§1 · Agent 1 · Literature and Table 1 audit · 2026-10-17T03:01:34Z

Proposition 1.7 · \(\SL_2/\SO_2\): sign local systems never occur in Satake objects (ℓ-adic proof)

proved audit: minor defect

Minor defect (second audit). Proof correct: Lang count, trace of the sign local system, weight separation (Chen–Yi Thm. 81 with the decomposition in the proof of Thm. 93; pointwise purity of each stalk follows since the perverse cohomology sheaves are pure and geometrically split). Over 𝔽_{q^r} the automorphism σ must be built from a non-square of 𝔽_{q^r}, as the phrase 'every odd q^r' implies. Consequence (ii) calls the parity obstruction open at the sheaf level; it had already been settled in the run (Lemma 2.6, Proposition 2.7: the odd constant-coefficient IC sheaves are not in the Satake span) — a stale status remark.

The function-level symmetry exchanging the two square classes, read through the trace of Frobenius and weight separation, kills the sign-isotypic part of every Satake object. So the standard object of a sign local system is not Satake-generated, and the sheaf statement fails for \(\SL_2/\SO_2\) in Chen–Yi's ℓ-adic setting.

Proposition 1.7.

Let \(X=\SL_2/T\) over \(\F_q\), \(q\) odd, with \(T\) split, \(\theta=\operatorname{Ad}\operatorname{diag}(1,-1)\) and \(G^\theta=T\) (connected), and let \(\mathcal C_\ell\) be Chen–Yi's category over \(\overline{\F}_q\). For every \(m\ge1\) and every dominant \(\lambda\), the sign-isotypic part of \(a_m^{*}(V_\lambda\ast\IC_0)\) vanishes. Hence \(\Delta_m^-:=a_{m!}\mathcal L_m^-\) lies in \(\mathcal C_\ell\) but not in \(\mathcal C_\ell^{\mathrm{Sat}}\), and \(\mathcal C_\ell^{\mathrm{Sat}}\ne\mathcal C_\ell\).

  1. Orbits and local systems. The orbits are \(\mathcal O_m\), \(m\ge0\), and \(\pi_0\Stab(x_m)=Z_{G^\theta}(S)=\{\pm I\}\) for \(m\ge1\) (Proposition III, step 4; split-forms family computation), with trivial Frobenius action. By Lang's theorem \(\mathcal O_m(\F_q)=Ky_m\sqcup K\sigma(y_m)\), the two square classes of Proposition J(2) (see Proposition 1.2, step 2, valid over \(\F_q(\!(t)\!)\) for \(q\) odd). The equivariant local systems on \(\mathcal O_m\) are \(\mathcal L_m^\pm\); with Weil structures normalised at \(y_m\), their traces on the two \(K\)-orbits are \((1,1)\) for \(\mathcal L^+\) and \((1,-1)\) for \(\mathcal L^-\) (Lang torsor).
  2. Traces. Write \(M_\lambda=V_\lambda\ast\IC_0\) and decompose \(H^i(a_m^{*}M_\lambda)=\mathcal L^+\otimes V^i_+\oplus\mathcal L^-\otimes V^i_-\) (Frobenius-stable isotypic decomposition). Over \(\F_{q^r}\),
$$t_{M_\lambda}(y)-t_{M_\lambda}(\sigma y)=2\sum_i(-1)^i\tr(\mathrm{Fr}^r\mid V^i_-).$$

By the dictionary (Theorem 1.6, step 3), \(t_{M_\lambda}\in\mathcal H_K\cdot\varphi_0\), which is \(\sigma\)-invariant (Proposition J(2), valid for every odd \(q^r\)). So \(\sum_i(-1)^i\tr(\mathrm{Fr}^r\mid V^i_-)=0\) for all \(r\ge1\).

  1. Weights. \(M_\lambda\simeq\bigoplus\IC_{\chi,\nu}(\delta/2)[2n](n)\) with each \(\IC_{\chi,\nu}\) pointwise pure of weight \(\delta(\nu)\) (Chen–Yi, Thm. 81, stated for all \(\chi\)); hence \(H^i(a_m^{*}M_\lambda)\) is pure of weight \(i\), and the Frobenius eigenvalues on \(V^i_-\) have absolute value \(q^{i/2}\). Linear independence of the functions \(r\mapsto\alpha^r\), together with the separation of weights between degrees, forces \(V^i_-=0\) for all \(i\).
  2. Non-generation. The subcategory \(\mathcal K_m\) of Lemma 1.5 (variant) is thick and contains every Satake object, hence \(\mathcal C_\ell^{\mathrm{Sat}}\); but \(H^0(a_m^{*}\Delta_m^-)=\mathcal L_m^-\ne0\), so \(\Delta_m^-\notin\mathcal K_m\).

Consequences. (i) The \(\SL_2/\SO_2\) row is "both fail" (ℓ-adically; for Betti sheaves over \(\C\) through the comparison of sign-isotypic multiplicities, the same transfer as open item 1). This is the support argument "in the spirit of the square-class obstruction" anticipated by the Summary, the square classes being read as the trace of \(\mathcal L^-\). (ii) The function-level failure is transported to the sheaf level; the parity obstruction of Proposition 1.2 (whether the odd constant-coefficient \(\IC_{(2j+1)\alpha^\vee}\) are summands of \(V_{k\alpha^\vee}\ast\IC_0\)) remains open at the sheaf level and is not needed. (iii) This is consistent with Chen–Yi, Remarks 13 and 89 (the relative Satake category is not semisimple and contains IC sheaves with nontrivial coefficients outside \(\Rep(\check G_X)\)). (iv) For \(\GL_2/T\) both levels hold (Proposition III; Proposition J(1)): the two rows differ in the group, not between the levels.

