§3 · Agent 3 · Symplectic family: fibres · 2026-10-17T02:40:26Z

Computation 3.2 · Convolution fibres over \(x_{(2,0)}\) and \(x_{(1,1)}\) for \(\lambda=\omega_2\) and \(\lambda=(2,0,0,0)\)

proved audit: minor defect

Minor defect (second audit). All fibres and counts are correct: recomputed at q = 2, 3 through an independent parametrization (dual Hermite normal forms of sublattices), with cell sizes 35/130 and 120/1080. Wording defects only: (i) the kernel of bar-omega_0 in item 2 is written <f_1, f_2> with f undefined (it is the span of e_1, e_2); (ii) the identification m^{-1}(x) = F_lambda(omega_x) holds for minuscule lambda; for lambda = (2,0,0,0) the fibre over Gr^{<= lambda} also contains the cell of (1,1,0,0), whereas item 3 counts exact relative position, which is what Lemma A(2) needs. Item 3 is used only at the function level (Theorem 3.9), so nothing is affected.

For \(\GL_4/\Sp_4\), computes the lattices on which a given form is perfect: these are the fibres of the convolution map and, counted over \(\F_q\), the values of Hecke translates of the basic function. Over the closed orbit \((1,1)\) the fibre for \(\omega_2\) is the Lagrangian Grassmannian \(LG(2,4)\), over the open orbit \((2,0)\) it is a point; for \((2,0,0,0)\) there are \(q(q+1)\) lattices over \((2,0)\) and none over \((1,1)\).

Computation 3.2.

Notation of Lemma 3.1 with \(m=2\). For \(\lambda\in\Lambda_T^+\) and a form \(\omega\) put $$F_\lambda(\omega):=\{\Lambda'\supseteq\Lambda_0\ \text{in relative position }-\lambda:\ \omega\ \text{perfect on }\Lambda'\}.$$ For \(g\in\Gr^{\le\lambda}\), \(g^{-1}x\in L^+X\) if and only if \(\omega_x\) is perfect on \(\Lambda'=g^{-T}\Lambda_0\); the node identifies the fibre \(m^{-1}(x)=\{gL^+G\in\Gr^{\le\lambda}:g^{-1}x\in L^+X\}\) with \(F_\lambda(\omega_x)\).

  1. \(\lambda=\omega_2\), \(x=x_{(1,1)}\) (\(\omega=t\omega_0\)): \(F=LG(2,4)\cong Q^3\), smooth irreducible of dimension 3, \(H^*(F)=k\oplus k[-2]\oplus k[-4]\oplus k[-6]\), \(\#F(\F_q)=(1+q)(1+q^2)\).
  2. \(\lambda=\omega_2\), \(x=x_{(2,0)}\) (\(\omega=t^2e_{12}^*+e_{34}^*\)): \(F\) is one point, so \(m\) is birational onto \(\overline{\cO_{(2,0)}}\).
  3. \(\lambda=(2,0,0,0)\): \(\#F(\omega_{(2,0)})=q(q+1)\) and \(F(t\omega_0)=\varnothing\); hence \(\supp(T_{(2,0,0,0)}\ast\varphi_0)=\cO_{(2,0)}\).
  4. \(\supp(V_{\omega_2}\ast\IC_0)=\overline{\cO_{(2,0)}}\).
  1. Reduction to lattices. \(\omega_{g^{-1}x}=g^{-1}\omega_xg^{-T}\), i.e. \(\omega_{g^{-1}x}(u,v)=\omega_x(g^{-T}u,g^{-T}v)\); for \(g\in Kt^\lambda K\) the lattice \(g^{-T}\Lambda_0\) is in relative position \(-\lambda\) to \(\Lambda_0\).
  1. The case \(\lambda=\omega_2=(1,1,0,0)\). Here \(\Lambda'=\Lambda_0+t^{-1}\widetilde W\) with \(W\in\Gr(2,t^{-1}\Lambda_0/\Lambda_0)\). The quotient \(\Lambda'/\Lambda_0\) has length 2 and \(\val\Pf(\omega|_{\Lambda_0})=2\) on the whole support, so the discriminant of \(\omega\) on \(\Lambda'\) is a unit as soon as \(\omega\) is integral on \(\Lambda'\): perfect is equivalent to integral. Writing \(\omega=\sum_rt^r\omega_r\), integrality on \(\Lambda_0+t^{-1}\widetilde W\) reads
$$\bar\omega_0(W,k^4)=0\qquad\text{and}\qquad\bar\omega_1|_W=0 .$$
  1. Item 1. For \(\omega=t\omega_0\): \(\bar\omega_0=0\) and \(\bar\omega_1=\omega_0\) is nondegenerate, so \(F\) is the set of Lagrangian planes. Its cohomology and point count are those of a smooth quadric threefold.
  1. Item 2. For \(\omega=t^2e_{12}^*+e_{34}^*\): \(W\) must lie in \(\ker\bar\omega_0\), the span of the images of \(e_1,e_2\) (written \(\langle\bar f_1,\bar f_2\rangle\) in the original), so \(W\) is that plane, and \(\bar\omega_1=0\) on it. One point.
  1. Item 3. \(\Lambda'=\Lambda_0+\mathcal O\,t^{-2}v\) with \(v\) primitive modulo \(t^2\) up to units, \(\Lambda'/\Lambda_0\cong\mathcal O/t^2\). Integrality holds if and only if \(v\in t^2\Lambda_0^\vee\), where \(\Lambda_0^\vee\) is the \(\omega\)-dual lattice. For \(\omega_{(2,0)}\), \(\Lambda_0^\vee=t^{-2}\langle e_1,e_2\rangle\oplus\langle e_3,e_4\rangle\), so \(v\) is a primitive vector of \(\langle e_1,e_2\rangle\otimes\mathcal O/t^2\) up to units: \(|\mathbb P^1(\mathcal O/t^2)|=q(q+1)\) choices. For \(t\omega_0\), \(\Lambda_0^\vee=t^{-1}\Lambda_0\) and \(v\in t\Lambda_0\) contradicts primitivity. As in item 4, a form perfect on a lattice of colength 2 containing \(\Lambda_0\) is integral on \(\Lambda_0\) with \(\val\Pf=2\), so the support lies in \(\cO_{(2,0)}\cup\cO_{(1,1)}\), and therefore equals \(\cO_{(2,0)}\).
  1. Item 4. If \(\omega\) is perfect on some \(\Lambda'\supseteq\Lambda_0\) of colength 2, then \(\omega\) is integral on \(\Lambda_0\) with \(\val\Pf=2\), so it has type \((2,0)\) or \((1,1)\); both occur by items 1 and 2.

