§2 · Agent 2 · Non-split forms · 2026-10-17T21:26:46Z
Summary 2.13 · Verdict: sheaves do not imply (SG) when \(T\) is non-split; nothing new under (R)
proved ★ key step
Unitary twists of the group case are counterexamples to IId: \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), (SG) fails over \(\mathbb F_q(\!(t)\!)\), and (C), D0–D3 and all of (R) except the splitness of \(T\) hold. The smallest are the norm torus and \(\mathrm{SL}_2(E)\) on hermitian matrices. The mechanism is that \(\mathcal H_K\) sees only \(\sigma\)-fixed coweights. Under (R) twisting is void, and what remains is a cancellation question at uncovered labels.
Summary 2.13.
(1) Counterexamples (proved). Let \(H\ne1\) be split connected reductive over \(\mathbb F_q\), \(q\) odd. The unitary twist \((\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}H,\ \vartheta\circ\sigma)\) satisfies:
- \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) over \(\overline{\mathbb F}_q\);
- (SG) fails over \(\mathbb F_q(\!(t)\!)\): \(\mathbf 1_{O_c}\in\mathcal H_K\ast\varphi_0\) only if \(c\in2\Lambda_H^+\);
- \((\mathrm{SG})_n\) holds exactly for even \(n\);
- (C) and D0–D3 hold, and so does (R) except "\(T\supseteq S\) split" and "\(T_0\) split".
Instances:
- \(\operatorname{Res}\mathbb G_m\) acting on \(\mathbb G_m\) through the norm, where the failure comes from an extra \(G(F)\)-orbit;
- \(\operatorname{Res}\mathrm{GL}_n\) acting on hermitian matrices, failing also inside the base orbit for \(n\ge2\);
- \(\mathrm{SL}_2(E)\) acting on hermitian matrices of determinant \(1\), with a single \(G(F)\)-orbit and every row of \(m^{(1)}\) nonzero.
The Galois twist \(\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\) satisfies (SG), so (SG) over \(\mathbb F_q(\!(t)\!)\) is not determined by \((G_k,X_k)\). For split \(\mathrm{GL}_2\) with a non-split torus \(H\), neither choice of \(H\) is a counterexample.
(2) Mechanism (proved).
- \(\mathcal H_K\) has basis \(T_\lambda\), \(\lambda\in(\Lambda_T^+)^\sigma\), and only these \(V_\lambda\) carry Weil structures.
- Induced Weil objects of the Satake category have zero trace at \(n=1\) on non-fixed \(\sigma\)-orbits.
- Thick generation uses the retracts \(V_\lambda\ast\mathrm{IC}_0\) of every \(\lambda\). In the unitary twist these are Frobenius-stable in \(\mathcal C_k\), with Weil structures that come from no Satake object.
- Forcing condition: \(\varphi\) injective on \((\Lambda_T^+)^\sigma\) with \(\varphi((\Lambda_T^+)^\sigma)\ne(\Lambda_S^+)^{\mathrm{Fr}}\), together with \(\mathcal M=\Lambda_S^+\) over \(k\).
(3) Proposition II's setting. Under (R) and (C) the twisting effects vanish. The forcing condition never produces a failure: either \(\varphi|_{\Lambda_T^+}\) is not injective modulo trivially acting cocharacters, or every label lifts and (SG) holds. By Lemma 4.8 of the run, a counterexample there needs an uncovered label \(\mu\notin\varphi(\Lambda_T^+)\) such that \(\mathrm{IC}_\mu\) is a summand but \(\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}\) is not a combination of the \(T_\lambda\ast\varphi_0\), whose leading terms cancel. This is open from this approach.
Where each claim is proved.
- Torus, both levels: Counterexample 2.3.
- Twisted forms over \(k\) and over \(F\): Setup 2.4.
- Sheaf level, for every form at once: Lemma 2.5.
- Function-level failure for unitary twists, by the leading-term lemma Lemma 2.2: Theorem 2.6.
- Explicit counts (hand-proved for all \(q\), checked for \(q=3,5\)): Computation 2.7.
- How the trace misses the summands: Lemma 2.8.
- Rows nonzero, columns deficient: Computation 2.9.
- Galois twist holds, so (SG) is not an invariant of \((G_k,X_k)\): Proposition 2.10.
- Split \(\mathrm{GL}_2\) with torus \(H\): Dead end 2.11.
- Mechanism, and the analysis under (R): Remark 2.12.
Which hypotheses fail in the examples. Only "\(T\supseteq S\) split" and "\(T_0\) split" in (R). Proposition IIc needs these only to give every \(V_\lambda\) a Weil structure (IIa); IIb survives because \(S\) is split. In these examples, what separates IIc from IId is the splitness of \(T\).
Answer to IId as posed. No. "\(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\Rightarrow\) (SG) over \(\mathbb F_q(\!(t)\!)\)" is false for spherical, indeed symmetric, \(X=G/H\) over \(\mathbb F_q\) with \(q\) odd, connected \(H\) and connected \(Z_{G^\theta}(S)\). Both levels are proved in characteristic \(p\). The question remains open under the full (R).
Builds on
- Counterexample 2.3 · Counterexample: \(\operatorname{Res}\mathbb G_m\) on \(\mathbb G_m\) through the norm
- Theorem 2.6 · Theorem: the unitary twist of \((H\times H)/H\) fails (SG) over \(\mathbb F_q(\!(t)\!)\)
- Computation 2.9 · Computation: unitary \(\mathrm{SL}_2\): every row of \(m^{(1)}\) nonzero, no spanning
- Proposition 2.10 · Proposition: the Galois twist \(\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\) satisfies (SG)
- Dead end 2.11 · Dead end: \(G=\mathrm{GL}_2\) split, \(H=T_E\) or \(T_E^1\) non-split
- Remark 2.12 · Remark: why the trace misses thick generation, and what (R) removes
Built on by
- Remark 6.1 · Registry after the first two returns: a non-split counterexample found twice; launch its audit