(H×H)/H · Proposition III family

Verdict

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

Proposition II checks H1–H5 for this family directly, on the affine Grassmannian of $H$.

Smallest case: $H=\mathrm{SL}_2$

$H=\mathbb G_m$ has no roots, so every label is dominant and lifts trivially; $H=\mathrm{SL}_2$ is the first case with a dominance condition. Write $(a,b)$ for $(a\alpha^\vee,b\alpha^\vee)$, so $\Lambda_T=\mathbb Z^2$:

$$\Lambda_T^+=\{(a,b):a\ge0\ge b\},\qquad\Lambda_S^+=\{(c,-c):c\ge0\},$$ $$\varphi(a,b)=(a-b,\,b-a).$$

The integer plane with coordinates a and b. The dominant quadrant a at least 0, b at most 0 is shaded and its points are blue. The antidiagonal line of points (c, minus c) is the split lattice; its points with c from 0 to 4 are green, all hit. Vertical arrows send (c, 0) to (c, minus c) for c equal to 1, 2, 3. A dashed segment joins the four dominant points (3,0), (2,-1), (1,-2), (0,-3) that map to (3,-3). — **Figure 1.** $(\mathrm{SL}_2\times\mathrm{SL}_2)/\mathrm{SL}_2$: every orbit label $(c,-c)$ is hit, by $(c,0)$ and by the other dominant points of its fibre (dashed, $c=3$).

Intuition on the smallest caseworked here

What $\varphi$ does. It keeps the difference $a-b$ and kills the cocharacters of the θ-fixed torus $T^\theta=\{(c,c)\}$: $$\varphi(a,b)=(a-b)\,(1,-1),\qquad\ker\varphi=\mathbb Z\,(1,1).$$
Why it is onto. On the quadrant $a\ge0\ge b$ the difference $a-b$ takes every value $c\ge0$, in exactly $c+1$ ways: $$\varphi^{-1}(c,-c)\cap\Lambda_T^+=\{(c+b,\,b):-c\le b\le0\}.$$
Geometry. Under $X\cong H$, $(g_1,g_2)\mapsto g_1g_2^{-1}$, the translate is a Cartan double coset of $\mathrm{SL}_2$: $$t^{(a,b)}x_0\longmapsto t^{(a-b)\alpha^\vee}\in L^+H\,t^{(a-b)\alpha^\vee}L^+H=\mathcal O_{(a-b,\,b-a)}.$$
For H5. $Z_{G^\theta}(S)\cong\mathbb G_m$ is connected, so the lift $(c,0)$ gives H5 at $(c,-c)$ by Proposition III, step 6.

1 · Model

$H$ is a connected complex reductive group with maximal torus $T_H$ and Borel subgroup $B_H\supseteq T_H$. The swap involution of $G=H\times H$ fixes the diagonal copy of $H$, which is connected:

$$G=H\times H,\qquad\theta(g_1,g_2)=(g_2,g_1),\qquad G^\theta=\{(h,h):h\in H\}.$$

For $H=\mathrm{GL}_m$, embed $G$ block-diagonally in $\mathrm{GL}_{2m}$. Then $\theta$ is conjugation by the block swap, where $1_m$ is the identity matrix:

$$\theta\begin{pmatrix}g_1&0\\0& g_2\end{pmatrix}=\begin{pmatrix}0&1_m\\1_m&0\end{pmatrix}\begin{pmatrix}g_1&0\\0& g_2\end{pmatrix}\begin{pmatrix}0&1_m\\1_m&0\end{pmatrix}=\begin{pmatrix}g_2&0\\0& g_1\end{pmatrix}.$$

$T$ is the product of two copies of $T_H$, and $S$ is its antidiagonal. For $\mathrm{GL}_m$ both are diagonal:

$$T=T_H\times T_H=\{(\operatorname{diag}(t_1,\dots,t_m),\operatorname{diag}(u_1,\dots,u_m))\},$$ $$S=\{(s,s^{-1}):s\in T_H\}=\{(\operatorname{diag}(s_1,\dots,s_m),\operatorname{diag}(s_1^{-1},\dots,s_m^{-1}))\}.$$

$\theta$ swaps the two diagonals, so it inverts $S$:

$$\theta(t,u)=(u,t),\qquad\theta(s,s^{-1})=(s^{-1},s)=(s,s^{-1})^{-1}.$$

$S$ has dimension $\operatorname{rank}H$, the rank of $X$, so it is a maximal θ-split torus.

