Proposition V as a paper · statement and complete proof (PDF) →

Definitions

• Function level and spherical generation

  \(F\) is a non-archimedean local field of either characteristic, with ring of integers \(\mathcal O\), uniformizer \(\varpi\), valuation \(\operatorname{val}\) and finite residue field \(k_F\) of cardinality \(q\). \(G\) is a split connected reductive group scheme over \(\mathcal O\) with split maximal torus \(T\), and \(t^\lambda=\lambda(\varpi)\) for \(\lambda\in\Lambda_T^+\). \(X\) is a smooth affine \(\mathcal O\)-scheme with a \(G\)-action and a base point \(x_0\in X(\mathcal O)\). With \(\operatorname{vol}K=1\):

  $$K=G(\mathcal O),\qquad\mathcal H_K=C_c(K\backslash G(F)/K),\qquad T_\lambda=\mathbf 1_{Kt^\lambda K},$$ $$(f\ast_K\varphi)(x)=\sum_{g\in G(F)/K}f(g)\,\varphi(g^{-1}x),\qquad C_K=C_c^\infty(X(F))^K,\qquad\varphi_0=\mathbf 1_{X(\mathcal O)}.$$

  Spherical generation is the equality

  $$\text{(SG)}\qquad\mathcal H_K\ast_K\varphi_0=C_K .$$

  Example. For \(X=\mathrm{GL}_2/T\) the \(K\)-orbits are the distances \(d\ge0\) from an apartment of the tree, \(T_{(1,0)}\) is the adjacency operator, and \(T_{(1,0)}^d\ast_K\varphi_0\) has leading orbit \(d\): (SG) by triangularity (classical page, §6).

• Spheres: the quadric model and its shells

  For \(n\ge2\) let \(L=\mathcal O^{2n}\) with basis \(e_1,\dots,e_{2n}\) and the hyperbolic quadratic form \(N\); its polar form \(B\) is perfect in every characteristic:

  $$N(x)=\sum_{i=1}^nx_ix_{2n+1-i},\qquad B(x,y)=N(x+y)-N(x)-N(y),\qquad B(e_i,e_j)=\delta_{i+j,2n+1}.$$

  \(G=\mathrm{Spin}_{2n}\) acts on the quadric \(Q=\{N=1\}\), smooth over \(\mathbb Z\), through \(\mathrm{SO}(L)\), the kernel of the Dickson invariant \(\mathrm O(L)\to\mathbb Z/2\). The model of \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\) is \(X=Q\) with \(x_0=e_1+e_{2n}\). The shells are the sets of fixed pole order:

  $$O_m=\{x\in Q(F):\min_i\operatorname{val}x_i=-m\},\qquad x_m=\varpi^me_1+\varpi^{-m}e_{2n}\in O_m\qquad(m\ge0).$$

  Example. For \(n=2\), \(\mathrm{Spin}_4/\mathrm{Spin}_3\cong(\mathrm{SL}_2\times\mathrm{SL}_2)/\mathrm{SL}_2\) is the group case and \(Q\) is \(\{x_1x_4+x_2x_3=1\}\cong\mathrm{SL}_2\); the shell \(O_m\) is the double coset of \(\operatorname{diag}(\varpi^m,\varpi^{-m})\).

• \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\): pair model, labels, \(J\)-model

  Here \(1\le p\le q\) are block sizes with \(n=p+q\) (not the residue cardinality). The pair model \(X\) has \(R\)-points the idempotents \(\pi\in M_n(R)\) of rank \(q\), with \(\mathrm{GL}_n\) acting by conjugation. Over \(F\), \(\pi\) is the pair \((U,V)=(\operatorname{im}\pi,\ker\pi)\), \(F^n=U\oplus V\), \(\dim V=p\), and \(X(\mathcal O)\) consists of the pairs that split \(L_0=\mathcal O^n\):

  $$X(\mathcal O)=\{(U,V):L_0=(L_0\cap U)\oplus(L_0\cap V)\},\qquad U_0=\langle e_i+e_{n+1-i}\ (i\le p),\ e_j\ (p\lt j\le q)\rangle,\quad V_0=\langle e_{n+1-i}\ (i\le p)\rangle .$$

