§2 · Agent 2 · Non-split forms · 2026-10-17T21:21:09Z

Setup 2.4 · Setup: the unitary and Galois $\mathbb F_q$-forms of the group case

proved audit: minor defect

For split $H$, $\operatorname{Res}H$ carries two involutions: $\sigma$ (Galois) and $\vartheta\circ\sigma$ (unitary, with $\vartheta$ a Chevalley involution). Both quotients become the group case over $k$. Their $K$-orbits on $X(F)$ are indexed by Cartan types: all of $\Lambda_H^+$ in the unitary case, only the self-dual types in the Galois case.

Setup 2.4.

Let $H$ be a split connected reductive group over $\mathbb F_q$ with split $T_H\subset B_H$. Let $\vartheta$ be an $\mathbb F_q$-involution of $H$ with $\vartheta(s)=s^{-1}$ on $T_H$ (a Chevalley involution). Put $G=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}H$ with Galois involution $\sigma$, $\theta_u=\vartheta\circ\sigma$, $\theta_g=\sigma$, $X_u=G/G^{\theta_u}$, and $X_g=G/G^{\theta_g}=G/H$. Write $E=\mathbb F_{q^2}(\!(t)\!)$.

Over $k$. $G_k=H\times H$, $\sigma(g_1,g_2)=(g_2,g_1)$, and $\mathrm{Fr}(g_1,g_2)=(\mathrm{Fr}\,g_2,\mathrm{Fr}\,g_1)$. The first coordinate of $\tau(g)=g\theta(g)^{-1}$ identifies $X_{u,k}$ and $X_{g,k}$ with $H$, with actions

$$(g_1,g_2)\cdot x=g_1x\vartheta(g_2)^{-1}\ \ (\text{unitary}),\qquad (g_1,g_2)\cdot x=g_1xg_2^{-1}\ \ (\text{Galois}),$$

base point $x_0=1$, $L^+X=L^+H$, and $\mathrm{IC}_0$ the constant sheaf on $L^+H$. Frobenius on $X_k$ is $x\mapsto\mathrm{Fr}(\vartheta(x))^{-1}$ (unitary) and $x\mapsto\mathrm{Fr}(x)^{-1}$ (Galois).

Over $F$. $X_u(F)=\{x\in H(E):\vartheta(\sigma x)=x^{-1}\}$ with $g\cdot x=gx\vartheta(\sigma g)^{-1}$, and $X_g(F)=\{x\in H(E):x\,\sigma(x)=1\}$ with $g\cdot x=gx\sigma(g)^{-1}$. In both cases $K=H(\mathcal O_E)$.
Orbits. The $L^+G$-orbits on $LX$ are $\mathcal O_c=L^+H\,t^c\,L^+H$, $c\in\Lambda_H^+$, with connected stabilizers $\cong J_c:=L^+H\cap t^cL^+Ht^{-c}$. Frobenius fixes every $\mathcal O_c$ in the unitary case and sends $\mathcal O_c$ to $\mathcal O_{-w_0c}$ in the Galois case.
$K$-orbits. Taking the Cartan type of $x\in H(E)$ gives bijections $K\backslash X_u(F)\cong\Lambda_H^+$ (with $t^c\in X_u(F)$) and $K\backslash X_g(F)\cong\{c\in\Lambda_H^+:c=-w_0c\}$.
Examples. For $H=\mathrm{GL}_n$ and $\vartheta(g)=g^{-T}$: $X_u(F)=\{x\in\mathrm{GL}_n(E):x^*=x\}$ with $g\cdot x=gxg^*$, and $X_g(F)\cong\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$. For $H=\mathbb G_m$ these are the two tori of Counterexample 2.3. For $H=\mathrm{SL}_2$, $X_u(F)$ is the set of hermitian matrices of determinant $1$.
Hypotheses. Both forms satisfy (C) and D0–D3. (R) fails: $T=Z_G(S)\cong\operatorname{Res}T_H$ is not split, and $G$ has no split maximal torus $T_0$. In the unitary form $S\cong T_H$ is split; in the Galois form $S$ is anisotropic.

1. Over $k$. $\mathbb F_{q^2}\otimes k=k\times k$ gives $G_k=H\times H$ with $\sigma$ the swap. Frobenius of $\operatorname{Res}$ is the swap composed with $\mathrm{Fr}$. Since $\vartheta$ is defined over $\mathbb F_q$, it commutes with $\sigma$, so $\theta_u(g_1,g_2)=(\vartheta g_2,\vartheta g_1)$ and $\theta_u^2=1$.

Fixed points: $G^{\theta_u}_k=\{(g,\vartheta g)\}\cong H$ and $G^{\theta_g}_k=\Delta H$, both connected.
Unitary: $\tau(g_1,g_2)=(g_1\vartheta(g_2)^{-1},\,g_2\vartheta(g_1)^{-1})=(x,\vartheta(x)^{-1})$.
Galois: $\tau(g_1,g_2)=(g_1g_2^{-1},g_2g_1^{-1})=(x,x^{-1})$.

