Proposition 4.5 · Proposition: \(K\backslash X(\mathbb F_q((t)))\simeq\Lambda_S^+\) by counting orbits

Inputs, valid over every test field \(E\) (characteristic \(0\) or \(p\)). (E1) Cartan decompositions \(G(E)=\bigsqcup_{\nu\in\Lambda_T^+}Kt^\nu K\) and \(H(E)=\bigsqcup_\eta H(\mathcal O)t^\eta H(\mathcal O)\), \(\eta\in X_*(T_H)\) dominant for a Borel subgroup of \(H\): \(H_\Lambda\) is split reductive with split maximal torus \(T_H\subseteq T\) (Computation 4.4); Bruhat–Tits (any non-archimedean local field). (E2) \(X(\mathcal O_E)=Kx_0\): S4-4 step 6 (Hensel for the smooth \(H\), Lang for its connected special fibre); smoothness (Edixhoven) and connectedness (Steinberg, Thm 8.1) need only \(p\ne2\), so this holds over \(\mathbb F_q[\![t]\!]\). (E3) \(X(E)=G(E)x_0\): by S4-4 step 3 (scheme-theoretic, any \(\mathcal O\) with \(2\in\mathcal O^\times\)), \(\mathring X=B/(B\cap H)\) with \(B\cap H\) a Borel subgroup scheme of \(\mathrm{Spin}_8\), smooth, connected, split solvable, so \(H^1(E,B\cap H)=1\) and \(\mathring X(E)=B(E)x_0\). \(G(E)\)-orbits in \(X(E)\) are open (the orbit maps are smooth) and \(\mathring X(E)\) is dense in the analytic manifold \(X(E)\), so every \(G(E)\)-orbit meets \(B(E)x_0\). (No \(H^1(E,F_4)\) is needed.) (E4) \(\theta(K)=K\), \(\tau(kx)=k\tau(x)\theta(k)^{-1}\), \(\tau(x_\kappa)=t^\kappa\), \(\tau(X(\mathcal O))\subseteq K\).

Proof.