$T^\theta=\{(c,c):c\in T_H\}$ is the diagonal copy of $T_H$, a connected torus. Every element of $T$ is an element of $T^\theta$ times an element of $S$, so $T=(T^\theta)^\circ\cdot S$:

$$(t,u)=(c,c)\cdot(s,s^{-1}),\qquad s_i^2=t_i/u_i,\qquad c_i=t_i/s_i.$$

2 · Lattices

Write a cocharacter of $T$ as a pair $(a,b)$ with $a,b\in\Lambda_H=X_*(T_H)$. For $\mathrm{GL}_m$, $\Lambda_H=\mathbb Z^m$, and $a$ is the cocharacter $s\mapsto\operatorname{diag}(s^{a_1},\dots,s^{a_m})$.

$$\Lambda_T=\Lambda_H\oplus\Lambda_H,\qquad\theta=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad\theta(a,b)=(b,a).$$

The blocks are identity maps of $\Lambda_H$; for $\mathrm{GL}_m$, $\theta$ is the $2m\times2m$ block swap of section 1. Its $(-1)$-eigenlattice is the antidiagonal:

$$\Lambda_S=\{(a,-a):a\in\Lambda_H\},\qquad\varphi=1-\theta=\begin{pmatrix}1&-1\\-1&1\end{pmatrix},\qquad\varphi(a,b)=(a-b,\,b-a).$$

3 · Positive roots

Take the Borel subgroup $B=B_H\times B_H^-$, with $B_H^-$ opposite to $B_H$. For $\mathrm{GL}_m$ it is upper triangular in the first factor and lower triangular in the second, and $\Phi_H^+=\{\varepsilon_i-\varepsilon_j:i\lt j\}$:

$$\Phi^+=\{(\alpha,0):\alpha\in\Phi_H^+\}\cup\{(0,-\alpha):\alpha\in\Phi_H^+\}.$$

No root vanishes on $S$: $(\alpha,0)$ and $(0,-\alpha)$ both pair with $(\alpha^\vee,-\alpha^\vee)\in\Lambda_S$ to $\langle\alpha,\alpha^\vee\rangle=2$. So $L=Z_G(S)=T$ and $\Phi_L=\varnothing$.

$\theta$ swaps the two components of a root, so $-\theta$ exchanges the two halves of $\Phi^+$:

$$-\theta(\alpha,0)=(0,-\alpha)\in\Phi^+,\qquad-\theta(0,-\alpha)=(\alpha,0)\in\Phi^+\qquad(\alpha\in\Phi_H^+).$$

Hence $B$ is itself a minimal θ-split parabolic:

$$B\cap\theta(B)=(B_H\cap B_H^-)\times(B_H^-\cap B_H)=T_H\times T_H=T.$$

4 · Centralizer

$(h,h)$ commutes with every $(s,s^{-1})$ exactly when $h$ commutes with $T_H$, that is, when $h\in T_H$. For $\mathrm{GL}_m$ the centralizer consists of the pairs $(\operatorname{diag}(t_1,\dots,t_m),\operatorname{diag}(t_1,\dots,t_m))$:

$$Z_{G^\theta}(S)=\{(t,t):t\in T_H\}\cong T_H.$$

It is a torus, hence connected.

5 · Coverage

A pair is dominant when $a$ is dominant for $B_H$ and $b$ for $B_H^-$. For $\mathrm{GL}_m$ this reads $a_1\ge\dots\ge a_m$ and $b_1\le\dots\le b_m$:

$$\Lambda_T^+=\{(a,b):a\in\Lambda_H^+,\ -b\in\Lambda_H^+\},\qquad\Lambda_S^+=\{(a,-a):a\in\Lambda_H^+\}.$$

Given $\mu=(a,-a)$ with $a\in\Lambda_H^+$, take $\lambda=(a,0)$. It is dominant, since $a\in\Lambda_H^+$ and $-0=0\in\Lambda_H^+$, and $\varphi$ returns $\mu$:

$$\lambda=(a,0)\in\Lambda_T^+,\qquad\varphi(a,0)=(a-0,\,0-a)=(a,-a)=\mu.$$