§1 · Agent 1 · Trace formalism · 2026-10-17T21:09:41Z

Proposition 1.8 · Under (R), (C), D0–D3: three mechanisms vanish; the obligation sits on uncovered labels

proved audit: minor defect ★ key step

Under (R) and (C), Frobenius fixes all labels, \(X(F)\) is a single \(G(F)\)-orbit, \(\mathcal H\)-linear group automorphisms act trivially on \(C_K\), and every column of \(m^{(n)}\) has a nonzero top entry for every \(n\). (SG) reduces to a spanning condition on the uncovered labels \(U=\Lambda_S^+\smallsetminus\varphi(\Lambda_T^+)\); coverage gives \((\mathrm{SG})_n\) for all \(n\), and \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) does not decide the rest.

Proposition 1.8.

Assume (R), (C), D0–D3, \(q\) odd.

(a) Frobenius acts trivially on \(\Lambda_T^+\) and \(\Lambda_S^+\); the label mechanism of Counterexample 1.5 is void.

(b) \(X(F_n)=G(F_n)\cdot x_0\) for every \(n\); the several-orbit mechanism (\(\mathrm{GL}_n/\mathrm O_n\), unitary instance of Counterexample 1.5) is void.

(c) An automorphism \(\epsilon\) of \((G,\theta)\) over \(\mathbb F_q\) with \(\epsilon^*V_\lambda\cong V_\lambda\) for all \(\lambda\) fixes every \(K_n\)-orbit, hence acts trivially on \(C_{K_n}\); the geometric form of criterion (b) of Proposition 1.6 is void, and so are the degree-0 automorphisms of \(\mathrm{id}_{\mathcal C_k}\) coming from \(\pi_0\) of stabilizers (the \(\varepsilon\) of \(\mathrm{SL}_2/\mathrm{SO}_2\)).

(d) For every \(\lambda\) and \(n\), $$m^{(n)}_{\varphi(\lambda),\lambda}=(-1)^{\delta(\varphi(\lambda))+\langle2\rho,\lambda\rangle}\sum_{\mu\le\lambda}a_{\lambda\mu}(q^n)\,(T_\mu\ast\varphi_0)(t^\lambda x_0)\neq0,$$ with \(a_{\lambda\lambda}=1\) and \(a_{\lambda\mu}\in\mathbb N[x]\) the Kazhdan–Lusztig polynomials of Zhu (5.6.2).

(e) Choose \(\lambda_\mu\in\varphi^{-1}(\mu)\) for each covered \(\mu\). Then \(\{T_{\lambda_\mu}\ast\varphi_0\}_{\mu\notin U}\cup\{\mathbf 1_{\mathcal O_\nu}\}_{\nu\in U}\) is a basis of \(C_K\); with \(p_U\) the projection onto \(\mathbb C^{(U)}\) along the first family, $$(\mathrm{SG})\iff\{p_U(T_\lambda\ast\varphi_0):\lambda\in\Lambda_T^+\}\ \text{spans}\ \mathbb C^{(U)} .$$ In particular \(U=\varnothing\Rightarrow(\mathrm{SG})_n\) for all \(n\) (Corollary C of the classical page).

(f) If for some \(\mu,\lambda\) every eigenvalue of \(A_{\mu,i}(\lambda)\) equals \(q^{(w+i-w_\mu)/2}\) (no root-of-unity factor) and \(E_{\mu,i}(\lambda)\neq0\) only for \(i\) of one parity, then \(m^{(n)}_{\mu\lambda}\neq0\) for every \(n\). Hence a triangular summand system — a map \(\mu\mapsto\lambda_\mu\) and an order \(\prec\) with finite lower sets, with \(M_{\nu\lambda_\mu}\neq0\Rightarrow\nu\preceq\mu\) and \(E_{\mu,\bullet}(\lambda_\mu)\neq0\) of this split Tate type — gives \((\mathrm{SG})_n\) for all \(n\).

(g) The exact remaining obligation for "\(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\Rightarrow(\mathrm{SG})\)" here is the spanning condition (e) at the given \(q\); by Proposition 1.6(S) it fails iff (given finite generation of \(C_K\)) some Hecke eigenfunctional vanishes at \(\varphi_0\). The mechanisms not excluded by (a)–(c) are gluing (two spectral points over one character, Remark 1.10) and cancellation at \(\mathrm{ev}_1\) inside the multiplicity spaces (an alternating sum \(\sum_i(-1)^i\operatorname{tr}A_{\mu,i}(\lambda)\) vanishing at \(n=1\): root-of-unity factors, e.g. Frobenius permuting top-dimensional components of a convolution fibre as in Example 1.4(E4), non-Tate classes, or coincidences between degrees of opposite parity; all excluded under the hypothesis of (f)).

Proofs.

