§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T23:00:57Z

Audit 4.18 · Audit: under (R), (C), D0–D3 the twisting mechanisms vanish

proved

Proposition 1.8 is re-derived. Under (C), \(T^\theta\) is connected and split, and \(S[2]\subseteq T^\theta\), so \(X(F_n)=G(F_n)x_0\). The top entries \(m^{(n)}_{\varphi(\lambda),\lambda}\) are nonzero by positivity of Kazhdan–Lusztig coefficients, with the stated sign (Zhu (5.6.2) checked). The basis and projection criterion (e) is correct. Item (c) is stated slightly too broadly.

Audit 4.18.

Assume (R), (C), D0–D3, \(q\) odd. (a) Frobenius is trivial on \(\Lambda_T\), \(\Lambda_S\). (b) \(T^\theta\) is a connected split torus, the maximal torus of \(Z_{G^\theta}(S)\); \(S[2]\subseteq T^\theta\); hence \(x_\mu\in G(F_n)x_0\) for all \(\mu\), and \(X(F_n)=G(F_n)x_0\). (c) An automorphism \(\epsilon\) of \((G,\theta)\) over \(\mathbb F_q\) with \(\epsilon^*V_\lambda\cong V_\lambda\) for all \(\lambda\) fixes every \(K_n\)-orbit. Under (C) no degree-\(0\) automorphisms of \(\mathrm{id}_{\mathcal C_k}\) come from \(\pi_0\) of stabilizers. (d) \(m^{(n)}_{\varphi(\lambda),\lambda}=(-1)^{\delta(\varphi(\lambda))+\langle2\rho,\lambda\rangle}\sum_{\mu\le\lambda}a_{\lambda\mu}(q^n)\,(T_\mu\ast\varphi_0)(t^\lambda x_0)\ne0\). (e) With \(U\) the uncovered labels and \(\lambda_\mu\in\varphi^{-1}(\mu)\cap\Lambda_T^+\), the family \(\{T_{\lambda_\mu}\ast\varphi_0\}_{\mu\notin U}\cup\{\mathbf 1_{\mathcal O_\nu}\}_{\nu\in U}\) is a basis of \(C_K\), and (SG) \(\iff\) the \(p_U(T_\lambda\ast\varphi_0)\) span \(\mathbb C^{(U)}\). (f) Split Tate multiplicity spaces of one parity give \(m^{(n)}_{\mu\lambda}\ne0\) for all \(n\).

Outcome: holds with a minor defect. In (c), the phrase "the geometric form of criterion (b) of Proposition 1.6 is void" is proved only for autoequivalences induced by automorphisms \(\epsilon\) of \((G,\theta)\). Proposition 1.6(b) allows any \(\mathcal H\)-linear t-exact Frobenius-compatible \(\sigma\), and those are not excluded. (By (d) and (e), such a \(\sigma\) could only permute labels in a way that preserves the supports of all rows of \(M\).) Item (g) is a correct statement of the obligation, conditional on finite generation of \(C_K\) (Audit 4.16).

