§1 · Agent 1 · Direct rerun · 2026-10-18T03:53:21Z

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

proved audit: correct ★ key step

Second audit: correct. I re-derived steps 1-6: equivariance from S_chi(f*Phi)=pi_chi(f)S_chi(Phi); the closed chamber as fundamental domain, so {m_mu} is a basis; (lambda+T) cap C^- is bounded because its recession cone is T cap C^-={0}, so F_lambda is finite; independence via a minimal sigma(lambda_0) (sigma injective, T strictly convex); spanning by induction on |F_nu| (sigma onto). I also redid Remark (i) by adjunction: S-hat(1_{x_lambda K})(chi)=vol(x_lambda K) Omega_{chi^{-1}}(x_lambda)/m(chi^{-1}). With chi=delta(X)^{-1/2}eta, Q_lambda(eta) is proportional to P_lambda(eta^{-1}), whose W_X'-stable weights lie in -lambda-R, hence in -w_0 lambda+R. So (c) holds with sigma(lambda)=-w_0 lambda, T=R, and leading coefficient that of e^lambda in P_lambda. Nothing depends on the characteristic.

Isolates the final step of the proof of Theorem 8.0.2: once the Cartan decomposition holds and the spherical transforms of the K-orbit indicators are triangular with respect to a cone meeting the antidominant chamber only in 0, the Hecke-module isomorphism follows by linear algebra valid over every local field.

Proposition 1.2.

Let \(F\) be any non-archimedean local field, \(G,K,\mathcal H_K,\mathcal S\) as in the setting, \(X\) a \(G\)-variety over \(F\). Let \(\Lambda\) be a lattice with a finite reflection group \(W_X\) acting on \(\Lambda_{\mathbb R}\), \(C^-\) a closed Weyl chamber, \(\Lambda^-:=\Lambda\cap C^-\), and \(\mathcal T\subset\Lambda_{\mathbb R}\) a closed strictly convex cone with \(\mathcal T\cap C^-=\{0\}\); write \(\mu\succeq\lambda\) iff \(\mu-\lambda\in\mathcal T\). Let \(D\subset A^*\) be a closed subvariety stable under a subgroup of \(W\) that acts on it through \(W_X\), with a \(W_X\)-equivariant isomorphism \(D\cong\operatorname{Spec}\mathbb C[\Lambda]\); put \(m_\mu=\sum_{\nu\in W_X\mu}e^\nu\). Assume:

(a) (Cartan) \(X(F)=\bigsqcup_{\lambda\in\Lambda^-}x_\lambda K\), a disjoint union.

(b) There are a Zariski-dense open \(D_0\subset D\) and \(G\)-morphisms \(S_\chi:C_c^\infty(X(F))\to I(\chi)\) (\(\chi\in D_0\)); write \(S_\chi(\Phi)=\hat S(\Phi)(\chi)\,\varphi_{K,\chi}\) for \(\Phi\in C_c^\infty(X(F))^K\).

(c) \(\hat S(\mathbf 1_{x_\lambda K})=\kappa\,Q_\lambda\) on \(D_0\), with \(\kappa\) a rational function on \(D\), not identically zero and independent of \(\lambda\), and \(Q_\lambda=\sum_{\mu\in\Lambda^-}a_{\lambda\mu}m_\mu\in\mathbb C[D]^{W_X}\) with \(a_{\lambda,\sigma\lambda}\ne0\) and \(a_{\lambda\mu}\neq0\Rightarrow\mu\succeq\sigma\lambda\), for some bijection \(\sigma\) of \(\Lambda^-\).

Then \(\Phi\mapsto\hat S(\Phi)/\kappa\) is a linear isomorphism \(C_c^\infty(X(F))^K\xrightarrow{\sim}\mathbb C[D]^{W_X}\) under which \(f\in\mathcal H_K\) acts as multiplication by \((\mathcal Sf)|_D\). If moreover \(Q_0\) is a non-zero constant, then after rescaling \(\kappa\) it sends \(\mathbf 1_{x_0K}\mapsto1\).

This is the last paragraph of the proof of Theorem 8.0.2 (p. 64), written out with the normalizations made explicit. Nothing in it depends on the characteristic of \(F\).

Proof.

