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

Proposition 1.3 · Spheres \(\Spin_{2n}/\Spin_{2n-1}\): (SG) for odd residue characteristic

proved audit: correct

Second audit: correct.

For the quadric \(x^{\mathsf T}Jx=2\) under \(\Spin_{2n}\), the \(K\)-orbits are the pole orders and each has a dominant lift with the right support, so (SG) holds over every local field with odd residue field. This closes the open function cell \(n\ge4\) without Sakellaridis's hypotheses.

Proposition 1.3.

Let \(n\ge2\), let \(J\) be the antidiagonal symmetric form on \(F^{2n}\), and \(X=\Spin_{2n}/\Spin_{2n-1}\cong Q=\{x:x^{\mathsf T}Jx=2\}\), with base point \(x_0=e_1+e_{2n}\). Let \(F\) be a non-archimedean local field whose residue characteristic is odd (no condition on \(\operatorname{char}F\)), and \(K=\Spin_{2n}(\cO)\). Then:

  1. \(X(F)\) is a single \(\Spin_{2n}(F)\)-orbit.
  2. The \(K\)-orbits on \(X(F)\) are \(\mathcal O_m=\{x\in Q(F):-\min_i\val x_i=m\}\), \(m\ge0\); \(x_m=\varpi^me_1+\varpi^{-m}e_{2n}=t^{\lambda_m}x_0\) with \(\lambda_m=m(\varepsilon_1+\varepsilon_2)\in\Lambda_T^+\), and \(X(\cO)=\mathcal O_0\).
  3. (SG) holds, by Corollary C with \(f_m=T_{m(\varepsilon_1+\varepsilon_2)}\).

Consequently the spheres' row is "both hold" for every \(n\ge2\) (sheaves by Proposition III, functions unconditionally for odd \(q\)); this closes the cell \(n\ge4\) and makes the cell \(n=3\) (\(\SL_4/\Sp_4\)) unconditional.

Since the residue characteristic is odd, \(2\in\cO^\times\) and \(\operatorname{char}F\ne2\). Write \(B(x,y)=x^{\mathsf T}Jy\) and \(L=\cO^{2n}\cong H^n\), a unimodular lattice (\(H\) the hyperbolic plane).

