§2 · Agent 2 · SL₂/SO₂: central inertia · 2026-10-17T02:22:38Z

Theorem 2.4 · \(\Csat\ne\mathcal C_K\) for \(\SL_2/\SO_2\): the sign local system on the first orbit is not generated

proved audit: correct ★ key step

Second audit: correct.

The standard object of the sign local system on \(\mathcal O_1\) lies in the odd part \(\mathcal C_K^-\), while every Satake object lies in the even part. So the categorical statement fails for \(\SL_2/\SO_2\), by a purely topological argument over \(\C\).

Theorem 2.4.

For \(X=\SL_2/\SO_2\), \(G=\SL_2\), \(k=\C\): $$\Csat\ne\mathcal C_K .$$ Explicit witnesses: \(\Delta(\mathcal L_1^-):=a_{1!}\mathcal L_1^-\), and \(\IC(\overline{\mathcal O_m},\mathcal L_m^-)\) for every \(m\ge1\). The same argument applies to any \(G/G^\theta\) in which a central element of \(G\) lying in \(G^\theta\) represents a nontrivial class in some \(\pi_0\Stab(x_\mu)\).

  1. The witness is an object of \(\mathcal C_K\). \(\mathcal L_1^-\) is the rank-one equivariant local system on \(\mathcal O_1\) given by the sign character of \(\pi_0\Stab(x_1)=\langle-I\rangle\cong\Z/2\) (Computation 2.1). \(\Delta(\mathcal L_1^-)\) is bounded, constructible, equivariant and supported on \(\overline{\mathcal O_1}=\mathcal O_1\sqcup\mathcal O_0\), so it lies in \(\mathcal C_K\); it is nonzero.
  2. It is odd. By Lemma 2.2 (property on local systems), \(\varepsilon_{\mathcal L_1^-}=\rho(-I)=-\mathrm{id}\); since \(a_{1!}\) commutes with \(\varepsilon\), \(\varepsilon_{\Delta(\mathcal L_1^-)}=-\mathrm{id}\ne\mathrm{id}\) (a nonzero object, and \(2\ne0\)). So \(\Delta(\mathcal L_1^-)\in\mathcal C_K^-\setminus\{0\}\).
  3. It is not generated. \(\Csat\subseteq\mathcal C_K^+\) (Lemma 2.3) and \(\mathcal C_K^+\cap\mathcal C_K^-=0\) (Lemma 2.2). Hence \(\Delta(\mathcal L_1^-)\notin\Csat\). The same applies to \(\IC(\overline{\mathcal O_m},\mathcal L_m^-)\), on which \(\varepsilon=-\mathrm{id}\).

Checks. Retracts cannot escape, since \(\mathcal C_K^+\) is closed under retracts. Only \(2\ne0\) in the coefficients is used. The argument is topological over \(\C\): no weights, no decomposition theorem, no transfer from ℓ-adic sheaves; it does not use H1–H5Hypotheses of the pages' Proposition I (H1–H5 imply categorical generation): H1 finite orbit filtration; H2 equivariant recollement; H3 constant coefficients on orbits; H4 the Hecke action preserves the category; H5 every orbit carries a Satake object with nonzero top splitting. or Propositions I and III. It decides the Summary's row "both conditions of Proposition III fail, undecided here" as fails. The generalisation stated above needs only the central element: it acts trivially on \(X\) and on \(\Gr\), stabilizers on \(\Gr\) are connected, and \(\IC_0\) is pulled back from a point.

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

  • Proposition 2.7 · The Satake span is exactly the even-label, constant-coefficient part, strictly inside the even part
  • Summary 2.9 · Verdict for \(\SL_2/\SO_2\): both levels fail, and the sheaf argument is the geometric form of Proposition J(2)
  • Audit 7.2 · Audit: the \(\SL_2/\SO_2\) verdict holds at both levels; recursions and fibre counts re-derived
  • Audit 7.7 · Audit: three proofs that sign local systems are not Satake-generated are consistent; the inertia proof needs no purity
  • Audit X2.7 · No \(\mathrm{SL}_2/\mathrm{SO}_2\) conclusion of the run rests on the false §8.2 remark
  • Summary X2.8 · Re-derivation of Summary 4.13, Summary 7.19, Summary 7.25, Summary 7.32 from the repaired steps
Original record, as the agent wrote it at 02:22 UTC

C^Sat is not C: the sign local system on the first orbit is not generated

Theorem. For \(X=\mathrm{SL}_2/\mathrm{SO}_2\), \(G=\mathrm{SL}_2\), \(k=\mathbb C\): \(\mathcal C^{\mathrm{Sat}}\neq\mathcal C\). Explicit witnesses: \(\Delta(\mathcal L_1^-):=a_{1!}\mathcal L_1^-\) and \(\mathrm{IC}(\overline{\mathcal O_m},\mathcal L_m^-)\), \(m\ge1\). Proof.

  1. \(\mathcal L_1^-\) is the rank-one equivariant local system on \(\mathcal O_1\) given by the sign character of \(\pi_0\operatorname{Stab}(x_1)=\mathbb Z/2=\langle -I\rangle\) (A2-1). \(\Delta(\mathcal L_1^-)\) is bounded, constructible, equivariant and supported on \(\overline{\mathcal O_1}=\mathcal O_1\sqcup\mathcal O_0\), so it is an object of \(\mathcal C\), and it is nonzero.
  2. By A2-2.5, \(\varepsilon_{\mathcal L_1^-}=\rho(-I)=-\mathrm{id}\); by A2-2.3, \(\varepsilon_{\Delta(\mathcal L_1^-)}=a_{1!}(-\mathrm{id})=-\mathrm{id}\neq\mathrm{id}\) (nonzero object, \(2\ne0\)). So \(\Delta(\mathcal L_1^-)\in\mathcal C^-\setminus0\).
  3. \(\mathcal C^{\mathrm{Sat}}\subseteq\mathcal C^+\) (A2-3) and \(\mathcal C^+\cap\mathcal C^-=0\). Hence \(\Delta(\mathcal L_1^-)\notin\mathcal C^{\mathrm{Sat}}\). \(\square\)

Checks of terms. Thick span includes retracts: \(\mathcal C^+\) is closed under retracts (A2-2(a)), so retracts cannot escape. Coefficients \(\mathbb C\): only \(2\neq0\) is used. The argument is purely topological over \(\mathbb C\) (no weights, no decomposition theorem, no transfer from \(\ell\)-adic): it does not depend on H1–H5 or on Proposition I/III, and so decides the Summary row 'both conditions of Prop III fail → undecided here' as fails. The same argument applies verbatim to any \(G/G^\theta\) in which a central element of \(G\) lying in \(G^\theta\) represents a nontrivial class of some \(\pi_0\operatorname{Stab}(x_\mu)\).

Sources