§3 · Agent 3 · Hypotheses in char p · 2026-10-18T02:57:58Z

Audit 3.5 · Delorme–Sécherre over $\mathbb F_q((t))$, $p$ odd: $G(F)=H(F)T(F)K$

proved audit: correct ★ key step

Second audit: correct. I read DS arXiv v1 §§1-4 in full. The field hypothesis is 'non Archimedean locally compact field of residue characteristic different from 2' throughout, and char k = 0 appears only in Lemma 3.6(iii), which §4 does not use. Each step of Prop. 2.1/2.4 (odd cardinality and pro-p indices), Lemma 3.9, Lemmas 4.1, 4.3, 4.6, 4.7, Prop. 4.4 and Thm 4.8 holds over F_q((t)), p odd: it uses |2| = 1, the scalar automorphisms of G_a, Borel 8.2(c) and 14.2, Humphreys 27.5(e), and building facts (BT I-II, Landvogt; not opened, but their standing hypotheses cover F_q((t))). Thm 4.8's proof needs x in the apartment of S_DS, which Audit 3.5's statement correctly includes. The replacement of Helminck-Helminck Lemma 2.2 holds: h^-1 S' h contains S, lies in L = S.Spin8, equals S.T_8' with T_8' Spin8(F)-conjugate to T_8 (Borel-Tits), and Spin8(F) is in H(F). With K = Stab(hyperspecial vertex) = G(O) and N(F)K = T(F)K, this gives G(F) = H(F)T(F)K.

Reads Delorme–Sécherre (arXiv:math/0612545v1): their field hypothesis, 'non-archimedean locally compact field of residue characteristic $\ne2$', includes $\mathbb F_q(\!(t)\!)$ with $p$ odd, and the proofs of Prop. 2.4, Lemma 3.9, Prop. 4.4 and Thm 4.8 use no characteristic-0 input. The only external lemma in Thm 4.8 (Helminck–Helminck, Lemma 2.2) is replaced for $E_6/F_4$ by a direct argument. With Proposition 3.4 this gives $G(F)=H(F)T(F)K$.

Audit 3.5.

Theorem (Delorme–Sécherre, Thm 0.4 = Thm 4.8, arXiv v1). Let $k$ be a non-archimedean locally compact field of residue characteristic $\ne2$, $G$ a connected reductive $k$-split group, $\sigma$ an involutive $k$-automorphism, $H$ open in $G^\sigma$, $A$ a maximal $(\sigma,k)$-split torus, $S_{\mathrm{DS}}\supseteq A$ a $\sigma$-stable maximal $k$-split torus, $\{A_j\}_{j\in J}$ representatives of the $H_k$-classes of maximal $(\sigma,k)$-split tori, $y_jAy_j^{-1}=A_j$, $x$ a special point of the apartment of $S_{\mathrm{DS}}$, $K$ its stabilizer. Then $G_k=\bigcup_{j}H_k\,y_j\,S_{\mathrm{DS},k}\,K$.

Claim. (i) The standing hypothesis of the paper (abstract, introduction, §§2–4) is exactly the one above, which includes $\mathbb F_q(\!(t)\!)$, $p$ odd. (ii) The proofs of Prop. 2.4, Lemma 3.9, Prop. 4.4 and Thm 4.8 use only: Bruhat–Tits theory and Landvogt's results on the building of a split group over a local field; that the groups $U_\Omega^\pm$ are pro-$p$ with $p$ odd; $2\in k^\times$ with $|2|=1$; structure theory from Borel's and Humphreys' books; and Helminck–Helminck, Lemma 2.2. None of these uses $\mathrm{char}\,k=0$. (iii) For $G=E_6$, $\sigma=\theta_0$, $H=G^\theta$, $S_{\mathrm{DS}}=T$, $A=S$, $x$ the hyperspecial vertex fixed by $K=G(\mathcal O)$: $J=\{1\}$ by Proposition 3.4(d), and Helminck–Helminck's lemma is replaced by a direct argument. Hence, over $F=\mathbb F_q(\!(t)\!)$, $$G(F)=H(F)\,T(F)\,K=K\,T(F)\,H(F).$$

