§7 · Agent 7 · Adversarial audit · 2026-10-17T02:55:57Z

Audit 7.1 · Audit: central inertia for \(\SL_2/\SO_2\) is a natural automorphism, and convolution is bimultiplicative

proved audit: correct

Second audit: correct. Claims true. It does not notice that the cofibre step of Lemma 2.2(a) needs ε² = id (minor; recorded at Lemma 2.2).

Re-checks the stabilizer computation, the construction of the central inertia \(\varepsilon\) and its multiplicativity under convolution. All hold, using only the standing realisation of \(\mathcal C_K\) as equivariant constructible sheaves.

Audit 7.1.

The stabilizer computation Computation 2.1, the construction of \(\varepsilon\) in Lemma 2.2, and steps 1–3 of Lemma 2.3 hold. They use only the pages' standing realisation of \(\mathcal C_K\) as \(L^+G\)-equivariant constructible sheaves whose orbit hearts are \(\Rep(\pi_0\Stab(x_\mu))\) (Proposition III, discussion of H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf.); no weights, no Frobenius and no transfer from \(\overline{\F}_q\) are involved. Severity: none.

  1. Computation 2.1. Recomputed: \(\SO(x_m)(F)=\{g_{a,b}\}\) as stated; integrality forces \(b\in t^m\cO\) and \(a=\pm\sqrt{1-b^2}\), so \(\pi_0=\Z/2\), generated by \(-I\), for \(m\ge1\), and the stabilizer is connected for \(m=0\). The model \(\{A=A^{\mathsf T},\det A=1\}\) is \(\SL_2/\SO_2\), and \(X(F)\) is a single \(\SL_2(F)\)-orbit, since \(H^1(F,\SO_2)=H^1(F,\mathbb G_m)=0\) (\(i\in F\)); so \(LX=LG\cdot x_0\) and there is no component issue.
  2. Lemma 2.2. A central \(z\in H\) acting trivially on \(Y\) is a 2-automorphism of \(\mathrm{id}_{[Y/H]}\). Any sheaf theory functorial in 2-morphisms — for instance equivariant sheaves on the finite-type models \(G(\cO/t^N)\backslash Z^{(N)}\), on which \(-I\) is still central and acts trivially — gives \(\varepsilon\in\operatorname{Aut}(\mathrm{id}_{\mathcal C_K})\). A point left implicit and checked here: \(\varepsilon\) commutes with the truncation functors (ordinary and perverse), because \(\tau^{\le n}F\to F\) is a counit and \(\Hom(\tau^{\le n}F,\tau^{\le n}F)\to\Hom(\tau^{\le n}F,F)\) is bijective. This is what makes legitimate the statement that all cohomology sheaves of \(V^-\) have \(\varepsilon=-1\) (Lemma 2.3, step 2) and the perverse-constituent statement of Proposition 2.7. Not every endofunctor commutes with \(\varepsilon\): tensoring with the sign character on \(\Rep(\Z/2)\) does not.
  3. Lemma 2.3, step 3. Traced through the descent: on \(LG\times^{L^+G}LX\), the left inertia of \(z=-I\) on the stalk \(V_g\otimes F_x\) is the left equivariance of \(V\) at \(z\) composed with the inverse of the descent datum for the middle element \(z\), that is \([(\text{left}_z)\circ(\text{right}_z)^{-1}\text{ on }V_g]\otimes\varepsilon_z(F)_x\). The right inertia of \(V\) is trivial, because \(V\) is pulled back from \(\Gr=LG/L^+G\) with its canonical right structure; so the bracket is \(\varepsilon_z(V)_g\). Then \(m_!\) commutes with \(\varepsilon\), and \(\varepsilon_{V\ast F}=\varepsilon_V\ast\varepsilon_F\). Step 2 (\(\varepsilon=\mathrm{id}\) on \(\mathcal H\)) needs only that \(L^+G\cap\operatorname{Ad}_{t^\lambda}L^+G\) is connected, which holds (pro-unipotent kernel over a parabolic subgroup of \(G(k)\)).

