§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:40:19Z

Theorem 4.11 · \(\SL_2/\SO_2\): the sign-twisted IC sheaves are not in the Satake span, so \(\Csat\ne\cC_K\) over \(\overline{\F}_q\)

proved audit: error

Error. Step 2 asserts that each tr_n IC_{triv,μ} is σ-invariant 'being constant on each O_ν(F_{q^n})'. Since σ swaps the two K_n-orbits of every O_ν, ν ≠ 0, this reason restates the claim, and nothing in the node excludes sign local systems in the cohomology sheaves of IC_{triv,μ} along lower orbits; step 3 (all sign coefficients vanish) depends on it, so the proof fails as written. The claim is true and the theorem holds: σ is geometrically the action of h ∈ L+SL2(k), so σ*IC_{χ,ν} ≅ IC_{χ,ν}, the two Weil structures differ by a scalar (End = scalars), and the top orbit shows the scalar is χ(−1); hence tr_n IC_{χ,ν}∘σ = χ(−1)·tr_n IC_{χ,ν} (Correction 10.8) — the Weil form of the central inertia of Theorem 2.4. Minor points: σ 'induced by conjugation by s' is defined on the split model SL2/T of Proposition J(2) (conjugation by s does not preserve SO(x²+y²)); Chen–Yi's Remark 89 is about non-semisimplicity, the nontrivial-coefficient remark is Remark 13. No run node repairs the step: Theorem 2.4 (02:22, before) proves Csat ≠ C over C independently, Proposition 1.7 (03:01, after) gives an ℓ-adic proof working with sign-isotypic parts on the orbit that avoids the gap. The conclusion still holds by another argument.
Used later by: Summary 4.13 · Verdict: one direction is a theorem, the other fails formally; Table 1 rows decided under D0–D5The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C..
Corrected by: Correction 10.8 · Correction: the square-class automorphism multiplies \(\tr_n\IC_{\chi,\nu}\) by \(\chi(-1)\), Lemma X2.1 · \(\sigma_n\) is translation by \(h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)\) with \(h^{-1}\mathrm{Fr}^n(h)=-I\), Lemma X2.2 · Weil inertia lemma: \(\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\), Theorem (repair of A4-11) X2.4 · Repair of A4-11: \(\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}\) in both \(\mathbb F_q\)-forms, \(q\) odd.

A square-class automorphism σ of \(X(\F_{q^n}(\!(t)\!))\) fixes every Hecke translate of \(\varphi_0\) but swaps the two \(K_n\)-orbits inside each nonzero geometric orbit; read through the trace dictionary over all \(\F_{q^n}\), this forces the sign local systems to have zero multiplicity in every Satake object. Hence \(\Csat\ne\cC_K\) for \(\SL_2/\SO_2\), deciding a row that Table 1 left undecided.

Theorem 4.11.

Let \(X=\SL_2/\SO_2\) over \(\F_q\), \(q\) odd. Assume D0–D2The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C. of Setup 4.1 in the form with disconnected stabilizers: \(\pi_0\Stab(x_\mu)=\{\pm1\}\) for \(\mu\ne0\), simple perverse sheaves \(\IC_{\chi,\mu}\) with \(\chi\in\{\mathrm{triv},\mathrm{sgn}\}\), and \(V_\lambda\ast\IC_0\simeq\bigoplus_b\IC_{\chi_b,\nu_b}[s_b]\) (Proposition III, step 5(b), without H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf.; the proof of Chen–Yi's Theorem 93 covers all symmetric varieties). Then \(\IC_{\mathrm{sgn},\mu}\notin\Csat\) for every \(\mu\ne0\); in particular \(\Csat_{\overline{\F}_q}\ne\cC_{\overline{\F}_q}\), and the same over \(\C\) under D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p).. Table 1's "undecided here" becomes "not equal" (conditional on D0–D2The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C.).

  1. Function level, every \(n\). Let \(F_n=\F_{q^n}(\!(t)\!)\), \(u\in\cO_n^\times\) a non-square unit, \(s=\operatorname{diag}(u,1)\), and \(\sigma\) the automorphism of \(X(F_n)\) induced by conjugation by \(s\). As in Proposition J(2), \(\sigma\) fixes \(\varphi_0\) and normalises \(K_n\) and every \(K_nt^\lambda K_n\), so \(\cH_{K_n}\varphi_0\subseteq(C_{K_n})^\sigma\).
  1. Orbits. By Lemma 4.2, \(\cO_\mu(\F_{q^n})\) consists of two \(K_n\)-orbits for \(\mu\ne0\) (the classes \(\pm1\in H^1(\F_{q^n},\{\pm1\})\)) and of one for \(\mu=0\). \(\sigma\) preserves each \(\cO_\mu(\F_{q^n})\) (\(L^+\GL_2\) is connected and permutes the \(L^+\SL_2\)-orbits preserving codimension) and swaps the two \(K_n\)-orbits: \(s\cdot x_\mu=h\cdot x_\mu\) with \(h=\operatorname{diag}(\sqrt u,1/\sqrt u)\in L^+\SL_2(k)\) (the centre acts trivially), and \(\mathrm{Fr}(h)^{-1}h=\operatorname{diag}(-1,-1)\), the nontrivial class, because \(\sqrt u^{\,q^n}=-\sqrt u\). Hence
$$(C_{K_n})^\sigma=\operatorname{span}\{\one_{\cO_\mu(\F_{q^n})}\}_\mu=\operatorname{span}\{\tr_n\IC_{\mathrm{triv},\mu}\}_\mu ,$$

