§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:59:57Z

Audit 4.16 · Audit: vanishing eigenfunctionals, multiplicity one, extra endomorphisms, row relations

proved

Every part of Proposition 1.6 is correct. The converse of (S) assumes that \(C_K\) is finitely generated over \(\mathcal H_K\); the packet never proves this for any example in characteristic \(p\). It is proved here for the unitary twist of every split \(H\), so criterion (S) applies there.

Audit 4.16.

  1. (S) as stated holds, and its converse holds under the hypothesis that \(C_K\) is a finitely generated \(\mathcal H_K\)-module. That hypothesis is not established anywhere in the audited packet for \(F=\mathbb F_q(\!(t)\!)\). Proposition 1.8(g) correctly states it as a proviso.
  2. Lemma (finite generation for the unitary twist). For the unitary twist of a split connected reductive \(H\), let \(g_1,\dots,g_r\) generate the monoid \(\Lambda_H^+\) (Gordan's lemma) and \(E=\{\sum\epsilon_ig_i:\epsilon_i\in\{0,1\}\}\). Then the \(2^r\) indicators \(\mathbf 1_{O_\epsilon}\), \(\epsilon\in E\), generate \(C_K\) over \(\mathcal H_K\). Hence, by (S), when (SG) fails there is a Hecke eigenfunctional vanishing at \(\varphi_0\).
  3. (d), (b), (c) hold as stated, for every homogeneous \(X\) and every non-archimedean \(F\) ((d) needs an invariant measure for the statements about \(C^\infty(X(F))\), as stated).

Outcome: holds (with the finite-generation proviso made explicit and verified in the example class of Audit 4.5).

  1. (S) converse. \(\mathcal H_K\) is a finitely generated commutative \(\mathbb C\)-algebra (Satake, valid for unramified \(G\) over \(\mathbb F_q(\!(t)\!)\)). If \(Q=C_K/\mathcal H_K\ast\varphi_0\ne0\) is finitely generated, choose a maximal ideal \(\mathfrak m\) in its support. Nakayama gives \(Q/\mathfrak mQ=Q_{\mathfrak m}/\mathfrak mQ_{\mathfrak m}\ne0\); elements outside \(\mathfrak m\) act invertibly on \(Q/\mathfrak mQ\), and \(\mathcal H_K/\mathfrak m=\mathbb C\). A nonzero linear form on \(Q/\mathfrak mQ\) gives \(\Lambda\) with character \(\chi_{\mathfrak m}\) and \(\Lambda(\varphi_0)=0\). Without finite generation this step can fail, e.g. if \(Q=\mathfrak mQ\) for all \(\mathfrak m\).
  2. Proof of the Lemma. Every \(c\in\Lambda_H^+\) is \(\epsilon+2a\) with \(\epsilon\in E\), \(a\in\Lambda_H^+\): write \(c=\sum n_ig_i\) and split \(n_i=\epsilon_i+2\lfloor n_i/2\rfloor\). For dominant \(a\) and \(\epsilon\), \(kt^ak'\cdot t^\epsilon=k\,t^a\big(k't^\epsilon\vartheta(\sigma k')^{-1}\big)t^a\vartheta(\sigma k)^{-1}\in Kt^aKt^\epsilon Kt^aK\subseteq\bigsqcup_{\nu\le2a+\epsilon}Kt^\nu K\). So \(T_a\ast\mathbf 1_{O_\epsilon}\) is supported on \(\{O_\nu:\nu\le2a+\epsilon\}\) (Audit 4.3: the \(K\)-orbit is the Cartan type). Its value at \(t^{2a+\epsilon}=t^a\cdot t^\epsilon\) is \(\ge1\), from the coset \(t^aK\). Induction on the finite lower sets \(\{\nu\le c\}\) then puts every \(\mathbf 1_{O_c}\) in the submodule generated by the \(\mathbf 1_{O_\epsilon}\), \(\epsilon\in E\). For \(H=\mathbb G_m\) this is the rank-\(2\) module \(C_c(\mathbb Z)\) over \(\mathbb C[T_1^{\pm1}]\) of Audit 4.11.
  3. (d). Under (SG), \(\Lambda\mapsto\Lambda(\varphi_0)\) is injective on \(\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\). For irreducible unramified \(\pi\), restriction \(\ell\mapsto\ell|_{C_K}\) is injective because \(\ell(C_K)=e_K\ell(\mathcal S)=\pi^K\ne0\) for \(\ell\ne0\). For \(A\in\operatorname{Hom}_G(\pi,C^\infty(X(F)))\), \(\Omega=A(v_0)\) determines \(A\). With \(G\) unimodular and an invariant measure, \(\Lambda_\Omega(f\ast\varphi)=\chi_\pi(f^\vee)\Lambda_\Omega(\varphi)\), and \(\Omega\mapsto\Lambda_\Omega\) is injective (\(\Lambda_\Omega(\mathbf 1_{Kx})=\operatorname{vol}(Kx)\Omega(x)\)). The bound and \(\int_{X(\mathcal O)}\Omega\ne0\) follow from (S).
  4. (b). A cyclic module over a commutative ring has only the endomorphisms given by the ring. Sheaf-level source: a t-exact Frobenius-compatible \(\mathcal H\)-linear \(\sigma\) fixing \((\mathrm{IC}_0,\mathrm{can})\) fixes every Weil object \(V_\lambda\ast\mathrm{IC}_0\). It carries the \(\mathrm{IC}_\nu\)-isotypic part of \({}^p\mathcal H^i\) to the \(\mathrm{IC}_{s(\nu)}\)-isotypic part. So \(E_{s(\nu),i}(\lambda)\cong E_{\nu,i}(\lambda)\) with Frobenius rescaled by \(c_\nu\), giving \(M_{s(\nu)\lambda}=[c_\nu]^{\pm1}M_{\nu\lambda}\); the functional \(e^*_{s(\nu)}-c_\nu^{\pm n}e^*_\nu\ne0\) kills all columns. Correct.
  5. (c). The algebraic dual of \(\bigoplus\overline{\mathbb Q}_\ell\) is \(\prod\overline{\mathbb Q}_\ell\); columns are finitely supported; the finite-block version extends a vector of \(W^\perp\) by zero. Correct.
  6. Citations. Sakellaridis Cor. 8.0.3 (geometric endomorphisms \(\cong\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\)) and Cor. 8.0.4 (under surjectivity of the restriction map) say what is claimed (sak.txt, lines 3909–3950).

Builds on

  • Proposition 1.6 · Forcing criteria: vanishing eigenfunctions, multiplicity two, extra endomorphisms

Built on by

  • Summary 4.20 · Verdict of the second audit: the mechanism claims hold, with minor repairs
  • Theorem 6.3 · Verdict: sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\)

Sources