§2 · Agent 2 · Non-split forms · 2026-10-17T21:22:28Z

Theorem 2.6 · Theorem: the unitary twist of $(H\times H)/H$ fails (SG) over $\mathbb F_q(\!(t)\!)$

proved audit: correct ★ key step

For every nontrivial split $H$, the Hecke translates of $\varphi_0$ on $X_u$ have leading orbits $2a$. By the leading-term lemma an orbit indicator is generated only if its label is even, so (SG) fails, while $\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k$. For $\mathrm{GL}_n$ ($n\ge2$) and $\mathrm{SL}_2$ the failure persists inside the base $G(F)$-orbit.

Theorem 2.6.

Let $H\ne1$ be split connected reductive over $\mathbb F_q$ (any $q$, in particular odd), let $(G,\theta_u,X_u)$ be its unitary twist (Setup 2.4), and $F=\mathbb F_q(\!(t)\!)$. Write $O_c=K\cdot t^c$ and $T_a=\mathbf 1_{H(\mathcal O_E)t^aH(\mathcal O_E)}$ for $a\in\Lambda_H^+\cong(\Lambda_T^+)^\sigma$.

$\operatorname{supp}(T_a\ast\varphi_0)\subseteq\bigcup_{c\le2a}O_c$ and $(T_a\ast\varphi_0)(t^{2a})\ge1$.
$\mathbf 1_{O_c}\in\mathcal H_K\ast\varphi_0\Rightarrow c\in2\Lambda_H^+$. For every finite lower set $L\subseteq\Lambda_H^+$, $\dim\big(\mathcal H_K\ast\varphi_0\cap C(L)\big)=\#\{a:2a\in L\}$.
$\Lambda_H^+\ne2\Lambda_H^+$, so (SG) fails over $F$. With Lemma 2.5: $\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k$ and (SG) fails. This answers IId(1) negatively by an example satisfying (C), D0–D3 and every clause of (R) except the splitness of $T$ and $T_0$.
$(\mathrm{SG})_n$ fails for odd $n$ and holds for even $n$.
Instances:
- $H=\mathbb G_m$ gives Counterexample 2.3.
- $H=\mathrm{GL}_n$: $\operatorname{Res}\mathrm{GL}_n$ acting on nondegenerate hermitian matrices by $x\mapsto gxg^*$. Here $X(F)$ has two $G(F)$-orbits, distinguished by the parity of $v(\det x)$. For $n\ge2$ generation also fails on the base orbit: for $L_d=\{c\le(2d,0,\dots,0)\}$, which lies in the base orbit, $\operatorname{codim}_{C(L_d)}(\mathcal H_K\ast\varphi_0\cap C(L_d))=p_{\le n}(2d)-p_{\le n}(d)\ge1$ for $d\ge1$, where $p_{\le n}$ counts partitions into at most $n$ parts.
- $H=\mathrm{SL}_2$: $\mathrm{SL}_2(E)$ acting on hermitian matrices of determinant $1$. Here $X(F)$ is one $G(F)$-orbit and $\mathbf 1_{K\cdot\operatorname{diag}(t,t^{-1})}\notin\mathcal H_K\ast\varphi_0$.

Hecke basis. $G$ is unramified and $\sigma$ swaps the factors of $\Lambda_T=\Lambda_H\oplus\Lambda_H$. For the $\sigma$-stable Borel $B_H\times B_H$ we get $(\Lambda_T^+)^\sigma=\{(a,a)\}\cong\Lambda_H^+$, and $Kt^{(a,a)}K=H(\mathcal O_E)t^aH(\mathcal O_E)$. By the Cartan decomposition of $H(E)$ (Zhu (2.1.2)), the $T_a$ form a basis of $\mathcal H_K$.
Support and top value. By Lemma A(2) of the classical page, $\operatorname{supp}(T_a\ast\varphi_0)=Kt^aX(\mathcal O)$. For $k\in K$ and $y\in X(\mathcal O)\subset H(\mathcal O_E)$, using $\sigma(t^a)=t^a$ and $\vartheta(t^a)=t^{-a}$:

$$kt^a\cdot y=kt^a\,y\,\vartheta(\sigma(kt^a))^{-1}=k\,t^a\,y\,t^a\,\vartheta(\sigma k)^{-1}\in Kt^aKt^aK\subseteq\bigcup_{\nu\le2a}Kt^\nu K.$$

The last inclusion holds because the convolution map sends $\operatorname{Gr}_{\le a}\mathbin{\tilde\times}\operatorname{Gr}_{\le a}$ into $\operatorname{Gr}_{\le2a}$ (Zhu, proof of Lemma 5.3.1); for $\mathrm{GL}_n$, Cauchy–Binet on minors gives it too. By Setup 2.4(4) the $K$-orbit of a point is its Cartan type, which proves the support claim. For the top value, $t^a\cdot1=t^a\vartheta(t^a)^{-1}=t^{2a}$, so the coset $t^aK$ contributes $1$ at $t^{2a}$, and all contributions are $\ge0$.

