§5 · Agent 5 · Audit: split candidate · 2026-10-17T23:20:36Z

Audit 5.1 · Audit of A3-1: (SG) forces \(\dim\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\le 1\)

proved

A3-1 holds over every non-archimedean local field, in particular \(\mathbb F_q(\!(t)\!)\); the only defect is that the change of variables \(y\mapsto y^{-1}\) must be made in the integral form of the Hecke action.

Audit 5.1.

Outcome: holds with a minor defect. Let \(F\) be a non-archimedean local field of any characteristic, \(X=G/H\) with \(H\) smooth and unimodular and \(X(F)=G(F)x_0\). If \(C_K=\mathcal H_K\ast\varphi_0\), then \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\le1\) for every character \(\chi\), and \(\dim\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\le1\) for every irreducible unramified \(\pi\).

Re-derivation.

  1. (a). If \(C_K=\mathcal H_K\ast\varphi_0\), a functional \(\Lambda\) with \(\Lambda(h\ast\varphi)=\chi(h)\Lambda(\varphi)\) is determined by \(\Lambda(\varphi_0)\). This holds without using commutativity. Commutativity of \(\mathcal H_K\) (Satake) is used only to show that every endomorphism is \(h_0\ast(-)\). Correct.
  2. Periods give eigenfunctions. For \(gx_0=g'x_0\) we have \(g'=gh\) with \(h\in H(F)\), so \(\Psi_\ell(g'x_0)=\ell(\pi(h)^{-1}\pi(g)^{-1}v_0)=\Psi_\ell(gx_0)\). \(K\)-invariance comes from \(\pi(k)v_0=v_0\). Writing \((f\ast\Psi)(gx_0)=\int_Gf(y)\Psi(y^{-1}gx_0)\,dy=\ell(\pi(g)^{-1}\pi(f)v_0)\) and using \(\pi(f)v_0=\chi_\pi(f)v_0\) gives the eigen-equation. The map \(\ell\mapsto\Psi_\ell\) is injective because \(\pi(G)v_0\) spans the irreducible \(\pi\). Correct.
  3. Eigenfunctions give eigenfunctionals (the defect). Step 4 of A3-1 substitutes \(y\mapsto y^{-1}\) in \(\sum_{yK}f(y)\int_X\varphi(x)\Psi(yx)\,dx\). The summand \(y\mapsto f(y)\Psi(yx)\) is not right \(K\)-invariant before integrating against \(\varphi\), so the substitution must be made in the integral form:
$$\Lambda_\Psi(f\ast\varphi)=\int_G f(y)\int_X\varphi(x)\Psi(yx)\,dx\,dy=\int_X\varphi(x)\int_G f(y^{-1})\Psi(y^{-1}x)\,dy\,dx=\Lambda_{f^\vee\ast\Psi}(\varphi),$$

using the \(G(F)\)-invariant measure on \(X(F)\), unimodularity of \(G(F)\) and \(\operatorname{vol}K=1\). The integrand has compact support, because \(f\) and \(\varphi\) do. With this repair the identity holds.

  1. Bijection. \(C_K\) has the basis \(\mathbf 1_{Kx}\) of indicators of compact open \(K\)-orbits, and \(\Lambda_\Psi(\mathbf 1_{Kx})=\operatorname{vol}(Kx)\Psi(x)\), so \(\Psi\mapsto\Lambda_\Psi\) is a linear bijection onto \(\operatorname{Hom}_{\mathbb C}(C_K,\mathbb C)\). It carries \(\chi\)-eigenfunctions to \(\chi\circ\vee\)-eigenfunctionals, and \(\chi\circ\vee\) is a character because \(\vee\) is an anti-automorphism of the commutative \(\mathcal H_K\). Hence (c).

Validity in characteristic \(p\). Only facts about \(\ell\)-groups are used:

None of these needs characteristic \(0\). Sakellaridis's Cor. 8.0.4(2) proves the same implication under his assumptions, which include characteristic 0; A3-1 does not need them.

Builds on

  • Lemma 3.1 · (SG) forces \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\le1\) over every local field

Built on by

  • Audit 5.5 · Audit of A3-5: two periods on \(\pi\boxtimes\pi\), so \(\mathcal H_K\ast\varphi_0\ne C_K\)
  • Summary 5.12 · Verdict: (SG) fails, proved; \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) modulo (O1)–(O3)

Sources