  A finite \(\mathcal O\)-module \(E\simeq\bigoplus_i\mathcal O/\varpi^{b_i}\) has type \(b_1\ge b_2\ge\cdots\), read off from

  $$\dim_{k_F}\varpi^{j-1}E/\varpi^jE=\#\{i:b_i\ge j\}.\qquad(\ast)$$

  The label of a pair relative to a lattice \(L\), with \(\pi_V\) the projection onto \(V\) along \(U\), has at most \(p\) parts and is \(\mathrm{GL}_n(F)\)-equivariant in \((U,V;L)\):

  $$\ell(U,V;L)=\operatorname{type}\big(\pi_V(L)/(L\cap V)\big),\qquad\ell=\ell(\,\cdot\,;L_0),\qquad O_a=\{\ell=a\}.$$

  \(J\)-model. The pages realize \(X=\mathrm{GL}_n/\mathrm{GL}_n^\theta\), \(\theta=\operatorname{Ad}J\), with \(J\) the permutation matrix of \(i\leftrightarrow n+1-i\) (\(i\le p\)): the involutions conjugate to \(J\). If \(2\in\mathcal O^\times\), \(\varepsilon\mapsto(1+\varepsilon)/2\) identifies involutions over any \(\mathcal O\)-algebra with idempotents.

  Example. For \(p=q=1\), \(X(F)\) is the set of ordered pairs of distinct lines in \(F^2\); \(\ell(U,V)=a\) when \(L_0\) meets \(U\) and \(V\) in lattices whose sum has index \(q^a\).

• Symmetric-space data over \(\mathbb F_q(\!(t)\!)\) (part (c))

  \(q\) odd; \(G\), \(\theta\) and \(X=G/G^\theta\) are defined over \(\mathbb F_q\), with \(G^\theta\) connected and \(T\supseteq S\) split; \(S\) is a maximal \(\theta\)-split torus. \(F=\mathbb F_q(\!(t)\!)\), so \(K=L^+G(\mathbb F_q)\), \(X(F)=LX(\mathbb F_q)\), \(X(\mathcal O)=L^+X(\mathbb F_q)\). The geometric orbits over \(\overline{\mathbb F}_q\) and the leading-term map are

  $$LX=\bigsqcup_{\mu\in\Lambda_S^+}\mathcal O_\mu,\qquad\mathcal O_\mu=L^+G\cdot x_\mu,\qquad\tau(x_\mu)=t^\mu,\qquad\varphi(\lambda)=\lambda-\theta(\lambda),$$

  with \(\tau(gG^\theta)=g\theta(g)^{-1}\) the Cartan embedding, which is injective, and \(\Lambda_S^+\) ordered by dominance.

• Sakellaridis's restriction map (part (d))

  \(A^*=\operatorname{Hom}(\Lambda_T,\mathbb C^\times)\), \(A_X^*=\operatorname{Hom}(\Lambda_S,\mathbb C^\times)\hookrightarrow A^*\) via \(y\mapsto y\circ\varphi\), \(L=Z_G(S)\) with half-sum of positive roots \(\rho_L\), \(\delta(X)^{1/2}=q^{\rho_L}\), and \(W_X\) the little Weyl group. For \(f\in\mathbb C[A^*]^W\):

  $$r(f)(y)=f(q^{\rho_L}y),\qquad r:\mathbb C[A^*]^W\longrightarrow\mathbb C[\Lambda_S]^{W_X}.$$

  The sign of the shift does not affect whether \(r\) is onto (A6-1).

Inputs of parts (c) and (d)

Parts (a) and (b) use none of these.

• D0 · Orbit geometry over \(\overline{\mathbb F}_q\).

  Chen–Yi's results over \(\overline{\mathbb F}_q\), \(p\ne2\): the orbits \(\mathcal O_\mu\), \(\mu\in\Lambda_S^+\) (Prop. 17), the closure order (Thm. 18), the support bound and the stabilizer lemma: $$L^+G\,t^\lambda L^+X\subseteq\bigsqcup_{\nu\le\varphi(\lambda)}\mathcal O_\nu,\qquad\pi_0\operatorname{Stab}_{L^+G}(x_\mu)\simeq\pi_0Z_{G^\theta}(\mu),\qquad Z_{G^\theta}(\mu)=Z_{G^\theta}(\mu)^\circ Z_{G^\theta}(S).$$
• D3 · Lang's theorem for the arc group.

