§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:10:18Z

Audit 4.1 · Audit: the \(\mathbb F_q\)-structures of the unitary and Galois twists

proved

Frobenius, the involutions, the fixed-point groups and the point sets claimed in Setup 2.4 are recomputed and hold; (C) holds and (R) fails exactly at the splitness of \(T\) and \(T_0\). The one clause used without proof, an \(\mathbb F_q\)-rational \(\theta\)-stable Borel pair, is true and is proved here; D2's purity clause must be restricted to Frobenius-stable \(\lambda\).

Audit 4.1.

Let \(H\) be split connected reductive over \(\mathbb F_q\) with split maximal torus \(T_H\), \(\vartheta\) an \(\mathbb F_q\)-involution of \(H\) with \(\vartheta(s)=s^{-1}\) on \(T_H\), \(G=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}H\) with Galois involution \(\sigma\), \(\theta_u=\vartheta\circ\sigma\), \(\theta_g=\sigma\), \(E=\mathbb F_{q^2}(\!(t)\!)\).

  1. Over \(k\): \(G_k=H\times H\), \(\sigma(g_1,g_2)=(g_2,g_1)\), \(\mathrm{Fr}(g_1,g_2)=(\mathrm{Fr}\,g_2,\mathrm{Fr}\,g_1)\); \(\theta_u\) and \(\theta_g\) are involutions defined over \(\mathbb F_q\).
  2. \(\tau(g)=g\theta_u(g)^{-1}\) identifies \(X_u\) with \(P_u=\{z:\theta_u(z)=z^{-1}\}\), which over \(k\) is the graph \(\{(x,\vartheta(x)^{-1})\}\cong H\). Hence \(X_u(F)=\{x\in H(E):\vartheta(\sigma x)=x^{-1}\}\) with \(g\cdot x=gx\,\vartheta(\sigma g)^{-1}\); for \(H=\mathrm{GL}_n\), \(\vartheta(g)=g^{-\mathsf T}\), this is the set of all nondegenerate hermitian matrices, \(g\cdot x=gxg^*\). The Frobenius of \(X_{u,k}\cong H\) is \(x\mapsto\mathrm{Fr}(\vartheta(x))^{-1}\), whose square is \(\mathrm{Fr}^2\). The Galois twist is the case \(\vartheta=1\): \(X_g(F)=\{x\in H(E):x\,\sigma(x)=1\}\), Frobenius \(x\mapsto\mathrm{Fr}(x)^{-1}\).
  3. \(G^{\theta_u}_k=\{(g,\vartheta g)\}\cong H\); \(S=\{(s,s)\}\) is a maximal \(\theta_u\)-split torus, split over \(\mathbb F_q\); \(Z_{G^{\theta_u}}(S)\cong T_H\) and \(Z_{G^{\theta_u}}(\mu)\cong Z_H(c)\) for \(\mu=(c,c)\). So (C) holds. For \(\theta_g\): \(G^{\theta_g}=\Delta H\), \(S=\{(s,s^{-1})\}\) is anisotropic, \(Z_{G^{\theta_g}}(S)=\Delta T_H\); (C) holds.
  4. Both twists have a \(\theta\)-stable Borel pair \((B_0,T_0)\) defined over \(\mathbb F_q\), and no maximal torus of \(G\) is split. So the clauses of (R) that fail are exactly "\(T\) split" and "\(T_0\) split".
  5. D0, D1, D3 hold. D2 holds over \(k\) for every \(\lambda\); its purity clause holds over \(\mathbb F_q\) for \(\sigma\lambda=\lambda\), and over \(\mathbb F_{q^2}\) for all \(\lambda\).

Outcome: holds with a minor defect. Item 4 is asserted in Setup 2.4(6) and Theorem 2.6(3) without proof (true, proof below); Setup 2.4(6) states D2 without the qualification of item 5.

