What this example is
\(X\) is the complex \((2n-1)\)-sphere, a symmetric space of rank one. With \(J\) and \(v=e_1+e_{2n}\) as in section 1, \(\mathrm{Spin}_{2n}\) moves \(v\) through its image \(\bar g\in\mathrm{SO}_{2n}\):
$$X=\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\xrightarrow{\ \sim\ }Q=\{x\in\mathbb C^{2n}:x^{\mathsf T}Jx=2\}$$ $$gG^\theta\longmapsto\bar g\,v.$$The map is well defined and injective because the stabilizer of \(v\) is \(\mathrm{Spin}(v^\perp)\), which contains the kernel \(\{\pm1\}\) of \(\mathrm{Spin}_{2n}\to\mathrm{SO}_{2n}\).
Involution. \(\theta\) is conjugation by the reflection \(r\) in \(v\):
$$\theta(g)=rgr^{-1},\qquad G^\theta=\mathrm{Spin}(v^\perp)\cong\mathrm{Spin}_{2n-1}.$$Orbits. \(S\) is one-dimensional, with image \(S_{\mathrm{SO}}\) in \(\mathrm{SO}_{2n}\), so the orbit labels form a ray. The orbit of a loop \(x\in Q(F)\) is its pole order:
$$S_{\mathrm{SO}}=\{\operatorname{diag}(s,1,\dots,1,s^{-1})\}$$ $$\Lambda_S^+=\{2m\varepsilon_1:m\ge0\}$$ $$\mathcal O_{2m\varepsilon_1}=\{x\in Q(F):\min_i\operatorname{val}(x_i)=-m\}$$ $$x_{2m\varepsilon_1}=t^me_1+t^{-m}e_{2n}.$$Why the pole order labels the orbits
- \(L^+G\) acts through \(\mathrm{SO}_{2n}(\mathcal O)\), \(\mathcal O=\mathbb C[\![t]\!]\), which preserves every lattice \(t^k\mathcal O^{2n}\) and hence \(\min_i\operatorname{val}(x_i)\).
- The dominant \(\lambda=(m,m,0,\dots,0)\) has \(\varphi(\lambda)=2m\varepsilon_1\) (section 5), so \(t^\lambda x_0\) is the orbit representative, and \(\bar t^{\lambda}=\operatorname{diag}(t^m,t^m,1,\dots,1,t^{-m},t^{-m})\): $$x_{2m\varepsilon_1}=t^\lambda x_0=\bar t^{\lambda}v=t^me_1+t^{-m}e_{2n}.$$
- Distinct orbits thus have representatives of distinct pole order, and \(LX\) is the disjoint union of the orbits, so the pole order separates them.
Low rank. The exceptional isomorphisms \(D_2=A_1\times A_1\) and \(D_3=A_3\) identify the first two members with other families:
$$\mathrm{Spin}_4/\mathrm{Spin}_3\cong(\mathrm{SL}_2\times\mathrm{SL}_2)/\mathrm{SL}_2$$ $$\mathrm{Spin}_6/\mathrm{Spin}_5\cong\mathrm{SL}_4/\mathrm{Sp}_4.$$The page proves both conditions of Proposition III for every \(n\ge2\).
Verdict
Both conditions of Proposition III hold: \(Z_{G^\theta}(S)\cong\mathrm{Spin}_{2n-2}\) is connected, and every \(\mu=2m\varepsilon_1\in\Lambda_S^+\) lifts to the dominant \(\lambda=m(\varepsilon_1+\varepsilon_2)\). So H1–H5 hold and \(\mathcal C^{\mathrm{Sat}}=\mathcal C\), as recorded in Proposition III, step 7.
Smallest case: \(n=2\)
For \(n=2\) the cocharacters live in the plane, and every lift used for larger \(n\) is \((m,m,0,\dots,0)\), so the plane \((x_1,x_2)\) shows the mechanism for all \(n\):
$$\Lambda_T=\{x\in\mathbb Z^2:x_1+x_2\ \text{even}\}$$ $$\Lambda_T^+=\{x_1\ge|x_2|\},\qquad\varphi(x)=2x_1\varepsilon_1.$$
Intuition on the smallest caseworked here
- What \(\varphi\) does. It reads the pole order of the translate, since \(\bar t^{\lambda}\) scales \(e_1\) and \(e_{2n}\) by \(t^{\pm x_1}\): $$t^\lambda x_0=t^{x_1}e_1+t^{-x_1}e_{2n},\qquad\varphi(\lambda)=2x_1\varepsilon_1.$$
- Why it is onto. \(x_1=m\) alone has odd sum when \(m\) is odd; the second coordinate repairs the parity without changing the pole order: $$\lambda=(m,m,0,\dots,0)\in\Lambda_T^+,\qquad\varphi(\lambda)=2m\varepsilon_1.$$
- Why \(\mathrm{Spin}\). For \(\mathrm{SO}_{2n}\) the lattice is \(\mathbb Z^n\), the label \(\varepsilon_1\) appears, and \(\varphi\) still only doubles; moreover \(G^\theta\cong\mathrm O_{2n-1}\) is disconnected: $$\Lambda_S(\mathrm{SO}_{2n})=\mathbb Z\varepsilon_1\supsetneq2\mathbb Z\varepsilon_1=\varphi\bigl(\Lambda_T(\mathrm{SO}_{2n})\bigr).$$
- For H5. \(Z_{G^\theta}(S)\cong\mathrm{Spin}_2\cong\mathbb G_m\) is connected, so the lift \((m,m)\) gives H5 at \(2m\varepsilon_1\) by Proposition III, step 6.
