§4 · Agent 4 · Close fields · 2026-10-18T02:49:29Z

Proposition 4.9 · Proposition: what the characteristic-\(p\) theorem needs for symmetric \(X\)

proved audit: minor defect

Minor defect (second audit). Re-derived the conditional theorem: x_kappa = kappa(sqrt varpi)x0 is rational (Galois acts by kappa(-1) in S[2], contained in H; E(sqrt varpi)/E separable for p odd) with tau(x_kappa) = t^kappa; a split maximal torus T_H of H lies in a split maximal torus gTg^-1, g in G(Lambda), so eta(varpi) is in K t^{eta'} K and Lemma 4.2(e) applies with Lambda-coefficients; by (T3) the label test is 'exists k: t^-l k^-1 t^kappa theta(k) t^{theta l} in K'; covering uses (T2), X(O) = Kx0 uses (T1) with Lang and Hensel, counting uses (T4), and the module transfer uses (T5). Defect: step 5 asserts that P_kappa is e-independent for a general symmetric X because the data of (8.1)/(7.4) 'are the same for all F'_e', without argument; (T5) does not give this, since the signs sigma of virtual weighted colours depend on the F'_e-rationality of type-T tori (p. 59), and a torus split by a ramified quadratic field such as Q_q(sqrt p) splits over Q_q(p^{1/e}) only for e even. The conclusion holds anyway: under (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. over Lambda (the 'e.g.' of (T5)) the tori are defined over the discrete valuation ring and the data are e-independent, and in general the data take finitely many values, so the transfer can be run along infinitely many e with constant data. The run's audit Audit 5.7 graded this step correct without addressing the point.

The argument of Proposition 4.5, Proposition 4.6, Theorem 4.8 is isolated for a general symmetric variety. In characteristic \(p\) it needs only that \(G^\theta\) is split with connected fibres, that \(X(E)=G(E)x_0\), and that \(\tau\) detects \(\mathcal O\)-points; everything else (the Cartan decomposition of \(X\), all of Sakellaridis's axioms, the theorem itself) is needed only over \(p\)-adic fields of unbounded ramification with residue field \(\mathbb F_q\). This is the exact list of further hypotheses the transfer route uses.

Proposition 4.9.

Let \(p\) be odd, \(\Lambda=W(\mathbb F_q)\), \(G\) split reductive over \(\Lambda\) with connected fibres and simply connected derived group, \(\theta\) an involution of \(G_\Lambda\) stabilizing a split maximal torus \(T\) that contains a maximal \(\theta\)-split torus \(S\) (split over \(\Lambda\)), \(H=G^\theta\), \(X=G/H\), \(x_0=eH\), \(\Lambda_S^+=X_*(S)\cap\Lambda_T^+\), \(x_\kappa:=\kappa(\sqrt\varpi)x_0\) (rational because \(S[2]\subseteq H\), and \(E(\sqrt\varpi)/E\) is separable for \(p\) odd; \(\tau(x_\kappa)=t^\kappa\)). Assume: (T1) \(H\) has connected fibres and is split over \(\Lambda\); (T2) \(X(E)=G(E)x_0\) for \(E=F\) and \(E=F'_e\), \(e\gg0\) (e.g. from \(\mathring X(E)=B(E)x_0\), part of (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). over \(E\), by the density argument of Proposition 4.5 (E3)); (T3) \(X(\mathcal O_E)=\{y\in X(E):\tau(y)\in G(\mathcal O_E)\}\) for these \(E\) (e.g. if \(\tau\) is a closed immersion over \(\Lambda\), as in Audit 4.3); (T4) \(X(F'_e)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa\) for \(e\gg0\) (e.g. Delorme–Sécherre with one \(H(F'_e)\)-class of maximal \((\theta,F'_e)\)-split tori and \(W_X\) realized in \(K\)); (T5) Theorem 8.0.2 holds over \(F'_e\) for \(e\gg0\), with Sakellaridis's labelling of \(Kx_\kappa\) independent of \(e\) (e.g. (A1)–(A5)Sakellaridis's standing assumptions for Theorem 8.0.2: (A1) characteristic-0 field and split group; (A2) integral model; (A3a) rank-one models over O, (A3b) Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) spherical roots and colours; (A5) homogeneous, affine, one open Borel orbit. over \(\mathcal O'_e\), with \(\psi'\) of conductor \(\mathcal O'_e\) as in Audit 4.7(a); (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). and (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. over \(\mathcal O'_e\) follow from their versions over \(\Lambda\) by flat base change, Audit 4.7 step 8). Then over \(F=\mathbb F_q(\!(t)\!)\): \(X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa\); \((T_\lambda*\mathbf 1_{Kx_\mu})(x_\nu)\) equals its value over \(F'_e\) for \(e\ge D(\lambda,\mu)\); and \(\mathbf 1_{Kx_\kappa}\mapsto P_\kappa\) (the images over \(F'_e\), given by (8.1) and (7.4), which depend only on \(q\) and the combinatorial data of \(X\)) is an isomorphism \(C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) compatible with \(\mathcal H_K\) and the Satake isomorphism, with \(\mathbf 1_{X(\mathcal O)}\mapsto1\).

