§2 · Agent 2 · Non-split forms · 2026-10-17T21:20:30Z

Counterexample 2.3 · Counterexample: $\operatorname{Res}\mathbb G_m$ on $\mathbb G_m$ through the norm

proved audit: correct ★ key step

The smallest twist. Over $k$ the pair is $\mathbb G_m^2$ acting on $\mathbb G_m$ by $(a,b)x=abx$, and every orbit is a Satake translate of $\mathrm{IC}_0$. Over $\mathbb F_q(\!(t)\!)$, Hecke translates of $\varphi_0$ reach only even valuations. The Galois twist $R/\mathbb G_m$ of the same $k$-pair satisfies (SG).

Counterexample 2.3.

Let $R=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m$, $z\mapsto\bar z$ its Galois involution, $N(z)=z\bar z$, $\theta(z)=\bar z^{-1}$, $H=R^\theta=\ker N$ (the norm-one torus), and $X=R/H\cong\mathbb G_m$ via $N$. For every prime power $q$:

over $k$, $\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k$;
over $F=\mathbb F_q(\!(t)\!)$,

$$\mathcal H_K\ast\varphi_0=\bigoplus_{m\in\mathbb Z}\mathbb C\,\mathbf 1_{t^{2m}\mathcal O^\times}\ \subsetneq\ C_K=\bigoplus_{c\in\mathbb Z}\mathbb C\,\mathbf 1_{t^{c}\mathcal O^\times},$$

so (SG) fails; $(\mathrm{SG})_n$ fails for odd $n$ and holds for even $n$;

for the Galois twist $X'=R/\mathbb G_m$ (involution $z\mapsto\bar z$), $X'(F)=X'(\mathcal O)$ is a single $K$-orbit and (SG) holds.

For $X$: (C) and D0–D3 hold. (R) holds except that $T=R$, the only maximal torus, is not split.

Over $k$.

Lattices. $R_k=\mathbb G_m^2$ with Frobenius $(a,b)\mapsto(b^q,a^q)$, $\theta(a,b)=(b^{-1},a^{-1})$, $X_k=\mathbb G_m$, $(a,b)\cdot x=abx$, and Frobenius $x\mapsto x^q$ on $X_k$. Hence $\Lambda_T=\mathbb Z^2$, $\sigma(a,b)=(b,a)$, $\theta(a,b)=(-b,-a)$, $\Lambda_S=\mathbb Z(1,1)$, and $\varphi(a,b)=(a+b)(1,1)$. There are no roots, so $\Lambda_T^+=\Lambda_T$.
Orbits. $LX(k)=k(\!(t)\!)^\times=\bigsqcup_{c\in\mathbb Z}\mathcal O_c$ with $\mathcal O_c=t^ck[\![t]\!]^\times$, each open and closed in $(LX)_{\mathrm{red}}$. $\operatorname{Stab}(t^c)=\{(a,a^{-1})\}\cong L^+\mathbb G_m$, which is connected.
Satake objects. $(\operatorname{Gr}_{\mathbb G_m^2})_{\mathrm{red}}=\mathbb Z^2$ and $V_{(a,b)}=\delta_{t^{(a,b)}}$. The map $m$ sends $\{t^{(a,b)}\}\mathbin{\tilde\times}L^+X$ isomorphically onto $\mathcal O_{a+b}$, so $V_{(a,b)}\ast\mathrm{IC}_0\cong\mathrm{IC}_{a+b}$, the constant sheaf on $\mathcal O_{a+b}$.
Generation. An object of $\mathcal C_k$ is the finite direct sum of its restrictions to orbits. On $\mathcal O_c=L^+R/\operatorname{Stab}$, with connected stabilizer, the cohomology sheaves are constant, so truncation triangles place the object in $\operatorname{Thick}\{\mathrm{IC}_c\}$ (Proposition I with H1–H3). Hence $\mathcal C_k=\operatorname{Thick}\{\mathrm{IC}_c:c\in\mathbb Z\}=\mathcal C_k^{\mathrm{Sat}}$ by step 3.

Over $F$.

