Audit 5.5 · Audit of A3-5: two periods on \(\pi\boxtimes\pi\), so \(\mathcal H_K\ast\varphi_0\ne C_K\)

Re-derivation.

1. Applying A3-4 on both sides. Apply A3-4 to \(V_i=P_i\oplus W_i^0\), with the trivial factor on \(P_i=W_i^{0\perp}\) and \(\sigma\) on \(W_i^0\). This gives nonzero \(f_j:\pi\to\sigma_{c_j}\) that are \(\mathrm{Sp}(P_1)\)-invariant and \(\mathrm{Sp}(W_1^0)\)-equivariant, and similarly \(f_j'\) for the second copy. Since both copies are the same \(\pi\), the same \(c_j\) occur: \(c_1=\chi_1\chi_2^{-1}\), \(c_2=\chi_1\chi_2\).
2. Identifications. \(\mathrm{Sp}(W_2^0)\) is identified with \(\mathrm{Sp}(W_1^0)\) by \(g\mapsto\varphi_0^{-1}g\varphi_0\), and \(\varphi_0\) is an isometry. Different identifications of \(\mathrm{Sp}(W)\) with \(\mathrm{SL}_2\) differ by \(\mathrm{GL}_2\)-conjugation, which preserves the isomorphism class of a principal series. So the targets of \(f_j\) and \(f_j'\) are the same \(\sigma_{c_j}\).
3. Invariance. \(\sigma_{c_j}\) is irreducible, so \(\sigma_{c_j}^\vee\cong\sigma_{c_j^{-1}}\cong\sigma_{c_j}\) by the standard intertwining operator, which is characteristic-free. Hence there is an invariant pairing \(\beta_j\). For \((h_1,h_2,(g,\varphi_0g\varphi_0^{-1}))\in H(F)\):

$$\ell_j\big(\pi(h_1g)v\otimes\pi(h_2\varphi_0g\varphi_0^{-1})w\big)=\beta_j\big(\sigma(g)f_j(v),\sigma(g)f_j'(w)\big)=\ell_j(v\otimes w).$$

4. Independence. \(F_\bullet=(f_1,f_2):\pi\to\sigma_{c_1}\oplus\sigma_{c_2}\) is onto, by Goursat: the image projects onto two non-isomorphic irreducibles, so it is everything. A tensor product of surjections is surjective, so \(\Pi=\pi\otimes\pi\twoheadrightarrow(\sigma_{c_1}\oplus\sigma_{c_2})^{\otimes2}\). The pairings \(\beta_1\circ\mathrm{pr}_{11}\) and \(\beta_2\circ\mathrm{pr}_{22}\) are supported on different summands, hence independent, and pulling back along a surjection preserves independence.
5. Conclusion. \(\Pi\) is irreducible, admissible and unramified, with \(\Pi^K=\pi^K\otimes\pi^K\). By Audit 5.1 (A3-1(b)), \(\ell_1,\ell_2\) give two independent \(\chi_\Pi\)-eigenfunctions, hence two independent \(\chi_\Pi\circ\vee\)-eigenfunctionals on \(C_K\). By A3-1(a), \(C_K\) is not a cyclic \(\mathcal H_K\)-module. The argument is uniform in \(F\), so it applies to every \(F_n=\mathbb F_{q^n}(\!(t)\!)\).

Consistency with Sakellaridis (characteristic 0, heuristic only). Over \((\chi,\chi)\), with \(x=\chi_1(\varpi)\), \(y=\chi_2(\varpi)\), \(\alpha^2=xy\), \(\beta^2=x/y\), the \(X\)-parameters are \((a,b,c)\) with \((a,b)\in\{(\alpha,\beta),(\beta,\alpha)\}\) and \((b,c)\in\{(\alpha,\beta),(\beta,\alpha)\}\), all up to \(W_X\). Generically there are exactly two, \((\alpha,\beta,\alpha)\) and \((\beta,\alpha,\beta)\). So the fibre of \(\mathbb C[A_X^*]^{W_X}\) (Thm 8.0.2) has dimension 2, matching the remark after Cor. 8.0.4 ("at those points the multiplicity ... is greater than one"). For generic \((\chi_1,\chi_2)\) the fibre is empty, which is consistent with rank \(X=3<4\). This comparison is not used in the proof.

What was not needed. No input of Sakellaridis's (A1)–(A5), nothing from characteristic 0, and none of the sheaf theory: the failure of (SG) over \(\mathbb F_q(\!(t)\!)\) is proved.