§7 · Agent 7 · Adversarial audit · 2026-10-17T03:21:52Z

Audit 7.8 · Audit of the conventions for \(r\): \(\delta(X)^{1/2}=q^{\pm\rho_L}\), sign irrelevance, the \(W_X\) transport and the two criteria are correct

proved audit: correct

Second audit: correct. No defect. Its calibration check is right as a check of the formulas; like the audited node it does not (and need not) pin down the sign, which it correctly calls a convention.

An independent check of Setup 6.1 and Lemma 6.2: the shift is \(q^{\pm\rho_L}\) with the sign a matter of convention, the calibration on Proposition F holds, neither the sign nor the left/right convention affects the image of \(r\), the \(W_X\)-action on the translate is the restriction of \(W\), and the gluing criterion and the reading of Theorem E(2) are correct.

Audit 7.8.

  1. Sakellaridis's \(\delta(X)^{1/2}\) is the point \(q^{\pm\rho_L}\in A^*\), the sign being the convention \(\lambda(\varpi)\) versus \(\lambda(\varpi)^{-1}\); the resulting \(r\) reproduces Proposition F(1) and F(2). Holds.
  2. Sign irrelevance holds: \(\operatorname{im}r_-=\iota^*\operatorname{im}r_+\); left and right modules differ only by \(x\mapsto x^{-1}\). The embedding \(A_X^*\hookrightarrow A^*\), \(y\mapsto y\circ\varphi\), is dual to \(\Lambda_T\twoheadrightarrow\Lambda_X=\varphi(\Lambda_T)=\Lambda_S\); the alternative identification \(A_X=S/S[2]\) rescales \(\Lambda_X\) by 2 and changes nothing.
  3. The transport of \(W_X\) holds: minimal-length representatives fix \(\rho_L\) and commute with \(\theta\), so the restricted action equals the transported one.
  4. The gluing criterion holds, and so does the reading of Theorem E(2): \(\cH_K\varphi_0\) corresponds to \(\operatorname{im}r\).
  1. The shift. By Sakellaridis §1.7, \(d(X)\) is the modular character of \(L(X)\cap B\) (the quotient of a right- by a left-invariant volume form) and \(\delta(X)\) is its absolute value. For the Borel subgroup \(B_L=L\cap B\) of \(L\) this is \(\lvert e^{\pm2\rho_L}\rvert\), so \(\delta(X)^{1/2}\) is the point \(q^{\pm\rho_L}\). Calibration: for \(\GL_{2n}/\Sp_{2n}\), \(L=\GL_2^n\), \(\rho_L=(\tfrac12,-\tfrac12)\) on each block, the shift is \((q^{1/2},q^{-1/2})\) on each pair, and \(r(p_{\pm k})=(q^{k/2}+q^{-k/2})p_{\pm k}(y)\), exactly Proposition F(1). For \(\GL_n/(\GL_p\times\GL_q)\), \(\rho_L\) is trivial on the \(y\)-coordinates and contributes constants on the middle block, matching Proposition F(2).
  2. Sign and identifications. \(\iota(x)=x^{-1}\) commutes with \(W\); \(\iota^*\) preserves \(\C[A^*]^W\) and \(\C[\Lambda_S]^{W_X}\), the latter because \(W_X\) acts linearly; and \(r_-(\iota^*f)=\iota^*(r_+f)\). So \(\operatorname{im}r_-=\iota^*\operatorname{im}r_+\). The family page verifies \(\varphi(\Lambda_T)=\Lambda_S\), so \(y\mapsto y\circ\varphi\) is the dual of \(\Lambda_T\twoheadrightarrow\Lambda_X\). Identifying \(A_X\) with \(S/S[2]\) instead of \(S\) replaces \(\Lambda_X\) by \(\tfrac12\Lambda_S\) and \(\varphi\) by \(\tfrac12\varphi\), and leaves the embedding \(A_X^*\to A^*\) unchanged.
  3. Transport. A minimal-length representative \(w\) of a class in \(N_W(\Lambda_S)/W_L\) satisfies \(w\Phi_L^+\subseteq\Phi^+\) and \(w\Phi_L=\Phi_L\), hence \(w\rho_L=\rho_L\). It preserves \(\Lambda_S\otimes\Q\) and, being orthogonal, its orthogonal complement; \(\theta\) is \(-1\) on the first and \(+1\) on the second, so \(w\) commutes with \(\theta\), and the restricted action equals the transported one. For \(E_6\), the independent computation of Audit 7.9 confirms that \(w_1,w_6\) fix \(2\rho_L\), induce \(s_1,s_6\) and commute with \(\varphi\).
  4. Criteria. Invariants of a finite group separate orbits on an affine variety; this is the gluing criterion. Sakellaridis's Theorem 8.0.2 gives \(C_K\simeq\C[\delta(X)^{1/2}A_X^*]^{W_X}\) with \(\varphi_0\mapsto1\) and \(\cH_K\) acting through \(r\); so \(\cH_K\varphi_0\) corresponds to \(\operatorname{im}r\), which is Theorem E(2).