The field. The abstract and introduction say 'non Archimedean locally compact field of residue characteristic different from 2'; §1 'non Archimedean non discrete locally compact field'; §§2–4 'residue characteristic different from 2'. The characteristic of $k$ is invoked once, in Lemma 3.6(iii) ('characteristic different from 2', for $[k^\times:k^{\times2}]<\infty$), which holds for $\mathbb F_q(\!(t)\!)$ with $p$ odd and is not used in §4.
Prop. 2.1 and 2.4 (every point of the building lies in a $\sigma$-stable apartment). For $\Omega=\{y,\sigma y\}$, $y$ in a chamber, $P_\Omega/N_\Omega$ is a projective limit of finite $\sigma$-sets $P_\Omega/P_\Omega^i$ whose cardinalities are powers of $p$ (Lemma 2.3, from the Iwahori decomposition $P_\Omega=U_\Omega^-U_\Omega^+N_\Omega$ and pro-$p$-ness of $U_\Omega^\pm$), hence odd, and a $\sigma$-set of odd cardinality has a fixed point (Lemma 2.2). Inputs: Landvogt, Cor. 12.6, Cor. 13.7; Bruhat–Tits I, 7.4. For a split group over $\mathbb F_q(\!(t)\!)$ these hold as for $p$-adic fields; the root-group filtrations are $t^m\mathbb F_q[[t]]$, pro-$p$.
Lemma 3.9. Prop. 2.4 applied to $Z_G(A)^\circ$. No further input.
Prop. 4.4. Among $\sigma$-stable apartments through the special point $x$, take one with $\dim S_{\mathrm{DS}}^-$ maximal. If $S_{\mathrm{DS}}^-$ is not maximal $(\sigma,k)$-split, Lemma 4.6 (Humphreys 27.5(e); Lemma 4.3 from Borel 8.2, 14.2) gives a $\sigma$-invariant root $a$ with $U_a\not\subseteq H$; Lemma 4.7: $\sigma=-1$ on $U_a\simeq\mathbb G_a$ (automorphisms of $\mathbb G_a$ are scalars in every characteristic); Lemma 4.1 writes $w=v^2$ in $U_{-a}(k)\simeq(k,+)$, i.e. $v=w/2$, with the same filtration index because $|2|=1$, and builds $c\in U_x$ with $c^{-1}\sigma(c)$ the affine reflection of $s_a$ fixing $x$ (Landvogt 7.4(ii), 7.7); then $c\cdot S_{\mathrm{DS}}$ is a $\sigma$-stable torus through $x$ with larger split part. Uses $p\ne2$, never $\mathrm{char}\,k=0$.
Thm 4.8. $g\cdot x$ lies in a $\sigma$-stable apartment $\mathcal A(S')$ with $S'^-$ maximal (Prop. 4.4); $S'^-=hy_jAy_j^{-1}h^{-1}$; Helminck–Helminck, Lemma 2.2 gives $S'=hy_jS_{\mathrm{DS}}$; Bruhat–Tits 1.13(2) (two points of an apartment conjugate under $G_k$ are conjugate under $N_k$) and $N_kK=S_{\mathrm{DS},k}K$ ($x$ special) give $g\in Hy_jS_{\mathrm{DS}}K$.
Replacing Helminck–Helminck for $E_6/F_4$. Here $J=\{1\}$, $y_1=1$. If $S'^-=hSh^{-1}$ with $h\in H(F)$ (Proposition 3.4(d)), then $h^{-1}S'h$ is a $\theta$-stable maximal $F$-split torus containing $S$, hence contained in $Z_G(S)=L=S\cdot\mathrm{Spin}_8$, so it equals $S\cdot T_8'$ with $T_8'$ a maximal $F$-split torus of the split group $\mathrm{Spin}_8$. Maximal $F$-split tori of $\mathrm{Spin}_8$ are $\mathrm{Spin}_8(F)$-conjugate (Borel–Tits, every field), and $\mathrm{Spin}_8(F)\subseteq H(F)$. So $S'=(hm)T(hm)^{-1}$ with $hm\in H(F)$.
Data. The hyperspecial vertex $x$ of $\mathcal A(T)$ has stabilizer $G(\mathcal O)=K$ ($G$ simply connected: stabilizer = parahoric, Bruhat–Tits II); $N(F)=T(F)\cdot N(\mathcal O)$ with Tits lifts in $G(\mathbb Z)$, so $N(F)K=T(F)K$. Hence $G(F)=H(F)T(F)K$; inverting, $G(F)=KT(F)H(F)$.

Status. Proved, modulo Delorme–Sécherre (read in arXiv v1; the statement had been quoted in S4-12 without its proof being checked) and the building-theoretic references inside it (Bruhat–Tits I–II, Landvogt), whose standing hypotheses do not involve the characteristic of $k$; those were not opened.

Builds on

Proposition 3.4 · Maximal $\theta$-split tori: dimension 2, one $H(F)$-class (residue char. $\ne2$)

Built on by

Theorem 3.8 · (A3b) for $\mathcal M_\theta$ over $\mathbb F_q((t))$: $K\backslash X(F)\simeq\Lambda_X^+$
Summary 3.12 · (Q3) verdict: (A1)–(A3c), (A5), (A4)(i,ii,iv) hold over $\mathbb F_q((t))$
Audit 5.9 · Audit of A3-4, A3-5, A3-8: θ-split tori, Delorme–Sécherre, (A3b) in char p

Sources

Delorme–Sécherre, An analogue of the Cartan decomposition for p-adic reductive symmetric spaces, arXiv:math/0612545v1 (2006): abstract; Lemmas 2.2, 2.3, Props 2.1, 2.4; Lemmas 3.6, 3.9; Lemmas 4.1, 4.3, 4.6, 4.7, Prop. 4.4, Thm 4.8 (read from the arXiv PDF; published Pacific J. Math. 251 (2011))
Helminck–Helminck, Trans. AMS 350 (1998), Lemma 2.2 (not opened; replaced)
Bruhat–Tits I, IHÉS 41 (1972), 7.4; Landvogt, LNM 1619 (cited inside DS, not opened)
S4-12

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Audit 3.5 · Delorme–Sécherre over \(\mathbb F_q((t))\), \(p\) odd: \(G(F)=H(F)T(F)K\)

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