§3 · Agent 3 · Hypotheses in char p · 2026-10-18T03:00:41Z

Audit 3.9 · Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points. for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\): S4-5 re-checked

proved audit: correct

Second audit: correct. I traced the dominant-lift case element by element: the Iwahori factorisations, the open cells over O (monomorphisms of smooth O-schemes of equal relative dimension, theta(U_P) = U_P^-, theta(U_Q) = U_Q^-), and J cap L acting through A_X(O) since L^theta = [L,L]. My lattice computation confirms that the labels without a dominant lift are exactly 3k omega1^v and 3k omega6^v, k odd (b = 0 forces mu = n1 omega1^v), and it confirms phi(mu_1) = 3 omega1^v = (4,3,5,6,4,2), min <b,mu_1> = 0 on the 16 roots of U_Q and -1 on the 8 roots of U_{P cap M}, zeta = 3 omega1^v in X_*(Z(M)^0) cap X_*(S), and S[2] in T_8. The half-translation works: F(sqrt t)/F is separable (p odd), z^-1 sigma(z) = zeta(-1) lies in H, R_z is M(F)-equivariant, and tau(z^k x0) = phi(mu_k)(t) gives mu_k(t)x0 = R_z^k(x0); U_{P cap M}(O) is absorbed into the Levi part. delta commutes with theta0 (delta C = C delta on the pinning, delta(sigma_L) = sigma_L), so Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points. holds at every label.

Re-checks each step of S4-5's proof of Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points. over \(\mathcal O=\mathbb F_q[[t]]\), \(p\) odd: Iwahori factorizations, the open cells \(U_P\times A_X\) and \(U_Q\times M/M^\theta\) (with the scheme-theoretic stabilizers of Proposition 3.2), the action of \(J\cap L\) through \(T(\mathcal O)\), and the half-translation \(R_z\), defined over \(F(\sqrt t)\), separable because \(p\) is odd. All steps hold; the numerical bounds were recomputed.

Audit 3.9.

For every \(\kappa\in\Lambda_S^+\) and every point \(x\in A_X(F)x_0\) with label \(\kappa\): \(Jx\subseteq B(\mathcal O)x\). Equivalently, with right actions, \(xJ\subseteq xB(\mathfrak o)\) for \(x\in A_X^+\): Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points. holds for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\), \(p\) odd.

  1. Reduction. \(x=ax_\kappa\) with \(a\in\mathrm{im}\,T(\mathcal O)\subseteq J\cap B(\mathcal O)\), and \(Jax_\kappa=aJx_\kappa\subseteq aB(\mathcal O)x_\kappa=B(\mathcal O)ax_\kappa\); so \(x=x_\kappa=\mu(t)x_0\) suffices.
  2. \(J=U_P(\mathcal O)(J\cap L)U_P^-(\mathfrak p)\): the big cell \(U_P^-\times L\times U_P\to G\) is an open immersion over \(\mathbb Z\); an element of \(J\) reduces into \(B(\mathbb F_q)\), inside the big cell, so lies in its \(\mathcal O\)-points with \(U_P^-\)-component \(\equiv1\bmod t\). Characteristic-free.
  3. \(t^{-\mu}u_{-\beta}(c)t^{\mu}=u_{-\beta}(t^{\langle\beta,\mu\rangle}c)\). Formal.
  4. \(U_P\times A_X\to X\) is an open immersion over \(\mathcal O\) (Proposition 3.2(4)); \(u'x_0\) with \(u'\in U_P^-(\mathfrak p)\) reduces to \(\bar x_0\in\mathring X(\mathbb F_q)\), so lies in \(\mathring X(\mathcal O)\) and equals \(uax_0\) with \(u\in U_P(\mathfrak p)\), \(a\in A_X(\mathcal O)=\mathrm{im}\,T(\mathcal O)\).
  5. \(t^\mu U_P(\mathfrak p)t^{-\mu}\subseteq U_P(\mathfrak p)\) for \(\langle\beta,\mu\rangle\ge0\) on \(\Phi(U_P)\). Formal.
  6. \(L^\theta=\mathrm{Spin}_8=[L,L]\) (Proposition 3.2) is normal in \(L\), so \(L\) acts on \(Lx_0\simeq L/[L,L]\) through a torus and \(J\cap L\subseteq L(\mathcal O)\) acts on \(T(F)x_0\) through \(A_X(\mathcal O)\).
  7. So \(J\mu(t)x_0\subseteq B(\mathcal O)\mu(t)x_0\) when \(\kappa\) has a dominant lift \(\mu\).

