§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:32:17Z

Audit 5.16 · Audit of A1-2: triangular spherical transforms give C_K ≅ C[D]^{W_X}

proved audit: correct

Second audit: correct. Steps 1–6 of Proposition 1.2 re-derived (a closed convex set with recession cone {0} is bounded; independence by a ⪰-minimal leading index; spanning by induction on |F_ν|). Remark (i) checked by adjunction: Ŝ(1_{xK})(w_L^{-1}ψ) = vol(xK)Ω_{ι(ψ)}(x)/m(ι(ψ)) with ι(ψ) = w_Lψ^{-1}, ι is inversion of A_X^* about δ(X)^{1/2}, and s_ν∘ι = s_{−w_0ν}, so P_λ∘ι ∝ P_{σλ} and the cone −w_0R = R. The convention claim (difference f ↦ f^∨) is right in substance: with the standard Satake transform, pairing with Ω_χ intertwines f with Sf(χ^{-1}) = Sf(ιχ); it passes over the vol(x_λK) factor, which Proposition 1.2 Remark (i) already treats, and changes nothing. The (A3b′) input via Theorem 3.8's dictionary (Sakellaridis's x_0·e^λ(ϖ) = e^{−λ}(t)x_0) is consistent, and injectivity of A_X^* → A^* holds for M_θ (T^θ connected: Smith normal form of 1−θ on the weight lattice is [1,1]).

The last step of Theorem 8.0.2 is correctly isolated as a characteristic-free statement. The independence, spanning and equivariance arguments are complete. The transport from \(D'\) to \(D\) in Remark (i) is consistent once read as the involution \(\iota(\psi)=w_L\psi^{-1}\) of \(D\).

Audit 5.16.

Outcome: holds. Steps 1–6 prove the statement under hypotheses (a)–(c), over any non-archimedean local field. Finiteness of \(F_\lambda\) uses \(\mathcal T\cap C^-=\{0\}\). Independence uses a \(\succeq\)-minimal leading index. Spanning is an induction on \(|F_\nu|\). Remark (i), made precise. Equivariance is preserved if \(D'\) is transported to \(D\) by \(\chi\mapsto w_L\chi\), since \(\mathcal Sf\) is \(W\)-invariant. In that coordinate, $$\hat S(\mathbf 1_{x_\lambda K})(w_L^{-1}\psi)=\operatorname{vol}(x_\lambda K)\,\Omega_{\iota(\psi)}(x_\lambda)/m(\iota(\psi)),\qquad \iota(\psi):=w_L\psi^{-1}.$$ Here \(\iota\) is inversion of \(A_X^*\) about the \(W_X'\)-fixed base point \(\delta(X)^{1/2}\) (\(w_L\) fixes \(A_X^*\) pointwise and inverts \(\delta(X)\)). So \(Q_\lambda\propto P_\lambda\circ\iota\), with leading index \(\sigma\lambda=-w_0^X\lambda\) and cone \(-w_0^X\mathcal T\), exactly as the Remark says. Convention (checklist item 8). With A1's left action \((g\Phi)(x)=\Phi(g^{-1}x)\) and \(\pi_\chi(f)\varphi_K=(\mathcal Sf)(\chi)\varphi_K\), the resulting module map is \(\mathbf 1_{x_\lambda K}\mapsto\operatorname{vol}(x_\lambda K)\,P_\lambda\circ\iota\) up to a constant. Sakellaridis writes \(\mathbf 1_{x_\lambda K}\mapsto P_\lambda\) in his right-action conventions. For \(\mathcal M_\theta\) the two cannot both hold with the same action convention: the module isomorphism with \(1\mapsto1\) is unique (A4-8 step 4), while \(\iota\) moves the leading index \(\lambda\mapsto-w_0^X\lambda\ne\lambda\). They therefore differ by \(f\mapsto f^\vee\). This changes neither surjectivity of \(r\) nor (SG).

  1. Equivariance. \(S_\chi(f*\Phi)=\pi_\chi(f)S_\chi(\Phi)\) for \(f*\Phi=\int f(g)\,g\Phi\,dg\). On \(K\)-invariants, \(\pi_\chi(f)\varphi_{K,\chi}=(\mathcal Sf)(\chi)\varphi_{K,\chi}\). Both sides of \(\hat S(f*\Phi)=(\mathcal Sf)\hat S(\Phi)\) are rational, so equality on \(D_0\) gives equality on \(D\).
  2. Basis. A closed chamber of a finite reflection group is a fundamental domain, so \(\{m_\mu\}_{\mu\in\Lambda^-}\) is a basis of \(\mathbb C[\Lambda]^{W_X}\). This needs \(D\cong\operatorname{Spec}\mathbb C[\Lambda]\) \(W_X\)-equivariantly. For \(D=\delta(X)^{1/2}A_X^*\) take the \(W_X'\)-fixed base point \(\delta(X)^{1/2}\), and use injectivity of \(A_X^*\to A^*\) (true for \(\mathcal M_\theta\), since \(T\to A_X\) has connected kernel \(T_8\)).
  3. Finiteness. \((\lambda+\mathcal T)\cap C^-\) is closed and convex, with recession cone \(\mathcal T\cap C^-=\{0\}\), so it is bounded.
  4. Independence. Take \(\lambda_0\) with \(\sigma\lambda_0\) minimal among the \(\sigma\lambda\) with \(c_\lambda\neq0\). If \(a_{\lambda,\sigma\lambda_0}\neq0\) then \(\sigma\lambda_0\succeq\sigma\lambda\), hence \(\lambda=\lambda_0\) by minimality and injectivity of \(\sigma\). So the coefficient of \(m_{\sigma\lambda_0}\) is \(c_{\lambda_0}a_{\lambda_0,\sigma\lambda_0}\ne0\).
  5. Spanning. \(m_{\sigma\lambda}=a_{\lambda,\sigma\lambda}^{-1}\big(Q_\lambda-\sum_{\mu\in F_{\sigma\lambda}\smallsetminus\{\sigma\lambda\}}a_{\lambda\mu}m_\mu\big)\), with \(|F_\mu|<|F_{\sigma\lambda}|\) for each \(\mu\) on the right; induct. Every \(\nu\in\Lambda^-\) is some \(\sigma\lambda\).
  6. Inputs for \(\mathcal M_\theta\). (a) is (A3b′): Proposition 4.5 or Theorem 3.8, read with antidominant labels through \(\kappa\mapsto-\bar\mu\). This needs no valuation-cone statement, because \(C^-\) is the \(W_X'\)-chamber. (b) is Proposition 1.5. (c) needs \(\kappa\not\equiv0\) independent of \(\lambda\), and leading coefficients \(\ne0\). \(\kappa\) is \(c\beta/m\) transported. \(m\not\equiv0\) because \(\Omega_\chi\not\equiv0\) (Audit 5.15 step 7). The leading coefficients are checked in Audit 5.17.
  7. Remarks (ii), (iii). The scalar \(m\) relating \(\Phi_{K,\chi}\) to \(\varphi_{K,\chi}\) is absorbed into \(\kappa\), so (4.5) and [CS80] are not needed. Theorem 3.2.2 is not used.

Builds on

  • Proposition 1.2 · Triangular spherical transforms give \(C_K\cong\mathbb C[D]^{W_X}\), any characteristic

Built on by

  • Summary 5.21 · Verdict of the second audit: A1-1 … A1-9 (direct rerun over F_q((t)))
  • Lemma S1.3 · Positivity of \(\hat S\) forces \(c\neq0\) and \(a_J/c>0\) over every local field

Sources