§3 · Agent 3 · Hypotheses in char p · 2026-10-18T03:01:49Z

Remark 3.11 · Not proved here: the analytic part of Statement 6.3.1 over \(\mathbb F_q((t))\)

blocked audit: correct

Second audit: correct. Checked against the text: Thm 6.3.2 is proved through Prop. 6.4.2, which rests on Thm 6.11.1 (Akhiezer and Wasserman classifications over k-bar), and through Prop. 6.4.1 via Prop. 6.5.1. The type-G case of 6.5.1 uses Example 5.4.1 (b = -e^a(1-q^-1 e^-a)/(1-q^-1 e^a), as restated in its proof), the type-U functional equations, and (6.10) at the P_a\SL3 vertices, which are exactly the vertices of this path. The q^-4 formula agrees with (6.3) (<gamma-check, rho_P(X)> = 4) and with (6.18) for n = 5. Since Statement 6.3.1 depends only on the first edge (p. 46), the explicit path of Proposition 3.10 legitimately replaces Prop. 6.4.2. The remark proves nothing and claims nothing beyond this bookkeeping; its list of open local identities (a)-(d) and of unopened characteristic-free citations is accurate.

Records the one part of (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5. that is not a property of the variety: Statement 6.3.1 is a formula for Sakellaridis's cocycles \(B_{w_\gamma}\), defined by integration over \(F\), and its proof (Thm 6.3.2) rests on rank-one and rank-two local computations. For \(\mathcal M_\theta\) the combinatorial reduction is done in Proposition 3.10; the remaining local identities must be re-derived over \(\mathbb F_q(\!(t)\!)\) within (Q1). Also lists the characteristic-free external results used but not opened.

Remark 3.11.

Remaining obligation for (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5.(iii). For the Brion path of Proposition 3.10, over \(F=\mathbb F_q(\!(t)\!)\): (a) the Mackey morphisms \(S_\chi,\Delta_\chi\) and the scalars \(b_w\) of §§3–5, with the normalizations of §3.3 and the Fourier transform for \(\psi\) of conductor \(\mathcal O\) (§3.8); (b) the type-\(U\) functional equation of §5 used in the proof of Prop. 6.5.1; (c) Example 5.4.1 (two orthogonal roots lowering an orbit: \(b=-e^{\check\alpha}\frac{1-q^{-1}e^{-\check\alpha}}{1-q^{-1}e^{\check\alpha}}\)); (d) identity (6.10), \(e^{-\check\alpha}(\chi)=q^{-1}e^{-w_\alpha\check\beta}(\chi)\), for \(P_\alpha\backslash\mathrm{SL}_3\). Given (a)–(d) over \(F\), the induction of Prop. 6.5.1 yields $$B_{w_\gamma}(\tilde\chi)=-e^{\check\gamma}\,\frac{1-q^{-4}e^{-\check\gamma}}{1-q^{-4}e^{\check\gamma}}\qquad(\gamma\in\Delta_X,\ \langle\check\gamma,\rho_{P(X)}\rangle=4).$$ These are statements about function theory over \(F\), part of (Q1), and are not verified here.

Cited, characteristic-free in their standing hypotheses, not opened locally: Bruhat–Tits III (1987), Thm 4.7; Steinberg (1968), Thm 7.5; Richardson (1977) or Haboush (1975) for the affineness of \(G/H\); Prasad–Yu (2002) Thm 2.1 or CGP A.8.12 for reductivity of \((G^\theta)^\circ\); Bruhat–Tits I–II and Landvogt inside Delorme–Sécherre; Jantzen II.2 (dominance of highest weights); Popov / Fossum–Iversen (\(\mathrm{Pic}\,G=0\)).

  1. Why (iii) is not a hypothesis on \(X\) alone. Theorem 6.3.2 is stated under 'no type \(N\), \(L_\gamma\) classical' and proved through Props 6.4.1 and 6.4.2. Prop. 6.4.2 (existence of a suitably glued Brion path) is combinatorial and relies on the characteristic-0 classification Thm 6.11.1 (Akhiezer, Wasserman); for \(\mathcal M_\theta\) it is replaced by the explicit path of Proposition 3.10. Prop. 6.4.1 rests on Prop. 6.5.1, whose proof uses (b)–(d), computed in §§5.2–5.4 and 6.6–6.10 by integration over the local field.
  2. Dead end. Statement 6.3.1 cannot be derived from the geometry of \(X\): it is an identity between rational functions of \(\chi\) defined by intertwining integrals. Nothing visible in (b)–(d) uses \(\mathrm{char}\,F=0\) (Haar measures, \(|\cdot|\), \(\psi\)), but by the rules of this run 'carries over verbatim' does not count, and the verification belongs to (Q1).
  3. Effect. Theorem 7.2.1 for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\) needs 6.3.1 and 7.1.5; 7.1.5 is proved (Proposition 3.10), so the \(\beta\)-formula is conditional on (a)–(d), on (Q1)'s version of §§3–4, and on (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. (Q2).

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