1 · The question
Proposition III reduces \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) to three checks at each finite level \((N,M)\) of \(LX\):
$$\underbrace{\text{I1 finitely many pieces},\qquad\text{I2 connected stabilizers},\qquad\text{I3 each piece reached by a Hecke translate}}_{\text{Proposition III}}\ \Longrightarrow\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I .$$Proposition IV runs the checks on two \(X\), one subpage each: the group, where I3 fails, and \(\mathbb G_m\), the smallest positive case. \(\mathbb A^n\) under \(\mathrm{GL}_n\), where generation holds although every orbit closure is infinite, is proved directly as Proposition IIa.
2 · Proposition IV
Proposition IV · Case studiesProved
- IV.2\(X=H\) under \(H\times H\), \(W\ne1\): I3 fails at every closed piece \(I\omega I\), \(\omega\in\Omega\). Consistently, \(\mathcal C_I^{\mathrm{Iwa}}\ne\mathcal C_I\) (Proposition II).
- IV.3\(X=\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\): I1, I2 and I3 hold, so \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\).
3 · Side by side
| Case | Pieces of level \((N,M)\) | I3 | \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) |
|---|---|---|---|
| \(\mathbb A^n\) (IIa) | a chain \(P_{nM}<P_{nM-1}<\cdots<P_{-nN}\) | holds: the translate \(G_{\tau^k}=\underline k_{\tau^k\mathcal O^n}\) per piece | yes |
| \(H\) (IV.2) | the double cosets \(IwI\), for \(M\gg0\) | fails at the closed \(I\omega I\) | no (Proposition II) |
| \(\mathbb G_m\) (IV.3) | the orbits \(\varpi^m\mathcal O^\times\), incomparable | holds | yes |
The difference is at the closed pieces, where only \(Y=\mathrm{IC}_0\) is allowed in I3: for \(\mathbb A^n\) the closed piece \(T_M=t^M\mathcal O^n\) is itself a translate of \(\mathrm{IC}_0\); for the group every translate of \(\mathrm{IC}_0\) is killed by the functor \(\Phi\) of Proposition II, and \(\Delta_\omega\) is not.