Proof. The proofs of Proposition 4.5, Proposition 4.6 and Theorem 4.8 (steps 1–3) go through with these changes.

  1. Cartan decomposition of \(H\). By (T1), \(H(E)=\bigsqcup_\eta H(\mathcal O)\eta(\varpi)H(\mathcal O)\), \(\eta\in X_*(T_H)\) for a split maximal torus \(T_H\subseteq H\) over \(\Lambda\), not necessarily inside \(T\). \(T_H\) lies in a split maximal torus \(T'\) of \(G_\Lambda\) (inside the smooth reductive centralizer \(Z_G(T_H)\), lift a split maximal torus of the special fibre containing \(T_{H,\mathbb F_q}\); a torus over the henselian \(\Lambda\) with split special fibre is split), and \(T'=gTg^{-1}\) with \(g\in G(\Lambda)\) (conjugacy of split maximal tori over a local ring, SGA 3 Exp. XXVI 7.16; not opened). So \(\eta(\varpi)=g\,t^{\eta'}g^{-1}\in Kt^{\eta'}K\) with \(\eta'\in X_*(T)\): the matrix coefficients of \(\rho_i(\eta(\varpi))\) are polynomials in the fixed \(\Lambda\)-point \(g\) and in \(\varpi^{\pm1}\), so Lemma 4.2(e) applies, and the bound of Proposition 4.5 step 4 gives a finite set \(B(\lambda)\).
  2. \(X(\mathcal O)=Kx_0\). (T1) and Hensel–Lang, as in (E2).
  3. Labels. Membership \(x_\kappa\in Kt^\lambda X(\mathcal O)\) is tested, by (T3), as \(\exists k\in K:\operatorname{val}\rho_i(t^{-\lambda}k^{-1}t^\kappa\theta(k)t^{\theta\lambda})=0\ \forall i\), a condition of type Lemma 4.2(e); it replaces Proposition 4.5 step 6, which used \(x_\kappa\in T(F)x_0\). Separation (step 1 of Proposition 4.5) uses only \(\tau(x_\kappa)=t^\kappa\).
  4. Covering uses (T2); counting uses (T4) on the \(p\)-adic side; Proposition 4.6 uses only (H-Cartan) on both sides, \(\theta\) over \(\Lambda\) and \(\tau(x_\nu)=t^\nu\).
  5. Module transfer is Theorem 4.8 steps 1–3 with (T5); \(P_\kappa\) is independent of \(e\) because (8.1) and (7.4) express it through \(q\), \(\Theta^+\) with its signs and exponents, \(\check\Phi_X\) and \(\delta_{P(X)}\), which are the same for all \(F'_e\). The uniqueness step of Theorem 4.8 used the \(E_6/F_4\) conductor and is not claimed here. \(\square\)

Reading for (Q1). The characteristic-\(p\) proof needs, in characteristic \(p\), only (T1)–(T3) (and the Cartan and Iwasawa decompositions of \(G(F)\)); no statement over an algebraic closure of \(\mathbb F_q(\!(t)\!)\), no Knop–Brion–Luna theory and no axiom of Sakellaridis over \(\mathbb F_q[\![t]\!]\). The price is (T4), (T5) at infinitely many ramified \(p\)-adic places with residue field \(\mathbb F_q\). For \(\mathcal M_\theta\): (T1) by S4-4 and Computation 4.4; (T2) by Proposition 4.5 (E3); (T3) by Audit 4.3; (T4) by S4-12, S4-13; (T5) by Audit 4.7 under \((I_\Lambda)\). Since \(\Lambda_S=(1-\theta)X_*(T)\) for \(\mathcal M_\theta\), the proof of Proposition 4.5 there avoids (T3).

Builds on

  • Proposition 4.5 · Proposition: \(K\backslash X(\mathbb F_q((t)))\simeq\Lambda_S^+\) by counting orbits
  • Proposition 4.6 · Proposition: all structure constants and Satake coefficients agree over close fields
  • Theorem 4.8 · Theorem: Theorem 8.0.2 for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\); (SG) fails

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