§2 · Agent 2 · SL₂/SO₂: central inertia · 2026-10-17T02:33:26Z

Lemma 2.6 · Each Satake object is a single IC sheaf: $V_{n\alpha^\vee}\ast\IC_0\simeq\IC(\overline{\mathcal O_{2n}})$

proved audit: correct

Second audit: correct. Both slice computations re-derived and checked in exact Gaussian-rational arithmetic (sl2_slices_G1.py): the x₁-slice is exactly {e² = −c⁴}; the x₀-slice is the cone {PR = Q²} plus the planes P_±, meeting the cone along the two lines S⁰ ∩ O₁.

Because the convolution map is finite and an isomorphism over the top orbit, pushing forward the IC sheaf of the convolution space gives exactly the IC sheaf of $\overline{\mathcal O_{2n}}$. No decomposition theorem is needed; explicit slices confirm the stalk ranks predicted by the fibre counts.

Lemma 2.6.

For every $n\ge0$, $$V_{n\alpha^\vee}\ast\IC_0\simeq\IC(\overline{\mathcal O_{2n}},\underline k)=:\IC_{2n}$$ up to the normalisation shift. Likewise, for the $\PGL_2$-action of Lemma 2.2, $V^{\PGL_2}_{m\omega^\vee}\mathbin{\bar\ast}\IC_0\simeq\IC_m$ for every $m\ge0$.

Reduction to a pushforward. $Z_n$ is Zariski-locally $\Gr^{\le n\alpha^\vee}\times L^+X$ (the torsor $LG\to\Gr$ is locally trivial) and $L^+X$ is formally smooth, so $V_{n\alpha^\vee}\mathbin{\widetilde\boxtimes}\IC_0=\IC(\Gr^{\le n\alpha^\vee})\mathbin{\widetilde\boxtimes}\underline k_{L^+X}=\IC(Z_n)$, extended by zero. Since $m_n$ is proper, $V_{n\alpha^\vee}\ast\IC_0=m_{n*}\IC(Z_n)$.
A general fact. Let $\nu:Z\to Y$ be finite with $Z$ irreducible, and an isomorphism over a dense open $U\subseteq Y$. Then $\nu_{*}\IC_Z\simeq\IC_Y$. Indeed $\nu_{*}$ is t-exact for finite $\nu$ and $\nu_{*}\IC_Z\vert_U=\IC_Y\vert_U$. For $i:D=Y\setminus U\hookrightarrow Y$, $i':\nu^{-1}D\hookrightarrow Z$ and a perverse sheaf $\mathcal G$ on $D$, adjunction and base change give $\Hom(\nu_{*}\IC_Z,i_{*}\mathcal G)=\Hom(\IC_Z,i'_{*}\nu'^!\mathcal G)$, which vanishes because $\nu'^!\mathcal G$ is in perverse degrees $\ge0$ and $\IC_Z=j'_{!*}$ has no nonzero quotient supported on $Z\setminus\nu^{-1}U$; dually there is no nonzero subobject supported on $D$ (characterisation of $j_{!*}$, Beilinson–Bernstein–Deligne, §1.4). Apply this with $\nu=m_n$, $U=\mathcal O_{2n}$ (Computation 2.5, parts 3 and 1). On reducible slice models it holds componentwise (Computation 2.5: distinct components go birationally to distinct components), and the IC sheaf of a reducible variety is the sum of those of its components.

Consistency checks. With Chen–Yi's slices $S^\lambda=\tau^{-1}(W^\lambda\cap\tau(LX))$, $W^\lambda=(L^{<0}G\cap\operatorname{Ad}_{t^\lambda}L^{<0}G)\,t^\lambda$ (Prop. 47: $S^\lambda\cap\mathcal O_\lambda=\{x_\lambda\}$ and $L^+G\times S^\lambda\to LX$ is formally smooth), $\IC_{2n}$ restricted to $S^\mu$ is the IC sheaf of the slice, whose stalk at $x_\mu$ must have rank $s_n(\mu)$ by the lemma.

