§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:01:44Z

Objection 5.4 · Audit of A4-6: structure constants transfer; the Satake support bound is supplied

proved audit: correct

Second audit: correct. Re-derived Proposition 4.6: the incidence formula tau(g^{-1}x_nu) = t^{-lambda}k^{-1}t^nu theta(k) t^{theta lambda}, K_lambda containing K(c(lambda)), the support bound, and n_{lambda nu} as a coset count with delta_B^{1/2}(t^nu) = q^{-<rho,nu>}. Proposition 4.6's sentence 'only nu with nu^+ <= lambda occur' states a true standard fact without proof, and some field-independent bound on nu is indeed needed for a uniform D'(lambda). Objection 5.4's primitive-row lemma (the valuation of r rho_i(t^lambda) lies between <lambda,w0 omega_i> and <lambda,omega_i> and equals <nu,w0 omega_i> by Lemma 4.2(d)) is valid and gives a finite box. The classification 'holds with a repair' is right.

Parts (a), (b), (d) of Proposition 4.6 hold. In (c), the uniform \(D'(\lambda)\) needs a bound, independent of the field, on the \(\nu\) with \(n_{\lambda\nu}\ne0\). Proposition 4.6 asserts it without proof, and Lemma 4.2(d) gives only an upper bound. A two-sided bound follows from a primitive-row argument.

Objection 5.4.

Outcome: holds with a repair at step 6 (part (c)). (a), (b), (d) are correct as written. In (c) the sentence "only \(\nu\) with \(\nu^+\preceq\lambda\) occur" is used to make \(D'(\lambda)\) independent of \(\nu\), but it is not proved. Lemma 4.2(d) yields only \(\langle\lambda-\nu,\omega_j\rangle\ge0\), which does not bound \(\nu\) from below, while the determinacy bound for "\(\operatorname{val}(\ell_i\rho_i(kt^\lambda))=\langle\nu,w_0\omega_i\rangle\)" grows with \(\langle\nu,w_0\omega_i\rangle\). Repair (Lemma). For \(E\) any test field, \(k\in K\) and \(\lambda\in\Lambda_T^+\), the \(\nu\) with \(kt^\lambda\in U(E)t^\nu K\) satisfies $$\langle\lambda,w_0\omega_i\rangle\le\langle\nu,w_0\omega_i\rangle\le\langle\lambda,\omega_i\rangle\qquad(i=1,\dots,6).$$ So \(n_{\lambda\nu}(E)\ne0\) only for \(\nu\) in a finite set \(\mathrm{Box}(\lambda)\) independent of \(E\), and \(D'(\lambda):=\max_{\nu\in\mathrm{Box}(\lambda)}D(\lambda,\nu)\) works.

Proof of the repair.

  1. Put \(r=\ell_i\rho_i(k)\), a row of \(\rho_i(k)\in\mathrm{GL}_n(\mathcal O)\). Then \(r\) is primitive: its entries lie in \(\mathcal O\) and some entry \(r_{\chi_0}\) is a unit.
  2. \(\ell_i\rho_i(kt^\lambda)=r\,\rho_i(t^\lambda)=(r_\chi\varpi^{\langle\lambda,\chi\rangle})_\chi\). Its valuation is at least \(\min_\chi\langle\lambda,\chi\rangle=\langle\lambda,w_0\omega_i\rangle\) and at most \(\langle\lambda,\chi_0\rangle\le\max_\chi\langle\lambda,\chi\rangle=\langle\lambda,\omega_i\rangle\), because \(\lambda\) is dominant.
  3. By Lemma 4.2(d) this valuation equals \(\langle\nu,w_0\omega_i\rangle\). Since the \(w_0\omega_i\) span \(X^*(T)_{\mathbb Q}\), \(\nu\) lies in a bounded region of \(\Lambda_T\), hence in a finite set \(\mathrm{Box}(\lambda)\) that does not depend on \(E\). (The sharper \(\nu\in\operatorname{conv}(W\lambda)\) is standard but not needed.) \(\square\)

Re-derivation of the rest.

  1. Orbit test (step 1). Under (H-Cartan): \(y\in Kx_\nu\iff\tau(y)\in Kt^\nu K\), using \(\tau(kx)=k\tau(x)\theta(k)^{-1}\), \(\tau(x_\nu)=t^\nu\) and disjointness of Cartan double cosets.
  2. Cosets (step 2). \(Kt^\lambda K/K\cong K/K_\lambda\) with \(K_\lambda=K\cap t^\lambda Kt^{-\lambda}\supseteq K(c(\lambda))\) (Iwahori factorization of \(K(c)\), \(c\ge1\)). Membership in \(K_\lambda\) is "\(\operatorname{val}\rho_i(t^{-\lambda}kt^\lambda)=0\) for all \(i\)".
  3. Incidence (step 3). For \(g=kt^\lambda\), \(\tau(g^{-1}x_\nu)=g^{-1}t^\nu\theta(g)=t^{-\lambda}k^{-1}t^\nu\theta(k)t^{\theta\lambda}\). The resulting condition is right \(K_\lambda\)-invariant and of type Lemma 4.2(e).
  4. Support (a). \(\tau(Kt^\lambda Kx_\mu)\subseteq Kt^\lambda K\,t^\mu\,Kt^{-\theta\lambda}K\). Two applications of Lemma 4.2(c), with \(\preceq\) additive and transitive, give \(\nu\preceq\lambda+\mu+(-\theta\lambda)^+\).
  5. Satake (c). \(\int_{U(E)}\mathbf 1_{Kt^\lambda K}(t^\nu u)\,du=n_{\lambda\nu}\), because two \(u\) give the same coset \(t^\nu uK\) iff they differ by \(U(E)\cap K=U(\mathcal O)\). The condition "\(kt^\lambda K\subseteq U(E)t^\nu K\)" is right \(K_\lambda\)-invariant: \(r\cdot\mathrm{GL}_n(\mathcal O)\) has the same valuation. With the box above, \(n_{\lambda\nu}(E)=n_{\lambda\nu}(E')\) for all \(\nu\) once \(E,E'\) are \(D'(\lambda)\)-close.
  6. Conventions (checklist item 8). The constants are defined by the same formulas over \(F\) and \(F'_e\): left action \((f*\varphi)(x)=\sum_{gK}f(g)\varphi(g^{-1}x)\), and \(\mathcal S\) normalized by \(\delta_B^{1/2}\) with \(\operatorname{vol}U(\mathcal O)=1\). So the transfer is insensitive to which convention the paper uses, provided the same one is used over both fields (see Audit 5.6 step 5).

Builds on

  • Proposition 4.6 · Proposition: all structure constants and Satake coefficients agree over close fields

Built on by

  • Summary 5.13 · Verdict of audit A5: transfer route A4-2…A4-9 and char-p hypotheses A3-1…A3-10

Sources