§3 · Agent 3 · Hypotheses in char p · 2026-10-18T02:56:12Z

Proposition 3.2 · (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7)., (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7). for \(\mathcal M_\theta\) over any complete DVR with 2 invertible

proved audit: correct

Second audit: correct. I re-derived each item. Conrad Lemma 2.2.4 and Thm 2.3.1 (checked in the local text, valid over any base) apply to G' = G x| Z/2 with Z/2 = mu_2 (2 invertible): Z_{G'}(mu_2) = H x mu_2, so H is smooth with Lie H = g^theta, and G'/(H x mu_2) = G/H is smooth and quasi-affine. My own computations confirm dim g^theta = 52, the theta-fixed roots = Phi_L, T^theta = T_8 (elementary divisors (1,1)), Z_G(T_8) = T and an F4 root system on Lie H. An independent Frenkel-Kac model of the E6 Lie algebra confirms the S4-11 input that theta0 is the identity on Spin8 and theta0^2 = id. The connectedness argument (Aut F4 inner by Conrad 7.1.9(3), z in Z_G(T_8) cap H = T_8), the big-cell computation of B cap H, the equality L^theta = Spin8 (S[2] in T_8, checked mod 2), the fibrewise-flatness open-immersion argument and the purity argument (every component of the complement of the affine open cell has codimension 1, and none can be the whole special fibre) all hold. The invariant forms (<2rho - 2rho_L, alpha_j^v> = 0 for j = 2..5) and X(O) = Kx0 (torsor, Hensel, Lang) also hold; only 2 in O^x is used.

Proves the group and integral-model hypotheses of Sakellaridis's §1.7 for \(\mathcal M_\theta\) over \(\mathcal O=\mathbb F_q[[t]]\), \(p\) odd, by arguments valid over any complete discrete valuation ring in which 2 is a unit: \(H\) is smooth with connected fibres of type \(F_4\), \(X\) is smooth and quasi-affine, the open Borel orbit is \(U_P\times A_X\) over \(\mathcal O\), invariant forms exist and \(X(\mathcal O)=Kx_0\).

Proposition 3.2.

Let \(\mathcal O\) be a complete discrete valuation ring with \(2\in\mathcal O^\times\) and finite residue field \(\mathbb F_q\) (for instance \(\mathbb F_q[[t]]\), \(p\) odd), \(F\) its fraction field. Then:

  1. (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7). \(G\) is split reductive over \(\mathcal O\) with connected fibres and simply connected derived group.
  2. \(H=G^\theta\) is a smooth closed subgroup scheme with \(\mathrm{Lie}\,H=\mathfrak g^\theta\) of rank 52; each geometric fibre of \(H\) is connected reductive of type \(F_4\), with maximal torus \(T^\theta=T_8\).
  3. \(X=G/H\) is represented by a smooth quasi-affine \(\mathcal O\)-scheme of relative dimension 26, and \(\tau:X\to G\) is a monomorphism.
  4. \(T^\theta=T_8\) is a split torus; \(B\cap H=B_L\cap\mathrm{Spin}_8=:B_8\) and \(P\cap H=\mathrm{Spin}_8\) scheme-theoretically; \((u,a)\mapsto uax_0\) is an open immersion \(U_P\times A_X\hookrightarrow X\) whose image is \(\mathring X:=X\smallsetminus\overline{(X\smallsetminus\mathring X_F)}\); \(\mathring X\) is smooth and surjective over \(\mathcal O\) and \(\mathring X_{\mathbb F_q}\) is one \(B_{\mathbb F_q}\)-orbit with smooth stabilizer.
  5. \(X\) carries a \(G\)-invariant relative volume form and \(\mathring X\) a \(B\)-invariant one, over \(\mathcal O\); their absolute values give a \(G(F)\)-invariant measure on \(X(F)\) and a \(B(F)\)-invariant measure on \(\mathring X(F)\).
  6. \(X(\mathcal O)=Kx_0\), so \(\varphi_0=\mathbf 1_{X(\mathcal O)}=\mathbf 1_{Kx_0}\).

In particular (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7). and (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7). hold for \(\mathcal M_\theta\) over \(\mathbb F_q[[t]]\) for every odd \(p\), including \(p=3\).