  Lang's theorem holds for the pro-algebraic groups \(L^+G=\varprojlim_kG(\mathbb F_q[t]/t^k)\) and \(\operatorname{Stab}_{L^+G}(x_\mu)\): $$H^1(\mathbb F_q,L^+G)=1,\qquad H^1\big(\mathbb F_q,\operatorname{Stab}_{L^+G}(x_\mu)\big)=H^1\big(\mathbb F_q,\pi_0Z_{G^\theta}(\mu)\big).$$
• C1 · Connected centralizer.

  Condition (1) of Proposition III: $$Z_{G^\theta}(S)\ \text{is connected}.$$
• C2 · Dominant lifts.

  Condition (2) of Proposition III: $$\varphi(\Lambda_T^+)=\Lambda_S^+ .$$
• (A) · Sakellaridis's hypotheses.

  Part (d) assumes A1–A5, stated on the classical page; A1 forces \(\operatorname{char}F=0\). For the spheres A1 and A5 hold, while A2–A4 are not verified.

Proof

Step 1 states the triangular criterion with its orbit hypothesis made explicit. Steps 2–4 verify it for the spheres, steps 5–7 for the pair model, steps 8–9 over \(\mathbb F_q(\!(t)\!)\) under D0 and D3, and step 10 compares with Theorem E.

1. Method Triangular criterion with explicit orbits

   If \(X(F)=\bigsqcup_{\mu\in I}O_\mu\) is a partition into compact open \(K\)-orbits with a partial order of finite lower sets, and each \(\mu\) has a dominant \(\lambda_\mu\) with

   $$t^{\lambda_\mu}x_0\in O_\mu,\qquad Kt^{\lambda_\mu}X(\mathcal O)\subseteq\bigcup_{\nu\le\mu}O_\nu,$$

   then (SG) holds with \(f_\mu=T_{\lambda_\mu}\) (Corollary C). No hypothesis on \(F\) enters.

   Proof

   • Counting (Lemma A(2)). \(Kt^\lambda K/K\) is finite, and the convolution counts cosets: $$(T_\lambda\ast_K\varphi_0)(x)=\#\{gK\subseteq Kt^\lambda K:g^{-1}x\in X(\mathcal O)\},\qquad\operatorname{supp}(T_\lambda\ast_K\varphi_0)=Kt^\lambda X(\mathcal O),\qquad(T_\lambda\ast_K\varphi_0)(t^\lambda x_0)\ge1,$$ the last because the coset \(t^\lambda K\) contributes \(1\).
   • Spanning. A \(K\)-invariant \(\varphi\in C_c^\infty(X(F))\) is constant on each orbit, and its compact support meets finitely many of the open orbits; so the indicators \(\mathbf 1_{O_\mu}\) span \(C_K\).
   • Induction (Proposition B). By counting, \(T_{\lambda_\mu}\ast_K\varphi_0=c_\mu\mathbf 1_{O_\mu}+\sum_{\nu\lt\mu}c_{\mu\nu}\mathbf 1_{O_\nu}\) with \(c_\mu\ge1\). Induction on \(\#\{\nu\le\mu\}\) puts every \(\mathbf 1_{O_\mu}\) in \(\mathcal H_K\ast_K\varphi_0\).
   • What changed. The classical page's Lemma A(1) gets compact open orbits from Borel–Serre finiteness, which needs \(\operatorname{char}F=0\); here the orbit partition is a hypothesis, verified directly in steps 3, 5 and 8.
2. Method Spheres · a hyperbolic plane through each point

   For \(x\in O_m\), \(y=\varpi^mx\) is primitive with \(N(y)=\varpi^{2m}\), and Hensel's lemma gives \(z\) with

   $$N(z)=0,\qquad B(y,z)=1,\qquad P=\mathcal Oy\oplus\mathcal Oz\xrightarrow{\ \sim\ }P_0=\mathcal Oe_1\oplus\mathcal Oe_{2n},\quad y\mapsto y_m=\varpi^{2m}e_1+e_{2n},\ z\mapsto e_1 .$$
   Proof