Leading terms. Apply Lemma 2.2(b) with $I=\Lambda_H^+$ under the dominance order (finite lower sets), $J=\Lambda_H^+$, $f_a=T_a$ and $\psi(a)=2a$, which is injective. This gives 2.
Odd labels exist. Suppose every dominant coweight lay in $2\Lambda_H$. Take $\xi=2\rho^\vee$ (or any $\xi$ if $H$ is a torus). For $\lambda\in\Lambda_H$ and $N\gg0$, both $\lambda+2N\xi$ and $2N\xi$ are dominant, so $\lambda\in2\Lambda_H$. Then $\Lambda_H=2\Lambda_H$, so $\Lambda_H=0$ and $H=1$. Hence some dominant $c\notin2\Lambda_H$ exists: $c=1$ for $\mathbb G_m$, $c=(1,0,\dots,0)$ for $\mathrm{GL}_n$, $c=\alpha^\vee$ for $\mathrm{SL}_2$. Such $c$ is not of the form $2a$ with $a$ dominant, so $\mathbf 1_{O_c}\notin\mathcal H_K\ast\varphi_0$.
Other $n$.
- Odd $n$: $\mathbb F_{q^2}\otimes\mathbb F_{q^n}=\mathbb F_{q^{2n}}$ is a field, $G(F_n)=H(E_n)$, and steps 1–4 apply verbatim.
- Even $n$: $G_{F_n}\cong H\times H$ and $X_u(F_n)\cong H(F_n)$ with action $g_1x\vartheta(g_2)^{-1}$. Then $T_{(c,0)}\ast\varphi_0=\mathbf 1_{K_Ht^cK_H}$ (classical page, Proposition D, with $\vartheta$ on the second factor), so $(\mathrm{SG})_n$ holds.
$\mathrm{GL}_n$.
- Parity: $\det(gxg^*)=N(\det g)\det x$ and $v(N(y))=2v(y)$, so $v(\det x)\bmod2$ is $G(F)$-invariant; it is even on $X(\mathcal O)$.
- Base orbit $=\{v(\det)\text{ even}\}$: a diagonal $\operatorname{diag}(t^{c_i})$ reduces by $t^{2k}=N(t^k)$ and by $\operatorname{diag}(t,t)=gg^*$ with $g=\left(\begin{smallmatrix}a&1\\-1&\bar a\end{smallmatrix}\right)$, $N(a)=t-1$, using $gg^*=(N(a)+1)\cdot1$ and $a\in\mathcal O_E^\times$, which exists since $N(\mathcal O_E^\times)=\mathcal O^\times$.
- Counting: $c\le(2d,0,\dots,0)$ forces $c_n\ge0$, so $\#L_d=p_{\le n}(2d)$ and $\#\{a:2a\in L_d\}=p_{\le n}(d)$. The partition $(2d-1,1,0,\dots)$ is not even, so the difference is $\ge1$ for $n\ge2$, $d\ge1$.
$\mathrm{SL}_2$. The labels are $c\alpha^\vee$, $c\ge0$; the leading labels are the even $c$, so $\mathbf 1_{O_{\alpha^\vee}}$ is not generated. To see that there is a single $G(F)$-orbit:
- A binary hermitian form $N(u)d_1+N(v)d_2$ is a $4$-dimensional quadratic form over the local field $F$ ($p$ odd), so it represents every element of $F^\times$. Hence $h_x\cong\operatorname{diag}(1,\det x)=1$.
- The isometry can be taken in $\mathrm{SL}_2(E)$: compose with a diagonal unitary element, using that $\det:U(h_x)\to E^1$ is onto.
- Example: $\operatorname{diag}(t,t^{-1})$ represents $1$ at $(u,v)=(t^{-1},v_0)$ with $N(v_0)=t-1$.
Hypotheses: Setup 2.4(6) and Lemma 2.5.

Remark (heuristic, not used): the counts in step 6 grow in the ratio $p_{\le n}(2d)/p_{\le n}(d)\to2^{n-1}$ on each of the two parity classes. This matches the description of $C_K$ as a free $\mathcal H_K$-module of rank $2^n$ for $p$-adic hermitian forms, which I recall from Hironaka's work and have not checked against a source.

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Lemma 2.2 · Lemma: orbit and leading-term obstructions to $\mathcal H_K\ast\varphi_0=C_K$
Setup 2.4 · Setup: the unitary and Galois $\mathbb F_q$-forms of the group case
Lemma 2.5 · Lemma: $V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c$, so sheaves generate for all forms

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Computation 2.7 · Computation: $T_\lambda\ast\varphi_0$ on hermitian $2\times2$ matrices
Lemma 2.8 · Lemma: Satake objects without a Weil structure are invisible to the trace at $n=1$
Computation 2.9 · Computation: unitary $\mathrm{SL}_2$: every row of $m^{(1)}$ nonzero, no spanning
Proposition 2.10 · Proposition: the Galois twist $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$ satisfies (SG)
Remark 2.12 · Remark: why the trace misses thick generation, and what (R) removes
Summary 2.13 · Verdict: sheaves do not imply (SG) when $T$ is non-split; nothing new under (R)
Audit 4.5 · Audit: the unitary twist fails (SG) over $\mathbb F_q(\!(t)\!)$ for split $H\ne1$
Computation 4.6 · Computation: hermitian Hecke counts $T_\lambda\ast\varphi_0$ for $q=3,5$
Audit 4.7 · Audit: $G(F)$-orbits of the hermitian spaces, and failure on the base orbit
Remark 6.1 · Registry after the first two returns: a non-split counterexample found twice; launch its audit
Theorem 6.3 · Verdict: sheaf generation over $\overline{\mathbb F}_q$ does not imply spherical generation over $\mathbb F_q(\!(t)\!)$

Sources

zhu-satake-v2.pdf, (2.1.2), proof of Lemma 5.3.1 (convolution into Gr_{≤μ1+μ2})
spherical-generation/classical.html, Lemma A(2), Proposition B, Proposition D
trace-pages/trace-b.html, step 1

Theorem 2.6 · Theorem: the unitary twist of \((H\times H)/H\) fails (SG) over \(\mathbb F_q(\!(t)\!)\)

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