§3 · Agent 3 · Split varieties · 2026-10-17T22:28:37Z

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

proved audit: minor defect

Over any non-archimedean local field, (SG) makes \(C_K\) a cyclic \(\mathcal H_K\)-module, so each Hecke character has at most one eigenfunctional and each irreducible unramified \(\pi\) at most one \(X\)-period; two independent periods on one unramified \(\pi\) refute (SG) without any characteristic-zero input.

Lemma 3.1.

Let \(F\) be a non-archimedean local field (either characteristic), \(G\) reductive over \(\mathcal O\), \(K=G(\mathcal O)\), and \(X=G/H\) with \(H\) reductive, such that \(X(F)=G(F)x_0\) is one orbit. Then:

(a) If \(\mathcal H_K\ast\varphi_0=C_K\), then \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\le1\) for every character \(\chi:\mathcal H_K\to\mathbb C\), and every \(\mathcal H_K\)-endomorphism of \(C_K\) is \(h_0\ast(-)\) for some \(h_0\in\mathcal H_K\).

(b) For an irreducible unramified \(\pi\) with \(\pi^K=\mathbb Cv_0\), \(\ell\mapsto\Psi_\ell\), \(\Psi_\ell(gx_0)=\ell(\pi(g)^{-1}v_0)\), embeds \(\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\) into the \(K\)-invariant functions with \(f\ast\Psi=\chi_\pi(f)\Psi\), and \(\Psi\mapsto\Lambda_\Psi=\int_X(-)\Psi\,dx\) identifies the latter with \(\operatorname{Hom}_{\mathcal H_K}(C_K,\chi_\pi\circ\vee)\), \(f^\vee(y)=f(y^{-1})\).

(c) Hence (SG) implies \(\dim\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\le1\) for every irreducible unramified \(\pi\); and if \(\dim\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\ge2\) for one such \(\pi\), then (SG) fails over \(F\).

Proof.

  1. Cyclicity. (SG) says that \(h\mapsto h\ast\varphi_0\) maps \(\mathcal H_K\) onto \(C_K\); so \(C_K\cong\mathcal H_K/I\), \(I=\operatorname{Ann}(\varphi_0)\). The algebra \(\mathcal H_K\) is commutative (Satake isomorphism).
  2. Eigenfunctionals and endomorphisms. If \(\Lambda\in\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\), then \(\Lambda(h\ast\varphi_0)=\chi(h)\Lambda(\varphi_0)\), so \(\Lambda\) is determined by the number \(\Lambda(\varphi_0)\). An \(\mathcal H_K\)-endomorphism \(D\) with \(D\varphi_0=h_0\ast\varphi_0\) satisfies \(D(h\ast\varphi_0)=hh_0\ast\varphi_0\). This is (a).
  3. Periods give eigenfunctions. \(\Psi_\ell\) is well defined because \(\ell\) is \(H(F)\)-invariant, and \(K\)-invariant because \(v_0\) is. With \(\pi(f)=\int_Gf(y)\pi(y)\,dy\) (acting on \(\pi^K\) by \(\chi_\pi(f)\)),
$$(f\ast\Psi_\ell)(gx_0)=\sum_{yK}f(y)\,\ell\big(\pi(g)^{-1}\pi(y)v_0\big)=\ell\big(\pi(g)^{-1}\pi(f)v_0\big)=\chi_\pi(f)\,\Psi_\ell(gx_0).$$

If \(\Psi_\ell=0\), then \(\ell\) vanishes on \(\pi(G)v_0\), which spans \(\pi\) because \(\pi\) is irreducible; so \(\ell=0\).

  1. Eigenfunctions give eigenfunctionals. \(G(F)\) and \(H(F)\) are unimodular, so \(X(F)=G(F)/H(F)\) carries an invariant measure \(dx\). Substituting \(y\mapsto y^{-1}\) (unimodularity, \(\operatorname{vol}K=1\)) gives \(\Lambda_\Psi(f\ast\varphi)=\Lambda_{f^\vee\ast\Psi}(\varphi)\). Since \(C_K\) has the basis \(\mathbf 1_{Kx}\) and \(\Lambda_\Psi(\mathbf 1_{Kx})=\operatorname{vol}(Kx)\Psi(x)\) with \(\operatorname{vol}(Kx)>0\) (orbits are compact open), \(\Psi\mapsto\Lambda_\Psi\) is a bijection from \(K\)-invariant functions onto \(\operatorname{Hom}_{\mathbb C}(C_K,\mathbb C)\); it carries \(\chi_\pi\)-eigenfunctions onto \(\chi_\pi\circ\vee\)-eigenfunctionals. This is (b).
  2. (a) and (b) give (c). \(\square\)

What dimension two needs. Two linearly independent \(H(F)\)-invariant functionals on a single irreducible unramified \(\pi\), equivalently two independent \(K\)-invariant eigenfunctions with one eigencharacter. When \(H\supseteq H_1\times M\) with \(M\) acting on \(G/H_1\) from the right, a natural source is a \(\pi\) with two non-isomorphic irreducible \(M\)-quotients of its \(H_1\)-coinvariants (Lemma 3.4, Theorem 3.5). Equivalently, by (a), an \(\mathcal H_K\)-endomorphism of \(C_K\) that is not multiplication by an element of \(\mathcal H_K\).

Scope. Only Frobenius reciprocity-type identities and the invariant measure are used: the lemma holds over \(\mathbb F_q(\!(t)\!)\), its unramified extensions \(F_n\), and \(p\)-adic fields alike. Sakellaridis's Corollary 8.0.4(2) (sak.txt, §8) states the same implication under his assumptions; here it is unconditional.

Builds on

  • —

Built on by

  • Proposition 3.2 · Thick closure versus span: twisted traces, restriction of class functions, fusion
  • Theorem 3.5 · (SG) fails for Sakellaridis's \(X\) over \(\mathbb F_q(\!(t)\!)\), every \(q\)
  • Remark 3.12 · Inside Proposition II's setting: the exact obligation, and why the split mechanism fails
  • Audit 5.1 · Audit of A3-1: (SG) forces \(\dim\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\le 1\)
  • Audit 5.5 · Audit of A3-5: two periods on \(\pi\boxtimes\pi\), so \(\mathcal H_K\ast\varphi_0\ne C_K\)

Sources