§3 · Agent 3 · Hypotheses in char p · 2026-10-18T02:57:13Z

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

proved audit: correct ★ key step

Second audit: correct. (a): I re-derived 52 = (6-d) + c + (72-I)/2 (char != 2) and computed that a subspace spanned by at most three roots contains at most 12 roots, which excludes d >= 3. (b): the SL2 trick is valid (Hopf automorphisms of G_a are scalars; T_a is theta-split and k-split because 2 is invertible), and the identity |Phi(M)| = 20 + 2d is exact, since a nontrivial a|_A has infinite order. My enumeration of the 57 node subsets gives Levi sizes {0,...,14,20,24}, with 24 only for {2,3,4,5}, so d = 2 and M is the D4-Levi. (c): the two open H-orbits on G/P coincide (dim H.Q = 52 - 28 = 24). (d): DS Lemma 3.9 (I read its proof: Prop. 2.4 for Z_G(A')^0, independent of the characteristic) gives S'; the transporter is a torsor under the smooth N_H(S), the cocycle lies in Z_H(S)(F^s) = Spin8(F^s) because Int(h)|_S is a map of split tori, and H^1(F,Spin8) = 1 (Lemma 3.3); the unstated half-line that every theta-split torus lies in a theta-stable maximal torus (Steinberg 7.5) is standard.

Proves directly, over any field of characteristic \(\ne2\), that a maximal \(\theta\)-split torus has dimension 2 with centralizer a \(D_4\)-Levi whose derived group lies in \(H\), that all of them are \(H\)-conjugate over an algebraic closure, and that over a non-archimedean local field with \(p\) odd, of characteristic 0 or \(p\), the maximal \((\theta,F)\)-split tori form a single \(H(F)\)-class. This replaces Helminck–Wang's Prop. 10.3, Vust/Richardson and Kneser in S4-12.

Proposition 3.4.

Let \(k\) be a field, \(\mathrm{char}\,k\ne2\). A torus \(A\subseteq G_k\) is \(\theta\)-split if \(\theta(a)=a^{-1}\) on \(A\), and \((\theta,k)\)-split if moreover it is \(k\)-split.

  1. A dimension identity. Let \(T'\) be a \(\theta\)-stable maximal torus (split over the field considered), \(\theta'\) the induced involution of \(X^*(T')\), \(d=\dim(T'^{-\theta})^\circ\). Root spaces are lines and \(d\theta(\mathfrak g_a)=\mathfrak g_{\theta'a}\). On a \(\theta'\)-orbit \(\{a,\theta'a\}\) of size 2, \(d\theta\) swaps two lines and fixes a line; on a \(\theta'\)-fixed (imaginary) root space \(d\theta=\pm1\) (char. \(\ne2\)). With \(I\) the number of imaginary roots and \(c\le I\) the number on which \(d\theta=+1\):
$$52=\dim\mathfrak g^\theta=(6-d)+c+\tfrac{72-I}{2}\le42-d+\tfrac I2 .$$
  1. (a). Imaginary roots lie in the \((+1)\)-eigenspace of \(\theta'\), of dimension \(6-d\), so they form a root subsystem of rank \(\le6-d\). If \(d\ge3\) this rank is \(\le3\) and \(I\le12\) (\(A_3\) is the largest rank-3 subsystem of \(E_6\)), contradicting \(I\ge20+2d\ge26\).
  2. The \(\mathrm{SL}_2\) trick. Let \(A\), \(S'\) be as in (b), \(A\) maximal; then \(A=(S'^{-\theta})^\circ\). For \(a\in\Phi(M,S')\), \(a\) and \(a\circ\theta'\) agree on \((S'^{\theta})^\circ\) and are trivial on \(A\), so \(\theta'a=a\), \(\theta(U_a)=U_a\) and \(\theta(u_a(x))=u_a(\varepsilon x)\) with \(\varepsilon=\pm1\) (automorphisms of \(\mathbb G_a\) are scalars in every characteristic). If \(\varepsilon=-1\), then \(\theta\) fixes \(a^\vee(\mathbb G_m)\) pointwise and acts on \(G_a=\langle U_a,U_{-a}\rangle\) (\(\simeq\mathrm{SL}_2\) or \(\mathrm{PGL}_2\) over \(k\)) as \(\mathrm{Int}\,\mathrm{diag}(-1,1)\); the torus \(T_a=\{\begin{pmatrix}x&y\\y&x\end{pmatrix}:x^2-y^2=1\}\) is \(\theta\)-split and \(k\)-split (conjugate by \(\begin{pmatrix}1&1\\1&-1\end{pmatrix}\), invertible as \(2\in k^\times\)), commutes with \(A\), and meets \(A\) in a finite group, so \(A\cdot T_a\) is a larger \((\theta,k)\)-split torus. Hence \(\theta\) is trivial on every root group of \(M\), and \([M,M]\subseteq H\).
  3. (b). Apply step 1 to \(S'\): \(\theta\) is trivial on \((S'^\theta)^\circ\) and on \([M,M]\), inversion on \(A\); roots of \(M\) are imaginary with \(d\theta=+1\); a root \(a\notin\Phi(M)\) has \(a|_A\ne1\), hence \((\theta'a)|_A=a|_A^{-1}\ne a|_A\). So, with \(d=\dim A\),
$$52=(6-d)+|\Phi(M)|+\tfrac{72-|\Phi(M)|}{2},\qquad |\Phi(M)|=20+2d .$$

