§3 · Agent 3 · Split varieties · 2026-10-17T22:30:49Z

Theorem 3.5 · (SG) fails for Sakellaridis's \(X\) over \(\mathbb F_q(\!(t)\!)\), every \(q\)

proved audit: correct ★ key step

For a generic unramified \(\pi\) of \(\mathrm{Sp}_4(F)\), \(\pi\boxtimes\pi\) carries two independent \(H(F)\)-invariant functionals, one through each intermediate quotient \(\sigma_{c_1},\sigma_{c_2}\); by the cyclicity criterion \(\mathcal H_K\ast\varphi_0\ne C_K\). The proof is characteristic-free and uses neither Sakellaridis's (A1)–(A5) nor his Theorem 8.0.2.

Theorem 3.5.

Let \(X=G/H\) be as in Lemma 3.3 and \(F\) any non-archimedean local field with ring of integers \(\mathcal O\) — in particular \(F=\mathbb F_{q^n}(\!(t)\!)\) for any \(q\) and \(n\). For \(\chi\) outside a proper Zariski-closed subset of the unramified characters of the torus of \(\mathrm{Sp}_4\), let \(\pi=I(\chi)\) and \(\Pi=\pi\boxtimes\pi\). Then $$\dim\operatorname{Hom}_{H(F)}(\Pi,\mathbb C)\ge2,\qquad\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi_\Pi\circ\vee)\ge2,\qquad \mathcal H_K\ast\varphi_0\subsetneq C_K .$$ So \((\mathrm{SG})\) and \((\mathrm{SG})_n\) fail for \(X\) for every \(q\) and every \(n\).

Proof.

  1. Setup. \(H(F)=\mathrm{Sp}(P_1)\times\mathrm{Sp}(P_2)\times\mathrm{SL}_2^{(3)}\) with \(P_i=W_i^{0\perp}\) and \(\mathrm{SL}_2^{(3)}=\{(g,\varphi_0g\varphi_0^{-1})\}\subset\mathrm{Sp}(W_1^0)\times\mathrm{Sp}(W_2^0)\) (Lemma 3.3(a)); \(X(F)=G(F)/H(F)\) (Lemma 3.3(b)).
  2. Two quotients on each side. Apply Lemma 3.4 to \(V_i=P_i\oplus W_i^0\): for generic \(\chi\) there are nonzero maps \(f_j:\pi\to\sigma_{c_j}\) (\(j=1,2\)), invariant under \(\mathrm{Sp}(P_1)\) and equivariant for \(\mathrm{Sp}(W_1^0)\), and likewise \(f_j':\pi\to\sigma_{c_j}\) for the second copy, with \(\mathrm{Sp}(W_2^0)\) identified with \(\mathrm{Sp}(W_1^0)\) through \(\varphi_0\).
  3. Functionals. \(\sigma_{c_j}\) is irreducible and \(\sigma_{c_j}^\vee\cong\sigma_{c_j^{-1}}\cong\sigma_{c_j}\), so there is a nonzero \(\mathrm{SL}_2(F)\)-invariant pairing \(\beta_j\) on \(\sigma_{c_j}\). Put \(\ell_j(v\otimes w)=\beta_j(f_j(v),f_j'(w))\). For \((h_1,h_2,g)\in H(F)\): \(h_1,h_2\) act trivially after \(f_j,f_j'\), and \(g\) acts diagonally, preserved by \(\beta_j\). So \(\ell_j\in\operatorname{Hom}_{H(F)}(\Pi,\mathbb C)\).
  4. Independence. \(F_\bullet=(f_1,f_2):\pi\to\sigma_{c_1}\oplus\sigma_{c_2}\) is onto: its image is an \(\mathrm{SL}_2\)-subrepresentation projecting onto both factors, which are irreducible and non-isomorphic (Lemma 3.4). Likewise \(F'_\bullet\). So \(\Pi\twoheadrightarrow(\sigma_{c_1}\oplus\sigma_{c_2})^{\otimes2}\), and \(\ell_1,\ell_2\) are pullbacks of \(\beta_1\circ\mathrm{pr}_{11}\) and \(\beta_2\circ\mathrm{pr}_{22}\), which are nonzero on different direct summands. Pullback along a surjection preserves independence.
  5. Conclusion. \(\Pi\) is irreducible and unramified for \(G(F)\) (outer tensor product of irreducible unramified representations). By Lemma 3.1(b), \(\ell_1,\ell_2\) give two independent \(K\)-invariant \(\chi_\Pi\)-eigenfunctions and two independent eigenfunctionals on \(C_K\); by Lemma 3.1(c) \(C_K\) is not cyclic, so \(\mathcal H_K\ast\varphi_0\ne C_K\). The same argument runs over every \(F_n\). \(\square\)

Remarks. (i) This is the multiplicity two predicted by Sakellaridis's Example 8.0.1 and his remark after Corollary 8.0.4: the \(X\)-parameters \((a,b,a)\) and \((b,a,b)\) of Lemma 3.3(d) correspond to \(\sigma_{c_1}\) and \(\sigma_{c_2}\). (ii) The failure holds for every residue characteristic, so it lives in the same characteristic as the sheaf side over \(\overline{\mathbb F}_q\). (iii) It occurs on a 2-parameter family of eigencharacters (the fused divisor), not at isolated \(q\)-dependent points as for \(E_6/F_4\).

Builds on

  • Lemma 3.1 · (SG) forces \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\le1\) over every local field
  • Lemma 3.3 · A model of \(X=\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) and its fusion divisor
  • Lemma 3.4 · Generic \(I(\chi)\) of \(\mathrm{Sp}_4\) has two \(\mathrm{SL}_2\)-quotients

Built on by

  • Remark 3.11 · No trace-zero shape for \(X\); the failure is \(q\)- and \(n\)-independent
  • Remark 3.12 · Inside Proposition II's setting: the exact obligation, and why the split mechanism fails
  • Summary 3.13 · Verdict: \(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) fails (SG); sheaves given D1–D2
  • Audit 5.5 · Audit of A3-5: two periods on \(\pi\boxtimes\pi\), so \(\mathcal H_K\ast\varphi_0\ne C_K\)
  • Remark 6.2 · Registry after the split agent and the first audit: audit the split candidate and the mechanism
  • Theorem 6.3 · Verdict: sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\)

Sources