   • Start. \(B\) is perfect and \(y\) primitive, so \(B(y,\cdot):L\to\mathcal O\) is onto; take \(z_1\) with \(B(y,z_1)=1\). Then \(N(z_1+ay)=\varpi^{2m}a^2+a+N(z_1)\).
   • \(m\ge1\). Modulo \(\varpi\) the polynomial is \(a+\bar N(z_1)\), with a simple root; Hensel's lemma (\(\mathcal O\) complete, hence henselian; non-monic polynomials allowed) gives \(a\) with \(z_2=z_1+ay\) isotropic.
   • \(m=0\). Witt's extension theorem over \(k_F\), valid in every characteristic for a nondegenerate polar form, gives \(\eta\in\mathrm O(\bar L)\) with \(\eta(\bar e_1+\bar e_{2n})=\bar y\); then \(\bar s=\eta\bar e_2\), \(\bar t=\eta\bar e_{2n-1}\) is a hyperbolic pair in \(\bar y^\perp\). Lift it to \(s,t\in y^\perp\). For \(c,d\in\mathcal O\), $$N(z_1+cs+dt)\equiv\big(\bar c+\bar B(z_1,t)\big)\big(\bar d+\bar B(z_1,s)\big)+\bar N(z_1)-\bar B(z_1,s)\bar B(z_1,t)\pmod\varpi$$ takes every residue, so we may assume \(N(z_1)\in\varpi\mathcal O\). Then \(a^2+a+N(z_1)\) has the simple root \(0\) modulo \(\varpi\), and Hensel's lemma gives \(a\in\varpi\mathcal O\).
   • Normalize. \(B(y,z_2)=1+2a\varpi^{2m}=:u\in\mathcal O^\times\); put \(z=u^{-1}z_2\). The Gram matrix of \(P\) is \(\begin{pmatrix}2\varpi^{2m}&1\\1&0\end{pmatrix}\), of unit determinant, so \(L=P\oplus P^\perp\), and \(\alpha y+\beta z\mapsto\alpha y_m+\beta e_1\) preserves \(N\).
   • Correction. A5-2 used \(B(y,z_2)=1\) and a residue for \(m=0\) with no simple root; its audit fixed both. For an arbitrary pair \(\bar s,\bar t\) with \(\bar B(\bar s,\bar t)=1\) the quadratic part can be degenerate when \(q\) is odd; the hyperbolic pair from Witt's theorem removes that case.
3. Result Spheres · the \(K\)-orbits are the pole-order shells

   For every \(F\) and \(n\ge2\), the shells partition \(Q(F)\) into compact open sets, and each is one \(K\)-orbit:

   $$Q(F)=\bigsqcup_{m\ge0}O_m,\qquad O_0=Q(\mathcal O),\qquad O_m=Kx_m .$$
   Proof

   • Shells. \(x\in\varpi L\) would give \(N(x)\in\varpi^2\mathcal O\), so \(m\ge0\). \(O_m=Q(F)\cap(\varpi^{-m}L\setminus\varpi^{1-m}L)\) is open, closed and bounded in the closed set \(Q(F)\), and \(\mathrm{SO}(L)(\mathcal O)\) preserves each \(\varpi^kL\).
   • Complement. Witt's theorem over \(k_F\) extends \(\bar P\simeq\bar P_0\) to \(\bar L\), so \(\overline{P^\perp}\simeq\bar M_0\simeq H^{n-1}\), \(M_0=\bigoplus_{i=2}^{2n-1}\mathcal Oe_i\). Hyperbolic lifting (lift a hyperbolic pair, make it isotropic by Hensel, split it off, induct) gives \(P^\perp\simeq M_0\) over \(\mathcal O\), hence \(g\in\mathrm O(L)\) with \(g(y)=y_m\).
   • Orientation. For a hyperbolic pair \(f,f'\) of \(P^\perp\) and \(w=f+f'\), the reflection \(r_w\) fixes \(P\) and is \(-1\) times the swap \(f\leftrightarrow f'\), which has Dickson invariant \(1\). So \(g\) or \(g\circ r_w\) lies in \(\mathrm{SO}(L)(\mathcal O)\) and maps \(x\) to \(\varpi^{-m}y_m=x_m\).
   • Spinor norm. The image of \(K\) in \(\mathrm{SO}(L)(\mathcal O)\) is the kernel of the spinor norm to \(H^1_{\mathrm{fppf}}(\mathcal O,\mu_2)=\mathcal O^\times/\mathcal O^{\times2}\). For \(u\in\mathcal O^\times\) the element \(r_{e_2+e_{2n-1}}r_{e_2+ue_{2n-1}}\) acts as \(\operatorname{diag}(u,u^{-1})\) on \(\langle e_2,e_{2n-1}\rangle\), fixes \(x_m\), and has spinor norm \([u]\); so \(\mathrm{SO}(L)(\mathcal O)=\operatorname{im}(K)\cdot\operatorname{Stab}(x_m)\) and \(O_m=Kx_m\).
   • Earlier proof. A1-3 proves the same for residue characteristic \(\ne2\) from the classification of unimodular lattices; A7-20 confirms that residue characteristic \(2\) is covered.
4. Result Spheres · (SG) by dominant lifts and the pole-order bound

