§3 · Agent 3 · Symplectic family: fibres · 2026-10-17T02:48:26Z

Theorem 3.9 · Explicit $T_{\omega_k}\ast\varphi_0$ over every local field; (SG) for $\GL_4/\Sp_4$ without Sakellaridis's hypotheses

proved audit: correct ★ key step

Second audit: correct. Re-derived and recomputed: values 15/40, 1, 6/12, 0 at q = 2, 3 by an independent lattice parametrization (g2_hnf_lattices.py). The fibre computation uses only the diagonal representatives and holds over any complete DVR with finite residue field, including residue characteristic 2. The support bound for covered labels, the G(F)-equivariance of sigma, and the finiteness hypothesis of Proposition B (dominance within a fixed Pfaffian valuation) hold. No use of (A1)-(A5). The subtraction identity was also recomputed with T_(1,1,1,-1) in place of T_(2,0,0,0), which gives the same function.

Evaluates the minuscule Hecke translates of the basic function for $\GL_{2m}/\Sp_{2m}$ over any non-archimedean local field. For $\GL_4/\Sp_4$ it writes the indicator of the uncovered orbit $(1,1)$ as a combination of two translates, reaches every other uncovered orbit by the scalar shift, and concludes spherical generation over every non-archimedean local field, with no use of (A1)–(A5).

Theorem 3.9.

Let $F$ be a non-archimedean local field with ring of integers $\mathcal O$, uniformizer $t$ and residue field $\F_q$; $K=\GL_{2m}(\mathcal O)$; notation of Lemma 3.1–Theorem 3.6.

For $1\le k\le2m-1$,

$$T_{\omega_k}\ast\varphi_0=\sum_{\nu\in\{0,1,2\}^m,\ |\nu|=k}\ \prod_{i=1}^{n_1(\nu)}(1+q^i)\ \one_{\cO_\nu}.$$

For $\GL_4$: $T_{\omega_2}\ast\varphi_0=\one_{\cO_{(2,0)}}+(1+q)(1+q^2)\one_{\cO_{(1,1)}}$ and $T_{(2,0,0,0)}\ast\varphi_0=q(q+1)\one_{\cO_{(2,0)}}$. Hence

$$\one_{\cO_{(1,1)}}=\frac{1}{(1+q)(1+q^2)}\Big(T_{\omega_2}-\frac{1}{q(q+1)}T_{(2,0,0,0)}\Big)\ast\varphi_0\in\cH_K\varphi_0 .$$

(SG) holds for $\GL_4/\Sp_4$ over every non-archimedean local field.
For $m\ge3$ item 1 does not by itself isolate $\one_{\cO_{(1^k0^{m-k})}}$ when other uncovered labels (e.g. $(2,1,1,0,\dots)$ for $k=4$) lie in the support; (SG) for $\GL_{2m}/\Sp_{2m}$, $m\ge3$, stays as on the classical page: under (A1)–(A5) via Proposition F, $r(p_{\pm k})=(q^{k/2}+q^{-k/2})p_{\pm k}(y)$.

Values. Lemma A(2) gives $(T_\lambda\ast\varphi_0)(x)=\#F_\lambda(\omega_x)(\F_q)$, lattices in exact relative position $-\lambda$. The fibre lemma of Theorem 3.6 is a computation with the representatives $\bigoplus_jt^{\nu_j}J_2$ and holds over every non-archimedean local field ($t=\varpi$); counting $\F_q$-points of $LG(n_1,2n_1)$ gives item 1. The two $\GL_4$ formulas are Computation 3.2, items 1–3, and item 2 follows by subtraction.

Uncovered labels. The scalar shift $\sigma(\omega)=t\omega$ is $G(F)$-equivariant on $X(F)$, so $\sigma_*$ commutes with $\cH_K$; $\sigma_*\one_{\cO_a}=\one_{\cO_{a+(1,1)}}$ and $\sigma_*^2=T_{(1,1,1,1)}\ast(-)$. With $f$ the element of item 2, $f\ast\varphi_0=\one_{\cO_{(1,1)}}$, and $\sigma_*^a\varphi_0=f\ast\sigma_*^{a-1}\varphi_0$; by induction $\one_{\cO_{(a,a)}}=\sigma_*^a\varphi_0\in\cH_K\varphi_0$ for $a\ge0$, and for $a<0$ using $T_{(-1)^4}$.

Covered labels. For $(a_1,a_2)$ with $a_1\ne a_2$ take $f_\mu=T_\lambda$ for a dominant lift $\lambda$ (Corollary C). The support bound $Kt^\lambda X(\mathcal O)\subseteq\bigcup_{\nu\le\varphi(\lambda)}\cO_\nu$ is the double-coset argument of the criterion page, step 2, which is valid over every local field.

Conclusion. The orbit indicators span $C_K$ and dominance has finite lower sets within each Pfaffian component, so Proposition B gives (SG), with no use of Sakellaridis's hypotheses.

Larger rank. For $m\ge3$ the support of $T_{\omega_k}\ast\varphi_0$ contains several uncovered labels, so item 1 alone does not triangularize.