Builds on

  • Proposition 1.2 · \(\SL_2/T\) versus \(\GL_2/T\): one variety, two Hecke algebras
  • Lemma 1.5 · Summand criterion: \(\Csat=\mathcal C_K\) if and only if every simple IC sheaf is a summand of a Satake object

Built on by

  • Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide
  • Audit 7.7 · Audit: three proofs that sign local systems are not Satake-generated are consistent; the inertia proof needs no purity
  • Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\); Lang’s count holds in both
  • Audit X2.7 · No \(\mathrm{SL}_2/\mathrm{SO}_2\) conclusion of the run rests on the false §8.2 remark
Original record, as the agent wrote it at 03:01 UTC

SL2/SO2: the sheaf statement fails (l-adic); sign local systems never occur in Satake objects

Theorem (ℓ-adic over \(\overline{\mathbb F}_q\), \(q\) odd). For \(X=\mathrm{SL}_2/T\) (\(T\) split, \(\theta=\operatorname{Ad}\operatorname{diag}(1,-1)\), \(G^\theta=T\) connected), with \(\mathcal C_\ell\) Chen–Yi's category: for every \(m\ge1\) and \(\lambda\), the sign-isotypic part of \(a_m^*(V_\lambda\ast\mathrm{IC}_0)\) vanishes. Hence \(\Delta^-_m:=a_{m!}\mathcal L^-_m\in\mathcal C_\ell\setminus\mathcal C_\ell^{\mathrm{Sat}}\) and \(\mathcal C_\ell^{\mathrm{Sat}}\ne\mathcal C_\ell\). Proof. 1. Orbits \(\mathcal O_m\) (\(m\ge0\)), \(\pi_0\operatorname{Stab}(x_m)=Z_{G^\theta}(S)=\{\pm I\}\) for \(m\ge1\) (criterion.html step 4; families/split-forms.html §4); Frobenius acts trivially on \(\pi_0\). Lang: \(\mathcal O_m(\mathbb F_q)=Ky_m\sqcup K\sigma(y_m)\), the two square classes of Prop J(2) (A1-2 step 2, valid over \(\mathbb F_q(\!(t)\!)\), \(q\) odd). The equivariant local systems on \(\mathcal O_m\) are \(\mathcal L^\pm_m\); with Weil structures normalized at \(y_m\), \(t_{\mathcal L^+}=(1,1)\), \(t_{\mathcal L^-}=(1,-1)\) on the two \(K\)-orbits (Lang-torsor computation). 2. Write \(H^i(a_m^*M_\lambda)=\mathcal L^+\otimes V^i_+\oplus\mathcal L^-\otimes V^i_-\) (isotypic decomposition, Frobenius-stable). Then \(t_{M_\lambda}(y)-t_{M_\lambda}(\sigma y)=2\sum_i(-1)^i\operatorname{tr}(\mathrm{Fr}^r|V^i_-)\) over \(\mathbb F_{q^r}\). By the dictionary (A1-6 step 3) \(t_{M_\lambda}\in\mathcal H_K\cdot\varphi_0\), which is \(\sigma\)-invariant (Prop J(2), every odd \(q^r\)). So \(\sum_i(-1)^i\operatorname{tr}(\mathrm{Fr}^r|V^i_-)=0\) for all \(r\ge1\). 3. Purity: \(M_\lambda\simeq\bigoplus\mathrm{IC}_{\chi,\nu}(\delta/2)[2n](n)\) with each \(\mathrm{IC}_{\chi,\nu}\) pointwise pure of weight \(\delta(\nu)\) (Chen–Yi Thm 81, stated for all \(\chi\)); hence \(H^i(a_m^*M_\lambda)\) is pure of weight \(i\), and the eigenvalues of \(\mathrm{Fr}\) on \(V^i_-\) have absolute value \(q^{i/2}\). Linear independence of \(r\mapsto\alpha^r\) and separation of weights force \(V^i_-=0\) for all \(i\). 4. \(\mathcal K_m\) (A1-5, variant) is thick and contains all Satake objects, hence \(\mathcal C_\ell^{\mathrm{Sat}}\); \(H^0(a_m^*\Delta^-_m)=\mathcal L^-_m\ne0\), so \(\Delta^-_m\notin\mathcal K_m\). ∎

Consequences. (i) The \(\mathrm{SL}_2/\mathrm{SO}_2\) row is both-fail (ℓ-adic; Betti over \(\mathbb C\) by the comparison of the sign-isotypic multiplicities, the same open item 1). This is the pages' expected "support argument in the spirit of the square-class obstruction", with the square classes read as the Frobenius trace of \(\mathcal L^-\). (ii) The function-level failure is here transported to the sheaf level; the parity obstruction of A1-2 (odd trivial-coefficient \(\mathrm{IC}_{(2j+1)\alpha^\vee}\) as summands of \(V_{k\alpha^\vee}\ast\mathrm{IC}_0\)) remains open at sheaf level and is not needed. (iii) Consistent with Chen–Yi Remark 89 (\(\operatorname{Perv}\) not semisimple; non-trivial coefficients outside \(\operatorname{Rep}(\check G_X)\)) and Remark 13. (iv) \(\mathrm{GL}_2/T\): both hold (Prop III; Prop J(1)); the two rows diverge in the group, not between levels.

Sources