1. Separation. By (E4), \(\tau(Kx_\kappa)\subseteq Kt^\kappa K\); distinct dominant \(\kappa\) give disjoint double cosets (E1).
2. Covering. \(X(E)=G(E)x_0=\bigcup_\lambda Kt^\lambda Kx_0=\bigcup_\lambda Z_\lambda(E)\) by (E3), (E1), (E2). This proves (a).
3. Orbits in \(Z_\lambda\) as classes in \(K\). Every \(K\)-orbit in \(Z_\lambda\) contains a point \(t^\lambda kx_0\), \(k\in K\), and \(t^\lambda kx_0\in Kt^\lambda k'x_0\) iff there is \(h\in H(E)\) with \(t^\lambda khk'^{-1}t^{-\lambda}\in K\) (put \(a=t^\lambda khk'^{-1}t^{-\lambda}\) in \(at^\lambda k'x_0=t^\lambda kx_0\)).
4. Bounding \(h\). Then \(h\in Kt^{-\lambda}Kt^\lambda K\), so by Lemma 4.2(c) \(h\in Kt^cK\) with \(c\preceq\lambda+\lambda^*\), \(\lambda^*=-w_0\lambda\). Writing \(h\in H(\mathcal O)t^\eta H(\mathcal O)\) (E1) gives \(h\in Kt^{\eta^+}K\), \(\eta^+\) the dominant \(W\)-conjugate. So \(\eta\in B(\lambda):=\{\eta\in X_*(T_H)\ H\text{-dominant}:\eta^+\preceq\lambda+\lambda^*\}\), a finite set. Hence \(k\sim k'\) iff for some \(\eta\in B(\lambda)\) and \(h_1,h_2\in H(\mathcal O)\): \(\operatorname{val}\rho_i(t^\lambda kh_1t^\eta h_2k'^{-1}t^{-\lambda})=0\) for all \(i\) (Lemma 4.2(c)).
5. Determinacy of \(N_\lambda\). For each of the finitely many \(\eta\) this is a condition of type Lemma 4.2(e) on \((k,k',h_1,h_2)\in G(\mathcal O)^2\times H(\mathcal O)^2\). With \(D_1(\lambda)\) the largest of the resulting \(D\), \(\sim\) is the pullback of the equivalence relation \(\approx\) on \(G(\mathcal O/\varpi^{D_1})\): \(\bar k\approx\bar k'\) iff some \(\eta\in B(\lambda)\) and \(\bar h_1,\bar h_2\in H(\mathcal O/\varpi^{D_1})\) satisfy the reduced congruences. So \(N_\lambda(E)=\#\big(G(\mathcal O/\varpi^{D_1})/\!\approx\big)\), a function of \((\mathcal O/\varpi^{D_1},\bar\varpi)\).
6. Determinacy of \(M_\lambda\). Fix lifts \(\mu_\kappa\) with \((1-\theta)\mu_\kappa=\kappa\). By (E2), \(x_\kappa\in Z_\lambda\) iff \(t^{\mu_\kappa}h\in Kt^\lambda K\) for some \(h\in H(E)\); such \(h\) lies in \(Kt^{-\mu_\kappa}Kt^\lambda K\), hence in \(H(\mathcal O)t^\eta H(\mathcal O)\) with \(\eta^+\preceq(-\mu_\kappa)^++\lambda\); so membership is "\(\exists\eta,h_1,h_2:\operatorname{val}\rho_i(t^{\mu_\kappa}h_1t^\eta h_2)=\langle\lambda,w_0\omega_i\rangle\ \forall i\)", of type Lemma 4.2(e). Moreover \(x_\kappa\in Z_\lambda\) gives \(t^\kappa=\tau(x_\kappa)\in Kt^\lambda Kt^{-\theta\lambda}K\) (E4), so \(\kappa\preceq\lambda+(-\theta\lambda)^+\): \(M_\lambda(E)\) lies in a finite set independent of \(E\), and is determined modulo \(\varpi^{D_2(\lambda)}\). This proves (b) with \(D(\lambda)=\max(D_1,D_2)\).
7. Counting. Let \(E,E'\) be \(D(\lambda)\)-close; then \(N_\lambda(E)=N_\lambda(E')\) and \(M_\lambda(E)=M_\lambda(E')\). If every orbit in \(Z_\lambda(E')\) contains some \(x_\kappa\), then \(\kappa\mapsto Kx_\kappa\) is a bijection \(M_\lambda(E')\to K\backslash Z_\lambda(E')\) (onto by hypothesis, injective by step 1), so \(N_\lambda(E')=|M_\lambda(E')|\). Over \(E\), \(\kappa\mapsto Kx_\kappa\) injects \(M_\lambda(E)\) into \(K\backslash Z_\lambda(E)\), and \(|M_\lambda(E)|=|M_\lambda(E')|=N_\lambda(E')=N_\lambda(E)\); an injection between finite sets of equal size is onto. This proves (c).
8. Over \(\mathbb F_q(\!(t)\!)\). For each \(\lambda\) take \(E'=F'_e\), \(e\ge D(\lambda)\): it is \(D(\lambda)\)-close to \(F\) (Lemma 4.2(a)), and (H-Cartan) holds over it (S4-13, any ramification; Audit 4.3 item 3). By (c) every \(K\)-orbit in \(Z_\lambda(F)\) is some \(Kx_\kappa\); by step 2 and step 1, \(X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa\). As \(X(F)=G(F)x_0\) and \(x_\kappa\in T(F)x_0\), \(G(F)=KT(F)H(F)\). The same holds over \(\mathbb F_{q^n}(\!(t)\!)\) for every \(n\). \(\square\)

Characteristic-\(p\) inputs. Only (E1)–(E4). Not used: Knop, Luna–Vust, Gaitsgory–Nadler, Chen–Yi, Delorme–Sécherre or Bruhat–Tits III in characteristic \(p\), any statement over \(\overline{\mathbb F_q(\!(t)\!)}\), any part of Theorem 8.0.2. Delorme–Sécherre enters only over the \(p\)-adic fields \(F'_e\) (characteristic \(0\)). Since \(D(\lambda)\) grows with \(\lambda\), infinitely many \(F'_e\) are used, one for each \(\lambda\).