Points. $G(F)=E^\times$ with $E=\mathbb F_{q^2}(\!(t)\!)$, $K=\mathcal O_E^\times$, $X(F)=F^\times$, and $g\cdot x=N(g)x$. We have $N(\mathcal O_E^\times)=\mathcal O^\times$ (Lang for the connected $L^+H$, D3) and $N(t)=t^2$. So the $K$-orbits are $t^c\mathcal O^\times$, $c\in\mathbb Z$, and there are two $G(F)$-orbits, distinguished by the parity of $v$.
Hecke action. $(\Lambda_T)^\sigma=\mathbb Z(1,1)$, so $\mathcal H_K$ has basis $T_m=\mathbf 1_{t^m\mathcal O_E^\times}$, $m\in\mathbb Z$. With $\operatorname{vol}K=1$,

$$(T_m\ast\varphi_0)(x)=\varphi_0\big(N(t^m)^{-1}x\big)=\mathbf 1_{t^{2m}\mathcal O^\times}(x).$$

Failure. Statement 2 follows, and $\mathbf 1_{t\mathcal O^\times}\notin\mathcal H_K\ast\varphi_0$ for every $q$. This is the orbit obstruction of Lemma 2.2(a): $X(F)\ne G(F)x_0$, because $H^1(F,H)=F^\times/NE^\times\cong\mathbb Z/2$ while $H^1(F,R)=1$. On $G(F)x_0=\{v\text{ even}\}$ generation holds.
Other $n$. For odd $n$, $E\otimes_FF_n$ is a field, and steps 5–7 apply with $q^n$ in place of $q$. For even $n$, $R$ splits, $\mathcal H_{K_n}$ has basis $T_{(a,b)}$, and $T_{(a,b)}\ast\varphi_0=\mathbf 1_{t^{a+b}\mathcal O_n^\times}$, so $(\mathrm{SG})_n$ holds.

Galois twist.

Over $k$. $X'\cong R^1$ via $z\mapsto z/\bar z$. Over $k$, $X'_k=\mathbb G_m$ with $(a,b)y=ay/b$ and Frobenius $y\mapsto y^{-q}$. Frobenius sends $\mathcal O_c$ to $\mathcal O_{-c}$, so only $\mathcal O_0$ is Frobenius-stable. The sheaf level is as in step 4, with $V_{(a,b)}\ast\mathrm{IC}_0=\mathrm{IC}_{a-b}$.
Over $F$. $X'(F)=E^1\subset\mathcal O_E^\times$, since $v(N z)=2v(z)$. By Hilbert 90, $E^1=\{u/\bar u:u\in\mathcal O_E^\times\}=K\cdot1$. So $C_K=\mathbb C\varphi_0$, and (SG) holds.

Trace reading.

Weil structures. $\mathrm{Fr}^*\delta_{t^{(a,b)}}=\delta_{t^{(b,a)}}$, so $V_\lambda$ has a Weil structure exactly when $\lambda=(a,a)$. The rational columns give $\mathrm{IC}_{2a}$.
Odd rows. For odd $c$, every $\lambda$ with $\varphi(\lambda)=c$ has $\sigma\lambda\ne\lambda$. The Weil object $V_{(a,b)}\oplus V_{(b,a)}$ (with $a+b=c$) has convolution $\mathrm{IC}_c\otimes\overline{\mathbb Q}_\ell^{\,2}$, and Frobenius acts on the multiplicity space by $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$: trace $0$ for odd $n$, $2$ for even $n$. So row $c$ of $m^{(1)}$ is zero, although $\mathrm{IC}_c=V_{(a,b)}\ast\mathrm{IC}_0$ is a summand used by thick generation.
Contrast. In $X'$ the odd rows have no rational points at all, so they impose no condition on (SG).

Hypotheses for $X$.

(R): $G,\theta,X$ are over $\mathbb F_q$, and $S=\mathbb G_m\subset R$ is split; but $T=R$ and $T_0=R$ are not split.
(C): $H=R^1$ is a connected torus and $Z_H(S)=H$.
D0: steps 2–3.
D1: trivial, since $m$ is an isomorphism onto an orbit.
D2: step 3.
D3: $L^+R$ and $L^+R^1$ are connected.

Builds on

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

Built on by

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 4.1 · Audit: the $\mathbb F_q$-structures of the unitary and Galois twists
Audit 4.2 · Audit: $V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c$; sheaves generate for all forms
Audit 4.3 · Audit: $K$-orbits on $X(F)$ for the twists, one per Frobenius-stable orbit label
Audit 4.8 · Audit: Weil structures, rational columns and $m^{(1)}$ for unitary twists
Audit 4.9 · Audit: $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$ satisfies (SG) over $\mathbb F_q(\!(t)\!)$
Audit 4.11 · Audit: the torus criterion and the norm-torus counterexample

Sources

trace-pages/_definitions.html, Proposition I
trace-pages/_hypotheses.html, D3
spherical-generation/classical.html, Lemma A(2)

Counterexample 2.3 · Counterexample: \(\operatorname{Res}\mathbb G_m\) on \(\mathbb G_m\) through the norm

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