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

Theorem 1.6 · Over \(\F_q(\!(t)\!)\), (SG) implies \(\Csat=\mathcal C_K\) when stabilizers are connected

proved audit: minor defect

Minor defect (second audit). Proof correct (same skeleton as Theorem 4.6 and Proposition 3.10). Unstated hypotheses that hold in the cases at hand: G, θ and Chen–Yi's θ-stable Borel pair defined over 𝔽_q with split torus (Chen–Yi §8.2; needed for the canonical Weil structures and for the basis T_λ). Connectedness of all L⁺G-stabilizers is quoted from Proposition III, step 4, which is argued in characteristic 0; the argument holds for p odd (fixed points of an involution of a pro-unipotent group are connected when p ≠ 2, and maximal θ-split tori are conjugate when p ≠ 2), but this is not said. Pointwise purity (Chen–Yi Thm. 81) is assumed but not used. Remark (b): with one q, spreading out gives the Betti statement only if that p lies outside a finite set of bad primes, so 'Betti follows' needs (SG) for all large p (available in the rows where it is used) or the transfer assumption.

Through the trace of Frobenius, an expression of an orbit indicator by Hecke translates of \(\varphi_0\) forces the corresponding IC sheaf to be a summand of a Satake object. So, with connected stabilizers, the function statement over \(\F_q(\!(t)\!)\) implies the ℓ-adic sheaf statement; the converse is not implied.

Theorem 1.6.

Let \(X=G/G^\theta\) be defined over \(\F_q\), \(q\) odd, with \(G^\theta\) connected and \(Z_{G^\theta}(S)\) connected, so that every \(L^+G\)-stabilizer on \(LX\) is connected (Proposition III, step 4). Put \(F=\F_q(\!(t)\!)\), \(K=G(\F_q[\![t]\!])\), and let \(\mathcal C_\ell=\Shv_{\mathrm{cons}}(L^+G\backslash LX)\) be Chen–Yi's ℓ-adic category over \(\overline{\F}_q\). Assume Chen–Yi's pointwise purity (Thm. 81) and the decomposition of Satake objects (proof of Thm. 93). Then $$\text{(SG) over }\F_q(\!(t)\!)\quad\Longrightarrow\quad\mathcal C_\ell^{\mathrm{Sat}}=\mathcal C_\ell .$$ More precisely, if \(\mathbf 1_{\mathcal O_\mu}\in\mathcal H_K\cdot\varphi_0\), then \(\IC_\mu\) is a shifted direct summand of some \(V_\lambda\ast\IC_0\).

Write \(t_{\mathcal F}\) for the trace-of-Frobenius function of a Weil complex \(\mathcal F\) on \(LX(\F_q)=X(F)\).

  1. Orbit basis. By Lang's theorem (connected stabilizers, finite-type quotients), \(\mathcal O_\mu(\F_q)\) is a single \(K\)-orbit, so \(C_K\) has the basis \(\{\mathbf 1_{\mathcal O_\mu}\}_{\mu\in\Lambda_S^+}\).
  2. IC sheaves are unitriangular. Each \(\IC_\mu\) has a canonical Weil structure (Chen–Yi, §9.1), and \(t_{\IC_\mu}=c_\mu\mathbf 1_{\mathcal O_\mu}+\sum_{\nu<\mu}(\cdots)\mathbf 1_{\mathcal O_\nu}\) with \(c_\mu\ne0\).
  3. Dictionary. \(M_\lambda:=V_\lambda\ast\IC_0\) is \(m_!\) of a Weil complex, with \(m\) proper on the support (Proposition III, step 3); proper base change and the Grothendieck–Lefschetz trace formula give \(t_{M_\lambda}=t_{V_\lambda}\ast_K\varphi_0\). The functions \(t_{V_\lambda}\) are unitriangular in the basis \(T_\lambda\) (Lusztig–Kato), so \(\operatorname{span}_\C\{t_{M_\lambda}\}=\mathcal H_K\cdot\varphi_0\).
  4. Constituents. \(t_{M_\lambda}=\sum_i(-1)^it_{{}^pH^iM_\lambda}\). Each \({}^pH^iM_\lambda\) is a mixed perverse sheaf over \(\F_q\); its simple constituents over \(\F_q\) are geometrically isotypic for Frobenius-stable geometric simples \(\IC_\nu\) (all simples have trivial coefficients, by connectedness), hence are scalar twists \(\IC_\nu\otimes\alpha\). So \(t_{M_\lambda}\in\operatorname{span}\{t_{\IC_\nu}:\nu\in D\}\), where \(D\) is the set of \(\nu\) such that \(\IC_\nu\) is a geometric constituent of some \({}^pH^iM_\lambda\). By semisimplicity (decomposition theorem), \(D\) is the set of \(\nu\) such that \(\IC_\nu\) is a shifted summand of some \(M_\lambda\).
  5. Extraction. Suppose \(\mathbf 1_{\mathcal O_\mu}=\sum_{\nu\in D}c_\nu t_{\IC_\nu}\), a finite sum, and let \(\nu_0\) be maximal with \(c_{\nu_0}\ne0\). Evaluating on \(\mathcal O_{\nu_0}(\F_q)\), only \(\nu_0\) contributes, giving \(c_{\nu_0}c_{\nu_0}'\ne0\); hence \(\mathcal O_{\nu_0}=\mathcal O_\mu\) and \(\mu\in D\).
  6. Conclusion. Under (SG), \(D=\Lambda_S^+\), and the summand criterion (Lemma 1.5, direction \(\Leftarrow\)) gives \(\mathcal C_\ell^{\mathrm{Sat}}=\mathcal C_\ell\).