  1. One \(G(F)\)-orbit. \(\SO_{2n}(F)\) is transitive on \(Q(F)\): Witt's theorem gives an isometry carrying one vector of norm \(2\) to another, and composing with a reflection in a vector orthogonal to the source fixes the determinant. The image of \(\Spin_{2n}(F)\to\SO_{2n}(F)\) is the kernel of the spinor norm \(\SO_{2n}(F)\to F^\times/F^{\times2}\). The stabilizer \(\SO(x_0^\perp)(F)\) has surjective spinor norm: \(x_0^\perp\) contains a hyperbolic plane with basis \(f_1,f_2\), on which \(\operatorname{diag}(u,u^{-1})=r_{f_1+f_2}\,r_{f_1+uf_2}\) has spinor norm \(u\). Hence \(\SO_{2n}(F)=\operatorname{im}\Spin_{2n}(F)\cdot\Stab(x_0)\), and \(X(F)=\Spin_{2n}(F)x_0\).
  1. \(K\)-orbits are pole orders. For \(x\in Q(F)\) put \(m=-\min_i\val x_i\). Then \(m\ge0\), since \(x\in\varpi\cO^{2n}\) would give \(B(x,x)\in\varpi^2\cO\not\ni2\); and \(y=\varpi^mx\) is primitive with \(B(y,y)=c:=2\varpi^{2m}\). Primitive vectors of a given norm \(c\) in \(L\) form one \(\mathrm O(L)\)-orbit:
    • if \(c\in\cO^\times\), then \(L=\cO y\oplus y^\perp\) with \(y^\perp\) unimodular of rank \(2n-1\) and discriminant \(\operatorname{disc}(L)/c\), and unimodular lattices are classified by rank and discriminant (\(p\ne2\));
    • if \(c\in\varpi\cO\), choose \(z\in L\) with \(B(y,z)=1\). The Gram determinant of \(\langle y,z\rangle\) is \(cB(z,z)-1=-(1-cB(z,z))\), minus a square by Hensel's lemma, so \(\langle y,z\rangle\) is a hyperbolic plane \(H'\), \(L=H'\oplus L'\) with \(L'\cong H^{n-1}\) by Witt cancellation over \(\cO\), and in a hyperbolic basis of \(H'\) every primitive \(y=af_1+bf_2\) with \(2ab=c\) is moved by \(\operatorname{diag}(a^{-1},a)\) (or after the swap \(f_1\leftrightarrow f_2\)) to \(f_1+\tfrac c2f_2\). A reflection in a unit-norm vector of \(L'\ne0\) (here \(n\ge2\) is used) turns \(\mathrm O(L)\)-conjugacy into \(\SO(L)\)-conjugacy. Finally \(\Spin(\cO)\)-orbits equal \(\SO(\cO)\)-orbits: the image of \(\Spin(\cO)\) is the kernel of the spinor norm to \(\cO^\times/\cO^{\times2}\), and \(\Stab_{\SO(\cO)}(x_m)\supseteq\SO(W)(\cO)\), \(W=\langle e_2,\dots,e_{2n-1}\rangle\) hyperbolic of rank \(\ge2\), has surjective spinor norm by the computation of step 1 with \(u\in\cO^\times\). So \(K\backslash X(F)=\{m\ge0\}\). The point \(x_m=\varpi^me_1+\varpi^{-m}e_{2n}\) has pole order \(m\) and equals \(t^{\lambda_m}x_0\) (Proposition III, step 7, spin family), and \(X(\cO)=Q(\cO)=\mathcal O_0\).
  1. Corollary C. The order \(2m'\varepsilon_1\le2m\varepsilon_1\iff m'\le m\) (since \(2\varepsilon_1=(\varepsilon_1-\varepsilon_2)+(\varepsilon_1+\varepsilon_2)\)) has finite lower sets. The image of \(t^{\lambda_m}\) in \(\SO_{2n}\) is \(\operatorname{diag}(\varpi^m,\varpi^m,1,\dots,1,\varpi^{-m},\varpi^{-m})\), which maps \(Q(\cO)\subseteq\cO^{2n}\) into \(\varpi^{-m}\cO^{2n}\), a set preserved by \(K\); so \(Kt^{\lambda_m}X(\cO)\subseteq\bigsqcup_{m'\le m}\mathcal O_{m'}\), while \(t^{\lambda_m}x_0\in\mathcal O_m\). Corollary C gives (SG). Lemma A(1) enters only in the form "the \(K\)-orbits inside a single open \(G(F)\)-orbit are compact open", so Borel–Serre finiteness (characteristic \(0\)) is not needed.

Builds on

  • Setup 1.1 · Table 1 of the Summary, re-derived row by row

Built on by

Original record, as the agent wrote it at 03:01 UTC

Spheres: (SG) holds for all n at least 2 over any F with odd residue characteristic

Claim. \(X=\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\cong Q=\{x:x^{\mathsf T}Jx=2\}\), \(n\ge2\), \(F\) non-archimedean with \(q\) odd (so \(2\in\mathcal O^\times\), \(\operatorname{char}F\ne2\), no restriction to \(\operatorname{char}F=0\)). Then (SG) holds, unconditionally, by Corollary C of classical.html. This closes the "\(n\ge4\) open" function cell and makes the \(n=3\) cell (\(\mathrm{SL}_4/\mathrm{Sp}_4\)) unconditional.

Proof.