  1. (a) \(G\), \(S\) split.
  2. (b), \(T^\theta\). \(L=Z_G(S)\) is a connected Levi containing \(T\). On \(\operatorname{Lie}L=\operatorname{Lie}L^\theta\oplus\operatorname{Lie}L^{-\theta}\), maximality of \(S\) gives \(\operatorname{Lie}L^{-\theta}=\operatorname{Lie}S\). So \(L^\theta S\) (\(S\) central in \(L\)) is open in the connected \(L\), whence \(L=L^\theta S\) with \(L^\theta\cap S=S[2]\), and \(\operatorname{rk}(L^\theta)^\circ=\dim T-\dim S\). Similarly \(T=(T^\theta)^\circ S\), so \(T_\theta=(T^\theta)^\circ\) has dimension \(\dim T-\dim S\) and is a maximal torus of \(L^\theta=Z_{G^\theta}(S)\), which is connected and reductive by (C) and \(p\ne2\). Then \(T^\theta\subseteq Z_{L^\theta}(T_\theta)=T_\theta\): \(T^\theta\) is connected, and split as a subtorus of the split \(T\). \(S[2]\) is \(\theta\)-fixed and lies in \(T\), so \(S[2]\subseteq T^\theta\). (Same argument for \(Z_G(\mu)^\theta=Z_{G^\theta}(\mu)\): its maximal torus is \(T_\theta\), so it is split.)
  3. (b), orbits. The cocycle \(\gamma\mapsto\mu(\sqrt t)^{-1}\gamma(\mu(\sqrt t))=\mu(\pm1)\) has values in \(S[2]\subseteq T_\theta\), and \(H^1(F_n,T_\theta)=0\) (Hilbert 90). So \(\mu(\pm1)_\gamma=s^{-1}\gamma(s)\) with \(s\in T_\theta\), and \(g'=\mu(\sqrt t)s^{-1}\in G(F_n)\) satisfies \(g'x_0=x_\mu\) (as \(s\in H\)). With IIb step 1, \(X(F_n)=\bigsqcup_\mu K_nx_\mu=G(F_n)x_0\). Checks: \(\mathrm{GL}_2/T\), \(T^\theta=\) centre, \(S[2]=\{\pm1\}\subseteq\) centre; for \(\mathrm{SL}_2/\mathrm{SO}_2\) (outside (C)) \(T^\theta=\{\pm1\}\) is disconnected, as it must be.
  4. (c). \(\epsilon\) fixes each \(\operatorname{Gr}_{\le\lambda}\), so \(\epsilon(t^\mu)\in L^+Gt^\mu L^+G\). Since \(\tau(\epsilon x)=\epsilon\tau(x)\) and the double cosets of distinct dominant labels are disjoint, \(\epsilon(\mathcal O_\mu)=\mathcal O_\mu\), and the \(K_n\)-orbits \(\mathcal O_\mu(\mathbb F_{q^n})\) are preserved. The limitation is stated in the outcome.
  5. (d), sign and positivity. \(x_{\varphi(\lambda)}=t^\lambda x_0\) (\(\tau(t^\lambda x_0)=t^{\lambda-\theta\lambda}\), \(\varphi(\lambda)\) dominant, \(\tau\) injective). Evaluating \(\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\sum_\nu m_{\nu\lambda}\operatorname{tr}_n\mathrm{IC}_\nu\) at this point kills every \(\nu<\varphi(\lambda)\) (support), and \(\operatorname{tr}_n\mathrm{IC}_{\varphi(\lambda)}=(-1)^{\delta(\varphi(\lambda))}\) there. Zhu (5.6.2), checked in the local copy, gives \(f_\mu=f_{\mathrm{IC}_\mu[-\langle2\rho,\mu\rangle]}=c_\mu+\sum_{\lambda<\mu}a_{\mu\lambda}(q)c_\lambda\) with \(a_{\mu\lambda}\in\mathbb N[x]\). So \(\operatorname{tr}_nV_\lambda=(-1)^{\langle2\rho,\lambda\rangle}\sum a_{\lambda\mu}(q^n)T_\mu\). Each \((T_\mu\ast\varphi_0)(t^\lambda x_0)\ge0\) and the \(\mu=\lambda\) term is \(\ge1\) (Lemma A(2)). The sum is positive and the sign is as stated.
  6. (e), basis. \(\operatorname{supp}(T_{\lambda_\mu}\ast\varphi_0)\subseteq\{\nu\le\mu\}\) (D0, induction through the unitriangular \(\operatorname{tr}V_\lambda\), \(\varphi\) monotone), with value \(\ge1\) at \(x_\mu\). Induction on finite lower sets gives spanning and independence. \(\ker p_U\subseteq\mathcal H_K\ast\varphi_0\), so (SG) holds iff \(p_U(\mathcal H_K\ast\varphi_0)=\mathbb C^{(U)}\). \(U=\varnothing\) is Corollary C, and is the run's Lemma 4.8 cited in verdict.html, Theorem 4 (checked).
  7. (f). For indices of one parity with eigenvalues \(q^{(w+i-w_\mu)/2}\) (fixed \(q^{1/2}\)), \(m^{(n)}_{\mu\lambda}=(-1)^{i_0}q^{n(w+i_0-w_\mu)/2}\sum_i\dim E_{\mu,i}\,q^{n(i-i_0)/2}\ne0\). The triangular-system corollary is Proposition B in the IC basis.

Builds on

  • Proposition 1.8 · Under (R), (C), D0–D3: three mechanisms vanish; the obligation sits on uncovered labels

Built on by

  • Summary 4.20 · Verdict of the second audit: the mechanism claims hold, with minor repairs

Sources