Remarks. (a) The converse does not follow: \(\Csat=\mathcal C_K\) says that every \(\IC_\nu\) occurs, not that the matrix of multiplicities \((m_{\lambda,\nu})\) has full rank; so "sheaf holds, function fails" is not excluded by this theorem. (b) Only one \(q\) is needed, but the conclusion is then over \(\overline{\F}_q\) for that \(p\); the Betti category over \(\C\) follows by comparing decomposition multiplicities (spreading out) — the same transfer (open item 1) the pages assume for every positive sheaf row. (c) Applications: \((H\times H)/H\) (Proposition D over \(\F_q(\!(t)\!)\)) and the spheres (Proposition 1.3) recover Propositions II and III ℓ-adically; \(\GL_4/\Sp_4\) (Proposition 1.4) gives \(\mathcal C_\ell^{\mathrm{Sat}}=\mathcal C_\ell\) although H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. fails there — H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. is sufficient (Proposition I) but not necessary. (d) For \(\GL_{2m}/\Sp_{2m}\) (\(m\ge3\)), \(\SL_{2n}/\Sp_{2n}\) (\(n\ge3\)), \(E_6/F_4\) and \(\GL_n/(\GL_p\times\GL_q)\), the sheaf statement follows from (SG) over \(\F_q(\!(t)\!)\) for one odd \(q\); Theorem E does not supply this, since (A1)(A1) Field and group: F is a local field of characteristic 0 (p-adic), and G is split over the integers, with connected fibres and simply connected derived group. Its characteristic-p counterpart is (A1ₚ). requires \(\operatorname{char}F=0\).

Builds on

  • 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 7.14 · Audit: the three proofs that (SG) implies \(\Csat=\cC_K\) are one proof in three notations; inputs listed and compared
  • Audit 7.23 · Audit of the \(\GL_{2m}/\Sp_{2m}\) theorem: the induction holds (directly, not through Proposition B as stated); the \(\SL\) corollary and the sheaf consequence hold
  • Re-derivation X1.6 · The verdict inputs downstream of A1-4, A4-10, A5-6, A7-11: what changes
Original record, as the agent wrote it at 03:01 UTC

