7 · Example · contrast

The group \(\mathrm{SL}_2\): same orbits, no doubling

This is the classical case of geometric Satake. Its orbit poset is the same as in the running example, but its Hecke map is different. That difference is why odd orbits are never top orbits in the running example (Part III).

1 Setup
Definition \(G'=\mathrm{SL}_2\times\mathrm{SL}_2\) acts on \(X=\mathrm{SL}_2\) by \((g,h)\cdot x=g\,x\,h^{-1}\). This is the symmetric variety \(G'/\Delta\mathrm{SL}_2\) for \(\theta(g,h)=(h,g)\), with \(\tau(g,h)\) identified with \(gh^{-1}\). The arc group is \(K\times K\), and for \(\lambda=((m_1,-m_1),(m_2,-m_2))\) one has \(D_{\le\lambda}=D_{\le m_1}\times D_{\le m_2}\).
Proved (cited) The \(K\times K\)-orbits on \(LX=\mathrm{SL}_2(F)\) are the Cartan double cosets \(\mathcal O_a:=Kt^{(a,-a)}K\), and \(\overline{\mathcal O}_a=\mathcal O_0\sqcup\dots\sqcup\mathcal O_a\) [Zhu, Prop. 2.1.5]. The index \(a\) is the pole order of \(x\), exactly as for \(\mathrm{SL}_2/\mathrm{SO}_2\).

2 The picture: points are vertices
Proved here Send \(x\) to the vertex \(xv_0\). Then \(({\kappa}_1,{\kappa}_2)\cdot x\) goes to \({\kappa}_1xv_0\), at the same distance from \(v_0\), and by Serre's criterion (Part II, step 6) \[ x\in\mathcal O_a\iff \operatorname{dist}(v_0,xv_0)=2a . \]

Click a filled vertex to choose \(xv_0\). In the running example a point is a line, and its index is the distance to the line. Here a point gives a vertex, and its index is half the distance.

3 Hecke translation
Proved here For \(g\in D_{\le m_1}\), \(h\in D_{\le m_2}\) and \(x\in\overline{\mathcal O}_a\), apply the triangle inequality twice: \[ \operatorname{dist}(v_0,\,gxh^{-1}v_0)\le \underbrace{\operatorname{dist}(v_0,gv_0)}_{\le 2m_1}+\underbrace{\operatorname{dist}(v_0,xv_0)}_{\le 2a}+\underbrace{\operatorname{dist}(xv_0,xh^{-1}v_0)}_{=\operatorname{dist}(v_0,h^{-1}v_0)\,\le 2m_2}, \] so \((D_{\le m_1}\times D_{\le m_2})\cdot\overline{\mathcal O}_a\subset\overline{\mathcal O}_{a+m_1+m_2}\).

Every orbit is a top. Take \(\lambda=((a,-a),(0,0))\). Then \(t^\lambda\cdot x_0=t^{(a,-a)}\in\mathcal O_a\) for every \(a\), odd or even.

4 Side by side
Table 1. and a point of \(LX\) is and Hecke of size \(\lambda\) moves the index by
a point of \(LX\) isHecke of size \(\lambda\) moves the index bytops of \(D_{\le\lambda}\cdot x_0\)
\(\mathrm{SL}_2/\mathrm{SO}_2\)a line \(\ell_A\)\(2m\) (one \(g\), on both sides)even \(a\) only
group \(\mathrm{SL}_2\)a vertex \(xv_0\) (index = half the distance)\(m_1+m_2\) (two independent elements)every \(a\)

Both cases have the orbit poset \(0<1<2<\cdots\). The difference is the Hecke map \(\lambda\mapsto\overline\lambda\). In the group case \(g\) and \(h\) are independent, so the two sides of \(x\) can move by different amounts. In the symmetric case \(\tau(g\cdot x)=g\,\tau(x)\,\theta(g)^{-1}\) uses one \(g\) on both sides.