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

Objection 7.15 · Objection: the parity count of the \(\SL_2/T\) analysis holds, but '\(\IC_0\) is a summand of \(V_{(1,0)}\ast\IC_0\)' is false

proved audit: correct

Second audit: correct. Verified: the coset bijection, the support slip, and the simplicity of V_{(1,0)}∗IC₀ (finite map, two-to-one over O₀, isomorphism over O₁); the localisation triangle Δ₁[1] → IC₁ → IC₀^{⊕2}[1] is correct with the normalisation δ(O₀) = 0, δ(O₁) = 1.

Confirms that the \(\SL_2\) Hecke algebra reaches only even polynomials in the \(\GL_2\) adjacency operator, for every \(q\). Refutes the sheaf-level remark: \(V_{(1,0)}\ast\IC_0\) is a single IC sheaf, so \(\IC_0\) is not a summand; the remark's conclusion survives for another reason.

Objection 7.15.

  1. Main claim of Proposition 1.2 (holds for every \(q\) and \(F\)). \(T^{\SL}_{(1,-1)}=T^{\GL}_{(1,-1)}=T_{(2,0)}=T_1^2-(q+1)\) on \(K_{\GL}\)-invariant functions, and \(\mathcal H_{K_{\SL}}\varphi_0=\C[y^2]\not\ni y-2\leftrightarrow\mathbf 1_{\{d=1\}}\). This is consistent with Computation 2.8(c). The phrase "support exactly \(\{0,\dots,2\deg p\}\)" should read "support contains \(2\deg p\)"; harmless.
  2. The remark in step 4 of Proposition 1.2 fails (cosmetic). In the \(\GL_2\) category, \(V_{(1,0)}\ast\IC_0\simeq\IC(\overline{\mathcal O_1})\) is simple; \(\IC_0\) is not a summand.
  3. The remark's conclusion is nevertheless true. The \(\Z/2\)-grading does not lift to the \(\GL_2\) category, because \(\mathcal C^{\mathrm{Sat}}_{K,\GL}=\mathcal C_{\GL}\) and the localisation triangle \(\Delta_1[1]\to\IC_1\to\IC_0^{\oplus2}[1]\) mixes the would-be parities under cofibres, not under retracts. No verdict depends on the remark; the "blocked route" wording of Summary 1.9 should be corrected accordingly.
  1. Coset bijection. \(K_{\GL}=K_{\SL}\cdot\{\operatorname{diag}(u,1)\}\); these diagonal matrices commute with \(t=\operatorname{diag}(\varpi,\varpi^{-1})\) and normalise \(K_{\SL}\), so \(K_{\SL}tK_{\SL}/K_{\SL}\to K_{\GL}tK_{\GL}/K_{\GL}\) is surjective, and injective because \(K_{\GL}\cap\SL_2(F)=K_{\SL}\). Hence \(T^{\SL}_{(1,-1)}=T^{\GL}_{(1,-1)}=T_{(2,0)}=T_1^2-(q+1)\) on \(K_{\GL}\)-invariants (Hecke relation \(T_1^2=T_{(2,0)}+(q+1)T_{(1,1)}\), the centre acting trivially). Since \(C_{K_{\GL}}\cong\C[y]\) is free of rank one, \(\mathcal H_{K_{\SL}}\varphi_0=\C[y^2]\), which does not contain \(y-2\), the element corresponding to \(\mathbf 1_{\{d=1\}}=(T_1-2)\varphi_0\). The operator \(T=A^2-(q+1)\) of Computation 2.8 is this identity, and the recursions of Computation 2.8 follow from the adjacency recursions (Audit 7.2); both give dimension \(N+1\) against \(2N+1\).
  2. The summand claim. The map \(Y=\Gr^{(1,0)}\mathbin{\widetilde\times}L^+X\to\overline{\mathcal O_1}\) is finite: its fibre over \(x_0\) consists of the two \(T\)-fixed points of \(\mathbb P^1\) (reduced), and it is birational over \(\mathcal O_1\). By the lemma of Lemma 2.6, \(V_{(1,0)}\ast\IC_0\simeq\IC(\overline{\mathcal O_1})\), which is simple. The function shadow \(T_1f_0=2f_0+f_1=\pm\tr\IC_1\) (two branches along \(\mathcal O_0\)) confirms that there is no \(\IC_0\) summand.
  3. The conclusion. For \(\GL_2/T\) both conditions of Proposition III hold, so \(\mathcal C^{\mathrm{Sat}}_{K,\GL}=\mathcal C_{\GL}\) (compare the \(\PGL_2\) half of Proposition 2.7). The localisation triangle for \(\IC_1\) on \(\overline{\mathcal O_1}=\mathcal O_1\sqcup\mathcal O_0\) is \(\Delta_1[1]\to\IC_1\to\IC_0^{\oplus2}[1]\); the second map is nonzero, so a would-be odd object maps nontrivially to a would-be even one, and no grading compatible with cofibres exists. Severity: cosmetic.

