Audit 4.5 · Audit: the unitary twist fails (SG) over \(\mathbb F_q(\!(t)\!)\) for split \(H\ne1\)

1. Hecke basis. \(G(F)=H(E)\) with \(E=\mathbb F_{q^2}(\!(t)\!)\), and \(K=H(\mathcal O_E)\) is hyperspecial; the Cartan decomposition of \(H(E)\) gives the basis \(T_a\), \(a\in\Lambda_H^+\), with \(t^a\in T_H(F)\subseteq H(E)\). In \(\Lambda_T=\Lambda_H\oplus\Lambda_H\) these are the Frobenius-fixed \((a,a)\).
2. Support. By Lemma A(2), \(\operatorname{supp}(T_a\ast\varphi_0)=Kt^aX(\mathcal O)=Kt^aK\cdot1\). For \(k,k'\in K\),

$$kt^ak'\cdot1=k\,t^a\,\big(k'\vartheta(\sigma k')^{-1}\big)\,\vartheta(\sigma t^a)^{-1}\,\vartheta(\sigma k)^{-1},\qquad\vartheta(\sigma t^a)^{-1}=\vartheta(t^a)^{-1}=t^a,$$

so the point lies in \(Kt^aKt^aK\subseteq\bigsqcup_{\nu\le2a}Kt^\nu K\) (convolution \(\operatorname{Gr}_{\le a}\mathbin{\tilde\times}\operatorname{Gr}_{\le a}\to\operatorname{Gr}_{\le2a}\)). By Audit 4.3 the \(K\)-orbit of a point is its Cartan type.

3. Top value. \(t^a\cdot1=t^a\vartheta(\sigma t^a)^{-1}=t^{2a}\), so the coset \(t^aK\) contributes \(1\) at \(t^{2a}\); all contributions are \(\ge0\).
4. Leading terms. Audit 4.4(b) with \(I=J=\Lambda_H^+\), \(f_a=T_a\), \(\psi(a)=2a\) (injective) gives item 3; if \(c=2a\) with \(a\in\Lambda_H\) then \(a\) is dominant, so \(2\Lambda_H^+=\Lambda_H^+\cap2\Lambda_H\).
5. Odd labels exist. \(\Lambda_H^+\) generates \(\Lambda_H\): \(\lambda=(\lambda+N\xi)-N\xi\) with \(\xi=2\rho^\vee\) regular dominant and \(N\gg0\) (any \(\xi\) for a torus). So \(\Lambda_H^+\subseteq2\Lambda_H\) would give \(\Lambda_H=2\Lambda_H\), i.e. \(\Lambda_H=0\) and \(H=1\). Requested test cases:
   • torus \(\mathbb G_m\): \(c=1\) (Audit 4.11);
   • \(\mathrm{SL}_2\) (simply connected, not adjoint): \(\Lambda=\mathbb Z\alpha^\vee\), \(c=\alpha^\vee\); Computation 4.6 computes \(T_{\alpha^\vee}\ast\varphi_0=\mathbf 1_{2\alpha^\vee}+(q+1)\mathbf 1_{\alpha^\vee}+q(q+1)\mathbf 1_0\);
   • \(\mathrm{PGL}_2\) (adjoint): \(\Lambda=\mathbb Z\varpi^\vee\), \(\varpi^\vee=\alpha^\vee/2\), \(c=\varpi^\vee\). Here \(X_u(F)\) is the set of nondegenerate hermitian matrices modulo \(F^\times\) (a lift of \(x\) with \(x^*=\lambda x\) has \(N\lambda=1\), and Hilbert 90 rescales it to a hermitian matrix), the labels are \(n=c_1-c_2\ge0\), and the odd \(n\) form the second \(G(F)\)-orbit;
   • \(\mathrm{GL}_n\) (neither): \(c=(1,0,\dots,0)\). The proof uses only that \(\vartheta\) inverts \(T_H\) (so that \(t^c\) is rational in \(\mathcal O_c\)) and that every \(J_c\) is connected, which hold for every split \(H\).
6. Other degrees. Odd \(n\): \(E\otimes_FF_n=\mathbb F_{q^{2n}}(\!(t)\!)\) is a field and steps 1–5 repeat with \(q^n\). Even \(n\): \(G_{F_n}\cong H\times H\) and, since \(\mathrm{Fr}_X^2=\mathrm{Fr}^2\) (Audit 4.1 item 3), \(X_u(F_n)\cong H(F_n)\) with action \(g_1x\vartheta(g_2)^{-1}\); \(\varphi_0\) is the unit of \(\mathcal H_{K_H}\) and \(T_{(c,0)}\ast\varphi_0=\mathbf 1_{K_Ht^cK_H}\) (Proposition D), so \((\mathrm{SG})_n\) holds.
7. Conclusion. For every odd \(q\) (indeed every \(q\)) and every split \(H\ne1\), \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) (Audit 4.2) while \(\mathbf 1_{O_c}\notin\mathcal H_K\ast\varphi_0\) for each dominant \(c\notin2\Lambda_H\). The smallest instance with a single \(G(F)\)-orbit is hermitian \(\mathrm{SL}_2\) (Audit 4.7).