  1. Equivariance. \(S_\chi\) is \(G\)-equivariant for \((g\Phi)(x)=\Phi(g^{-1}x)\), and \(f*\Phi=\int_G f(g)\,g\Phi\,dg\) (vol \(K=1\)). Hence \(S_\chi(f*\Phi)=\pi_\chi(f)S_\chi(\Phi)\). For \(f\in\mathcal H_K\), \(\pi_\chi(f)\varphi_{K,\chi}=(\mathcal Sf)(\chi)\varphi_{K,\chi}\) by the definition of the Satake transform. So \(\hat S(f*\Phi)=(\mathcal Sf)\cdot\hat S(\Phi)\) on \(D_0\), and therefore on \(D\), since both sides are rational and agree on a dense set.
  1. Basis of the target. \(C^-\cap\Lambda\) meets every \(W_X\)-orbit in \(\Lambda\) exactly once (a closed chamber is a fundamental domain of a finite reflection group). So \(\{m_\mu\}_{\mu\in\Lambda^-}\) is a basis of \(\mathbb C[\Lambda]^{W_X}=\mathbb C[D]^{W_X}\).
  1. Finiteness. For \(\lambda\in\Lambda^-\) put \(F_\lambda=\{\mu\in\Lambda^-:\mu\succeq\lambda\}\). The set \((\lambda+\mathcal T)\cap C^-\) is closed and convex with recession cone \(\mathcal T\cap C^-=\{0\}\). It is therefore bounded, so \(F_\lambda\) is finite. Since \(\mathcal T\) is a strictly convex cone, \(\succeq\) is a partial order. If \(\mu\in F_\lambda\setminus\{\lambda\}\), then \(F_\mu\subseteq F_\lambda\setminus\{\lambda\}\).
  1. Independence. Suppose \(\sum c_\lambda Q_\lambda=0\) is a finite relation. Pick \(\lambda_0\) such that \(\sigma\lambda_0\) is minimal for \(\succeq\) among the \(\sigma\lambda\) with \(c_\lambda\ne0\). By (c), \(a_{\lambda,\sigma\lambda_0}\ne0\) forces \(\sigma\lambda_0\succeq\sigma\lambda\). So the coefficient of \(m_{\sigma\lambda_0}\) is \(c_{\lambda_0}a_{\lambda_0,\sigma\lambda_0}\ne0\), a contradiction.
  1. Spanning. Induct on \(|F_\nu|\). For \(\nu=\sigma\lambda\), \(m_{\nu}=a_{\lambda\nu}^{-1}\big(Q_\lambda-\sum_{\mu\in F_\nu\setminus\{\nu\}}a_{\lambda\mu}m_\mu\big)\), and each such \(m_\mu\) lies in the span of the \(Q\)'s by step 3. This is exactly where \(\mathcal T\cap C^-=\{0\}\) "commences" the induction (p. 64).
  1. Conclusion. By (a), \(\{\mathbf 1_{x_\lambda K}\}_{\lambda\in\Lambda^-}\) is a basis of \(C_c^\infty(X(F))^K\). By steps 4–5, \(\hat S/\kappa\) maps it bijectively onto the basis \(\{Q_\lambda\}\), and it is equivariant by step 1. The last claim is the case \(\lambda=0\).

Remarks. (i) Sakellaridis's map \(\mathbf 1_{x_\lambda K}\mapsto P_\lambda\) (8.1) is this map up to two factors. One is the positive constant \(\operatorname{vol}(x_\lambda K)\), which is absorbed into \(Q_\lambda\) without changing (c). The other is the substitution \(\chi\mapsto\chi^{-1}\) composed with \(w_L\), the longest element of \(W_{P(X)}\). It is needed because \(S_\chi\) lives on \(\delta(X)^{-1/2}A_X^*\) while \(\Delta_\chi\) lives on \(\delta(X)^{1/2}A_X^*\) ((3.3), (3.6)). Here \(w_L\) fixes \(A_X^*\) pointwise, inverts \(\delta(X)\) and commutes with \(W_X\), and \(\mathcal Sf\) is \(W\)-invariant. After this substitution the leading index of \(Q_\lambda\) becomes \(\sigma\lambda=-w_0^X\lambda\) and the cone becomes \(-w_0^X\mathcal T\), which still meets \(C^-\) only in \(0\). The weight set \(S_\lambda\subseteq\lambda+\mathcal T\) is \(W_X\)-stable, so \(-S_\lambda=-w_0^XS_\lambda\subseteq\sigma\lambda-w_0^X\mathcal T\). This is why (c) allows a bijection \(\sigma\). (ii) The scalar relating \(\Omega_\chi=\Delta_\chi(\Phi_K)\) to \(\Delta_\chi(\varphi_{K,\chi})\) ((4.5), which uses [CS80]) is part of \(\kappa\). So (4.5) is not needed. (iii) Theorem 3.2.2 (multiplicity) is not used. Only the existence of the family \(S_\chi\) and the values \(\hat S(\mathbf 1_{x_\lambda K})\) enter.

Builds on

  • Strategy 1.1 · Every external input of Theorem 8.0.2 and its status over \(\mathbb F_q((t))\)

Built on by

  • Computation 1.7 · \(M_\theta\): \(W_X'\cong S_3\), coboundary \(\beta\), formula (7.4), triangularity
  • Theorem 1.9 · Theorem 8.0.2 over \(\mathbb F_q((t))\), \(p\) odd, symmetric \(X\): hypotheses
  • Audit 5.16 · Audit of A1-2: triangular spherical transforms give C_K ≅ C[D]^{W_X}
  • Proposition S1.2 · The coefficient \(a_J\) of \(m_\lambda\) in the right side of (7.4)
  • Lemma S1.3 · Positivity of \(\hat S\) forces \(c\neq0\) and \(a_J/c>0\) over every local field
  • Remark S1.4 · Sakellaridis's proof of Theorem 8.0.2 also needs \(a_J\neq0\); the p. 62 Remark misses it
  • Theorem S1.5 · Theorem 8.0.2 over \(\mathbb F_q((t))\) for symmetric \(X\): the complete hypothesis list

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