Brute-force counts: 15 and 40 over \((1,1)\) at \(q=2,3\); 1 and 1 over \((2,0)\); 6 and 12 for item 3; 0 over \(t\omega_0\).

Verified by computation: g2_hnf_lattices.py

Builds on

  • Lemma 3.1 · The loop space of \(\GL_{2m}/\Sp_{2m}\): orbits, closure order, scalar shift and codimensions

Built on by

  • Proposition 3.3 · Decomposition of \(V_{\omega_2}\ast\IC_0\) for \(\GL_4/\Sp_4\): \(\IC_{(1,1)}\) is a retract with multiplicity \(k[2]\oplus k\oplus k[-2]\)
  • 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
  • Theorem 3.9 · Explicit \(T_{\omega_k}\ast\varphi_0\) over every local field; (SG) for \(\GL_4/\Sp_4\) without Sakellaridis's hypotheses
  • 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 02:40 UTC

Fibres of the convolution over x_(2,0) and x_(1,1) for lambda = omega_2 and (2,0,0,0)

Convention: \(m^{-1}(x)=\{gL^+G\in\mathrm{Gr}^{\le\lambda}:g^{-1}x\in L^+X\}\); \(g^{-1}x\in L^+X\iff\omega_x\) perfect on \(\Lambda':=g^{-T}\Lambda_0\). So $$m^{-1}(x)\cong F_\lambda(\omega_x):=\{\Lambda'\supseteq\Lambda_0\text{ in relative position }-\lambda:\ \omega_x|_{\Lambda'}\text{ perfect}\}.$$ For \(\lambda=\omega_2=(1,1,0,0)\): \(\Lambda'=\Lambda_0+W\), \(W\in\mathrm{Gr}(2,t^{-1}\Lambda_0/\Lambda_0)\). Since \([\Lambda':\Lambda_0]=q^2\) and \(v(\mathrm{Pf}\,\omega|_{\Lambda_0})=2\) on the whole support, perfect \(\iff\) integral, and integrality of \(\omega=\sum_r t^r\omega_r\) on \(\Lambda_0+W\) reads $$\bar\omega_0(W,k^4)=0\quad\text{and}\quad\bar\omega_1|_{W}=0 .$$

  1. \(x=x_{(1,1)}\), \(\omega=t\omega_0\): \(\bar\omega_0=0\), \(\bar\omega_1=\omega_0\) nondegenerate: \(F=\{W\text{ Lagrangian}\}=LG(2,4)\cong Q^3\), smooth irreducible, \(\dim 3\), \(H^*=k[0]\oplus k[-2]\oplus k[-4]\oplus k[-6]\), \(\#F(\mathbb F_q)=(1+q)(1+q^2)\).
  2. \(x=x_{(2,0)}\), \(\omega=t^2e_{12}^*+e_{34}^*\): \(W\subseteq\ker\bar\omega_0=\langle\bar f_1,\bar f_2\rangle\) forces \(W=\langle\bar f_1,\bar f_2\rangle\), and \(\bar\omega_1=0\): \(F=\) one point. So \(m\) is birational onto \(\mathcal O_{(2,0)}\).
  3. \(\lambda=(2,0,0,0)\): \(\Lambda'=\Lambda_0+\mathcal O t^{-2}v\), \(\Lambda'/\Lambda_0\cong\mathcal O/t^2\). Integrality \(\iff v\in t^2\Lambda_0^{\vee}\). For \(\omega_{(2,0)}\): \(\Lambda_0^\vee=t^{-2}\langle e_1,e_2\rangle\oplus\langle e_3,e_4\rangle\), so \(v\in\langle e_1,e_2\rangle\otimes\mathcal O/t^2\) primitive, modulo units: \(\#F=|\mathbb P^1(\mathcal O/t^2)|=q(q+1)\). For \(t\omega_0\): \(\Lambda_0^\vee=t^{-1}\Lambda_0\), \(v\in t\Lambda_0\) contradicts primitivity: \(F=\varnothing\). Hence \(\operatorname{supp}(T_{(2,0,0,0)}\ast\varphi_0)=\mathcal O_{(2,0)}\) only.
  4. Support of \(V_{\omega_2}\ast\mathrm{IC}_0\): \(\omega\) perfect on some \(\Lambda'\supseteq\Lambda_0\) of colength 2 \(\Rightarrow\) integral with \(v(\mathrm{Pf})=2\) \(\Rightarrow\) type \((2,0)\) or \((1,1)\), both attained: \(\operatorname{supp}=\overline{\mathcal O_{(2,0)}}\).

Brute-force check (python3 -I, scratchpad/A3/fibres.py): counts \(15,40\) for \((1,1)\) at \(q=2,3\); \(1,1\) for \((2,0)\); \(6,12\) for item 3 at \(q=2,3\); \(0\) over \(t\omega_0\). All as predicted.

Sources