   The lift \(\lambda_m=(m,m,0,\dots,0)\) reaches \(O_m\), and every translate stays in the lower shells:

   $$t^{\lambda_m}x_0=x_m,\qquad Kt^{\lambda_m}X(\mathcal O)\subseteq Q(F)\cap\varpi^{-m}L=\bigsqcup_{m'\le m}O_{m'} .$$
   Proof

   • Labels. \(\Lambda_S^+=\{2m\varepsilon_1\}\) corresponds to \(m\ge0\), ordered as integers, with finite lower sets.
   • Lift. \(\Lambda_T^+=\{x\in\mathbb Z^n:\sum x_i\text{ even},\ x_1\ge\dots\ge x_{n-1}\ge|x_n|\}\) contains \(\lambda_m\); its image in \(\mathrm{SO}(L)\) is \(\operatorname{diag}(\varpi^m,\varpi^m,1,\dots,1,\varpi^{-m},\varpi^{-m})\), which sends \(x_0\) to \(x_m\).
   • Bound. \(t^{\lambda_m}L\subseteq\varpi^{-m}L\) and \(K\) preserves \(\varpi^{-m}L\); \(X(\mathcal O)=Q(\mathcal O)\subseteq L\).
   • Conclusion. Step 3 and step 1 with \(f_m=T_{\lambda_m}\) give (SG) for every \(n\ge2\) and every \(F\). For \(n=2\) this is the group case \(\mathrm{SL}_2\); for \(n=3\), \(\mathrm{SL}_4/\mathrm{Sp}_4\).
5. Result Pair model · a lattice normal form classifies the orbits

   For \((U,V)\) with label \(a\) and \(h=\#\{i:a_i\gt0\}\) there are bases \(u_i\) of \(U\) and \(v_j\) of \(V\) with

   $$L_0=\bigoplus_{i\le h}\big(\varpi^{a_i}\mathcal Ou_i\oplus\mathcal O(u_i+v_i)\big)\oplus\bigoplus_{h\lt i\le q}\mathcal Ou_i\oplus\bigoplus_{h\lt j\le p}\mathcal Ov_j ,$$

   so \(\ell\) is a bijection \(K\backslash X(F)\to\Lambda_S^+\), every orbit is compact open, \(\ell(t^{\lambda_a}\pi_0)=a\) for \(\lambda_a=(a_1,\dots,a_p,0,\dots,0)\), and \(X(\mathcal O)=O_0=K\pi_0\).

   Proof

   • Goursat. With \(A=\pi_UL_0\supseteq A'=L_0\cap U\) and \(B=\pi_VL_0\supseteq B'=L_0\cap V\), \(\pi_Ul\mapsto\pi_Vl\) is an isomorphism \(\psi:A/A'\to B/B'\), and \(L_0=\{\alpha+\beta:\psi(\bar\alpha)=\bar\beta\}\).
   • Stacked bases. The Smith form over \(\mathcal O\) gives \(u_i\) with \(A'=\bigoplus\varpi^{a_i}\mathcal Ou_i\), the exponents being the type of \(B/B'\), that is \(a\). Lifts \(v_i\) of \(\psi(\bar u_i)\), completed by Nakayama and corrected into \(B'\), give \(B'=\bigoplus\varpi^{a_i}\mathcal Ov_i\); the condition \(\psi(\bar\alpha)=\bar\beta\) reads \(\alpha_i\equiv\beta_i\bmod\varpi^{a_i}\).
   • Injective. Two pairs with the same label differ by the \(g\) matching their bases; \(g\) preserves \(L_0\), so \(g\in K\).
   • Lift. \(t^{\lambda_a}U_0=\langle\varpi^{a_i}e_i+e_{n+1-i},e_j\rangle\), \(t^{\lambda_a}V_0=V_0\), and \(\pi_{V_0}(e_i)=-\varpi^{-a_i}e_{n+1-i}\); so the quotient is \(\bigoplus\mathcal O/\varpi^{a_i}\). A pair splits \(L_0\) iff its label is \(0\).
   • Open orbits. For \(\pi\) near \(\pi_1\), \(g(\pi)=\pi\pi_1+(1-\pi)(1-\pi_1)\) lies in \(K\) and conjugates \(\pi_1\) to \(\pi\).
6. Result Pair model · componentwise support bound

