GL(n)/(GL(p)×GL(q)) · Proposition III family

Verdict

Both conditions of Proposition III hold: $Z_{G^\theta}(S)\cong\mathbb G_m^p\times\mathrm{GL}_{q-p}$ is connected, and every $\mu=(a_1,\dots,a_p,0,\dots,0,-a_p,\dots,-a_1)\in\Lambda_S^+$ lifts to the dominant $\lambda=(a_1,\dots,a_p,0,\dots,0)$. So H1–H5 hold and $\mathcal C^{\mathrm{Sat}}=\mathcal C$, as recorded in Proposition III, step 7.

Smallest case: $(p,q)=(1,1)$

For $n=2$, $J$ swaps $e_1\leftrightarrow e_2$, and $G^\theta\cong\mathrm{GL}_1\times\mathrm{GL}_1$ is the torus that is diagonal in the basis $e_1\pm e_2$:

$$\theta(x_1,x_2)=(x_2,x_1),\qquad\Lambda_T^+=\{x_1\ge x_2\},$$ $$\Lambda_S^+=\{(a,-a):a\ge0\},\qquad\varphi(x)=(x_1-x_2)\,(1,-1).$$

The integer plane (x1, x2). The half-plane x1 at least x2 is shaded and its points are blue. The antidiagonal line is the split lattice, which is also the cocharacter lattice of SL2; its points (a, minus a) for a from 0 to 3 are green, all hit. Vertical arrows send (a, 0) to (a, minus a). A dashed line of slope one through (1,0) and (0,-1) is the fibre over (1,-1); it crosses the antidiagonal at (1/2, -1/2), marked with a red cross, which is not a lattice point. — **Figure 1.** $\mathrm{GL}_2/(\mathrm{GL}_1\times\mathrm{GL}_1)$: the fibres of $\varphi$ run along the centre, so $(a,0)$ lifts $(a,-a)$, while inside the $\mathrm{SL}_2$ line the fibre over $(1,-1)$ has no point.

Intuition on the smallest caseworked here

What $\varphi$ does. It keeps $x_1-x_2$ and kills the central direction: $$\ker\varphi=\mathbb Z\,(1,1)=X_*(Z(\mathrm{GL}_2)).$$
Why it is onto. The centre lets the lift leave the antidiagonal: $$(a,0)=\tfrac a2(1,-1)+\tfrac a2(1,1),\qquad\varphi(a,0)=(a,-a).$$
Contrast with $\mathrm{SL}_2$. Without the centre, $\Lambda_T=\mathbb Z(1,-1)=\Lambda_S$ and $\varphi$ doubles, as for $\mathrm{SL}_2/\mathrm{SO}_2$. Losing the centre also breaks $\mathrm{SL}_3/\mathrm{S}(\mathrm{GL}_1\times\mathrm{GL}_2)$ at $(1,0,-1)$ (Proposition III, remarks): $$\varphi(k,-k)=(2k,-2k),\qquad(1,-1)\notin\varphi\bigl(\Lambda_{T}(\mathrm{SL}_2)\bigr).$$
Larger $(p,q)$. The lift $(a_1,\dots,a_p,0,\dots,0)$ of section 5 works the same way, and $Z_{G^\theta}(S)=\mathbb G_m^p\times\mathrm{GL}_{q-p}$ stays connected, so H5 holds by Proposition III, step 6.

