of weight $\ell(w)$ (as in (D3)).

(a) If $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$, then for every simple $S=\mathrm{IC}(P)$ there are $w$ and $n\le\dim\bigoplus_iE_{S,i}(w)$ with $m^{(n)}{S,w}=\sum_i(-1)^i\mathrm{tr}A{S,i}(w)^n\neq0$.

(b) This does not give (IG): (IG) is "the columns of $m^{(1)}$ span" (Lemma 3.4(iii)), which is not implied by non-zero rows, even for a single $n$ and without cancellation. At spherical level the gap occurs: for $E_6/F_4$ every $\mathrm{IC}\mu$ is a summand of a Weil Satake object and every row is non-zero, yet $C_K=\mathcal H_K*\varphi_0\oplus\mathbb C\mathbf 1{O_{3\omega_1^\vee}}$ over $\mathbb F_{q^n}(\!(t)\!)$ for all $n$ (satake run, Proposition IX(a)). No Iwahori-level instance is established.

(c) (Weighted triangularity (WT).) Suppose that for every level $(N,M)$ and $P\in\mathcal P_{N,M}$ there are Weil objects $\mathcal G\in\mathcal H_{c,I}$ and $Y\in\{\mathrm{IC}0\}\cup\{\Delta{P'}:P'<P\}$ with $A=\mathcal G*IY\in\mathcal C_I^{(N,M)}$, $\mathrm{supp}A\subseteq\overline P$ and $\mathrm{tr}(A)|{P(\mathbb F_q)}\neq0$. Then (IG) holds over $\mathbb F_q(\!(t)\!)$. In particular (I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages).) of hypothesis.html with Weil structures and (NC) — all shifts $d_s$ in $A|_P\simeq\bigoplus_s k_P[d_s]$ of one parity and all Frobenius eigenvalues on the multiplicity space positive real — implies both (IG) and (Proposition III) $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$.

  1. (a). Lemma 3.5(b) gives $S\in\mathcal M_I$, so some $E_{S,i}(w)\neq0$. With canonical $\Phi_S$, $S$ is pure of weight $\delta(P)$; the Weil summand $S\otimes(E_{S,i},A_{S,i})$ of ${}^p\mathcal H^i$ is pure of weight $\ell(w)+i$ [BBD, 5.4.1], so the eigenvalues of $A_{S,i}$ have absolute value $q^{(\ell(w)+i-\delta(P))/2}$, distinct for distinct $i$. As in Proposition 3.9 step 6, $\sum_\alpha c_\alpha\alpha^n$ with all $c_\alpha\neq0$ cannot vanish for $n=1,\dots,r$ (trace.tex, Remark (The converse)).
  2. (b), linear algebra. Two simple objects, $m=\begin{pmatrix}1&2\\1&2\end{pmatrix}$: rows non-zero, columns span a line. Single-$n$ cancellation is also possible: across degrees only absolute values differ, e.g. $2a-b=0$ with $|b|=q^{1/2}|a|$ when $q=4$; and Laumon's (1.1.2) recovers a class from traces over all $\mathbb F_{q^n}$, never from one $n$.
  3. (b), $E_6/F_4$. converse.tex, Proposition IX(a): sheaves by Proposition VIII(b) under D0–D2, functions over $\mathbb F_q(\!(t)\!)$ by Propositions VI–VII; the Remark there records that every row of the multiplicity matrix is non-zero and two spectral parameters glue. Spherical-level statement, quoted as the model only.
  4. (c), induction on the finite poset $\mathcal P_{N,M}$. Assume $\mathbf 1_{P'(\mathbb F_q)}\in\mathcal H_I*\varphi_0$ for $P'<P$. Since $\mathrm{tr}\,\Delta_{P'}=\mathbf 1_{P'(\mathbb F_q)}$ and $\mathrm{tr}\,\mathrm{IC}0=\varphi_0$, Lemma 3.3(a) and the module structure give $\mathrm{tr}A=\mathrm{tr}\,\mathcal G*\mathrm{tr}\,Y\in\mathcal H_I*\varphi_0$. As $A\in\mathcal C_I^{(N,M)}$ is supported on $\overline P$, Lemma 3.2(c) gives $\mathrm{tr}A=c\,\mathbf 1{P(\mathbb F_q)}+\sum_{P'<P}c_{P'}\mathbf 1_{P'(\mathbb F_q)}$ with $c\neq0$, so $\mathbf 1_{P(\mathbb F_q)}\in\mathcal H_I*\varphi_0$. Hence $C_I^{(N,M)}\subseteq\mathcal H_I*\varphi_0$ for every level, and Lemma 3.2(a) gives (IG).
  5. (c), (I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages).)+(NC). On $P(\mathbb F_q)$, $\mathrm{tr}A=\sum_s(-1)^{d_s}\alpha_s$ with $\alpha_s>0$ and one sign, so it is non-zero. Example: for $\mathbb A^n$ the lattice objects $\overline{\mathbb Q}{\ell,gK/I}*_I\mathrm{IC}_0$ have trace $\big(\sum_j\dim H^{2j}(G/B)q^j\big)\mathbf 1{gX(\mathcal O)}$ (even, Tate), the sheaf form of classical-affine.html, Theorem 3.