Builds on

  • Computation 2.1 · Stabilizers for \(\SL_2/\SO_2\): \(\mu_2\) times a pro-unipotent group
  • Lemma 2.2 · Central inertia splits \(\mathcal C_K\) into even and odd parts; the even part is the \(\PGL_2\)-equivariant category
  • Lemma 2.3 · Hecke convolution preserves the central grading, so every Satake object is even

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 02:55 UTC

Central inertia for SL2/SO2: epsilon is a natural automorphism and convolution is bimultiplicative

Steps audited. A2-1 stabilizer; A2-2 construction of \(\varepsilon\); A2-3 steps 1–3.

A2-1 (holds). Recomputed: \(\mathrm{SO}(x_m)(F)=\{(a,bt^m;-bt^{-m},a)\}\), integrality forces \(b\in t^m\mathcal O\), \(a=\pm\sqrt{1-b^2}\); so \(\pi_0=\mathbb Z/2\) generated by \(-I\) for \(m\ge1\), connected for \(m=0\). The model \(X=\{A=A^{\mathsf T},\det A=1\}\) is \(\mathrm{SL}_2/\mathrm{SO}_2\) and \(X(F)\) is one \(\mathrm{SL}_2(F)\)-orbit (\(H^1(F,\mathrm{SO}_2)=H^1(F,\mathbb G_m)=0\) since \(i\in F\)), so \(LX=LG\cdot x_0\): no component issue.

A2-2 (holds). A central \(z\in H\) acting trivially on \(Y\) is a 2-automorphism of \(\mathrm{id}_{[Y/H]}\); any sheaf theory functorial in 2-morphisms (equivariant sheaves on finite-type models \(G(\mathcal O/t^N)\backslash Z^{(N)}\), where \(-I\) is still central and acts trivially) gives \(\varepsilon\in\mathrm{Aut}(\mathrm{id}_{\mathcal C})\). One point A2 leaves implicit and I checked: \(\varepsilon\) commutes with truncation functors (ordinary and perverse) because \(\tau^{\le n}F\to F\) is a counit and \(\mathrm{Hom}(\tau^{\le n}F,\tau^{\le n}F)\to\mathrm{Hom}(\tau^{\le n}F,F)\) is bijective; this is what makes 'all cohomology sheaves of \(V^-\) have \(\varepsilon=-1\)' (A2-3 step 2) and the perverse-constituent statement of A2-7 legitimate. (Not every endofunctor commutes with \(\varepsilon\): \(\otimes\,\mathrm{sgn}\) on \(\mathrm{Rep}(\mathbb Z/2)\) does not.)

A2-3 step 3 (holds, justified not merely asserted). Traced: on \(LG\times^{L^+G}LX\) the left inertia of \(z=-I\) at the stalk \(V_g\otimes F_x\) is (left-equivariance of \(V\) at \(z\))\(\circ\)(descent datum for the middle element \(z\))\(^{-1}\) \(=\big[(\text{left}_z)\circ(\text{right}_z)^{-1}\text{ on }V_g\big]\otimes\varepsilon_z(F)_x\). The right inertia of \(V\) is trivial because \(V\) is pulled back from \(\mathrm{Gr}=LG/L^+G\) with the canonical right structure, so the bracket is \(\varepsilon_z(V)_g\). Then \(m_!\) commutes with \(\varepsilon\) (A2-2.3). Hence \(\varepsilon_{V\ast F}=\varepsilon_V\otimes\varepsilon_F\). Step 2 (\(\varepsilon=\mathrm{id}\) on \(\mathcal H\)) needs only connectedness of \(L^+G\cap\mathrm{Ad}_{t^\lambda}L^+G\) (true: pro-unipotent kernel over a parabolic of \(G(k)\)).

Hypothesis. The pages' standing realisation of \(\mathcal C\) as \(L^+G\)-equivariant constructible sheaves with \(\mathcal C_\mu^\heartsuit=\mathrm{Rep}(\pi_0\mathrm{Stab}(x_\mu))\) (criterion.html H3 discussion). No weights, no Frobenius, no transfer from \(\overline{\mathbb F}_q\). Severity: none.

Sources