Lemma 1.7 · Close-field transfer: witnesses of (SG) persist over highly ramified \(p\)-adic fields

Proof.

1. \(x_\mu\) is rational and universal. For \(\gamma\in\mathrm{Gal}(F^{\mathrm{sep}}/F)\), \(\gamma(\mu(\sqrt\varpi))=\mu(\pm1)\mu(\sqrt\varpi)\) and \(\mu(-1)\in S[2]\subseteq H\) fixes \(x_0\), so \(x_\mu\in X(F)\); \(\tau(x_\mu)=\mu(\sqrt\varpi)\theta(\mu(\sqrt\varpi))^{-1}=\mu(\varpi)\). Writing \(x_0=\sum_\chi v_\chi\) in \(T\)-weight vectors, \(x_\mu=\sum_\chi\sqrt\varpi^{\langle\chi,\mu\rangle}v_\chi\); rationality forces \(\langle\chi,\mu\rangle\) even whenever \(v_\chi\neq0\). So the coordinates of \(x_\mu\) are Laurent monomials in \(\varpi\) with coefficients in \(\Lambda\), the same for every \(\mathcal O\).

2. Cosets. \(Kt^\lambda K/K\cong K/K_\lambda\), \(K_\lambda=K\cap t^\lambda Kt^{-\lambda}\). With \(c=\max_\alpha|\langle\alpha,\lambda\rangle|\) over roots, the congruence subgroup \(K(c)=\ker(G(\mathcal O)\to G(\mathcal O/\varpi^c))\) lies in \(K_\lambda\) (Iwahori factorization of \(K(c)\) into \(T(1+\varpi^c\mathcal O)\) and root groups \(U_\alpha(\varpi^c\mathcal O)\), conjugated by \(t^{-\lambda}\) into \(U_\alpha(\varpi^{c-\langle\alpha,\lambda\rangle}\mathcal O)\subseteq K\)). The subgroup \(K_\lambda/K(m)\subseteq G(\mathcal O/\varpi^m)\) (\(m\ge c\)) is cut out by divisibility of matrix coefficients by powers \(\varpi^{\langle\chi-\chi',\lambda\rangle}\) in a faithful representation, a condition on the ring \(\mathcal O/\varpi^m\) and \(\varpi\) only.

3. The incidence condition. For \(g=kt^\lambda\), \(g^{-1}x_\mu\in X(\mathcal O)\iff t^{-\lambda}k^{-1}x_\mu\in V(\mathcal O)\) (\(X\) is closed in \(V\)). Its coordinates are \(\varpi^{-\langle\chi,\lambda\rangle}\sum_{\chi'}\rho(k^{-1})_{\chi\chi'}\varpi^{\langle\chi',\mu\rangle/2}v_{\chi'}\); with \(b\) the largest exponent of \(\varpi^{-1}\) occurring, the condition is that finitely many polynomials over \(\Lambda\) in the entries of \(k^{-1}\) and in \(\varpi\) vanish modulo \(\varpi^b\). It depends on \(k\) modulo \(K(b)\) and on \((\mathcal O/\varpi^b,\varpi)\). So for \(m_0\ge\max(c,b)\), \(N_{\mathcal O}(\lambda,\mu)\) is the number of \(\bar k\in G(\mathcal O/\varpi^{m_0})/(K_\lambda/K(m_0))\) satisfying conditions defined over \((\mathcal O/\varpi^{m_0},\bar\varpi)\): (a).

4. Close rings. If \(e(\mathcal O')\ge m\), then \(p\in\varpi'^m\mathcal O'\), the Teichmüller section makes \(\mathcal O'/\varpi'^m\) an \(\mathbb F_q\)-algebra, and \(\mathcal O'/\varpi'^m=\mathbb F_q[\varpi']/(\varpi'^m)\cong\mathbb F_q[t]/(t^m)\), compatibly with \(\Lambda\to\mathbb F_q\) and \(\varpi'\mapsto t\).

5. (b). Let \(f'=\sum_{\lambda\in\Lambda_0}c_\lambda T_\lambda\ast\varphi_0'-\mathbf 1_{K'x'_\mu}\) over \(\mathcal O'\). Its support lies in \(\bigcup_\lambda K't^\lambda X(\mathcal O')\cup K'x'_\mu\subseteq G(F')X(\mathcal O')\). Since \(\tau(kx)=k\tau(x)\theta(k)^{-1}\) and \(\theta(K')=K'\), \(\tau\) maps \(K'\)-orbits into double cosets, \(\tau(K't^\lambda X(\mathcal O'))\subseteq K't^\lambda K't^{-\theta\lambda}K'\subseteq\bigcup_{\nu\le\lambda+(-\theta\lambda)^+}K't^\nu K'\) (convolution of double cosets, valid over any local field), and \(\tau(K'x'_\nu)\subseteq K't^\nu K'\) with distinct dominant \(\nu\) giving disjoint double cosets. With (H-Cartan), the support of \(f'\) is contained in the finite set of orbits \(K'x'_\nu\), \(\nu\in N_0=\{\nu:\nu\le\lambda+(-\theta\lambda)^+,\ \lambda\in\Lambda_0\}\cup\{\mu\}\). For \(\nu\in N_0\), \(f'(x'_\nu)=\sum c_\lambda N_{\mathcal O'}(\lambda,\nu)-\delta_{\mu\nu}=\sum c_\lambda N_{\mathbb F_q[\![t]\!]}(\lambda,\nu)-\delta_{\mu\nu}=0\) once \(e(\mathcal O')\) exceeds the \(m_0\) of (a) for all pairs in \(\Lambda_0\times N_0\). So \(f'=0\). The coefficients may be taken in \(\mathbb Q\) by IIb step 7.

6. (c) is the contrapositive of (b).

7. Scope. This is the precise bridge from a \(p\)-adic failure to a failure over \(\mathbb F_q(\!(t)\!)\): failures at bounded ramification do not transfer. Sakellaridis's Theorem 8.0.2 is stated for \(p\)-adic fields, but his axioms (A3) are proved only at almost all places of a global field; to transfer a failure of \(r\) by (c) one needs his theorem, with (A2)–(A4) and (H-Cartan), at places with residue field \(\mathbb F_q\) and unbounded ramification. No such case is established here. \(\square\)