by unitriangularity as in Lemma 4.4 (part 1) and because \(\tr_n\IC_{\mathrm{triv},\mu}\) is σ-invariant, being constant on each \(\cO_\nu(\F_{q^n})\). The function \(\tr_n\IC_{\mathrm{sgn},\mu}\) has top part \(\pm(\one_{O'_\mu}-\one_{O''_\mu})\), which is σ-anti-invariant; and the \(\tr_n\IC_{\chi,\mu}\) over all \((\chi,\mu)\) form a basis of \(C_{K_n}\otimes\overline{\Q}_\ell\).

  1. Coefficients. Expand as in Lemma 4.4 (part 3), with multiplicity spaces \(E_{(\chi,\nu),i}\); this is valid because \(\End\IC_{\chi,\nu}=\overline{\Q}_\ell\) and the \(\IC_{\chi,\nu}\) are pairwise non-isomorphic and Frobenius-fixed, \(\mathcal L_{\mathrm{sgn}}\) being defined over \(\F_q\):
$$\tr_n(V_\lambda\ast\IC_0)=\sum_{(\chi,\nu)}m^{(n)}_{(\chi,\nu),\lambda}\,\tr_n\IC_{\chi,\nu}.$$

By steps 1–2 and Lemma 4.3, \(\tr_n(V_\lambda\ast\IC_0)\in(C_{K_n})^\sigma\), so \(m^{(n)}_{(\mathrm{sgn},\mu),\lambda}=0\) for all \(n\ge1\), all \(\lambda\) and all \(\mu\ne0\).

  1. By Lemma 4.4 (part 4(ii)), \(E_{(\mathrm{sgn},\mu),i}(\lambda)=0\) for all \(i\) and \(\lambda\): \(\IC_{\mathrm{sgn},\mu}\) is a constituent of no \({}^pH^i(V_\lambda\ast\IC_0)\).
  1. Apply Lemma 4.5 (part 2) with \(\Sigma\) the set of all simple objects except \(\IC_{\mathrm{sgn},\mu}\). \(\square\)

Remarks. (i) Chen–Yi, Remarks 13 and 89, note that for \(\SL_2/\SO_2\) the relative Satake category contains IC complexes with nontrivial coefficients "not belonging to \(\Rep(\check G_X)\)"; the theorem shows that they are not even in the thick Satake span. (ii) The argument is the sheaf-level form of the square-class obstruction hoped for in §4 of the Summary. It needs Weil structures over all \(\F_{q^n}\), not a single \(q\), because of the eigenvalue cancellation discussed in Lemma 4.4. (iii) Whether the \(\IC_{\mathrm{triv},\mu}\) for odd \(\mu\) (the uncovered labels) lie in \(\Csat\) is not decided here.

Builds on

  • Lemma 4.2 · \(K\)-orbits on \(X(\F_q(\!(t)\!))\) via Lang's theorem: \(\Lambda_S^+\) exactly when \(Z_{G^\theta}(S)\) is connected
  • Lemma 4.4 · Expansion of \(\tr(V_\lambda\ast\IC_0)\) in the IC basis: the coefficients are traces of Frobenius on multiplicity spaces
  • Lemma 4.5 · \(\Csat=\cC_K\) if and only if every \(\IC_\mu\) is a direct summand of a Satake object

Built on by

Original record, as the agent wrote it at 02:40 UTC

SL2/SO2: the sign-twisted IC sheaves are not in the Satake span, so C^Sat is not C over F_q-bar

Theorem A4-11. \(X=\mathrm{SL}_2/\mathrm{SO}_2\) over \(\mathbb F_q\), \(q\) odd. Assume D0–D2 in the form with disconnected stabilizers: \(\pi_0\mathrm{Stab}(x_\mu)=\{\pm1\}\) for \(\mu\ne0\), simple perverse sheaves \(\mathrm{IC}_{\chi,\mu}\) (\(\chi\in\{\mathrm{triv},\mathrm{sgn}\}\)), and \(V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus\mathrm{IC}_{\chi_b,\nu_b}[s_b]\) (criterion.html Step 5 Lemma (b) without H3; Chen–Yi Thm 93 proof covers all symmetric varieties). Then \(\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C^{\mathrm{Sat}}\) for every \(\mu\ne0\); in particular \(\mathcal C^{\mathrm{Sat}}_{\overline{\mathbb F}_q}\ne\mathcal C_{\overline{\mathbb F}_q}\) (and over \(\mathbb C\) under D5). Table 1's "undecided here" becomes "not equal" (conditional on D).

