SL₂/SO₂, GLₙ/Oₙ and PGL₂/PO₂

Verdict

For $\mathrm{SL}_2/\mathrm{SO}_2$ both conditions fail: $Z_{G^\theta}(S)=\{\pm I\}$ is disconnected, and the label $\alpha^\vee$ has no dominant lift. For $\mathrm{GL}_n/\mathrm{O}_n$ and $\mathrm{PGL}_2/\mathrm{PO}_2$ the fixed group $G^\theta$ is disconnected, so Proposition III does not apply; $\varphi$ still doubles and misses $\varepsilon_1$, resp. $\omega^\vee$.

These are the last two rows of Proposition III, step 7.

Smallest cases

$\mathrm{SL}_2/\mathrm{SO}_2$ is the smallest symmetric space where coverage fails, and $\mathrm{GL}_1/\mathrm O_1$ ($n=1$) already shows the doubling for $\mathrm{GL}_n/\mathrm O_n$:

$$\theta=-1,\qquad\varphi=2,$$ $$\Lambda_S^+\setminus\varphi(\Lambda_T^+)=\{\alpha^\vee,3\alpha^\vee,5\alpha^\vee,\dots\}\ \ (\mathrm{SL}_2).$$

Two horizontal rows. The top row shows the dominant cocharacters 0, alpha check, up to 4 alpha check of SL2 in blue. Arrows send n alpha check to 2n alpha check on the bottom row of orbit labels m alpha check for m from 0 to 8. Even m are green, odd m are red rings. — **Figure 1.** $\mathrm{SL}_2/\mathrm{SO}_2$: $\varphi$ doubles, so every odd label $m\alpha^\vee$ has no lift.

Intuition on the smallest casesworked here

What $\varphi$ does. Through the Cartan embedding $\tau(g\,\mathrm{SO}_2)=gg^{\mathsf T}$, a diagonal translate is squared: $$\tau\bigl(t^{n\alpha^\vee}x_0\bigr)=\operatorname{diag}(t^{2n},t^{-2n}),\qquad\varphi(n\alpha^\vee)=2n\alpha^\vee.$$
Why odd labels are missed. The orbit $\mathcal O_{\alpha^\vee}$ exists (the doubling example writes down a point of it), but $\varphi(\Lambda_T)=2\Lambda_T$ has index $2$ in $\Lambda_S=\Lambda_T$.
Both conditions fail. The centralizer is finite and disconnected, so the orbits also carry nonconstant local systems: $$Z_{G^\theta}(S)=\mathrm{SO}_2\cap T=\{\pm I\}.$$
$\mathrm{GL}_1/\mathrm O_1$. $\tau(g)=g^2$ identifies $X$ with $\mathbb G_m$; the orbits are the valuations, and the Satake objects only reach the even ones, since $LG\cdot x_0$ consists of squares: $$LX=F^\times=\bigsqcup_{m\in\mathbb Z}t^m\mathcal O^\times$$ $$LG\cdot x_0=(F^\times)^2=\bigsqcup_{m\ \text{even}}t^m\mathcal O^\times.$$

1 · Model

Involution. All three groups carry the inverse transpose $\theta(g)=(g^T)^{-1}$, and $\theta(g)=g$ exactly when $gg^T=I$. In $\mathrm{PGL}_2$ the class of $g$ is fixed when $gg^T$ is scalar, and rescaling $g$ makes it $I$:

$$\mathrm{SL}_2^\theta=\mathrm{SO}_2,\qquad\mathrm{GL}_n^\theta=\mathrm{O}_n,\qquad\mathrm{PGL}_2^\theta=\mathrm{O}_2/\{\pm I\}=\mathrm{PO}_2.$$

Components. $\mathrm{SO}_2$ is connected. $\mathrm{O}_n$ has the two components $\det=\pm1$, and so does $\mathrm{PO}_2$, since $\det(-g)=\det g$ for $2\times2$ matrices.

Tori. $T$ is the diagonal torus, and $\theta$ inverts it. So $S=T$ is a maximal θ-split torus, $\dim S$ is the rank of $G$, and $T=(T^\theta)^\circ\cdot S$ trivially:

$$\theta(t)=t^{-1}\ \ (t\in T),\qquad\theta[\operatorname{diag}(t,1)]=[\operatorname{diag}(t^{-1},1)]\ \ \text{in }\mathrm{PGL}_2.$$

2 · Lattices

$\theta=-1$ on every cocharacter, so the split lattice is the whole lattice and $\varphi$ doubles:

$$\theta=-1,\qquad\Lambda_S=\Lambda_T,\qquad\varphi=1-\theta=2.$$

The lattices are $\Lambda_T=\mathbb Z\alpha^\vee$ for $\mathrm{SL}_2$, $\mathbb Z^n$ for $\mathrm{GL}_n$, and $\mathbb Z\omega^\vee$ for $\mathrm{PGL}_2$. Here $\alpha^\vee(t)=\operatorname{diag}(t,t^{-1})$, $\omega^\vee(t)=[\operatorname{diag}(t,1)]$, and $\alpha^\vee=2\omega^\vee$ in $\mathrm{PGL}_2$.

3 · Positive roots

$B$ is the upper-triangular Borel. Since $S=T$, the Levi $L=Z_G(S)$ is $T$ and $\Phi_L$ is empty; $\theta=-1$ on characters too, so $-\theta\alpha=\alpha\in\Phi^+$ for every positive root.

4 · Centralizer

$\mathrm{SL}_2/\mathrm{SO}_2$. A diagonal matrix $\operatorname{diag}(a,a^{-1})$ is orthogonal exactly when $a^2=1$. So the centralizer is $\mu_2$, and condition (1) fails:

$$Z_{G^\theta}(S)=\mathrm{SO}_2\cap T=\{I,-I\}.$$

$\mathrm{GL}_n/\mathrm{O}_n$ and $\mathrm{PGL}_2/\mathrm{PO}_2$. $G^\theta$ is disconnected, so these spaces lie outside the setting of Proposition III, which assumes $G^\theta$ connected. Their centralizers $G^\theta\cap T$ are the finite groups $\{\operatorname{diag}(\pm1,\dots,\pm1)\}$ and $\{[\operatorname{diag}(\pm1,1)]\}$.

5 · Coverage

Here $\Lambda_S^+=\Lambda_T^+$, and doubling preserves dominance. So the image is the even part of the cone:

$$\varphi(\Lambda_T^+)=2\Lambda_T^+\subsetneq\Lambda_T^+=\Lambda_S^+.$$

Labels without a lift. $\varphi$ misses every dominant label with an odd coordinate in these bases. The smallest are

$$\alpha^\vee\ \ (\mathrm{SL}_2),\qquad\varepsilon_1=(1,0,\dots,0)\ \ (\mathrm{GL}_n),\qquad\omega^\vee\ \ (\mathrm{PGL}_2).$$

For $\mathrm{SL}_2/\mathrm{SO}_2$, the doubling example computes $\varphi$ through the Cartan embedding and writes down a point of the orbit labelled $\alpha^\vee$.