§1 · Agent 1 · Literature and Table 1 audit · 2026-10-18T18:44:04Z

Lemma 1.4 · GL(2)/Sp(2) = \(\mathbb G_m\): hypotheses checked and every simple object is a summand

proved

For the torus-like row every \(I\)-orbit is a closed piece with connected stabilizer and a rational point, and every simple object is a nonzero graded multiple of a convolution \(\mathrm{IC}_{t^\lambda}\ast_I\mathrm{IC}_0\); both levels hold.

Lemma 1.4.

Let \(X=\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\), \(g\cdot x=\det(g)x\). In all three settings:

  1. \((LX)_{\mathrm{red}}=\bigsqcup_{m\in\mathbb Z}t^m L^+\mathbb G_m\), and the \(I\)-orbits are the \(t^m\mathcal O^\times\) (each open and closed). (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).) \(Y_N=\bigsqcup_{|m|\le N}t^m\mathcal O^\times\), \(Y_{N,M}=\bigsqcup_{|m|\le N}t^m(\mathcal O/t^M)^\times\); the pieces are the orbits.
  2. (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) The stabilizer of \(t^m\bmod t^M\) in \(I/I_{N,M}\) is \(\{g\in I:\det g\in1+t^M\mathcal O\}/I_{N,M}\), connected. (Frob) \(t^m\) is rational; \(I(\mathcal O)\backslash X(F)=F^\times/\mathcal O^\times\) for every \(F\).
  3. \(L^+X=\mathcal O^\times\) is one \(K\)-orbit, closed, pro-smooth; \(\mathrm{IC}_0=\underline k_{\mathcal O^\times}\).
  4. Simple objects: \(\underline k_{t^m\mathcal O^\times}\) (shifted), \(m\in\mathbb Z\). For \(\langle\det,\lambda\rangle=m\), \(\mathrm{IC}_{t^\lambda}\ast_I\mathrm{IC}_0\simeq IH^*(\overline{It^\lambda I/I})\otimes\underline k_{t^m\mathcal O^\times}\ne0\), so \(M_I\) is everything.
  5. Status: sheaves \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) (case-gm.html, IV.3); functions (IG) holds for every \(F\) (classical.html, Proposition G, \(n=1\)).

Proof.

  1. \(LX=L\mathbb G_m\) is non-reduced, but constructible sheaves see only \((L\mathbb G_m)_{\mathrm{red}}=\mathbb Z\times L^+\mathbb G_m\). \(\det(I)=\det(K)=\mathcal O^\times\), so \(I\) acts on \(t^m\mathcal O^\times\) transitively.
  2. \(\{g\in I:\det g\in 1+t^M\mathcal O\}=(I\cap\mathrm{SL}_2(\mathcal O))\rtimes\{\mathrm{diag}(1,u):u\in1+t^M\mathcal O\}\); the first factor is the Iwahori of \(\mathrm{SL}_2\) (torus times pro-unipotent), the second pro-unipotent. Its image modulo \(I_{N,M}\) is connected. This is the stabilizer computation behind case-gm.html, step 1, written out.
  3. \(\det:K\to\mathcal O^\times\) is surjective.
  4. The connected components of \(L\mathrm{GL}_2\) are the fibres of \(\operatorname{val}\det\), so \(\overline{It^\lambda I}\subseteq\{\operatorname{val}\det=m\}\) and \(m:[x,u]\mapsto\det(x)u\) maps \(\overline{It^\lambda I}\times^I\mathcal O^\times\) onto \(t^m\mathcal O^\times\); \([x,u]\mapsto(xI,\det(x)u)\) identifies it with \(\overline{It^\lambda I/I}\times t^m\mathcal O^\times\), \(m\) the projection. Base change.
  5. Cited.

Builds on

  • Setup 1.1 · Setting audit: D-modules, Betti and ℓ-adic give the same \(\mathcal C_I\) and \(\ast_I\)

Built on by

  • Lemma 1.6 · Dictionary inputs at Iwahori level: rational orbits, Lang, trace bases, convolution
  • Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument
  • Summary 1.10 · Verdict of the audit: rows 1–3 agree, the rest reduce to one module computation
Original record, as the agent wrote it at 18:44 UTC

GL(2)/Sp(2) = \(\mathbb G_m\): hypotheses checked and every simple object is a summand

Proof.

  1. \(LX=L\mathbb G_m\) is non-reduced, but constructible sheaves see only \((L\mathbb G_m)_{\mathrm{red}}=\mathbb Z\times L^+\mathbb G_m\). \(\det(I)=\det(K)=\mathcal O^\times\), so \(I\) acts on \(t^m\mathcal O^\times\) transitively.
  2. \(\{g\in I:\det g\in 1+t^M\mathcal O\}=(I\cap\mathrm{SL}_2(\mathcal O))\rtimes\{\mathrm{diag}(1,u):u\in1+t^M\mathcal O\}\); the first factor is the Iwahori of \(\mathrm{SL}_2\) (torus times pro-unipotent), the second pro-unipotent. Its image modulo \(I_{N,M}\) is connected. This is the stabilizer computation behind case-gm.html, step 1, written out.
  3. \(\det:K\to\mathcal O^\times\) is surjective.
  4. The connected components of \(L\mathrm{GL}_2\) are the fibres of \(\operatorname{val}\det\), so \(\overline{It^\lambda I}\subseteq\{\operatorname{val}\det=m\}\) and \(m:[x,u]\mapsto\det(x)u\) maps \(\overline{It^\lambda I}\times^I\mathcal O^\times\) onto \(t^m\mathcal O^\times\); \([x,u]\mapsto(xI,\det(x)u)\) identifies it with \(\overline{It^\lambda I/I}\times t^m\mathcal O^\times\), \(m\) the projection. Base change.
  5. Cited.

Sources