Proof. 1. (Function level, every \(n\).) Let \(F_n=\mathbb F_{q^n}(\!(t)\!)\), \(u\in\mathcal O_n^\times\) a non-square, \(s=\operatorname{diag}(u,1)\), \(\sigma\) the automorphism of \(X(F_n)\) induced by conjugation by \(s\). As in classical.html Prop. J(2): \(\sigma\) fixes \(\varphi_0\), normalises \(K_n\) and every \(K_nt^\lambda K_n\), so \(\mathcal H_{K_n}\varphi_0\subseteq(C_{K_n})^\sigma\).

  1. (Orbits.) By A4-2, \(\mathcal O_\mu(\mathbb F_{q^n})\) is two \(K_n\)-orbits for \(\mu\ne0\) (classes \(\pm1\in H^1(\mathbb F_{q^n},\{\pm1\})\)) and one for \(\mu=0\). \(\sigma\) preserves each \(\mathcal O_\mu(\mathbb F_{q^n})\) (\(L^+\mathrm{GL}_2\) is connected and permutes the \(L^+\mathrm{SL}_2\)-orbits preserving codimension) and swaps the two \(K_n\)-orbits: \(s\cdot x_\mu=h\cdot x_\mu\) with \(h=\operatorname{diag}(\sqrt u,1/\sqrt u)\in L^+\mathrm{SL}_2(k)\) (the centre acts trivially), and \(\mathrm{Fr}(h)^{-1}h=\operatorname{diag}(-1,-1)=-1\), the nontrivial class, since \(\sqrt u^{\,q^n}=-\sqrt u\). Hence \((C_{K_n})^\sigma=\operatorname{span}\{\mathbf 1_{\mathcal O_\mu(\mathbb F_{q^n})}\}_\mu=\operatorname{span}\{\operatorname{tr}_n\mathrm{IC}_{\mathrm{triv},\mu}\}_\mu\) (unitriangularity as in A4-4.1; \(\operatorname{tr}_n\mathrm{IC}_{\mathrm{triv},\mu}\) is \(\sigma\)-invariant, being constant on each \(\mathcal O_\nu(\mathbb F_{q^n})\)), while \(\operatorname{tr}_n\mathrm{IC}_{\mathrm{sgn},\mu}\) has top part \(\pm(\mathbf 1_{O'_\mu}-\mathbf 1_{O''_\mu})\), \(\sigma\)-anti-invariant; the \(\operatorname{tr}_n\mathrm{IC}_{\chi,\mu}\) over all \((\chi,\mu)\) form a basis of \(C_{K_n}\otimes\overline{\mathbb Q}_\ell\).
  2. (Coefficients.) Expanding as in A4-4.3 (with \(E_{(\chi,\nu),i}\), valid since \(\operatorname{End}\mathrm{IC}_{\chi,\nu}=\overline{\mathbb Q}_\ell\), the \(\mathrm{IC}_{\chi,\nu}\) are pairwise non-isomorphic and Fr-fixed, \(\mathcal L_{\mathrm{sgn}}\) being defined over \(\mathbb F_q\)): \(\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\sum m^{(n)}_{(\chi,\nu),\lambda}\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\). By 1–2 and A4-3, \(\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)\in(C_{K_n})^\sigma\), so \(m^{(n)}_{(\mathrm{sgn},\mu),\lambda}=0\) for all \(n\ge1\), \(\lambda\), \(\mu\ne0\).
  3. A4-4.4(ii): \(E_{(\mathrm{sgn},\mu),i}(\lambda)=0\) for all \(i,\lambda\): \(\mathrm{IC}_{\mathrm{sgn},\mu}\) is a constituent of no \({}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)\). 5. A4-5(ii) with \(\Sigma=\) all simples except \(\mathrm{IC}_{\mathrm{sgn},\mu}\). \(\square\)

Remarks. (i) Chen–Yi Remark 13/89 note that for \(\mathrm{SL}_2/\mathrm{SO}_2\) the relative Satake category contains IC-complexes with nontrivial coefficients "not belonging to \(\mathrm{Rep}(\check G_X)\)"; A4-11 shows they are not even in the thick Satake span. (ii) The argument is the sheaf-level form of the square-class obstruction the Summary §4 hoped for; it needs Weil structures over all \(\mathbb F_{q^n}\) (not a single \(q\)) because of the eigenvalue cancellation in A4-4. (iii) Whether \(\mathrm{IC}_{\mathrm{triv},\mu}\) for odd \(\mu\) (uncovered labels) lie in \(\mathcal C^{\mathrm{Sat}}\) is not decided.

Sources