Builds on

  • Proposition 1.2 · \(\SL_2/T\) versus \(\GL_2/T\): one variety, two Hecke algebras
  • Lemma 2.6 · Each Satake object is a single IC sheaf: \(V_{n\alpha^\vee}\ast\IC_0\simeq\IC(\overline{\mathcal O_{2n}})\)
  • Computation 2.8 · Function level for \(\SL_2/T\): Lang count of \(K\)-orbits, the sign-class dictionary, and a Hecke dimension count for every \(q\)

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

A1-2: parity count correct and consistent with A2-8; the remark 'IC_0 is a summand of V_(1,0)*IC_0' is false

Main claim (holds, every \(q\), every \(F\)). \(K_{\mathrm{GL}}=K_{\mathrm{SL}}\cdot\{\mathrm{diag}(u,1)\}\), \(D\) commutes with \(t=\mathrm{diag}(\varpi,\varpi^{-1})\) and normalises \(K_{\mathrm{SL}}\), so \(K_{\mathrm{SL}}tK_{\mathrm{SL}}/K_{\mathrm{SL}}\to K_{\mathrm{GL}}tK_{\mathrm{GL}}/K_{\mathrm{GL}}\) is bijective (injective since \(K_{\mathrm{GL}}\cap\mathrm{SL}_2(F)=K_{\mathrm{SL}}\)); hence \(T^{\mathrm{SL}}_{(1,-1)}=T^{\mathrm{GL}}_{(1,-1)}=T_{(2,0)}=T_1^2-(q+1)\) on \(K_{\mathrm{GL}}\)-invariants (Hecke relation \(T_1^2=T_{(2,0)}+(q+1)T_{(1,1)}\), centre acting trivially). \(C_{K_{\mathrm{GL}}}\cong\mathbb C[y]\) free of rank one, so \(\mathcal H_{K_{\mathrm{SL}}}\varphi_0=\mathbb C[y^2]\not\ni y-2\leftrightarrow\mathbf 1_{d=1}\). Consistent with A2-8(c): A2's \(T=A^2-(q+1)\) is this identity, and A2's recursions follow from the adjacency recursions (A7-2); both give \(\dim=N+1\) vs \(2N+1\). ('Support exactly \(\{0,\dots,2\deg p\}\)' should read 'contains \(2\deg p\)'; harmless.)

Remark in step 4 (fails; cosmetic). 'IC\(_0\) is a summand of \(V_{(1,0)}\ast\mathrm{IC}_0\) in the \(\mathrm{GL}_2\) category' is false: \(Y=\mathrm{Gr}^{(1,0)}\tilde\times L^+X\to\overline{\mathcal O_1}\) is finite (fibre over \(x_0\) = the two \(T\)-fixed points of \(\mathbb P^1\), reduced), birational over \(\mathcal O_1\), so by A2-6's lemma \(V_{(1,0)}\ast\mathrm{IC}_0\simeq\mathrm{IC}(\overline{\mathcal O_1})\) is simple; the function shadow \(T_1f_0=2f_0+f_1=\pm\mathrm{tr}\,\mathrm{IC}_1\) (two branches along \(\mathcal O_0\)) confirms there is no \(\mathrm{IC}_0\) summand. The conclusion of the remark (the \(\mathbb Z/2\)-grading does not lift to \(\mathcal C_{\mathrm{GL}}\)) is nevertheless true, for a different reason: \(\mathcal C^{\mathrm{Sat}}_{\mathrm{GL}}=\mathcal C_{\mathrm{GL}}\) (Prop. III; A2-7's PGL\(_2\) half) and the localisation triangle \(\Delta_1[1]\to\mathrm{IC}_1\to\mathrm{IC}_0^{\oplus2}[1]\) mixes the would-be parities under cofibres, not under retracts. Severity: cosmetic (no verdict depends on it; A1-9's 'blocked route' wording should be corrected accordingly).

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