Function level implies sheaf level over F_q((t)) when stabilizers are connected

Theorem (one implication between the levels, ℓ-adic). Let \(X=G/G^\theta\) be defined over \(\mathbb F_q\), \(q\) odd, \(G^\theta\) connected, with \(Z_{G^\theta}(S)\) connected (so all \(L^+G\)-stabilizers are connected, criterion.html step 4). Put \(F=\mathbb F_q(\!(t)\!)\), \(K=G(\mathbb F_q[\![t]\!])\), and let \(\mathcal C_\ell=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX)\) be Chen–Yi's ℓ-adic category over \(\overline{\mathbb F}_q\). Assume Chen–Yi Theorem 81 (pointwise purity) and the decomposition of Satake objects (Theorem 93's proof). Then $$\text{(SG) over }\mathbb F_q(\!(t)\!)\ \Longrightarrow\ \mathcal C_\ell^{\mathrm{Sat}}=\mathcal C_\ell .$$ More precisely, \(\mathbf 1_{\mathcal O_\mu}\in\mathcal H_K\cdot\varphi_0\) implies that \(\mathrm{IC}_\mu\) is a shifted summand of some \(V_\lambda\ast\mathrm{IC}_0\). Proof. 1. By Lang's theorem (connected stabilizers, finite-type quotients) \(\mathcal O_\mu(\mathbb F_q)\) is a single \(K\)-orbit, so \(C_K\) has basis \(\{\mathbf 1_{\mathcal O_\mu}\}_{\mu\in\Lambda_S^+}\). 2. Each \(\mathrm{IC}_\mu\) has a canonical Weil structure (Chen–Yi §9.1); its trace function is \(t_{\mathrm{IC}_\mu}=c_\mu\mathbf 1_{\mathcal O_\mu}+\sum_{\nu<\mu}(\dots)\mathbf 1_{\mathcal O_\nu}\), \(c_\mu\ne0\): unitriangular in the orbit basis. 3. Dictionary: \(M_\lambda:=V_\lambda\ast\mathrm{IC}_0\) is \(m_!\) of a Weil complex with \(m\) proper on the support (criterion.html step 3), so by proper base change and Grothendieck–Lefschetz \(t_{M_\lambda}=t_{V_\lambda}\ast_K\varphi_0\); the \(t_{V_\lambda}\) are unitriangular in the \(T_\lambda\) (Lusztig–Kato), hence \(\operatorname{span}_{\mathbb C}\{t_{M_\lambda}\}=\mathcal H_K\cdot\varphi_0\). 4. \(t_{M_\lambda}=\sum_i(-1)^it_{{}^pH^iM_\lambda}\); each \({}^pH^iM_\lambda\) is a mixed perverse sheaf over \(\mathbb F_q\) whose simple constituents over \(\mathbb F_q\) are geometrically isotypic for Frobenius-stable geometric simples \(\mathrm{IC}_\nu\) (all simples are \(\mathrm{IC}_\nu\) with trivial coefficients, by connectedness), hence are scalar twists \(\mathrm{IC}_\nu\otimes\alpha\); so \(t_{M_\lambda}\in\operatorname{span}\{t_{\mathrm{IC}_\nu}:\nu\in D\}\), \(D:=\{\nu:\mathrm{IC}_\nu\text{ a geometric constituent of some }{}^pH^iM_\lambda\}\). By semisimplicity (decomposition theorem), \(D=\{\nu:\mathrm{IC}_\nu\text{ a shifted summand of some }M_\lambda\}\). 5. If \(\mathbf 1_{\mathcal O_\mu}=\sum_{\nu\in D}c_\nu t_{\mathrm{IC}_\nu}\) (finite sum), let \(\nu_0\) be maximal with \(c_{\nu_0}\ne0\); evaluating on \(\mathcal O_{\nu_0}\) gives \(c_{\nu_0}c'_{\nu_0}\ne0\), so \(\nu_0=\mu\in D\). 6. Under (SG), \(D=\Lambda_S^+\), and A1-5 (⟸) gives \(\mathcal C_\ell^{\mathrm{Sat}}=\mathcal C_\ell\). ∎

Remarks. (a) The converse is not implied: \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) says every \(\mathrm{IC}_\nu\) occurs, not that the matrix \((m_{\lambda,\nu})\) of multiplicities has full rank; so "sheaf holds, function fails" is not excluded by this theorem. (b) Only one \(q\) is needed, but the conclusion is over \(\overline{\mathbb F}_q\) for that \(p\); Betti \(\mathcal C\) over \(\mathbb C\) follows by the comparison of decomposition multiplicities (spreading out), the same open item 1 the pages assume for every positive sheaf row. (c) Applications: \((H\times H)/H\) (Prop D over \(\mathbb F_q(\!(t)\!)\)) and the spheres (A1-3) recover Props II/III ℓ-adically; \(\mathrm{GL}_4/\mathrm{Sp}_4\) (A1-4) gives \(\mathcal C_\ell^{\mathrm{Sat}}=\mathcal C_\ell\) although H5 fails there — H5 is sufficient (Prop I) but not necessary. (d) For \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) (\(m\ge3\)), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (\(n\ge3\)), \(E_6/F_4\), \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\): the sheaf cell follows from (SG) over \(\mathbb F_q(\!(t)\!)\) for one odd \(q\); Sakellaridis's Theorem E does not supply it ((A1): \(\operatorname{char}F=0\)).

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