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

Lemma 4.8 · Under the two conditions of Proposition III, (SG) holds over \(\F_q(\!(t)\!)\) without Sakellaridis's hypotheses

proved audit: correct

Second audit: correct. Corollary C applies: X is homogeneous, the labels Λ_S^+ have finite lower sets, K\X(F) = Λ_S^+ with compact open orbits comes from Lemma 4.2 (replacing Lemma A's Borel–Serre step), t^λ x0 = x_{φ(λ)} is rational, and supp(T_λ ∗ φ0) = K t^λ X(O) ⊆ (L+G t^λ L+X)(F_q) lies in the closure of O_{φ(λ)} by the support bound. The listed rows satisfy both conditions of Proposition III (criterion page table; Table 1 of the Summary). Table 1's entries quoted are accurate (n=3 was 'under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.', not open — the quotation lists only n ≥ 4).

If \(Z_{G^\theta}(S)\) is connected and every orbit label has a dominant lift, the Hecke translates \(T_\lambda\ast\varphi_0\) are triangular over \(\F_q(\!(t)\!)\), so (SG) holds there with no use of (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit.. This settles the function column over \(\F_q(\!(t)\!)\) for the spheres (all \(n\)), \(\GL_n/(\GL_p\times\GL_q)\), \(\SL_4/\Sp_4\), \(\GL_2/\Sp_2\) and the group.

Lemma 4.8.

Assume D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers. and D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q. of Setup 4.1, \(q\) odd, \(Z_{G^\theta}(S)\) connected and \(\varphi(\Lambda_T^+)=\Lambda_S^+\). Then (SG) holds for \(F=\F_q(\!(t)\!)\), and for every \(\F_{q^n}(\!(t)\!)\).

Consequences (function column of Table 1, for \(F=\F_q(\!(t)\!)\), \(q\) odd). (SG) holds, with no use of (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit., for \(\Spin_{2n}/\Spin_{2n-1}\) for all \(n\ge2\) (Table 1: "\(n\ge4\) open"), \(\GL_n/(\GL_p\times\GL_q)\) (Table 1: "under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies."), \(\SL_4/\Sp_4\), \(\GL_2/\Sp_2\) and the group. This does not touch the \(p\)-adic statements (Objection 4.12).

Apply Corollary C with labels \(\Lambda_S^+\), orbits \(O_\mu:=\cO_\mu(\F_q)=Kx_\mu\) (Lemma 4.2), and the dominance order, whose lower sets are finite (H1H1 · Finite orbit filtration: each orbit closure is the union of the finitely many orbits below it, and every object lives on finitely many orbit closures.(a)).

  1. For \(\mu\in\Lambda_S^+\) pick \(\lambda\in\varphi^{-1}(\mu)\cap\Lambda_T^+\).
  2. Then \(t^\lambda x_0=x_{\varphi(\lambda)}\in O_\mu\) (criterion page, "Theorem (orbits)"); it is a rational point.
  3. By the geometric support bound (Proposition III, step 2; D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers.),
$$K\,t^\lambda X(\cO)=L^+G(\F_q)\,t^\lambda\,L^+X(\F_q)\subseteq\big(L^+G\,t^\lambda L^+X\big)(\F_q)\subseteq\bigsqcup_{\nu\le\varphi(\lambda)}\cO_\nu(\F_q).$$
  1. Corollary C gives (SG) with \(f_\mu=T_\lambda\). \(\square\)

This is the function-level shadow of Proposition I, with Lang's theorem supplying the orbit bijection that Proposition I obtains from H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf..

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

Built on by

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

Under the two conditions of Proposition III, (SG) holds over F_q((t)) without Sakellaridis's hypotheses

Lemma A4-8. Assume D0, D3, \(q\) odd, \(Z_{G^\theta}(S)\) connected and \(\varphi(\Lambda_T^+)=\Lambda_S^+\). Then (SG) holds for \(F=\mathbb F_q(\!(t)\!)\) (and every \(\mathbb F_{q^n}(\!(t)\!)\)).

Proof. Apply classical.html Corollary C with \(\Lambda_X^+:=\Lambda_S^+\), \(\mu\mapsto O_\mu:=\mathcal O_\mu(\mathbb F_q)=Kx_\mu\) (A4-2), the dominance order (finite lower sets: H1(a)). For \(\mu\) pick \(\lambda\in\varphi^{-1}(\mu)\cap\Lambda_T^+\). Then \(t^\lambda x_0=x_{\varphi(\lambda)}\in O_\mu\) (criterion.html Theorem (orbits), rational point), and \(K\,t^\lambda X(\mathcal O)=L^+G(\mathbb F_q)\,t^\lambda\,L^+X(\mathbb F_q)\subseteq\big(L^+G\,t^\lambda L^+X\big)(\mathbb F_q)\subseteq\bigsqcup_{\nu\le\varphi(\lambda)}\mathcal O_\nu(\mathbb F_q)\) by the geometric support bound (criterion.html Lemma Step 2, D0). Corollary C gives (SG) with \(f_\mu=T_\lambda\). \(\square\)

Consequences (function column of Table 1, for \(F=\mathbb F_q(\!(t)\!)\), \(q\) odd). \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\) for all \(n\ge2\) (Table 1: "\(n\ge4\) open"), \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) (Table 1: "under (A)"), \(\mathrm{SL}_4/\mathrm{Sp}_4\), \(\mathrm{GL}_2/\mathrm{Sp}_2\), the group: (SG) holds, with no use of (A1)–(A5). This does not touch the \(p\)-adic statements (A4-12). It is the function-level shadow of Proposition I with Lang's theorem supplying the orbit bijection that Proposition I gets from H3.

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