Point-count shadow of Proposition 3.3. $1+q+q^2+q^3=\tr(\IC_{(2,0)})_{x_{(1,1)}}+q^2(q^{-1}+1+q)$: $\IC_{(2,0)}$ has trace 1 on the closed orbit (rational smoothness) and the three $\IC_{(1,1)}$-summands carry $q,q^2,q^3$.

Verified by computation: g2_hnf_lattices.py

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Computation 3.2 · Convolution fibres over $x_{(2,0)}$ and $x_{(1,1)}$ for $\lambda=\omega_2$ and $\lambda=(2,0,0,0)$
Theorem 3.6 · $\GL_{2m}/\Sp_{2m}$: the labels $(1^k0^{m-k})$, $(2^{k-m}1^{2m-k})$ and their shifts lie in $\mathcal M$; $\GL_4/\Sp_4$ is complete

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Proposition 3.10 · Function level implies sheaf level: (SG) over $\F_q(\!(t)\!)$ forces $\mathcal M=\Lambda_S^+$
Summary 3.11 · Verdict for the symplectic family: no disagreement between the levels; $\GL_4/\Sp_4$ settled at both
Audit 7.6 · $\GL_4/\Sp_4$ lattice counts recomputed for $q=2,3$: the values of four agents agree

Original record, as the agent wrote it at 02:48 UTC

Function level: explicit T_omega_k * phi_0 for every local field; (SG) for GL_4/Sp_4 unconditionally

Lemma A(2): $(T_\lambda\ast\varphi_0)(x)=\#F_\lambda(\omega_x)(\mathbb F_q)$. From the fibre lemma of A3-6 (valid over any non-archimedean local field, $t=\varpi$): $$T_{\omega_k}\ast\varphi_0=\sum_{\nu\in\{0,1,2\}^m,\ |\nu|=k}\ \prod_{i=1}^{n_1(\nu)}(1+q^i)\ \mathbf 1_{\mathcal O_\nu}\qquad(1\le k\le 2m-1).$$ For $\mathrm{GL}_4$ (A3-2): $T_{\omega_2}\ast\varphi_0=\mathbf 1_{\mathcal O_{(2,0)}}+(1+q)(1+q^2)\mathbf 1_{\mathcal O_{(1,1)}}$, $T_{(2,0,0,0)}\ast\varphi_0=q(q+1)\mathbf 1_{\mathcal O_{(2,0)}}$. Hence $$\mathbf 1_{\mathcal O_{(1,1)}}=\frac{1}{(1+q)(1+q^2)}\Big(T_{\omega_2}-\frac{1}{q(q+1)}T_{(2,0,0,0)}\Big)\ast\varphi_0\in\mathcal H_K\varphi_0 ,$$ with no use of Sakellaridis's hypotheses. With $\sigma_*\mathbf 1_{\mathcal O_a}=\mathbf 1_{\mathcal O_{a+(1,1)}}$, $\sigma_*$ commuting with $\mathcal H_K$ ($\sigma$ is $G(F)$-equivariant) and $\sigma_*^{2}=T_{(1,1,1,1)}\ast$: $\mathbf 1_{\mathcal O_{(a,a)}}=\sigma_*^{a}\varphi_0\in\mathcal H_K\varphi_0$ for all $a\in\mathbb Z$ by induction ($\sigma_*^{a}\varphi_0=f\ast\sigma_*^{a-1}\varphi_0$ with $f\ast\varphi_0=\mathbf 1_{\mathcal O_{(1,1)}}$; negative $a$ via $T_{(-1)^4}$). Covered labels $(a_1,a_2)$, $a_1\ne a_2$: $f_\mu=T_\lambda$ for a dominant lift (Corollary C; the support bound $Kt^\lambda X(\mathcal O)\subseteq\bigcup_{\nu\le\varphi(\lambda)}\mathcal O_\nu$ is the double-coset argument of criterion.html step 2, valid for every local field). Proposition B then gives $$\boxed{\text{(SG) holds for }\mathrm{GL}_4/\mathrm{Sp}_4\text{ over every non-archimedean local field.}}$$ For $m\ge3$ the formula above does not by itself isolate $\mathbf 1_{\mathcal O_{(1^k0^{m-k})}}$ when other uncovered labels (e.g. $(2,1,1,0,\dots)$ for $k=4$) lie in the support; (SG) for $\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}$, $m\ge3$, stays as on the page: under (A1)–(A5) via Prop. F, $r(p_{\pm k})=(q^{k/2}+q^{-k/2})p_{\pm k}(y)$.

Point-count shadow of A3-3: $1+q+q^2+q^3=\mathrm{tr}(\mathrm{IC}_{(2,0)})_{x_{(1,1)}}+q^{2}(q^{-1}+1+q)$, i.e. $\mathrm{IC}_{(2,0)}$ has trace $1$ on the closed orbit (rational smoothness) and the three $\mathrm{IC}_{(1,1)}$-summands carry $q,q^2,q^3$.

Sources

classical.html Lemma A(2), Prop. B, Cor. C, Prop. F
criterion.html step 2
scratchpad fibres.py

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Theorem 3.9 · Explicit \(T_{\omega_k}\ast\varphi_0\) over every local field; (SG) for \(\GL_4/\Sp_4\) without Sakellaridis's hypotheses

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Function level: explicit T_omega_k * phi_0 for every local field; (SG) for GL_4/Sp_4 unconditionally

Sources