   Translates by \(\lambda_a\) only reach labels that are componentwise at most \(a\):

   $$Kt^{\lambda_a}X(\mathcal O)\subseteq\bigsqcup_{b\le a}O_b,\qquad b\le a\iff b_i\le a_i\ \ \forall i .$$
   Proof

   • Reduction. \(Kt^{\lambda_a}X(\mathcal O)=Kt^{\lambda_a}K\pi_0\). For \(g=k_1t^{\lambda_a}k_2\), equivariance gives \(\ell(g\pi_0)=\ell(\pi_0;L')\) with \(L'=k_2^{-1}t^{-\lambda_a}L_0\supseteq L_0\) and \(L'/L_0\simeq\bigoplus\mathcal O/\varpi^{a_i}\).
   • Quotient. \(\pi_0\) splits \(L_0\), so \(\pi_{V_0}(L_0)=L_0\cap V_0\subseteq L'\cap V_0\): the module \(\pi_{V_0}(L')/(L'\cap V_0)\) is a quotient of \(L'/L_0\).
   • Types. By \((\ast)\) a quotient has no more parts \(\ge j\) for each \(j\), so its type is componentwise at most \(a\).
   • Remark. C-4 bounds the minors of \(t^\lambda\pi t^{-\lambda}\) and proves \(Kt^\lambda X(\mathcal O)\subseteq\bigcup_{b\le_{\mathrm{dom}}\varphi(\lambda)}O_b\) for every dominant \(\lambda\); it is not needed here.
7. Result Pair model · (SG) over every \(F\); the \(J\)-model when \(2\) is a unit

   The componentwise order has finite lower sets, so steps 5–6 and step 1 with \(f_a=T_{\lambda_a}\) give (SG) for every \(1\le p\le q\) and every \(F\). If \(2\in\mathcal O^\times\):

   $$\varepsilon\longmapsto\tfrac12(1+\varepsilon),\qquad J\longmapsto\pi_J\in X(\mathcal O)=K\pi_0 .$$
   Proof

   • (SG). Labels \(a\) with the componentwise order; lift and bound from steps 5 and 6; Corollary C.
   • Models. If \(2\in\mathcal O^\times\), \(\varepsilon\mapsto(1+\varepsilon)/2\) and \(\pi\mapsto2\pi-1\) are inverse \(\mathrm{GL}_n\)-equivariant bijections between involutions of type \((q,p)\) and idempotents of rank \(q\) over every \(\mathcal O\)-algebra. So integral points and basic functions agree, and (SG) for the \(J\)-model is (SG) for the pair model.
   • Residue characteristic 2. The models differ: over \(\mathbb Q_2\), \(n=2\), the point \(t^{(1,0)}\pi_0\) has label \(1\) but Cartan label \(0\) (C-4). The statement is for the pair model.
   • Audit. A7-21 re-derived the normal form and the bound, and checked them by machine over \(\mathbb F_2[\![t]\!]\).
8. Result Over \(\mathbb F_q(\!(t)\!)\) · Lang's theorem gives one orbit per label

   Under D0, D3 and C1, the rational points of each geometric orbit form one compact open \(K\)-orbit:

   $$K\backslash X(F)\simeq\Lambda_S^+,\qquad O_\mu:=\mathcal O_\mu(\mathbb F_q)=Kx_\mu .$$
   Proof

