Audit 4.3 · Check: \(\tau\) is a closed immersion over \(W(\mathbb F_q)\), giving (H-int)

Proof of 1.

1. Monomorphism. For a \(\Lambda\)-algebra \(R\) and \(g_1,g_2\in G(R)\): \(g_1\theta(g_1)^{-1}=g_2\theta(g_2)^{-1}\iff g_2^{-1}g_1=\theta(g_2^{-1}g_1)\iff g_2^{-1}g_1\in H(R)\). So the fppf sheaf map \(G/H\to G\) is injective and \(\tau\) is a monomorphism.
2. Target. \(Q:=\{g:\theta(g)=g^{-1}\}\) is the fixed scheme of the involution \(\iota(g)=\theta(g)^{-1}\) of the smooth \(G_\Lambda\); as \(2\in\Lambda^\times\) it is smooth (Edixhoven 1992, Prop. 3.4, as used in S4-4). \(\tau\) lands in \(Q\) and is equivariant for the twisted action \(g\cdot q=gq\theta(g)^{-1}\) of \(G\) on \(Q\).
3. Fibres. Over an algebraically closed field \(k\), \(\operatorname{char}k\ne2\), let \(q\in Q(k)\) and \(\theta_q:=\operatorname{Int}(q)\circ\theta\), an involution since \(\theta(q)=q^{-1}\). The stabilizer of \(q\) is \(G^{\theta_q}\) (smooth). Writing tangent vectors at \(q\) as \(\frac{d}{d\epsilon}\exp(\epsilon Y)q\), one computes \(\iota(\exp(\epsilon Y)q)=\exp(-\epsilon\operatorname{Ad}(q)d\theta(Y))\,q\), so \(T_qQ\) is the \((-1)\)-eigenspace of \(d\theta_q\). Hence \(\dim G\cdot q=\dim\mathfrak g-\dim\mathfrak g^{\theta_q}=\dim T_qQ=\dim_qQ\): every twisted orbit is open in \(Q_k\), so each orbit is open and closed. At \(x_0\), \(d\tau=2\cdot\mathrm{id}\) on \(\mathfrak p=\mathfrak g/\mathfrak h=T_1Q\), and by equivariance \(d\tau\) is bijective everywhere; so \(\tau_k\) is an étale monomorphism onto the open and closed orbit \(G\cdot1\), i.e. an isomorphism onto it (EGA IV\(_4\) 17.9.1).
4. Open immersion over \(\Lambda\). \(X\) and \(Q\) are flat of finite presentation and \(\tau\) is flat on fibres, hence flat (EGA IV\(_3\) 11.3.10); it is unramified (a monomorphism of finite type, EGA IV\(_4\) 17.2.6), hence étale, and an étale monomorphism is an open immersion (17.9.1).
5. Closedness. Let \(y\in\overline{\tau(X)}\smallsetminus\tau(X)\). Since \(\tau(X)_\eta\) is closed in the open \(Q_\eta\), \(y\) lies in the special fibre, in a geometric twisted orbit \(\Omega\ne\tau(X)_{\bar{\mathbb F}_q}\). Pass to \(\Lambda_r=W(\mathbb F_{q^r})\) (finite étale, compatible with closures), pick a closed point \(y'\in\Omega(\mathbb F_{q^r})\), and lift it to \(q'\in Q(\Lambda_r)\) (Hensel, \(Q\) smooth). If \(q'_\eta=\tau(x)\) with \(x\in X(E_r)\), \(E_r=\Lambda_r[1/p]\), then by (H-Cartan) over the \(p\)-adic field \(E_r\) (S4-13) \(x\in K_rx_\kappa\), so \(\tau(x)\in K_rt^\kappa K_r\cap G(\Lambda_r)\), forcing \(\kappa=0\), \(x=kx_0\) and \(y'=\bar k\theta(\bar k)^{-1}\in\tau(X)\): contradiction. Otherwise the twisted orbit map \(g\mapsto g\cdot q'\) is smooth (fibrewise smooth onto open orbits, plus flatness), so its image \(U\) is open; \(U_\eta=G\cdot q'_\eta\) misses the open and closed \(G\)-stable \(\tau(X)_\eta\), and \(U_s=\Omega\) misses \(\tau(X)_s\). So \(U\ni y\) is an open set missing \(\tau(X)\): contradiction. Thus \(\tau(X)\) is closed in \(Q\), hence in \(G\).
6. Consequences. A closed subscheme of the affine \(G_\Lambda\) is affine. For \(\mathcal O\) a domain over \(\Lambda\), an \(\operatorname{Frac}\mathcal O\)-point of the closed \(X\subseteq G\) whose image lies in \(G(\mathcal O)\) is an \(\mathcal O\)-point of \(X\). Closed immersions are stable under base change, so this holds over \(\mathbb F_q[\![t]\!]\).

Proof of 2. \(\mathrm{GL}(W)\to\operatorname{End}W\oplus\operatorname{End}W\), \(A\mapsto(A,A^{-1})\), is a closed immersion; compose with \(X\hookrightarrow G\hookrightarrow\mathrm{GL}(W)\). Equivariance: \(\tau(gx)=g\tau(x)\theta(g)^{-1}\). On the matrix unit \(e_\chi e_{\chi'}^*\), \(t\in T\) acts by \(\chi(t)\chi'(\theta(t))^{-1}\). A closed immersion \(G_\Lambda\hookrightarrow\mathrm{GL}(W)\) exists for the split reductive \(G_\Lambda\) (affine flat of finite type over a discrete valuation ring; e.g. SGA 3, Exp. VI\(_B\), 11.11, or a fibrewise faithful sum of Weyl modules and SGA 3, Exp. XVI, 1.5; not opened), and \(T\) acts diagonally on \(W\) because it is split.

Proof of 3. S4-12 applies Delorme–Sécherre's Theorem 4.8 (stated for any non-archimedean locally compact field of residue characteristic \(\ne2\)) with \(|J|=1\) from Kneser's \(H^1(F',\mathrm{Spin}_8)=1\), valid for every finite extension of \(\mathbb Q_p\); S4-13 moves labels into \(\Lambda_S^+\) by \(\dot n_i\in H(\mathbb Z)\cap N_G(T)\). Neither uses \(e(F')=1\).

Remark (the pages' characteristic-\(p\) route). The pages obtain (H-Cartan) over \(\mathbb F_q(\!(t)\!)\) in IIb step 1 from input D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers. = Chen–Yi, Prop. 17(i). Chen–Yi work over \(\bar{\mathbb F}_p\), \(p\ne2\), but prove Prop. 17(i) by citing Gaitsgory–Nadler, Thm 8.2.9, and Nadler 2004, Thm 4.2, which are over \(\mathbb C\) (chen-yi-v4.pdf, p. 10). That is a characteristic-\(0\) citation used in characteristic \(p\) (checklist item 1); Proposition 4.5 gives an independent proof for \(\mathcal M_\theta\).

Status. 1 and 3 are proved from the cited inputs; 2 uses the cited existence of a closed immersion \(G_\Lambda\hookrightarrow\mathrm{GL}(W)\). The main line (Proposition 4.5, Proposition 4.6, Theorem 4.8) does not need 1 or 2.