$S^{\alpha^\vee}\cap\overline{\mathcal O_2}=\{h\,t^{\alpha^\vee}\}$ with

$$h=\begin{pmatrix}1+ct^{-1}+c^2t^{-2}+c^3t^{-3}&e\,t^{-1}\\ e\,t^{-3}&1-ct^{-1}\end{pmatrix},\qquad e^2=-c^4:$$

two smooth branches $e=\pm ic^2$ through $x_1$ (a tacnode). The IC sheaf of a curve is the pushforward from its normalisation, with stalk $k^2$ at $x_1$, equal to $s_1(1)=2$.

$S^0\cap\overline{\mathcal O_2}$ consists of the symmetric $h\in L^{<0}\SL_2$ with entries in $t^{-2}\cO$. In coordinates $h=\begin{pmatrix}1+a_1\tau+a_2\tau^2&b_1\tau+b_2\tau^2\\ b_1\tau+b_2\tau^2&1-a_1\tau+d_2\tau^2\end{pmatrix}$, $\tau=t^{-1}$, the equations are $a_2+d_2=a_1^2+b_1^2$, $a_1(d_2-a_2)=2b_1b_2$, $a_2d_2=b_2^2$. The solutions form a quadric cone $Y_{\mathrm{main}}\simeq\{xz=y^2\}$ (normal, unibranch) and two planes $P_\pm=\{(a_1,\pm ia_1,\nu,\pm i\nu,-\nu)\}$. Three components through $x_0$, each with IC stalk $k$, give $k^3=k^{s_1(0)}$; the normalisation of this slice is the slice model $Z_S$, with one point of $m_1^{-1}(x_0)$ on each component.
Function level (split form over $\F_q$, Computation 2.8): the trace of $V_{\alpha^\vee}\ast\IC_0$ is $(T+1)f_0=3f_0+2f_1+f_2$, the three fibre cardinalities.

Remark. The decomposition theorem is not used, nor rational smoothness of $\Gr^{\le n\alpha^\vee}$ (true for $\SL_2$). What makes $m_n$ finite is that $\mathcal O_0=L^+X$ is a line in a tree: the fibres are vertices of a line inside a ball.

Verified by computation: slices.py.

Builds on

Computation 2.5 · Fibres of the convolution map are finite sets of tree vertices; the map is finite and an isomorphism over the top orbit

Built on by

Proposition 2.7 · The Satake span is exactly the even-label, constant-coefficient part, strictly inside the even part
Audit 7.2 · Audit: the $\SL_2/\SO_2$ verdict holds at both levels; recursions and fibre counts re-derived
Objection 7.15 · Objection: the parity count of the $\SL_2/T$ analysis holds, but '$\IC_0$ is a summand of $V_{(1,0)}\ast\IC_0$' is false
Computation X2.3 · Stalks for $\mu=\alpha^\vee,2\alpha^\vee$: $\mathcal L_{\mathrm{sgn}}$ is Kummer on each slice branch
Theorem (repair of A4-11) X2.4 · Repair of A4-11: $\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}$ in both $\mathbb F_q$-forms, $q$ odd
Audit X2.7 · No $\mathrm{SL}_2/\mathrm{SO}_2$ conclusion of the run rests on the false §8.2 remark

Original record, as the agent wrote it at 02:33 UTC

Each Satake object is a single IC sheaf: V_n * IC_0 equals IC of the closure of O_2n

Lemma. $V_{n\alpha^\vee}\ast\mathrm{IC}_0\simeq\mathrm{IC}(\overline{\mathcal O_{2n}},\underline k)=:\mathrm{IC}_{2n}$ (up to the normalisation shift), for every $n\ge0$. Likewise, for the $\mathrm{PGL}_2$-action of A2-2, $V^{\mathrm{PGL}_2}_{m\omega^\vee}\bar\ast\mathrm{IC}_0\simeq\mathrm{IC}_m$ for every $m\ge0$. Proof.