  1. (a). \(G\) and \(S\) are split, so Frobenius is trivial on \(X_*(T)\) and \(X_*(S)\).
  1. (b). Put \(T_\theta=(T^\theta)^\circ\). Since \(L=Z_G(S)=S\cdot Z_{G^\theta}(S)\) with finite intersection \(S[2]\) (criterion page, step 4), \(\operatorname{rank}Z_{G^\theta}(S)=\dim T-\dim S=\dim T_\theta\); so \(T_\theta\) is a maximal torus of the connected (by (C)) reductive group \(Z_{G^\theta}(S)\), and \(T^\theta\subseteq Z_{Z_{G^\theta}(S)}(T_\theta)=T_\theta\): \(T^\theta\) is connected, and split as a subtorus of the split \(T\). For \(\mu\in\Lambda_S\), \(x_\mu=\mu(\sqrt t)x_0\) and the cocycle \(\gamma\mapsto\mu(\sqrt t)^{-1}\gamma(\mu(\sqrt t))=\mu(\pm1)\) takes values in \(S[2]\subseteq T^\theta=T_\theta\subseteq H\). Its class in \(H^1(F_n,H)\) is the image of a class in \(H^1(F_n,T_\theta)=0\) (Hilbert 90), so \(x_\mu\in G(F_n)x_0\). By IIb step 1, \(X(F_n)=\bigsqcup_\mu K_nx_\mu\); hence \(X(F_n)=G(F_n)x_0\).
  1. (c). \(\epsilon^*V_\lambda\cong V_\lambda\) forces \(\epsilon\) to fix every dominant coweight (the supports \(\mathrm{Gr}_{\le\lambda}\) are distinct). Then \(\tau(\epsilon x_\mu)=\epsilon(t^\mu)\in L^+Gt^\mu L^+G\), and since \(\tau(\mathcal O_\nu)\subseteq L^+Gt^\nu L^+G\) with disjoint double cosets, \(\epsilon(\mathcal O_\mu)=\mathcal O_\mu\). The \(K_n\)-orbits are the \(\mathcal O_\mu(\mathbb F_{q^n})\) (IIb step 1), so each is preserved. Under (C), \(\pi_0\operatorname{Stab}(x_\mu)=1\), so no degree-0 automorphism of \(\mathrm{id}_{\mathcal C_k}\) arises from stabilizers.
  1. (d). \(\varphi(\lambda)\) is the top label in the support (D0), and \(\mathrm{tr}_n\mathrm{IC}_\nu(x_{\varphi(\lambda)})=0\) for \(\nu<\varphi(\lambda)\), so evaluating at \(x_{\varphi(\lambda)}=t^\lambda x_0\): \(m^{(n)}_{\varphi(\lambda),\lambda}=(-1)^{\delta(\varphi(\lambda))}\big(\mathrm{tr}_nV_\lambda\ast\varphi_0\big)(t^\lambda x_0)\). By Zhu (5.6.2) over \(\mathbb F_{q^n}\), \(\mathrm{tr}_nV_\lambda=(-1)^{\langle2\rho,\lambda\rangle}\sum_{\mu\le\lambda}a_{\lambda\mu}(q^n)T_\mu\) with \(a_{\lambda\mu}\in\mathbb N[x]\), \(a_{\lambda\lambda}=1\). All \((T_\mu\ast\varphi_0)(t^\lambda x_0)\) are non-negative integers and the \(\mu=\lambda\) term is \(\ge1\) (Lemma A(2) of the classical page): the sum is positive.
  1. (e). By D0 and Lemma A(2), \(T_{\lambda_\mu}\ast\varphi_0=N_\mu\mathbf 1_{\mathcal O_\mu}+\sum_{\nu<\mu}(\ast)\mathbf 1_{\mathcal O_\nu}\), \(N_\mu\ge1\). Lower sets are finite, so by induction the stated family spans each \(\operatorname{span}\{\mathbf 1_{\mathcal O_\nu}:\nu\le\kappa\}\) and is independent: a basis. \(\mathcal H_K\ast\varphi_0\) contains \(\ker p_U\), so it equals \(C_K\) iff its image under \(p_U\) is \(\mathbb C^{(U)}\), and \(p_U(\mathcal H_K\ast\varphi_0)\) is spanned by the \(p_U(T_\lambda\ast\varphi_0)\) (those with \(\lambda=\lambda_\mu\) vanish). For \(U=\varnothing\) the same holds over every \(F_n\).
  1. (f). Let \(i_0\) be one index with \(E_{\mu,i_0}(\lambda)\neq0\). On the indices with \(E_{\mu,i}(\lambda)\neq0\), \((-1)^i=(-1)^{i_0}\) and \((i-i_0)/2\in\mathbb Z\), so \(m^{(n)}_{\mu\lambda}=(-1)^{i_0}\big(q^{1/2}\big)^{n(w+i_0-w_\mu)}\sum_i\dim E_{\mu,i}(\lambda)\,q^{n(i-i_0)/2}\), a nonzero number times a sum of positive terms. Given the triangular system, \(\mathrm{tr}_n(V_{\lambda_\mu}\ast\mathrm{IC}_0)=m^{(n)}_{\mu\lambda_\mu}\mathrm{tr}_n\mathrm{IC}_\mu+\sum_{\nu\prec\mu}(\ast)\mathrm{tr}_n\mathrm{IC}_\nu\) with nonzero leading coefficient, and induction on the finite lower sets of \(\prec\) puts every \(\mathrm{tr}_n\mathrm{IC}_\mu\) in the span (Proposition B of the classical page, in the IC basis).
  1. (g). Combine (e) with Proposition 1.6(S). Mechanisms (a)–(c) are excluded; Theorem 1.2(i) shows \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) is the non-vanishing of rows of \(M\), which constrains neither the rank of the \(U\)-reduced system nor the root-of-unity parts of the \(A_{\mu,i}(\lambda)\). For the known non-covered families: \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) satisfy (SG) (Theorem 4 of the run); \(E_6/F_4\) is excluded from this question and open over \(\mathbb F_q(\!(t)\!)\). \(\square\)

Builds on

  • Theorem 1.2 · Theorem A: decategorification is base change of one \(R\)-matrix
  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion
  • Proposition 1.6 · Forcing criteria: vanishing eigenfunctions, multiplicity two, extra endomorphisms

Built on by

  • Dead end 1.11 · Dead end: forcing \((\mathrm{SG})_1\) to fail inside (R)+(C) by cancellation or gluing
  • Summary 1.12 · Verdict on part (2), with consequences for (1) and (3)
  • Audit 4.18 · Audit: under (R), (C), D0–D3 the twisting mechanisms vanish
  • Theorem 6.3 · Verdict: sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\)

Sources