1 · Model

Let $n=p+q$ with $p\le q$. The involution $w$ of $\{1,\dots,n\}$ swaps $i\leftrightarrow n+1-i$ for $i\le p$ and fixes the middle indices $p\lt i\le q$; $J$ is its permutation matrix:

$$Je_i=e_{w(i)},\qquad J^2=1,\qquad\theta(g)=JgJ.$$

For example, for $p=1$ and $q=3$ the matrix swaps $e_1\leftrightarrow e_4$ and fixes $e_2,e_3$:

$$J=\begin{pmatrix}0&0&0&1\\0&1&0&0\\0&0&1&0\\1&0&0&0\end{pmatrix}.$$

$J$ has eigenvalue $+1$ on the span of the $e_i+e_{n+1-i}$ ($i\le p$) and the middle $e_i$, of dimension $q$, and eigenvalue $-1$ on the span of the $e_i-e_{n+1-i}$ ($i\le p$), of dimension $p$. So $G^\theta$, the centralizer of $J$, is a product of general linear groups and is connected:

$$G=\mathrm{GL}_n,\qquad G^\theta=Z_{\mathrm{GL}_n}(J)\cong\mathrm{GL}_q\times\mathrm{GL}_p.$$

$T$ is the diagonal torus, and $\theta$ permutes its entries by $w$. $S$ is the subtorus that $\theta$ inverts, of dimension $p$, the rank of $X$:

$$\theta(\operatorname{diag}(t_1,\dots,t_n))=\operatorname{diag}(t_{w(1)},\dots,t_{w(n)}),$$ $$S=\{\operatorname{diag}(s_1,\dots,s_p,1,\dots,1,s_p^{-1},\dots,s_1^{-1})\},\qquad\theta(s)=s^{-1}.$$

$T^\theta$ is the connected torus of diagonal matrices with $t_i=t_{n+1-i}$ for $i\le p$. Every element of $T$ is an element of $T^\theta$ times an element of $S$, so $T=(T^\theta)^\circ\cdot S$:

$$\operatorname{diag}(t_1,\dots,t_n)=\operatorname{diag}(c_1,\dots,c_p,t_{p+1},\dots,t_q,c_p,\dots,c_1)\cdot\operatorname{diag}(s_1,\dots,s_p,1,\dots,1,s_p^{-1},\dots,s_1^{-1}),$$ $$s_i^2=t_i/t_{n+1-i},\qquad c_i=t_i/s_i\qquad(i\le p).$$

2 · Lattices

A vector $x\in\mathbb Z^n$ is the cocharacter $s\mapsto\operatorname{diag}(s^{x_1},\dots,s^{x_n})$. $\theta$ permutes the coordinates by $w$, so its integer matrix is $J$ itself:

$$\Lambda_T=\mathbb Z^n,\qquad\theta=J,\qquad(\theta x)_i=x_{w(i)}.$$

The $(-1)$-eigenvectors vanish in the middle and change sign under $i\leftrightarrow n+1-i$:

$$\Lambda_S=\{(a_1,\dots,a_p,0,\dots,0,-a_p,\dots,-a_1):a_i\in\mathbb Z\},\qquad\varphi=1_n-J,$$ $$\varphi(x)=(x_1-x_n,\dots,x_p-x_{q+1},0,\dots,0,x_{q+1}-x_p,\dots,x_n-x_1).$$

3 · Positive roots

$B$ is the upper triangular Borel subgroup, with $\Phi^+=\{\varepsilon_i-\varepsilon_j:i\lt j\}$. $S$ acts on $e_i$ by $s_i$, $1$ and $s_{n+1-i}^{-1}$ for $i\le p$, $p\lt i\le q$ and $i\gt q$, so $L=Z_G(S)$ keeps $2p$ diagonal entries and the middle block:

$$L=\{\operatorname{diag}(t_1,\dots,t_p,g,u_p,\dots,u_1):g\in\mathrm{GL}_{q-p}\}\cong\mathbb G_m^{2p}\times\mathrm{GL}_{q-p},$$ $$\Phi_L=\{\varepsilon_i-\varepsilon_j:p\lt i,j\le q,\ i\ne j\}.$$

$\theta\varepsilon_i=\varepsilon_{w(i)}$, so $\theta$ fixes $\Phi_L$. Every other positive root $\varepsilon_i-\varepsilon_j$, $i\lt j$, falls in one of five cases, and in each $-\theta(\varepsilon_i-\varepsilon_j)=\varepsilon_{w(j)}-\varepsilon_{w(i)}$ is positive because $w(j)\lt w(i)$:

Table 1. Case and $-\theta(\varepsilon_i-\varepsilon_j)$ and Positive since
Case	$-\theta(\varepsilon_i-\varepsilon_j)$	Positive since
$i\le p\lt j\le q$	$\varepsilon_j-\varepsilon_{n+1-i}$	$j\le q\lt n+1-i$
$i\lt j\le p$	$\varepsilon_{n+1-j}-\varepsilon_{n+1-i}$	$n+1-j\lt n+1-i$
$i\le p,\ j\gt q$	$\varepsilon_{n+1-j}-\varepsilon_{n+1-i}$	$n+1-j\le p\lt n+1-i$
$p\lt i\le q\lt j$	$\varepsilon_{n+1-j}-\varepsilon_i$	$n+1-j\le p\lt i$
$q\lt i\lt j$	$\varepsilon_{n+1-j}-\varepsilon_{n+1-i}$	$n+1-j\lt n+1-i$

The remaining pairs $i\lt j$ lie in the middle block, in $\Phi_L$. So $B$ lies in a minimal θ-split parabolic.

4 · Centralizer

$Z_{G^\theta}(S)=G^\theta\cap L=L^\theta$. $\theta$ swaps the entries $t_i\leftrightarrow u_i$ of $L$ and fixes $g$, so the centralizer is

$$Z_{G^\theta}(S)=\{\operatorname{diag}(t_1,\dots,t_p,g,t_p,\dots,t_1):t_i\in\mathbb G_m,\ g\in\mathrm{GL}_{q-p}\}\cong\mathbb G_m^p\times\mathrm{GL}_{q-p}.$$

It is connected, as a product of connected groups. For $p=q$ the middle block is empty and the centralizer is the torus $\mathbb G_m^p$.

5 · Coverage

A cocharacter is dominant when its entries weakly decrease, so a label is dominant exactly when $a_1\ge\dots\ge a_p\ge0\ge-a_p\ge\dots\ge-a_1$:

$$\Lambda_T^+=\{x\in\mathbb Z^n:x_1\ge x_2\ge\dots\ge x_n\},$$ $$\Lambda_S^+=\{(a_1,\dots,a_p,0,\dots,0,-a_p,\dots,-a_1):a_1\ge\dots\ge a_p\ge0\}.$$

Given such a label $\mu$, take $\lambda=(a_1,\dots,a_p,0,\dots,0)$. Its entries weakly decrease, since $a_p\ge0$, and $\theta$ moves $a_1,\dots,a_p$ to the last $p$ coordinates in reverse order:

$$\lambda=(a_1,\dots,a_p,0,\dots,0)\in\Lambda_T^+,\qquad\theta\lambda=(0,\dots,0,a_p,\dots,a_1),$$ $$\varphi(\lambda)=(a_1,\dots,a_p,0,\dots,0,-a_p,\dots,-a_1)=\mu.$$

Case	\(-\theta(\varepsilon_i-\varepsilon_j)\)	Positive since
\(i\le p\lt j\le q\)	\(\varepsilon_j-\varepsilon_{n+1-i}\)	\(j\le q\lt n+1-i\)
\(i\lt j\le p\)	\(\varepsilon_{n+1-j}-\varepsilon_{n+1-i}\)	\(n+1-j\lt n+1-i\)
\(i\le p,\ j\gt q\)	\(\varepsilon_{n+1-j}-\varepsilon_{n+1-i}\)	\(n+1-j\le p\lt n+1-i\)
\(p\lt i\le q\lt j\)	\(\varepsilon_{n+1-j}-\varepsilon_i\)	\(n+1-j\le p\lt i\)
\(q\lt i\lt j\)	\(\varepsilon_{n+1-j}-\varepsilon_{n+1-i}\)	\(n+1-j\lt n+1-i\)