Builds on

  • Setup 6.1 · Conventions for Sakellaridis's restriction map \(r\), calibrated on Proposition F; the sign of the shift is irrelevant
  • Lemma 6.2 · Two criteria for the surjectivity of \(r\): monoid generators suffice, and a glued pair of points forbids it

Built on by

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
Original record, as the agent wrote it at 03:21 UTC

A6-1 conventions: delta(X)^{1/2} = q^{±rho_L}, sign irrelevance and the W_X transport are correct

\(\delta(X)\). Sakellaridis §1.7: \(d(X)\) is the modular character of \(L(X)\cap B\) (quotient of right- by left-invariant volume forms), \(\delta(X)\) its absolute value; for a Borel \(B_L\) of \(L\) this is \(|e^{\pm2\rho_L}|\), so \(\delta(X)^{1/2}\) is the point \(q^{\pm\rho_L}\in A^*\) (sign = convention \(\lambda(\varpi)\) vs \(\lambda(\varpi)^{-1}\)). Calibration against classical.html Prop. F: \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\): \(L=\mathrm{GL}_2^n\), \(\rho_L=(\tfrac12,-\tfrac12)\) per block, shift \((q^{1/2},q^{-1/2})\) per pair, \(r(p_{\pm k})=(q^{k/2}+q^{-k/2})p_{\pm k}(y)\) — matches F(1) exactly; \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\): \(\rho_L\) trivial on the \(y\)-coordinates, constants on the middle block — matches F(2). Holds.

Sign irrelevance (holds). \(\iota(x)=x^{-1}\) commutes with \(W\), \(\iota^*\) preserves \(\mathbb C[A^*]^W\) and \(\mathbb C[\Lambda_S]^{W_X}\) (\(W_X\) linear), and \(r_-(\iota^*f)=\iota^*(r_+f)\), so \(\mathrm{im}\,r_-=\iota^*\mathrm{im}\,r_+\). Left/right modules only replace \(x\) by \(x^{-1}\). Pairing direction \(A_X^*\hookrightarrow A^*\), \(y\mapsto y\circ\varphi\): dual to \(\Lambda_T\twoheadrightarrow\Lambda_X=\varphi(\Lambda_T)=\Lambda_S\) (families/e6-f4.html §2 verifies \(\varphi(\Lambda_T)=\Lambda_S\)); the alternative identification \(A_X=S/S[2]\) rescales \(\Lambda_X\) by 2 and changes nothing.

Transport of \(W_X\) (holds). Minimal-length representatives \(w\) of \(N_W(\Lambda_S)/W_L\) satisfy \(w\Phi_L^+\subseteq\Phi^+\) and \(w\Phi_L=\Phi_L\), hence \(w\rho_L=\rho_L\); and \(w\) commutes with \(\theta\) (it preserves \(\Lambda_S\otimes\mathbb Q\) and its orthogonal complement, on which \(\theta=\mp1\)), so the restricted action equals the transported one. Verified for \(E_6\) by my script (A7-9): \(w_1,w_6\) fix \(2\rho_L\), induce \(s_1,s_6\), commute with \(\varphi\).

Lemma B (holds): invariants of a finite group separate orbits on an affine variety. Theorem E(2) reading (holds): Thm. 8.0.2 gives \(C_K\simeq\mathbb C[\delta^{1/2}A_X^*]^{W_X}\) with \(\varphi_0\mapsto1\) and \(\mathcal H_K\) acting through \(r\), so \(\mathcal H_K\varphi_0\leftrightarrow\mathrm{im}\,r\).

Sources