  1. Frobenius and \(\sigma\). \(\mathbb F_{q^2}\otimes_{\mathbb F_q}k\cong k\times k\) by \(a\otimes b\mapsto(\iota_1(a)b,\iota_2(a)b)\), \(\iota_2=\mathrm{Fr}\circ\iota_1\). In the coordinates \((x,y)\) the map \(1\otimes\mathrm{Fr}\) is \((x,y)\mapsto(y^q,x^q)\) (because \(x^q=\iota_2(a)b^q\) and \(y^q=\iota_1(a)b^q\)), and \(\sigma\otimes1\) is \((x,y)\mapsto(y,x)\). Applying \(H\) gives item 1. \(\vartheta\) acts coordinatewise and commutes with \(\mathrm{Fr}\); so \(\theta_u(g_1,g_2)=(\vartheta g_2,\vartheta g_1)\), \(\theta_u^2=1\) and \(\mathrm{Fr}\,\theta_u=\theta_u\,\mathrm{Fr}\).
  2. The point sets. \(P_{u,k}=\{(z_1,z_2):(\vartheta z_2,\vartheta z_1)=(z_1^{-1},z_2^{-1})\}=\{(x,\vartheta(x)^{-1})\}\) is connected and \(\cong H\). The map \(\tau(g_1,g_2)=(g_1\vartheta(g_2)^{-1},\,g_2\vartheta(g_1)^{-1})\) is constant on \(G^{\theta_u}\)-cosets and induces \((H\times H)/\{(h,\vartheta h)\}\xrightarrow{\sim}H\), \((g_1,g_2)\mapsto g_1\vartheta(g_2)^{-1}\). So \(X_u\cong P_u\) over \(\mathbb F_q\), \(X_u(F)=P_u(F)\), and the transported action is \(g\cdot z=gz\theta_u(g)^{-1}\). For \(\mathrm{GL}_n\): \(\vartheta(\sigma x)=x^{-1}\iff\sigma(x)^{\mathsf T}=x\), and \(gz\theta_u(g)^{-1}=gz\,\sigma(g)^{\mathsf T}=gzg^*\).
  3. Frobenius on \(X_k\). \(x=g_1\vartheta(g_2)^{-1}\mapsto\mathrm{Fr}(g_2)\,\vartheta(\mathrm{Fr}\,g_1)^{-1}=\mathrm{Fr}\big(\vartheta(x)^{-1}\big)\). With \(\psi=\mathrm{Fr}\circ\vartheta\circ\mathrm{inv}\), \(\psi^2=\mathrm{Fr}^2\vartheta^2=\mathrm{Fr}^2\); so every rational point of \(LX_u\) lies in \(LH(\mathbb F_{q^2})=H(E)\), and on \(H(E)\) the condition \(\psi(x)=x\) reads \(\vartheta(\sigma x)=x^{-1}\) (on \(E\)-points \(\mathrm{Fr}\) is \(\sigma\)). This matches item 2.
  4. Fixed points and centralizers. \((g_1,g_2)=(\vartheta g_2,\vartheta g_1)\iff g_2=\vartheta g_1\). \(\theta_u(s,s)=(s^{-1},s^{-1})\), \(\dim S=\operatorname{rk}H=\operatorname{rk}X_u\), and \(\mathrm{Fr}(s,s)=(\mathrm{Fr}\,s,\mathrm{Fr}\,s)\) with \(T_H\) split, so \(S\cong T_H\) is split. \((g,\vartheta g)\) centralizes \(S\) iff \(g\in Z_H(T_H)=T_H\); it centralizes \(\mu=(c,c)\) iff \(g\in Z_H(c)\) (note \(\vartheta Z_H(c)=Z_H(-c)=Z_H(c)\)), a Levi subgroup, connected. In the Galois twist \(\mathrm{Fr}(s,s^{-1})=(\mathrm{Fr}\,s^{-1},\mathrm{Fr}\,s)\), i.e. Frobenius is \(-1\) on \(X_*(S)\).
  5. A rational \(\theta_u\)-stable Borel pair (the missing step). \(B_1\times B_2\) is \(\theta_u\)-stable iff \(B_1=\vartheta B_2\), and Frobenius-stable iff \(B_1=\mathrm{Fr}\,B_2\). Both hold iff \(B_1=\mathrm{Fr}\,B_2\) and \(B_2\) is stable under the Steinberg endomorphism \(F'=\vartheta\circ\mathrm{Fr}\) of \(H\) (a Frobenius for another \(\mathbb F_q\)-form of \(H\)). By the Lang–Steinberg theorem \(F'\) fixes a Borel subgroup \(B_2\) and a maximal torus \(T_2\subseteq B_2\). Then \(B_0=\mathrm{Fr}\,B_2\times B_2\supseteq T_0=\mathrm{Fr}\,T_2\times T_2\) is a \(\theta_u\)-stable Borel pair over \(\mathbb F_q\) (\(\theta_u T_0=\vartheta T_2\times\vartheta\mathrm{Fr}\,T_2=T_0\)). For \(\theta_g\) take \(B_H\times B_H\supseteq T_H\times T_H\).
  6. No split maximal torus. A maximal torus of \(G_k\) is \(T'_1\times T'_2\); if it is Frobenius-stable, Frobenius maps \(X_*(T'_1)\) onto \(X_*(T'_2)\) and back, so the invariants have rank \(\operatorname{rk}H<2\operatorname{rk}H\) (\(H\ne1\) has positive rank). Hence "\(T\supseteq S\) split" and "\(T_0\) split" fail, all other clauses of (R) hold.
  7. D0–D3. Over \(k\) both twists are the group case with the second factor twisted by \(\vartheta\) or \(1\), so D0 is the Cartan decomposition of \(LH\) with the stabilizers of Audit 4.3; D1: \(LX\), \(L^+G\) and the convolution diagram over a Frobenius-stable \(\operatorname{Gr}_{\le\lambda}\) are defined over \(\mathbb F_q\); D3: \(L^+G\) and all stabilizers are connected (Audit 4.3). D2: a Weil structure on \(V_\lambda\) over \(\mathbb F_q\) exists only for \(\sigma\lambda=\lambda\) (Audit 4.8), so "\(V_\lambda\ast\mathrm{IC}_0\) is pure" is meaningful over \(\mathbb F_q\) only for those \(\lambda\); the decomposition over \(k\) holds for all \(\lambda\) (Audit 4.2). No downstream step uses purity of a non-rational \(V_\lambda\ast\mathrm{IC}_0\).
  8. The hypothesis lists of Counterexample 2.3 and Counterexample 1.5 (steps 3, 5, 8) are the cases \(H=\mathbb G_m\) and \(H=\mathrm{GL}_m\) of items 1–6; for \(\mathbb G_m\), \(T=T_0=B_0=R\) is the only maximal torus.

Builds on

  • Setup 2.4 · Setup: the unitary and Galois \(\mathbb F_q\)-forms of the group case
  • Counterexample 2.3 · Counterexample: \(\operatorname{Res}\mathbb G_m\) on \(\mathbb G_m\) through the norm
  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion

Built on by

  • Summary 4.12 · Verdict of the audit: the unitary twist is a valid counterexample; the repairs are minor

Sources