\(\Phi(M)\) is a Levi subsystem of rank \(\le\mathrm{rank}\,H=4\) (\([M,M]\subseteq H\); Proposition 3.2 step 3). Levi subsystems of \(E_6\) of rank \(\le4\) have \(0,2,4,6,8,10,12,14,20\) or \(24\) roots, and 24 only for the \(D_4\) on \(\{2,3,4,5\}\) (exact enumeration of the 57 subsets of at most four nodes). \(d=0\) would give \(M=G\) and \(\theta=\mathrm{id}\). So \(d=2\), \(|\Phi(M)|=24\), \(M\) is a \(D_4\)-Levi, and \(Z(M)^\circ\) (dimension \(6-4=2\)) contains \(A\), hence equals \(A\).

  1. (c). Over \(\bar k\), let \(A\) be maximal \(\theta\)-split and \(M=Z_G(A)\) as in (b) (a \(\theta\)-stable maximal torus of \(M\) exists by Steinberg 1968, Thm 7.5, applied to \(\theta|_M\)). For generic \(\lambda\in X_*(A)\), \(Q:=P_G(\lambda)\) has Levi \(M\), \(\theta(Q)=P_G(\lambda^{-1})\) is opposite to \(Q\), and \(Q\cap\theta(Q)=M\). Since \(\{2,3,4,5\}\) is the only \(D_4\) subdiagram, \(Q\) is \(G\)-conjugate to \(P\), i.e. \(Q\in G/P\). \(H\cap Q\subseteq Q\cap\theta(Q)=M\), so \(H\cap Q=M^\theta=A[2]\cdot[M,M]\), of dimension 28, and the orbit \(H\cdot Q\subseteq G/P\) has dimension \(52-28=24=\dim G/P\): it is open. Likewise \(H\cap P=\mathrm{Spin}_8\) (Proposition 3.2) and \(H\cdot P\) is open. \(G/P\) is irreducible, so the two open orbits meet and coincide: \(Q=hPh^{-1}\) with \(h\in H(\bar k)\). Then \(M=Q\cap\theta(Q)=h(P\cap\theta(P))h^{-1}=hLh^{-1}\) (\(\theta(h)=h\); \(P\cap\theta(P)=L\), S4-11) and \(A=Z(M)^\circ=hSh^{-1}\).
  2. (d). Let \(A'\) be a maximal \((\theta,F)\)-split torus. Delorme–Sécherre's Lemma 3.9 (a consequence of their Prop. 2.4; both checked for \(\mathbb F_q(\!(t)\!)\) in steps 2–3 of Audit 3.5, which do not use the present node) gives a \(\theta\)-stable maximal \(F\)-split torus \(S'\supseteq A'\). By (b) \(\dim A'=2\), so by (a) \(A'_{\bar F}\) is maximal \(\theta\)-split and by (c) \(A'=hSh^{-1}\) with \(h\in H(\bar F)\). The transporter \(\mathrm{Tr}=\{h\in H:hSh^{-1}=A'\}\) is a closed \(F\)-subscheme of \(H\), a torsor under \(N_H(S)\); \(N_H(S)^\circ=Z_H(S)=L\cap H=\mathrm{Spin}_8\) is smooth (Conrad, Lemma 2.2.4) and \(N_H(S)/Z_H(S)\) embeds in the constant group \(\mathrm{Aut}(S)\simeq\mathrm{GL}_2(\mathbb Z)\), so \(N_H(S)\) and \(\mathrm{Tr}\) are smooth and \(\mathrm{Tr}(F^s)\ne\emptyset\). For \(\sigma\in\mathrm{Gal}(F^s/F)\), \(\mathrm{Int}(h)|_S:S\to A'\) is a homomorphism of \(F\)-split tori, hence Galois-invariant, so \(c_\sigma:=h^{-1}\sigma(h)\) lies in \(Z_H(S)(F^s)=\mathrm{Spin}_8(F^s)\) and is a cocycle. \(H^1(F,\mathrm{Spin}_8)=1\) (Lemma 3.3), so \(c_\sigma=z\sigma(z)^{-1}\), and \(h'=hz\in H(F)\) satisfies \(h'Sh'^{-1}=A'\).

Remarks. The cocycle never lands in the disconnected \(N_H(S)\), whose cohomology involves \(H^1(F,S_3)\), infinite for \(p=3\); the bad prime causes no difficulty. Helminck–Wang's Prop. 10.3 (field hypotheses not checked) and the conjugacy theorems of Vust and Richardson are not used.

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