§1 · Agent 1 · Direct rerun · 2026-10-18T03:56:05Z

Lemma 1.4 · Finiteness of \(B(F)\backslash X(F)\) for symmetric \(X\) over \(\mathbb F_q((t))\), \(p\) odd

proved audit: minor defect ★ key step

Minor defect (second audit). I re-derived the argument. Y(F)/B(F)=ker(H^1(F,B_y)->H^1(F,B)) with H^1(F,B)=1. M_y is a closed subgroup of T, hence G_m^a x mu_2^b. Each twisted unipotent radical equals U_H(lambda) for a cocharacter of a split maximal F-torus that is positive on its weights, so it is F-split; I could not open CGP 2.1.10, but Milne, Algebraic Groups 22.86, states the same over any infinite field. Hence H^1(F,B_y) injects into (F^x/F^x2)^b, and |F^x/F^x2|=4. The filtration (orbits by decreasing dimension, each B(F)-orbit open and closed in Y(F)) is right. For (d) I computed (1+w_L)P=Z w1+Z w6, saturated in P, so T^theta is connected and X°(F)=x0B(F). Defect: step 5 justifies openness by 'the implicit function theorem over a complete non-discretely valued field', but F_q((t)) is discretely valued. The theorem actually needed holds over every complete non-trivially valued field, so this slip in the wording changes no claim.

Proves that a symmetric variety over F_q((t)) with p odd has finitely many B(F)-orbits, each open in its geometric orbit, by showing every Borel stabilizer is a split torus times μ_2's extended by a split unipotent group; this is the orbit finiteness [Sa08] uses implicitly.

Lemma 1.4.

Let \(F=\mathbb F_q((t))\) with \(p\) odd, \(G\) split reductive over \(F\), \(B=T\ltimes U\) a split Borel subgroup, \(\theta\) an involution of \(G\) over \(F\), \(X=G/G^\theta\). Assume Springer's results in char \(\neq2\): \(B_{\bar F}\) has finitely many orbits on \(X_{\bar F}\), and the lemma of Lemma 1.3(e). Then for every \(y\in X(F)\), with \(Y=By\):

(a) \(B\to Y\), \(b\mapsto by\), is smooth. Every \(B(F)\)-orbit in \(Y(F)\) is open and closed in \(Y(F)\).

(b) \(Y(F)/B(F)\hookrightarrow H^1(F,M_y)\), where \(M_y:=B_y/(B_y\cap U)\) is a closed subgroup of the split torus \(T\) of the form \(\mathbb G_m^a\times\mu_2^b\). Hence \(|Y(F)/B(F)|\le4^b\), and \(Y(F)\) is a single \(B(F)\)-orbit when \(M_y\) is connected.

(c) \(B(F)\backslash X(F)\) is finite, and \(X(F)\) has a finite filtration by \(B(F)\)-stable open subsets whose successive differences are single \(B(F)\)-orbits.

(d) For \(X=M_\theta\): \(B\cap H=T^\theta\ltimes(U\cap L)\) with \(T^\theta\) connected, so \(X̊(F)=x_0B(F)\).

The argument needs the stabilizers to have no \(p\)-torsion. For instance, a \(B_{\bar F}\)-orbit with stabilizer \(\mathbb Z/p\) (resp. \(\mu_p\)) would carry infinitely many \(B(F)\)-orbits, since \(H^1(F,\mathbb Z/p)=F/\wp(F)\) and \(H^1(F,\mu_p)=F^\times/F^{\times p}\) are infinite.

This replaces the implicit finiteness of Borel orbits and of Galois cohomology in [Sa08], used for Theorems 3.2.2 and 3.7.1 (checklist 6).

Proof.

