§3 · Agent 3 · Hypotheses in char p · 2026-10-18T03:00:21Z

Theorem 3.8 · (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits. for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\): \(K\backslash X(F)\simeq\Lambda_X^+\)

proved audit: correct ★ key step

Second audit: correct. The dictionary is checked against Sakellaridis §2.1 p. 8 (A_X^+ = X-anti-dominant points e^lambda(varpi)): x0.e^lambda(varpi) corresponds to e^{-lambda}(t)x0, and with Proposition 3.7(3) lambda lies in C^- exactly when kappa = phi(mu) is dominant, a sign that matches Proposition 3.7's arc valuation v_kappa = -mu-bar. x_kappa does not depend on the lift, since two lifts differ by X_*(T_8) and T_8(F) is in H. Covering: X(F) = G(F)x0 = K T(F) x0 (Lemma 3.3, Audit 3.5), T(F) = T(O){mu(t)}, and n_i in H(Z) cap N(T), inside K, commute with theta and so act on Lambda_S by s_gamma_i, which moves every label into the dominant chamber. Separation: tau(kx) = k tau(x) theta(k)^-1 with theta(K) = K, tau(mu(t)x0) = kappa(t), kappa is G-dominant, and the Cartan decomposition of split G(F) holds over every non-archimedean local field; equal labels give one orbit via im T(O), so both clauses of Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. hold over F_q((t)).

Combines \(G(F)=KT(F)H(F)\) (Audit 3.5), \(X(F)=G(F)x_0\) and \(A_X(F)x_0=T(F)x_0\) (Lemma 3.3), the little Weyl group inside \(K\) (S4-13, characteristic-free), \(\Lambda_X^+=\Lambda_X\cap\mathcal V\) (Proposition 3.7) and the Cartan decomposition of \(G(F)\) to prove both clauses of Sakellaridis's Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\), \(p\) odd.

Theorem 3.8.

Let \(F=\mathbb F_q(\!(t)\!)\), \(p\) odd, \(K=G(\mathbb F_q[[t]])\). For \(\kappa\in\Lambda_S\) put \(x_\kappa:=\mu(t)x_0\) for any \(\mu\in X_*(T)\) with \(\varphi(\mu)=\kappa\) (well defined up to \(A_X(\mathcal O)=\mathrm{im}\,T(\mathcal O)\subseteq K\)). Then

  1. \(X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}K\,x_\kappa\);
  2. in Sakellaridis's right-action terms, \(A_X^+\) (points \(x_0\cdot a\), \(a\in A_X(F)\), with label in \(\Lambda_X^+\)) contains a complete set of representatives of \(X(F)/K\), and points of \(A_X^+\) with distinct labels lie in distinct \(K\)-orbits. So Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. holds, and \(X(F)/K\simeq\Lambda_X^+\simeq\Lambda_S^+\).
  1. Dictionary (checklist item 8). Sakellaridis's point \(x_0\cdot e^{\lambda}(\varpi)\) is our \(e^{-\lambda}(t)x_0=\mu(t)x_0\) with \(\bar\mu=-\lambda\in\Lambda_X\). By Proposition 3.7(3), \(\lambda\in\Lambda_X^+=\Lambda_X\cap C^-\) iff \(\bar\mu\in-C^-\) iff \(\kappa=\varphi(\mu)\in\Lambda_S^+\) (\(\langle\varphi(\mu),\gamma_i\rangle=2\langle\bar\mu,\gamma_i\rangle\)). So \(A_X^+\) corresponds to \(\{x_\kappa:\kappa\in\Lambda_S^+\}\) up to \(A_X(\mathcal O)\).
  2. Every \(K\)-orbit meets \(A_X^+\). For \(x\in X(F)\): \(x=gx_0\) (Lemma 3.3(2)), \(g=kt'h\) (Audit 3.5), so \(x=kt'x_0\) with \(t'\in T(F)=T(\mathcal O)\cdot\{\mu(t)\}\), \(T(\mathcal O)\subseteq K\): \(x\in K\mu(t)x_0\). The elements \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H(\mathbb Z)\cap N_G(T)\subseteq K\) (S4-13) satisfy \(\dot n_i\mu(t)x_0=(s_{\alpha_i}s_{\beta_i}\mu)(t)x_0\) and \(\varphi(s_{\alpha_i}s_{\beta_i}\mu)=s_{\gamma_i}\varphi(\mu)\) (Proposition 3.6(2)); the group they generate acts on \(\Lambda_S\) as \(W_X\simeq\langle s_{\gamma_1},s_{\gamma_6}\rangle\), which moves every \(\kappa\in\Lambda_S\) into \(\Lambda_S^+\).
  3. Distinct labels, distinct orbits. \(\tau\) is injective on \(F\)-points and \(\tau(kx)=k\tau(x)\theta(k)^{-1}\) with \(\theta(K)=K\), so \(Kx\mapsto K\tau(x)K\) is well defined; \(\tau(\mu(t)x_0)=\varphi(\mu)(t)\), and \(\kappa\in\Lambda_S^+\) is \(G\)-dominant (\(\langle\alpha,\kappa\rangle=\langle\alpha|_S,\kappa\rangle\ge0\) for \(\alpha\in\Phi^+\)). The Cartan decomposition \(G(F)=\bigsqcup_{\nu\in\Lambda_T^+}Kt^\nu K\) holds over every non-archimedean local field (Tits, Corvallis 3.3.3; Bruhat–Tits I; for \(k(\!(t)\!)\), \(k\) algebraically closed, Zhu (2.1.2)), so distinct \(\kappa\) give distinct double cosets, hence distinct \(K\)-orbits. Points with the same label differ by \(A_X(\mathcal O)=\mathrm{im}\,T(\mathcal O)\subseteq K\) (Lemma 3.3(3)).
  4. Hence the \(K\)-orbits are exactly the \(Kx_\kappa\), \(\kappa\in\Lambda_S^+\).

Every step is characteristic-free given \(p\) odd; the characteristic-0 inputs of S4-12/S4-13 (Kneser, Vust/Richardson, Helminck–Wang) are replaced by Bruhat–Tits III (Lemma 3.3) and Proposition 3.4.

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