Proposition X4.5 · \(E_6/F_4\), \(p\) odd: Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points., half of Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits., types in 2.2.1

Proof of 1. (a) \(\tau:X\to G\) is injective and \(\tau(kx)=k\tau(x)\theta(k)^{-1}\); since \(\theta(K)=K\), the double coset \(K\tau(x)K\) depends only on \(Kx\). (b) \(\tau(t^\mu x_0)=t^{\varphi(\mu)}\), and \(\kappa\in\Lambda_S^+\subseteq\Lambda_T^+\) is dominant, so by the Cartan decomposition \(G(F)=\bigsqcup_{\nu\in\Lambda_T^+}Kt^\nu K\) distinct labels give distinct double cosets, hence distinct \(K\)-orbits. (c) Two points of \(A_X(F)x_0\) with the same label differ by \(A_X(\mathcal O)\), which is the image of \(T(\mathcal O)\subseteq K\) because the kernel \(T^\theta\) of \(T\to A_X\) is a split torus (\(H^1(\mathcal O,T^\theta)=0\)).

Proof of 2, covered labels (\(\kappa=\varphi(\mu)\) with \(\mu\in\Lambda_T^+\); by the family-page Lemma these are all labels except \(3k\omega_1^\vee,3k\omega_6^\vee\), \(k\) odd). By 1(c) and \(T(\mathcal O)\subseteq J\cap B(\mathcal O)\) it suffices to take \(x_\kappa=t^\mu x_0\).

1. Iwahori factorization: \(J=U_P(\mathcal O)\,(J\cap L)\,U_P^-(\mathfrak p)\).
2. \(\langle\beta,\mu\rangle\ge0\) for \(\beta\in\Phi(U_P)\), so \(t^{-\mu}U_P^-(\mathfrak p)t^\mu\subseteq U_P^-(\mathfrak p)\) and \(U_P^-(\mathfrak p)t^\mu x_0\subseteq t^\mu U_P^-(\mathfrak p)x_0\).
3. Open cell over \(\mathcal O\): \(U_P\times A_X\to X\), \((u,a)\mapsto uax_0\), is an open immersion (as in Proposition X4.4, step 3, with \(P\cap H=L^\theta\)). A point \(u'x_0\), \(u'\in U_P^-(\mathfrak p)\), reduces to \(\bar x_0\), so it lies in the cell: \(u'x_0=uax_0\) with \(u\in U_P(\mathfrak p)\) and \(a\in A_X(\mathcal O)=\operatorname{im}T(\mathcal O)\). Hence \(U_P^-(\mathfrak p)x_0\subseteq U_P(\mathfrak p)T(\mathcal O)x_0\).
4. \(t^\mu U_P(\mathfrak p)t^{-\mu}\subseteq U_P(\mathfrak p)\), so \(U_P^-(\mathfrak p)t^\mu x_0\subseteq U_P(\mathfrak p)T(\mathcal O)t^\mu x_0\).
5. \(L^\theta=\mathrm{Spin}_8=[L,L]\) is normal in \(L\), so \(L\) acts on \(Lx_0=L/L^\theta\) through the torus \(L/[L,L]\) and \(J\cap L\) acts on \(T(F)x_0\) through \(T(\mathcal O)\).
6. \(Jt^\mu x_0\subseteq U_P(\mathcal O)(J\cap L)U_P(\mathfrak p)T(\mathcal O)t^\mu x_0=U_P(\mathcal O)U_P(\mathfrak p)(J\cap L)T(\mathcal O)t^\mu x_0\subseteq U_P(\mathcal O)T(\mathcal O)t^\mu x_0\subseteq B(\mathcal O)t^\mu x_0\).

Why a second argument is needed. A representative \(\mu\) with \(\varphi(\mu)=\kappa\) and \(\langle\beta,\mu\rangle\ge0\) on \(\Phi(U_P)\) exists iff \(\kappa\) has a dominant lift, since \(W_L\) permutes \(\Phi(U_P)\) and fixes \(\Lambda_S\); exact search over \(a,b\le18\) (c2b_iwahori.py) finds the failures exactly at \(3k\omega_1^\vee,3k\omega_6^\vee\), \(k\) odd.

Proof of 2, uncovered labels \(\kappa=3k\omega_1^\vee\), \(k\) odd (the labels \(3k\omega_6^\vee\) follow with nodes \(1\leftrightarrow6\), \(3\leftrightarrow5\); representatives found by the same computation).