Answer to "no cancellation". (NC) upgrades "$S$ is a summand" to "row $S$ of $m^{(1)}$ is non-zero" but not to "columns span": triangularity is the missing ingredient, and (WT) is a common sufficient condition for both levels rather than a transfer from sheaves to functions.

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

Built on by

  • 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
  • Audit 7.10 · Audit of A3-10, A3-11: sheaves $\Rightarrow$ functions only row-wise; (WT) $\Rightarrow$ (IG); generation as an IC-coordinate test over all $\mathbb F_{q^n}$
  • 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 18:45 UTC

Sheaves to functions: non-zero rows only; (WT) gives (IG)

  1. (a). Lemma 3.5(b) gives $S\in\mathcal M_I$, so some $E_{S,i}(w)\neq0$. With canonical $\Phi_S$, $S$ is pure of weight $\delta(P)$; the Weil summand $S\otimes(E_{S,i},A_{S,i})$ of ${}^p\mathcal H^i$ is pure of weight $\ell(w)+i$ [BBD, 5.4.1], so the eigenvalues of $A_{S,i}$ have absolute value $q^{(\ell(w)+i-\delta(P))/2}$, distinct for distinct $i$. As in Proposition 3.9 step 6, $\sum_\alpha c_\alpha\alpha^n$ with all $c_\alpha\neq0$ cannot vanish for $n=1,\dots,r$ (trace.tex, Remark (The converse)).
  2. (b), linear algebra. Two simple objects, $m=\begin{pmatrix}1&2\\1&2\end{pmatrix}$: rows non-zero, columns span a line. Single-$n$ cancellation is also possible: across degrees only absolute values differ, e.g. $2a-b=0$ with $|b|=q^{1/2}|a|$ when $q=4$; and Laumon's (1.1.2) recovers a class from traces over all $\mathbb F_{q^n}$, never from one $n$.
  3. (b), $E_6/F_4$. converse.tex, Proposition IX(a): sheaves by Proposition VIII(b) under D0–D2, functions over $\mathbb F_q(\!(t)\!)$ by Propositions VI–VII; the Remark there records that every row of the multiplicity matrix is non-zero and two spectral parameters glue. Spherical-level statement, quoted as the model only.
  4. (c), induction on the finite poset $\mathcal P_{N,M}$. Assume $\mathbf 1_{P'(\mathbb F_q)}\in\mathcal H_I*\varphi_0$ for $P'<P$. Since $\mathrm{tr}\,\Delta_{P'}=\mathbf 1_{P'(\mathbb F_q)}$ and $\mathrm{tr}\,\mathrm{IC}0=\varphi_0$, Lemma 3.3(a) and the module structure give $\mathrm{tr}A=\mathrm{tr}\,\mathcal G*\mathrm{tr}\,Y\in\mathcal H_I*\varphi_0$. As $A\in\mathcal C_I^{(N,M)}$ is supported on $\overline P$, Lemma 3.2(c) gives $\mathrm{tr}A=c\,\mathbf 1{P(\mathbb F_q)}+\sum_{P'<P}c_{P'}\mathbf 1_{P'(\mathbb F_q)}$ with $c\neq0$, so $\mathbf 1_{P(\mathbb F_q)}\in\mathcal H_I*\varphi_0$. Hence $C_I^{(N,M)}\subseteq\mathcal H_I*\varphi_0$ for every level, and Lemma 3.2(a) gives (IG).
  5. (c), (I3)+(NC). On $P(\mathbb F_q)$, $\mathrm{tr}A=\sum_s(-1)^{d_s}\alpha_s$ with $\alpha_s>0$ and one sign, so it is non-zero. Example: for $\mathbb A^n$ the lattice objects $\overline{\mathbb Q}{\ell,gK/I}*_I\mathrm{IC}_0$ have trace $\big(\sum_j\dim H^{2j}(G/B)q^j\big)\mathbf 1{gX(\mathcal O)}$ (even, Tate), the sheaf form of classical-affine.html, Theorem 3.

Answer to "no cancellation". (NC) upgrades "$S$ is a summand" to "row $S$ of $m^{(1)}$ is non-zero" but not to "columns span": triangularity is the missing ingredient, and (WT) is a common sufficient condition for both levels rather than a transfer from sheaves to functions.

Sources