  1. Torsor description. \(Y\) is the orbit of an \(F\)-point, a locally closed \(F\)-subvariety. \(B_y\) is smooth: smoothness is checked over \(\bar F\), where it is Lemma 1.3(e). So \(B/B_y\xrightarrow{\sim}Y\) and \(B\to Y\) is smooth. By [Serre, Galois Cohomology, I.5.4, Cor. 1], \(Y(F)/B(F)\) is in bijection with \(\ker(H^1(F,B_y)\to H^1(F,B))\). Since \(H^1(F,T)=1\) (Hilbert 90, \(T\) split) and \(H^1(F,U)=1\) (\(U\) split unipotent, \(H^1(F,\mathbb G_a)=0\)), \(H^1(F,B)=1\). Hence \(Y(F)/B(F)\cong H^1(F,B_y)\).
  1. Unipotent part. Let \(U_y:=B_y\cap U\). Over \(\bar F\), after \(B(\bar F)\)-conjugation, \(B_y=T_1\ltimes U_1\) with \(U_1=B_y\cap U\) (Lemma 1.3(e)). So \(U_y\) is smooth, connected and unipotent. \(M_y=B_y/U_y\hookrightarrow T\) is a closed subgroup of a split torus. It is therefore split diagonalizable, and it is smooth because its geometric component group has exponent 2.
  1. \(U_y\) and its twists are split. Let \(c\) be a cocycle with values in \(B_y(F_s)\). The inner twist \({}_cB_y^\circ\) is a smooth connected solvable \(F\)-group. Let \(T''\) be a maximal \(F\)-torus of it. Since \(M_y\) is commutative, inner twisting does not change it, so \(T''\hookrightarrow M_y\subseteq T\) and \(T''\) is split. Over \(\bar F\), \(T''\) is conjugate to \(T_1^\circ\). By Lemma 1.3(e) its weights on \(\operatorname{Lie}({}_cU_y)\) are non-zero and lie in an open half-space. Choose \(\lambda\in X_*(T'')\), defined over \(F\) since \(T''\) is split, positive on all of them. Then \({}_cU_y=U_{{}_cB_y^\circ}(\lambda)\), which is \(F\)-split unipotent by [Conrad–Gabber–Prasad, Prop. 2.1.10] (cited, not opened). Hence \(H^1(F,{}_cU_y)=1\).
  1. Injectivity. By [Serre, I.5.5, Cor. 2], the fibres of \(H^1(F,B_y)\to H^1(F,M_y)\) are quotients of the sets \(H^1(F,{}_cU_y)\), which are trivial by step 3. So the map is injective. Write \(M_y\cong\mathbb G_m^a\times\mu_2^b\). Then \(H^1(F,M_y)=(F^\times/F^{\times2})^b\), and \(|F^\times/F^{\times2}|=4\) for \(p\) odd, with classes \(1,\varepsilon,t,\varepsilon t\). This proves (b).
  1. (a), openness. A smooth morphism of smooth \(F\)-varieties induces a submersion of \(F\)-analytic manifolds, so \(B(F)\to Y(F)\) is open (implicit function theorem over a complete non-discretely valued field, valid in every characteristic). Hence each of the finitely many \(B(F)\)-orbits in \(Y(F)\) is open, and therefore also closed, in \(Y(F)\).
  1. (c). There are finitely many \(F\)-rational \(B_{\bar F}\)-orbits (Springer). Order them by decreasing dimension: \(\overline{Y}(F)\setminus Y(F)\) lies in orbits of smaller dimension. Refining by the finitely many \(B(F)\)-orbits of step 5 gives the filtration.
  1. (d). For \(M_\theta\), \(T^\theta\) is the torus generated by \(\alpha_2^\vee,\dots,\alpha_5^\vee\), which is connected (S4-3 step 2; family page §4). With Lemma 1.3(c), \(M_{x_0}=T^\theta\) is connected, so \(b=0\) in (b).
  1. Why no \(p\)-torsion matters. If \(B_y=\mathbb Z/p\), step 1 gives \(Y(F)/B(F)\cong H^1(F,\mathbb Z/p)\). Artin–Schreier theory gives \(|F/\wp F|=\infty\) for \(F=\mathbb F_q((t))\), and \(F^\times/F^{\times p}\) is infinite since \(t^{1/p}\notin F\). For symmetric varieties with \(p\) odd these stabilizers do not occur, by Lemma 1.3(e).

Consequence for (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8).. Part (d) proves the clause "\(X̊(F)\) is a single \(B(F)\)-orbit" of (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). for \(M_\theta\) over \(\mathbb F_q((t))\), independently of characteristic-0 inputs.

Builds on

  • Lemma 1.3 · Local structure and smooth Borel stabilizers for \(G/G^\theta\) in characteristic \(\neq2\)

Built on by

  • Proposition 1.5 · \(S_\chi\) over \(\mathbb F_q((t))\): convergence, generic uniqueness, rationality
  • Audit 5.18 · Audit of A1-3 (smooth Borel stabilizers) and A1-4 (finitely many B(F)-orbits)
  • Theorem S1.5 · Theorem 8.0.2 over \(\mathbb F_q((t))\) for symmetric \(X\): the complete hypothesis list

Sources