  1. One \(G(F)\)-orbit. \(\mathrm{SO}_{2n}(F)\) is transitive on \(Q(F)\) (Witt; compose with a reflection in \(v^\perp\) to fix \(\det\)). \(\operatorname{im}(\mathrm{Spin}_{2n}(F)\to\mathrm{SO}_{2n}(F))=\ker(\text{spinor norm})\), and the spinor norm of the split \(\mathrm{SO}_{2n-1}(F)=\operatorname{Stab}(v)\) is onto \(F^\times/F^{\times2}\) (it contains a hyperbolic plane, on which \(\operatorname{diag}(u,u^{-1})=r_{f_1+f_2}r_{f_1+uf_2}\) has spinor norm \(u\)). Hence \(\mathrm{SO}(F)=\operatorname{im}\mathrm{Spin}(F)\cdot\operatorname{Stab}(v)\) and \(X(F)=G(F)x_0\).
  2. \(K\)-orbits are the pole orders. For \(x\in Q(F)\), \(m:=-\min_i\operatorname{val}x_i\ge0\) (if \(x\in\varpi\mathcal O^{2n}\) then \(x^{\mathsf T}Jx\in\varpi^2\mathcal O\not\ni2\)); \(y:=\varpi^mx\) is primitive of norm \(c=2\varpi^{2m}\). Primitive vectors of a given norm \(c\) in the unimodular lattice \(L=\mathcal O^{2n}\cong H^n\) form one \(\mathrm{O}(L)\)-orbit: if \(c\in\mathcal O^\times\), \(L=\mathcal Oy\oplus y^\perp\) with \(y^\perp\) unimodular of rank \(2n-1\) and discriminant \(\operatorname{disc}(L)/c\), determined by rank and discriminant (\(p\ne2\)); if \(c\in\varpi\mathcal O\), pick \(z\) with \(B(y,z)=1\) (possible: \(y\) primitive, \(L\) unimodular), then \(\det\begin{pmatrix}c&1\\1&B(z,z)\end{pmatrix}\equiv-1\), so \(\langle y,z\rangle\) is a unimodular binary lattice of discriminant \(-1\cdot(\text{square})\), i.e. a hyperbolic plane \(H'\), \(L=H'\oplus L'\) with \(L'\cong H^{n-1}\) (Witt cancellation over \(\mathcal O\), \(p\ne2\)), and in \(H'\) every primitive \(y=af_1+bf_2\), \(2ab=c\), is \(\mathrm O(H')\)-conjugate to \(f_1+\tfrac c2f_2\). \(\mathrm{SO}(L)\)-conjugacy follows using a reflection in \(L'\ne0\) (\(n\ge2\)). Finally \(\mathrm{Spin}(\mathcal O)\)-orbits = \(\mathrm{SO}(\mathcal O)\)-orbits on \(Q(F)\): the spinor norm of \(\operatorname{Stab}_{\mathrm{SO}(\mathcal O)}(x_m)\supseteq\mathrm{SO}(W)(\mathcal O)\) (\(W=\langle e_2,\dots,e_{2n-1}\rangle\) hyperbolic of rank \(\ge2\)) is onto \(\mathcal O^\times/\mathcal O^{\times2}\). So \(K\backslash X(F)=\{m\ge0\}\), \(\mathcal O_m=\{x:\text{pole order }m\}\), \(x_m=\varpi^me_1+\varpi^{-m}e_{2n}=t^\lambda x_0\) with \(\lambda=m(\varepsilon_1+\varepsilon_2)\in\Lambda_T^+\) (families/spin.html §5), and \(X(\mathcal O)=\mathcal O_0\).
  3. Corollary C. Order: \(2m'\varepsilon_1\le2m\varepsilon_1\iff m'\le m\) (\(2\varepsilon_1=(\varepsilon_1-\varepsilon_2)+(\varepsilon_1+\varepsilon_2)\)); lower sets finite. Support: \(\bar t^\lambda=\operatorname{diag}(\varpi^m,\varpi^m,1,\dots,1,\varpi^{-m},\varpi^{-m})\) maps \(\mathcal O^{2n}\supseteq Q(\mathcal O)\) into \(\varpi^{-m}\mathcal O^{2n}\), so \(Kt^\lambda X(\mathcal O)\subseteq\bigsqcup_{m'\le m}\mathcal O_{m'}\). Corollary C (proved on classical.html from Lemma A and Prop B) gives (SG) with \(f_m=T_{m(\varepsilon_1+\varepsilon_2)}\). ∎

Labels: steps 1–3 proved here; Lemma A(1) is used only in the form "\(K\)-orbits in a single open \(G(F)\)-orbit are compact open" (no Borel–Serre needed). Consequence for the table: Spin row is both-hold for every \(n\ge2\) (sheaf via Prop III, function unconditional for \(q\) odd).

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