   • Rational orbits. \(X(F)=\bigsqcup_\mu\mathcal O_\mu(\mathbb F_q)\) by D0; each \(\mathcal O_\mu\) is Frobenius-stable, since \(\tau\) is injective and \(t^\mu\in S(F)\) makes \(x_\mu\) rational.
   • Cocycles. For rational \(y=gx_\mu\), \(\mathrm{Fr}(g)^{-1}g\in\operatorname{Stab}(x_\mu)\). By D3 this gives a bijection $$K\backslash\mathcal O_\mu(\mathbb F_q)\simeq H^1\big(\mathbb F_q,\operatorname{Stab}(x_\mu)\big)=H^1\big(\mathbb F_q,\pi_0Z_{G^\theta}(\mu)\big),$$ which is trivial under C1 because \(Z_{G^\theta}(\mu)=Z_{G^\theta}(\mu)^\circ Z_{G^\theta}(S)\).
   • Topology. The orbit map is smooth, so \(K\)-orbits are open; they are compact as images of \(K\).
   • Audit. A4-2 also claims the converse (disconnected \(Z_{G^\theta}(S)\) splits an orbit), proved only by its audit; it is not used. What is proved is the bijectivity Sakellaridis expects after his Axiom 2.4.2, not Axiom 2.4.1 itself.
9. Result Over \(\mathbb F_q(\!(t)\!)\) · (SG) under D0, D3, C1, C2

   For \(\mu\in\Lambda_S^+\) take \(\lambda\) with \(\varphi(\lambda)=\mu\) (C2). Then \(t^\lambda x_0=x_\mu\) and, by D0,

   $$Kt^\lambda X(\mathcal O)\subseteq\big(L^+G\,t^\lambda L^+X\big)(\mathbb F_q)\subseteq\bigsqcup_{\nu\le\mu}\mathcal O_\nu(\mathbb F_q).$$
   Proof

   • Order. Dominance on \(\Lambda_S^+\) has finite lower sets (H1(a)).
   • Lift. \(\tau(t^\lambda x_0)=t^\lambda\theta(t^\lambda)^{-1}=t^{\varphi(\lambda)}\) and \(\tau\) is injective, so \(t^\lambda x_0=x_\mu\in O_\mu\).
   • Conclusion. Step 8 and step 1 give (SG); replacing \(q\) by \(q^r\) changes nothing.
   • Scope. Spheres and \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) satisfy C1 and C2, so this reproves (a) and (b) over \(\mathbb F_q(\!(t)\!)\), \(q\) odd, at the cost of D0 and D3. \(E_6/F_4\) fails C2 (Proposition VIII). (SG) over \(\mathbb F_q(\!(t)\!)\) cannot be fed into Theorem E, which needs \(\operatorname{char}F=0\).
10. Result Consistency · Sakellaridis's restriction map is onto for the spheres

    With \(u=y(2\varepsilon_1)\), the orbit sum of the roots restricts to a positive multiple of \(u+u^{-1}\):

    $$r(m_{\varepsilon_1+\varepsilon_2})=c_n(q)(u+u^{-1})+\mathrm{const},\qquad\operatorname{im}r=\mathbb C[u+u^{-1}] .$$
    Proof

    • Data. \(\Lambda_S=\mathbb Z\cdot2\varepsilon_1\), \(W_X=\mathbb Z/2\), \(\Phi_L=\{\pm\varepsilon_i\pm\varepsilon_j:2\le i\lt j\}\) of type \(D_{n-1}\), \(\rho_L=(0,n-2,\dots,1,0)\), and \(x^\mu\mapsto q^{\langle\rho_L,\mu\rangle}u^{\mu_1}\).
    • Restriction. The roots \(\pm\varepsilon_1\pm\varepsilon_j\) give \(q^{\pm(n-j)}u^{\pm1}\); the roots without \(\varepsilon_1\) give constants. Summing over \(j=2,\dots,n\) gives \(c_n(q)u+c_n(q)u^{-1}\).
    • Onto. \(\operatorname{im}r\) is a subring containing \(u+u^{-1}\), which generates \(\mathbb C[u^{\pm1}]^{\mathbb Z/2}\).
    • Status. Theorem E turns this into (SG) only under A1–A5 with \(\operatorname{char}F=0\); A1 and A5 hold for the spheres, A2–A4 are not verified (A6-3). The answer agrees with step 4, which needs none of them.
11. Conclusion Proposition V

    Steps 4 and 7 prove (a) and (b) over every non-archimedean local field with no hypothesis; step 9 proves (c) under D0, D3, C1, C2; step 10 proves (d).

    $$\text{(SG)}:\quad\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\ (n\ge2),\quad\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\ \text{(pair model)},\quad\text{every }F .$$

    In the verdict's table these are the function cells "holds, every \(F\)" of the two rows, new for \(n\ge3\) and for the \(\mathrm{GL}\) family, which was only known under (A). Both rows also satisfy \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) by Proposition III, so they are not divergences.