Iwahori generation · Proposition IV.3

\(\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\)

The smallest positive case: every piece is a closed orbit and a translate of \(\mathrm{IC}_0\).

1 · The case

\(X=\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\). \(\mathrm{GL}_2\) acts through \(\det\), and \(\det(I)=\det(K)=\mathcal O^\times\):

$$LX=F^\times,\qquad I\backslash LX=K\backslash LX=F^\times/\mathcal O^\times\simeq\mathbb Z,\qquad\Delta_{t^\lambda}\ast_I\mathrm{IC}_0=\underline k_{\varpi^{\langle\det,\lambda\rangle}\mathcal O^\times}\ \text{(up to shift)} .$$

Every piece is closed and every closed piece is a translate of \(\mathrm{IC}_0\), so the conditions of Proposition III hold with no induction.

2 · Proposition IV.3

Proposition IV.3 · GL(2)/Sp(2) ≃ 𝔾ₘProved

For \(X=\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\), I1, I2 and I3 hold; hence

$$\boxed{\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I .\ }$$

Proof

Step 1 checks I1 and I2; step 2 checks I3.

  1. Result I1 and I2: every orbit is a closed piece with connected stabilizer.

    $$\mathcal P_{N,M}=\{\varpi^m\mathcal O^\times\}\ \text{at every level }M\ge1,\qquad\text{pairwise incomparable, each closed, each with connected stabilizer.}$$

    Proof
    • \(\mathrm{GL}_2\) acts through \(\det\), and \(\det(I)=\det(K)=\mathcal O^\times\). Every \(I\)-orbit \(\varpi^m\mathcal O^\times\) is closed with connected stabilizer and is a piece at every level \(M\ge1\); I1 and I2 hold, and the pieces of a level are pairwise incomparable.
  2. Result I3: one translate of \(\mathrm{IC}_0\) per orbit.

    $$\langle\det,\lambda\rangle=m:\qquad\Delta_{t^\lambda}\ast_I\mathrm{IC}_0\simeq\underline k_{\varpi^m\mathcal O^\times}\ \text{(up to shift)},\qquad Y=\mathrm{IC}_0 .$$

    Proof
    • For \(\varpi^m\mathcal O^\times\) take \(\mathcal G=\Delta_{t^\lambda}\) with \(\langle\det,\lambda\rangle=m\) and \(Y=\mathrm{IC}_0\); \(\mathcal G\ast_I\mathrm{IC}_0\) is the pushforward of a constant sheaf along \(It^\lambda I\times^I\mathcal O^\times\to\varpi^m\mathcal O^\times\), whose fibres are affine spaces: a nonzero constant sheaf on that single orbit.
Remarksno induction · beyond tori