  1. (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7).. Chevalley–Demazure group scheme over \(\mathbb Z\), base-changed to \(\mathcal O\).
  2. Smoothness of \(H\). Let \(G':=G\rtimes(\mathbb Z/2)_{\mathcal O}\), the generator acting by \(\theta\) (an automorphism of order 2 over \(\mathbb Z\), S4-11). Since \(2\in\mathcal O^\times\), \((\mathbb Z/2)_{\mathcal O}\simeq\mu_{2,\mathcal O}\) is of multiplicative type, so by Conrad, Lemma 2.2.4, its centralizer \(Z_{G'}(\mu_2)\) is smooth with Lie algebra \(\mathrm{Lie}(G')^{\mu_2}\). An element \(g\varepsilon\) centralizes the generator iff \(\theta(g)=g\), so \(Z_{G'}(\mu_2)=H\times\mu_2\) and \(H\), the fibre over the identity section of the étale \(\mu_2\), is smooth with \(\mathrm{Lie}\,H=\mathfrak g^\theta\). Over \(\mathbb Z[1/2]\) the idempotent \((1+d\theta)/2\) splits \(\mathfrak g\), and the rank of \(\mathfrak g^\theta\) is \(4+24+24=52\): the \(+1\)-eigenspace of \(-w_L\) on \(\mathfrak t\) is the black coroot span; \(\theta\) is the identity on \(\mathrm{Spin}_8\) (S4-11); the 48 other roots form 24 orbits \(\{\beta,\theta\beta\}\) with \(\theta\beta\ne\beta\) (exact computation: the \(\theta\)-fixed roots are exactly \(\Phi_L\), no root has \(\theta\beta=-\beta\)).
  3. Fibres are \(F_4\) and connected. Let \(k\) be an algebraically closed field of characteristic \(\ne2\). (a) The character group of \(T^\theta=\ker(\theta-1)\) is \(X^*(T)/(1-\theta^*)X^*(T)\); the elementary divisors of \(1-\theta^*\) are \((1,1)\) (computed), so the quotient is free of rank 4: \(T^\theta\) is a torus, equal to \(T_8\), in every characteristic. (b) No root of \(E_6\) is orthogonal to all of \(\alpha_2^\vee,\dots,\alpha_5^\vee\) (computed), so \(Z_G(T_8)=T\). (c) Hence \(T_8\) is a maximal torus of \(H_k^\circ\) (its centralizer there lies in \(T\cap H=T_8\)). (d) \(H_k^\circ\) is reductive: identity component of the fixed points of an automorphism of order prime to \(\mathrm{char}\,k\) of a reductive group (Prasad–Yu 2002, Thm 2.1; CGP, Prop. A.8.12, as quoted in Conrad, Remark 3.1.5; Steinberg 1968; all characteristic-free, not opened). Its \(T_8\)-roots on \(\mathfrak g^\theta\) are the 24 roots of \(\Phi_L\) and the 24 restrictions \(\beta|_{T_8}\) of the pairs, each with a one-dimensional space: the root system \(F_4\). So \(H_k^\circ\simeq F_4\). (e) Let \(g\in H(k)\). \(\mathrm{Int}(g)\) preserves \(H_k^\circ\), and every automorphism of \(F_4\) is inner (Conrad, Thm 7.1.9: automorphisms modulo \(G_{\mathrm{ad}}\) are automorphisms of the based root datum, trivial for \(F_4\)); so \(\mathrm{Int}(g)|_{H^\circ}=\mathrm{Int}(h)\) with \(h\in H^\circ(k)\), and \(z=h^{-1}g\) centralizes \(T_8\). By (b) \(z\in T\cap H=T^\theta=T_8\subseteq H^\circ\). Thus \(H(k)=H^\circ(k)\), and \(H_k\), being smooth, is connected. (This replaces Steinberg's Thm 8.1, which is characteristic-free in any case.)
