§2 · Agent 2 · Non-split forms · 2026-10-17T21:20:00Z

Lemma 2.2 · Lemma: orbit and leading-term obstructions to $\mathcal H_K\ast\varphi_0=C_K$

proved audit: correct

Every Hecke translate of $\varphi_0$ is supported on $G(F)\cdot X(\mathcal O)$. If the translates have injective leading terms with a support bound, an orbit indicator is generated only when its label is a leading term. This is the converse partner of the pages' Proposition B, and it decides (SG) in the twisted examples.

Lemma 2.2.

Let $F$ be non-archimedean and $X=G/H$ homogeneous, with compact open $K$-orbits $\{O_\nu\}_{\nu\in I}$ and a partial order on $I$ whose lower sets are finite. For a lower set $L\subseteq I$ let $C(L)=\bigoplus_{\nu\in L}\mathbb C\,\mathbf 1_{O_\nu}$.

(a) Orbit obstruction. Every $f\in\mathcal H_K\ast\varphi_0$ is supported in $G(F)\cdot X(\mathcal O)$. If $F=\mathbb F_q(\!(t)\!)$, $H$ is connected and $H^1(\mathbb F_q,L^+H)=1$, then $X(\mathcal O)=Kx_0$. So (SG) implies $X(F)=G(F)x_0$; for instance (SG) fails when $\ker\big(H^1(F,H)\to H^1(F,G)\big)\ne1$.

(b) Leading-term obstruction. Let $\{f_j\}_{j\in J}\subset\mathcal H_K$ be such that the $f_j\ast\varphi_0$ span $\mathcal H_K\ast\varphi_0$. Let $\psi:J\to I$ be injective with $$\operatorname{supp}(f_j\ast\varphi_0)\subseteq\bigcup_{\nu\le\psi(j)}O_\nu,\qquad (f_j\ast\varphi_0)|_{O_{\psi(j)}}\ne0\qquad(j\in J).$$ Then for every lower set $L\subseteq I$ $$\mathcal H_K\ast\varphi_0\ \cap\ C(L)=\operatorname{span}\{f_j\ast\varphi_0:\psi(j)\in L\}.$$ Consequently $\mathbf 1_{O_\mu}\in\mathcal H_K\ast\varphi_0\Rightarrow\mu\in\psi(J)$, and $\text{(SG)}\iff\psi(J)=I$.

Proof of (a).

$(f\ast\varphi_0)(x)=\sum_{gK}f(g)\,\varphi_0(g^{-1}x)$. This is nonzero only if $g^{-1}x\in X(\mathcal O)$ for some $g$, so $x\in G(F)\cdot X(\mathcal O)$.
$G\to X$ is smooth and surjective because $H$ is smooth. Hence $L^+G\to L^+X$ is surjective on $k$-points: lift modulo $t$, then lift step by step using smoothness. So $L^+X(k)=L^+G(k)\cdot x_0\cong L^+G/L^+H$.
The $\mathbb F_q$-points of a homogeneous space $L^+G/L^+H$ form $\ker\big(H^1(\mathbb F_q,L^+H)\to H^1(\mathbb F_q,L^+G)\big)$-many $K$-orbits. This is the argument of IIb, step 1, with $\mu=0$. With $H^1(\mathbb F_q,L^+H)=1$ (D3, $H$ connected) we get $X(\mathcal O)=L^+X(\mathbb F_q)=Kx_0$.
By steps 1 and 3, $\mathcal H_K\ast\varphi_0$ is supported in $G(F)x_0$. The $G(F)$-orbits on $X(F)$ are indexed by $\ker(H^1(F,H)\to H^1(F,G))$. If this kernel is nontrivial, the indicator of a $K$-orbit outside $G(F)x_0$ is not generated.

Proof of (b).

Independence. The $f_j\ast\varphi_0$ are linearly independent. A nontrivial relation would have a term whose $\psi(j)$ is maximal among the terms present, and by step 3 below that term would be the only one nonzero on $O_{\psi(j)}$.
Setup. Let $f=\sum_jc_jf_j\ast\varphi_0$ be a finite sum lying in $C(L)$. Put $M=\{\psi(j):c_j\ne0\}$, a finite set, and let $m\in M$ be maximal.
Top value. $f_j\ast\varphi_0$ vanishes on $O_m$ unless $m\le\psi(j)$. For $c_j\ne0$ this forces $\psi(j)=m$ by maximality, hence $j=j_0:=\psi^{-1}(m)$ by injectivity. So $f|_{O_m}=c_{j_0}(f_{j_0}\ast\varphi_0)|_{O_m}\ne0$, and therefore $m\in L$.
All terms. Every element of the finite set $M$ lies below a maximal one, and $L$ is a lower set, so $M\subseteq L$. This gives "$\subseteq$". The inclusion "$\supseteq$" is the support bound.
One orbit. Take $f=\mathbf 1_{O_\mu}$ and $L=\{\nu\le\mu\}$, which is finite. Step 3 applied to a maximal $m$ gives $\mathbf 1_{O_\mu}|_{O_m}\ne0$, so $m=\mu\in\psi(J)$.
(SG). If (SG) holds, step 5 puts every $\mu$ in $\psi(J)$. Conversely, if $\psi$ is onto, then $f_\mu:=f_{\psi^{-1}(\mu)}$ satisfies the hypotheses of the classical page's Proposition B (support in $\{\nu\le\mu\}$, nonzero on $O_\mu$), and Proposition B gives (SG).

Remarks.

Part (b) needs $\psi$ to be injective. When several Hecke labels share one leading orbit, combinations can cancel leading terms and reach labels that are not leading terms. Remark 2.12(4) shows that under (R) and (C), either $\varphi|_{\Lambda_T^+}$ is not injective modulo cocharacters acting trivially on $X$, or every label lifts and (SG) holds. So (b) never yields a failure in Proposition II's setting.
In the trace setting the hypotheses of (b) hold automatically for $\psi=\varphi$ restricted to rational labels: the support bound comes from D0 by induction on the unitriangular $\operatorname{tr}V_\lambda$ (IIa), and the top value from the coset $t^\lambda K$ (Lemma A(2)). Only injectivity is extra.

Builds on

Strategy 2.1 · Strategy: change the $\mathbb F_q$-structure of a pair whose sheaf statement is known

Built on by

Counterexample 2.3 · Counterexample: $\operatorname{Res}\mathbb G_m$ on $\mathbb G_m$ through the norm
Theorem 2.6 · Theorem: the unitary twist of $(H\times H)/H$ fails (SG) over $\mathbb F_q(\!(t)\!)$
Proposition 2.10 · Proposition: the Galois twist $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$ satisfies (SG)
Dead end 2.11 · Dead end: $G=\mathrm{GL}_2$ split, $H=T_E$ or $T_E^1$ non-split
Audit 4.4 · Audit: the orbit obstruction and the injective leading-term lemma

Sources

spherical-generation/classical.html, Lemma A(2) and Proposition B
trace-pages/trace-b.html, step 1 (cocycle and Lang)
trace-pages/_hypotheses.html, D3

Lemma 2.2 · Lemma: orbit and leading-term obstructions to \(\mathcal H_K\ast\varphi_0=C_K\)

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