Projecting to $x$ is $G_k$-equivariant for the stated actions, sends $eH$ to $1$, and sends $L^+G\cdot x_0$ onto $L^+H$. For Frobenius in the unitary case, $x=g_1\vartheta(g_2)^{-1}$ goes to $\mathrm{Fr}(g_2)\vartheta(\mathrm{Fr}\,g_1)^{-1}=\mathrm{Fr}\big(\vartheta(\vartheta(g_2)g_1^{-1})\big)=\mathrm{Fr}(\vartheta(x))^{-1}$. The Galois case is the same with $\vartheta=1$.

2. Over $F$. $G(F)=H(E)$ embeds in $G(k(\!(t)\!))=H\times H$ by $e\mapsto(e,\sigma e)$, so the first coordinate of $\tau$ of an $F$-point is the point itself. The defining equations and actions follow from 1, and the Cartan type of $x\in H(E)$ is the label of the corresponding $k$-point.

3. Orbits and stabilizers. $\vartheta(L^+H)=L^+H$, so the $L^+G$-orbit of $x$ is $L^+HxL^+H$, and these are the $\mathcal O_c$ by the Cartan decomposition over $k(\!(t)\!)$ (Zhu (2.1.2)). The condition $g_1t^c\vartheta(g_2)^{-1}=t^c$ means $\vartheta(g_2)=t^{-c}g_1t^c$, so $\operatorname{Stab}(t^c)\cong J_c$, which is connected (Zhu §5.1). For the labels: $\vartheta(L^+t^cL^+)=L^+t^{-c}L^+$ because $\vartheta(t^c)=t^{-c}$, so $\vartheta(x)^{-1}$ has type $c$ again; $\mathrm{Fr}$ preserves the double coset of the rational $t^c$. In the Galois case $x^{-1}$ has type $-w_0c$.

4. $K$-orbits.

Unitary: $\vartheta(\sigma(t^c))=t^{-c}=(t^c)^{-1}$, so $t^c\in X_u(F)$ is a rational point of $\mathcal O_c$.
Galois: a Frobenius-stable $\mathcal O_c$ is a homogeneous space of the connected group $L^+G$, so it has a rational point by Lang; if $c\ne-w_0c$, $\mathcal O_c$ has no rational point.

With connected stabilizers, the cocycle argument of IIb, step 1 (Lang, D3) shows that the rational points of each Frobenius-stable orbit form a single $K$-orbit. Since $LX(\mathbb F_q)=X(F)$ is the union of the rational points of the orbits, 4 follows.

5. Examples.

$H=\mathrm{GL}_n$: $\vartheta(\sigma x)=(x^*)^{-1}$, so the condition reads $x^*=x$, and $gx\vartheta(\sigma g)^{-1}=gxg^*$. In the Galois case, Hilbert 90 for $\mathrm{GL}_n$ gives $X_g(F)=\{g\sigma(g)^{-1}\}\cong\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$.
$H=\mathbb G_m$: $\vartheta(z)=z^{-1}$, $X_u(F)=F^\times$, and $g\cdot x=g\bar gx$.
$H=\mathrm{SL}_2$, $\vartheta(g)=g^{-T}$: hermitian matrices of determinant $1$.

6. Hypotheses.

Unitary. $S=\{(s,s)\}$ is $\theta_u$-split, since $\theta_u(s,s)=(s^{-1},s^{-1})$; it is defined and split over $\mathbb F_q$, because Frobenius maps $(s,s)$ to $(\mathrm{Fr}\,s,\mathrm{Fr}\,s)$. $Z_{G^\theta}(S)=\{(t,\vartheta t):t\in T_H\}\cong T_H$, connected.
Galois. $S=\{(s,s^{-1})\}$, on which Frobenius acts by $-1$ on $X_*(S)$, so $S$ is anisotropic. $Z_{G^\theta}(S)=\Delta T_H$, connected.
In both cases $Z_G(S)=T_H\times T_H=\operatorname{Res}T_H$. Frobenius swaps the two factors of $X_*(T)$, so $T$ is not split, and $G$ has $\mathbb F_q$-rank $\operatorname{rk}H<\operatorname{rk}G$, so it has no split maximal torus.
D0 for the group case: Cartan decomposition; closure order given by dominance (Zhu Prop. 2.1.5); support bound from the convolution map $\operatorname{Gr}_{\le a}\mathbin{\tilde\times}\operatorname{Gr}_{\le b}\to\operatorname{Gr}_{\le a+b}$ (Zhu, proof of Lemma 5.3.1); $J_c=Z_H(c)\ltimes$(pro-unipotent).
D1 and D2: Lemma 2.5.
D3: $L^+G$ and $J_c$ are connected.

Builds on

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

Built on by

Lemma 2.5 · Lemma: $V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c$, so sheaves generate for all forms
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)
Audit 4.1 · Audit: the $\mathbb F_q$-structures of the unitary and Galois twists
Audit 4.3 · Audit: $K$-orbits on $X(F)$ for the twists, one per Frobenius-stable orbit label

Sources

zhu-satake-v2.pdf, (2.1.2), Prop. 2.1.5, §5.1 (connectedness of L+G ∩ t^μ L+G t^-μ)
trace-pages/trace-b.html, step 1
trace-pages/_hypotheses.html, (R), (C), D0-D3

Setup 2.4 · Setup: the unitary and Galois \(\mathbb F_q\)-forms of the group case

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