  4. \(X\) quasi-affine. Conrad, Thm 2.3.1, applied to the smooth affine \(G'\) and its multiplicative-type subgroup \(\mu_2\): \(G'/Z_{G'}(\mu_2)\) is a smooth quasi-affine \(\mathcal O\)-scheme (built from Artin's theorem, Knutson's representability for quasi-finite separated algebraic spaces and Zariski's Main Theorem). Since \(G'=G\cdot\mu_2\) and \(G\cap(H\times\mu_2)=H\), it is \(G/H\). Relative dimension \(78-52=26\). \(\tau\) is a monomorphism: \(g\theta(g)^{-1}=g'\theta(g')^{-1}\iff g^{-1}g'\in H\) on points with values in any \(\mathcal O\)-algebra.
  5. The open orbit. For \(g=ul\in B\cap H\) (\(u\in U_P\), \(l\in B_L\)): \(\theta(u)=u\,l\theta(l)^{-1}\in\theta(U_P)\cap P=U_P^-\cap P=1\) (\(\theta\) maps \(\Phi^+\smallsetminus\Phi_L\) into \(\Phi^-\), computed; \(U_P^-\times P\to G\) is an open immersion over \(\mathbb Z\)), so \(u=1\) and \(l\in B_L\cap L^\theta\). \(L=S\cdot\mathrm{Spin}_8\) with \(S\cap\mathrm{Spin}_8=S[2]\simeq\mu_2^2\) étale, \(\theta(sm)=s^{-1}m\), so \(L^\theta=S[2]\cdot\mathrm{Spin}_8=\mathrm{Spin}_8\) on geometric fibres (\(S[2]\subseteq T_8\), family page §4); \(L^\theta\) is smooth (step 2) and contains the smooth \(\mathrm{Spin}_8\) with the same fibres, so they are equal (EGA IV 17.9.5). Hence \(B\cap H=B_8\) (smooth, connected fibres, relative dimension 16) and \(P\cap H=\mathrm{Spin}_8\). The orbit map \(B/B_8\to X\) is a monomorphism of smooth \(\mathcal O\)-schemes of relative dimension \(42-16=26\); on each fibre it is unramified between smooth varieties of equal dimension, hence étale and an open immersion; by the fibrewise flatness criterion it is flat, hence an open immersion (EGA IV 17.9.1). \(B/B_8\simeq U_P\times B_L/B_8=U_P\times T/T_8=U_P\times A_X\) is affine, and the purity argument of S4-4 step 4 (EGA IV 21.12.7 in the regular 27-dimensional \(X\)) gives \(\mathring X=U_P\times A_X\). The stabilizer \(B_8\) is smooth, so the orbit map of the special fibre is separable.
  6. Forms. \(\Omega^{26}_{X/\mathcal O}=G\times^H\det(\mathfrak g/\mathfrak h)^\vee\) and \(H\) has no non-trivial characters (connected semisimple fibres), so the bundle is \(G\)-equivariantly trivial. On \(\mathring X=B/B_8\), \(B_8\) acts on \(\det(\mathfrak b/\mathfrak b_8)\) by \(2\rho-2\rho_L\) restricted to \(T_8\), which pairs to \(1\cdot2-1\cdot2=0\) with \(\alpha_j^\vee\), \(j=2,\dots,5\) (\(\langle\rho_L,\alpha_j^\vee\rangle=(-3,1,1,1,1,-3)\), computed). Weil's construction of measures from volume forms works over every non-archimedean local field.
  7. \(X(\mathcal O)=Kx_0\). \(G\to X\) is an \(H\)-torsor; \(H^1(\mathcal O_{\text{ét}},H)=1\) for smooth affine \(H\) with connected fibres over the henselian \(\mathcal O\) with finite residue field (Hensel reduces to \(H^1(\mathbb F_q,H_{\mathbb F_q})\), which vanishes by Lang): the argument recorded in Conrad after Remark 7.2.13 for any non-archimedean local field. So every \(x\in X(\mathcal O)\) lifts to \(G(\mathcal O)\).

Characteristic. Only \(2\in\mathcal O^\times\) is used (steps 2, 5, 6); no step uses \(\mathrm{char}\,F=0\).

Builds on

  • Audit 3.1 · Where the \(p\)-adic verification (S4, S4b) used characteristic 0, and what replaces it

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