7. Let \(M\) be the standard Levi of nodes \(2,\dots,6\) (type \(D_5\)) and \(Q=MU_Q\supseteq P\); \(\Phi(U_Q)\) is the \(16\) positive roots with \(\alpha_1\)-coefficient \(>0\). \(\theta=-w_L\) negates \(\alpha_1\)-coefficients and preserves \(\Phi_M\), so \(\theta(U_Q)=U_Q^-\), \(Q\cap H=M^\theta\), and \(U_Q\times M/M^\theta\to X\) is an open immersion over \(\mathcal O\) (dimensions \(16+(46-36)=26\); same argument as step 3). Likewise \(U_{P\cap M}\times A_X\to M/M^\theta\).
8. \(\zeta:=3\omega_1^\vee=4\alpha_1^\vee+3\alpha_2^\vee+5\alpha_3^\vee+6\alpha_4^\vee+4\alpha_5^\vee+2\alpha_6^\vee\) is the primitive cocharacter of \(Z(M)^\circ\subseteq S\): \(\langle\beta,\zeta\rangle=0\) on \(\Phi_M\), \(=3\) on \(\Phi(U_Q)\), and \(\varphi(\zeta)=2\zeta\). Put \(\mu_1:=2\alpha_1^\vee+\alpha_3^\vee+\alpha_4^\vee+\alpha_5^\vee+\alpha_6^\vee\); then \(\varphi(\mu_1)=3\omega_1^\vee\), \(\min_{\Phi(U_Q)}\langle\beta,\mu_1\rangle=0\) and \(\min_{\Phi(U_{P\cap M})}\langle\beta,\mu_1\rangle=-1\). For \(\mu_k:=\mu_1+\tfrac{k-1}2\zeta\): \(\varphi(\mu_k)=3k\omega_1^\vee\) and the same two bounds hold.
9. A rational half-translation. Choose \(z\in Z(M)^\circ(\bar F)\) with \(z^2=\zeta(\varpi)\) and define \(R_z(mx_0):=mzx_0\) for \(m\in M(F)\). It is well defined (\(z\) is central in \(M\)), \(M(F)\)-equivariant, and lands in \(X(F)\): for \(\sigma\in\operatorname{Gal}\), \(z^{-1}\sigma(z)\in\zeta(\mu_2)\subseteq S[2]\subseteq H\). Since \(\theta(z)=z^{-1}\), \(\tau(z^kx_0)=z^{2k}=t^{k\zeta}=t^{\varphi(\mu_k)}=\tau(t^{\mu_k}x_0)\), so \(t^{\mu_k}x_0=R_z^k(x_0)\).
10. The Levi part. \((J\cap M)t^{\mu_k}x_0=R_z^k\big((J\cap M)x_0\big)\subseteq R_z^k\big(B_M(\mathcal O)x_0\big)=B_M(\mathcal O)t^{\mu_k}x_0\), the inclusion being the covered case \(\kappa=0\) inside \(M/M^\theta\) (steps 1–6 for \(P\cap M\), \(\mu=0\)).
11. The unipotent part. By step 8, \(U_Q^-(\mathfrak p)t^{\mu_k}x_0\subseteq t^{\mu_k}U_Q^-(\mathfrak p)x_0\subseteq t^{\mu_k}U_Q(\mathfrak p)\,\mathcal Y\subseteq U_Q(\mathfrak p)\,t^{\mu_k}\mathcal Y\), where \(\mathcal Y=\{y\in(M/M^\theta)(\mathcal O):\bar y=\bar x_0\}\) (open cell of step 7). Each \(y\in\mathcal Y\) is \(uax_0\) with \(u\in U_{P\cap M}(\mathfrak p)\), \(a\in\operatorname{im}T(\mathcal O)\), and \(t^{\mu_k}ut^{-\mu_k}\in\prod_\beta U_\beta(\varpi^{\langle\beta,\mu_k\rangle}\mathfrak p)\subseteq U_{P\cap M}(\mathcal O)\) because \(\langle\beta,\mu_k\rangle\ge-1\). So \(t^{\mu_k}\mathcal Y\subseteq B_M(\mathcal O)t^{\mu_k}x_0\).
12. With \(J=U_Q(\mathcal O)(J\cap M)U_Q^-(\mathfrak p)\): \(Jt^{\mu_k}x_0\subseteq U_Q(\mathcal O)(J\cap M)U_Q(\mathfrak p)B_M(\mathcal O)t^{\mu_k}x_0=U_Q(\mathcal O)(J\cap M)t^{\mu_k}x_0\subseteq U_Q(\mathcal O)B_M(\mathcal O)t^{\mu_k}x_0=B(\mathcal O)t^{\mu_k}x_0\).

Proof of 3. (a) \(W(X)=W_X\ltimes W_{P(X)}\) has order \(6\cdot|W(D_4)|=1152\), and Knop's action is transitive on maximal-rank orbits (Sakellaridis p. 9), giving \(45\). (b) If \((Y,\alpha)\) had type \(T\) or \(N\), then \(Y\) is the open \(B\)-orbit of \(YP_\alpha\) and the rank-one quotient pulls back a \(B\)-eigenfunction of weight \(\alpha\) or \(2\alpha\), so \(\alpha\in\mathbb Q\otimes\mathcal X(Y)=w(\mathbb Q\omega_1\oplus\mathbb Q\omega_6)\) for the \(w\) with \(Y=w\mathring X\) (Knop's action transports the weight lattices: Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv. 70 (1995); cited, not checked). But \(W\) preserves roots and no root lies in \(\mathbb Q\omega_1\oplus\mathbb Q\omega_6\): \((a\omega_1+b\omega_6)^2=\tfrac43(a^2+ab+b^2)\ne2\) for integers, and indeed no root \(\beta\) has \(\theta\beta=-\beta\) (c1_a4.py).

Field and model: 1 uses only \(\theta(K)=K\) (any \(p\) for which \(\theta\) is defined over \(\mathcal O\)); 2 uses \(p\ne2\) and \(\mathcal M_\theta\); 3 is a statement over \(\bar F\).