1 · Model
Let \(V=\mathbb C^{2n}\) carry the symmetric form \(x^{\mathsf T}Jy\), where \(J_{ij}=1\) if \(i+j=2n+1\) and \(J_{ij}=0\) otherwise. In this basis \(\mathrm{SO}_{2n}=\mathrm{SO}(V)\) has the diagonal maximal torus:
$$T_{\mathrm{SO}}=\{\operatorname{diag}(t_1,\dots,t_n,t_n^{-1},\dots,t_1^{-1})\}.$$\(\theta\) is conjugation by the reflection \(r\) in \(v=e_1+e_{2n}\). Since \(v^{\mathsf T}Jv=2\), the reflection \(r(x)=x-(x^{\mathsf T}Jv)\,v\) sends \(e_1\mapsto-e_{2n}\) and \(e_{2n}\mapsto-e_1\) and fixes \(e_2,\dots,e_{2n-1}\); in blocks of sizes \(1,2n-2,1\):
$$r=\begin{pmatrix}0&0&-1\\0&1_{2n-2}&0\\-1&0&0\end{pmatrix},\qquad r^2=1,\qquad\theta(g)=rgr^{-1}.$$Conjugation by a lift of \(r\) to \(\mathrm{Pin}_{2n}\) lifts \(\theta\) to \(G=\mathrm{Spin}_{2n}\). Since \(\mathrm{Spin}_{2n}\) is simply connected, \(G^\theta\) is connected (Steinberg 1968, Theorem 8.1); it is the spin group of \(v^\perp\), with image \(\mathrm{SO}(v^\perp)=\{g\in\mathrm{SO}_{2n}:gv=v\}\):
$$G=\mathrm{Spin}_{2n},\qquad G^\theta=\mathrm{Spin}(v^\perp)\cong\mathrm{Spin}_{2n-1}.$$\(\theta\) inverts the first entry of \(T_{\mathrm{SO}}\) and fixes the others:
$$\theta\operatorname{diag}(t_1,t_2,\dots,t_2^{-1},t_1^{-1})=\operatorname{diag}(t_1^{-1},t_2,\dots,t_2^{-1},t_1).$$So \(\theta\) inverts \(S_{\mathrm{SO}}\) below, and every element of \(T_{\mathrm{SO}}\) is a θ-fixed element times an element of \(S_{\mathrm{SO}}\):
$$S_{\mathrm{SO}}=\{\operatorname{diag}(s,1,\dots,1,s^{-1})\},$$ $$\operatorname{diag}(t_1,t_2,\dots,t_2^{-1},t_1^{-1})=\operatorname{diag}(1,t_2,\dots,t_2^{-1},1)\cdot\operatorname{diag}(t_1,1,\dots,1,t_1^{-1}).$$In \(\mathrm{Spin}_{2n}\), \(T\) is the preimage of \(T_{\mathrm{SO}}\) and \(S\) is the identity component of the preimage of \(S_{\mathrm{SO}}\). \(\theta\) inverts \(S\), and \(\dim S=1\) is the rank of \(X\).
The connected group \((T^\theta)^\circ\cdot S\) has the Lie algebra of \(T\), the sum of the θ-fixed and θ-inverted parts. So \(T=(T^\theta)^\circ\cdot S\).