7–12. Labels without dominant lift (\(3k\omega_1^\vee\), \(3k\omega_6^\vee\), \(k\) odd; family page §5). \(Q=MU_Q\), \(M\) the Levi of \(\{2,\dots,6\}\); \(\theta\) negates \(\alpha_1\)-coefficients, so \(\theta(U_Q)=U_Q^-\), \(Q\cap H=M^\theta\) scheme-theoretically, and \(U_Q\times M/M^\theta\to X\), \(U_{P\cap M}\times A_X\to M/M^\theta\) are open immersions over \(\mathcal O\) (monomorphisms of smooth \(\mathcal O\)-schemes of equal relative dimensions \(16+10\) and \(8+2\); \(M^\theta\) smooth since \(2\in\mathcal O^\times\)). Recomputed exactly: \(\varphi(\mu_1)=3\omega_1^\vee=4\alpha_1^\vee+3\alpha_2^\vee+5\alpha_3^\vee+6\alpha_4^\vee+4\alpha_5^\vee+2\alpha_6^\vee\) for \(\mu_1=2\alpha_1^\vee+\alpha_3^\vee+\alpha_4^\vee+\alpha_5^\vee+\alpha_6^\vee\); \(\min_{\Phi(U_Q)}\langle\beta,\mu_1\rangle=0\) (16 roots), \(\min_{\Phi(U_{P\cap M})}\langle\beta,\mu_1\rangle=-1\) (8 roots); \(\langle\beta,3\omega_1^\vee\rangle=0\) on \(\Phi_M\), \(=3\) on \(\Phi(U_Q)\). Half-translation: \(z:=\zeta(s)\), \(s^2=t\), \(\zeta=3\omega_1^\vee\), lies in \(Z(M)^\circ(F')\), \(F'=F(\sqrt t)\); as \(p\) is odd, \(F'/F\) is a separable quadratic extension with \(\sigma(s)=-s\), and \(z^{-1}\sigma(z)=\zeta(-1)\in S[2]\subseteq T_8\subseteq H\). So \(R_z(mx_0):=mzx_0\) is a well-defined \(M(F)\)-equivariant map with values in \(X(F')^{\sigma}=X(F)\). \(\tau(z^kx_0)=z^{2k}=\varphi(\mu_k)(t)\), \(\mu_k=\mu_1+\tfrac{k-1}2\zeta\), so \(\mu_k(t)x_0=R_z^k(x_0)\). The Levi part \((J\cap M)\mu_k(t)x_0=R_z^k((J\cap M)x_0)\subseteq B_M(\mathcal O)\mu_k(t)x_0\) (label 0 inside \(M/M^\theta\), steps 1–6 for \(M\)); the unipotent part uses \(\langle\beta,\mu_k\rangle\ge-1\) on \(\Phi(U_{P\cap M})\); \(J=U_Q(\mathcal O)(J\cap M)U_Q^-(\mathfrak p)\) gives \(J\mu_k(t)x_0\subseteq B(\mathcal O)\mu_k(t)x_0\). The labels \(3k\omega_6^\vee\) follow from the pinned diagram automorphism \(\delta\) (\(1\leftrightarrow6\), \(3\leftrightarrow5\)), which commutes with \(C\), fixes \(\sigma_L\) and hence commutes with \(\theta_0\), and preserves \(T,B,K,J\).

Characteristic. Beyond \(2\in\mathcal O^\times\), the residue characteristic enters only through the separability of \(F(\sqrt t)/F\), which fails exactly for \(p=2\).

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