2 · Lattices
\(X_*(T_{\mathrm{SO}})=\mathbb Z^n\), where \(x\) is the cocharacter \(s\mapsto\operatorname{diag}(s^{x_1},\dots,s^{x_n},s^{-x_n},\dots,s^{-x_1})\); \(\varepsilon_i\) denotes a coordinate vector, of cocharacters and of characters alike, paired by the dot product. Since \(\mathrm{Spin}_{2n}\) is simply connected, \(\Lambda_T\) is the coroot lattice of type \(D_n\):
$$\Lambda_T=\{x\in\mathbb Z^n:x_1+\dots+x_n\ \text{even}\},\qquad\theta=\operatorname{diag}(-1,1,\dots,1).$$\(\theta\) negates \(x_1\), which keeps the sum even. Its \((-1)\)-eigenvectors in \(\Lambda_T\) are the even multiples of \(\varepsilon_1\):
$$\Lambda_S=\mathbb Z\cdot2\varepsilon_1,\qquad\varphi=1-\theta=\operatorname{diag}(2,0,\dots,0),\qquad\varphi(x)=2x_1\varepsilon_1.$$3 · Positive roots
The roots are \(\pm\varepsilon_i\pm\varepsilon_j\) with \(i\lt j\). \(B\) is the preimage of the upper triangular matrices in \(\mathrm{SO}_{2n}\); for \(i\lt j\) the root spaces below are upper triangular, with \(E_{k,l}\) the matrix units:
$$\mathfrak g_{\varepsilon_i-\varepsilon_j}=\mathbb C\,(E_{i,j}-E_{2n+1-j,\,2n+1-i}),\qquad\mathfrak g_{\varepsilon_i+\varepsilon_j}=\mathbb C\,(E_{i,\,2n+1-j}-E_{j,\,2n+1-i}),$$ $$\Phi^+=\{\varepsilon_i-\varepsilon_j,\ \varepsilon_i+\varepsilon_j:i\lt j\}.$$\(S_{\mathrm{SO}}\) acts on \(e_1\), on \(W=\operatorname{span}(e_2,\dots,e_{2n-1})\) and on \(e_{2n}\) by \(s\), \(1\) and \(s^{-1}\). So \(L=Z_G(S)\) is the preimage of \(\{\operatorname{diag}(c,h,c^{-1}):h\in\mathrm{SO}(W)\}\), and its roots are those without \(\varepsilon_1\):
$$\Phi_L=\{\pm\varepsilon_i\pm\varepsilon_j:2\le i\lt j\le n\}\qquad(\text{type }D_{n-1},\ \text{empty for }n=2).$$\(\theta\) fixes \(\varepsilon_2,\dots,\varepsilon_n\), so \(\theta\alpha=\alpha\) on \(\Phi_L\). The other positive roots are \(\varepsilon_1\pm\varepsilon_j\), and \(\theta\varepsilon_1=-\varepsilon_1\):
$$-\theta(\varepsilon_1-\varepsilon_j)=\varepsilon_1+\varepsilon_j\in\Phi^+,\qquad-\theta(\varepsilon_1+\varepsilon_j)=\varepsilon_1-\varepsilon_j\in\Phi^+.$$So \(B\) lies in a minimal θ-split parabolic.
4 · Centralizer
Work first in \(\mathrm{SO}_{2n}\). An element of \(\mathrm{SO}(v^\perp)\) that commutes with \(S_{\mathrm{SO}}\) preserves \(\mathbb Ce_1\), \(W\) and \(\mathbb Ce_{2n}\) and fixes \(v=e_1+e_{2n}\), so it is the identity on \(e_1\) and \(e_{2n}\):
$$Z_{\mathrm{SO}(v^\perp)}(S_{\mathrm{SO}})=\left\{\begin{pmatrix}1&0&0\\0& h&0\\0&0&1\end{pmatrix}:h\in\mathrm{SO}(W)\right\}\cong\mathrm{SO}_{2n-2}.$$In \(\mathrm{Spin}_{2n}\), an element \(g\in G^\theta\) commutes with \(S\) exactly when its image commutes with \(S_{\mathrm{SO}}\), since \(s\mapsto gsg^{-1}s^{-1}\) maps the connected \(S\) into the kernel \(\{\pm1\}\) and sends \(1\) to \(1\). So the centralizer is the preimage of \(\mathrm{SO}(W)\):
$$Z_{G^\theta}(S)=\mathrm{Spin}(W)\cong\mathrm{Spin}_{2n-2}.$$It is connected: \(\dim W=2n-2\ge2\), so \(\mathrm{Spin}(W)\) contains the kernel \(-1\) in its identity component.
5 · Coverage
Dominance means \(\langle\varepsilon_i-\varepsilon_{i+1},x\rangle\ge0\) for \(i\lt n\) and \(\langle\varepsilon_{n-1}+\varepsilon_n,x\rangle\ge0\):
$$\Lambda_T^+=\{x\in\Lambda_T:x_1\ge\dots\ge x_{n-1}\ge|x_n|\},\qquad\Lambda_S^+=\{2m\varepsilon_1:m\ge0\}.$$Given \(\mu=2m\varepsilon_1\) with \(m\ge0\), take \(\lambda=m(\varepsilon_1+\varepsilon_2)\). It lies in \(\Lambda_T^+\), since its sum \(2m\) is even and \(x_1=x_2=m\ge0\) with all other coordinates \(0\), and \(\theta\) negates only its first coordinate:
$$\lambda=(m,m,0,\dots,0)\in\Lambda_T^+,\qquad\theta\lambda=(-m,m,0,\dots,0),\qquad\varphi(\lambda)=(2m,0,\dots,0)=\mu.$$The dominant lifts of \(2m\varepsilon_1\) are exactly the dominant \(x\) with \(x_1=m\), such as \(m\varepsilon_1\) for even \(m\) and \(m\varepsilon_1+\varepsilon_2\) for odd \(m\).
The simply connected form matters. For \(\mathrm{SO}_{2n}\) itself, \(G^\theta\cong\mathrm O_{2n-1}\) is disconnected, and \(\Lambda_T=\mathbb Z^n\) gives \(\Lambda_S=\mathbb Z\varepsilon_1\), which \(\varphi(x)=2x_1\varepsilon_1\) misses at \(\varepsilon_1\).