$Z_n$ is Zariski-locally $\mathrm{Gr}^{\le n\alpha^\vee}\times L^+X$ (the torsor $LG\to\mathrm{Gr}$ is locally trivial) and $L^+X$ is formally smooth, so $V_{n\alpha^\vee}\mathbin{\widetilde\boxtimes}\mathrm{IC}_0=\mathrm{IC}(\mathrm{Gr}^{\le n\alpha^\vee})\mathbin{\widetilde\boxtimes}\underline k_{L^+X}=\mathrm{IC}(Z_n)$, extended by zero to $\mathrm{Gr}^{\le n\alpha^\vee}\mathbin{\widetilde\times}LX$. Hence $V_{n\alpha^\vee}\ast\mathrm{IC}_0=m_{n*}\mathrm{IC}(Z_n)$ ($m_n$ proper).
General fact: if $\nu:Z\to Y$ is finite, $Z$ irreducible, and $\nu$ is an isomorphism over a dense open $U\subset Y$, then $\nu_*\mathrm{IC}_Z\simeq\mathrm{IC}_Y$. Indeed $\nu_*$ is t-exact for finite $\nu$, $\nu_*\mathrm{IC}_Z|_U=\mathrm{IC}_Y|_U$, and for $i:D=Y\setminus U\hookrightarrow Y$, $i':\nu^{-1}D\hookrightarrow Z$: $\operatorname{Hom}(\nu_*\mathrm{IC}_Z,i_*\mathcal G)=\operatorname{Hom}(\mathrm{IC}_Z,i'_*\nu'^!\mathcal G)=0$ and dually $\operatorname{Hom}(i_*\mathcal G,\nu_*\mathrm{IC}_Z)=\operatorname{Hom}(i'^*\nu^*\ldots)$, both zero because $\mathrm{IC}_Z=j'_{!*}$ has no sub- or quotient object supported on $Z\setminus\nu^{-1}U$ (BBD §1.4, characterisation of $j_{!*}$). Apply with $\nu=m_n$, $U=\mathcal O_{2n}$ (A2-5(i),(iii)); on reducible slice models the same holds componentwise (A2-5(ii)), and $\mathrm{IC}$ of a reducible variety is the sum of the $\mathrm{IC}$ of its components. $\square$

Consistency checks (computed, numerically verified). With Chen–Yi's slices $S^\lambda=\tau^{-1}(W^\lambda\cap\tau(LX))$, $W^\lambda=(L^{<0}G\cap\mathrm{Ad}_{t^\lambda}L^{<0}G)t^\lambda$ (Prop. 47: $S^\lambda\cap\mathcal O_\lambda=\{x_\lambda\}$, $L^+G\times S^\lambda\to LX$ formally smooth), $\mathrm{IC}_{2n}|_{S^\mu}$ is $\mathrm{IC}$ of the slice, whose stalk at $x_\mu$ must have rank $s_n(\mu)$ by the lemma.

$S^{\alpha^\vee}\cap\overline{\mathcal O_2}=\{h\,t^{\alpha^\vee}\}$ with $h=\begin{pmatrix}1+c t^{-1}+c^2t^{-2}+c^3t^{-3}&e\,t^{-1}\\ e\,t^{-3}&1-ct^{-1}\end{pmatrix}$, $e^2=-c^4$: two smooth branches $e=\pm ic^2$ through $x_1$ (tacnode). $\mathrm{IC}$ of a curve is the pushforward from its normalisation, stalk $k^2$ at $x_1$: equals $s_1(1)=2$. ✓
$S^0\cap\overline{\mathcal O_2}=\{h\in L^{<0}\mathrm{SL}_2\text{ symmetric, entries in }t^{-2}\mathcal O\}$, coordinates $h=\begin{pmatrix}1+a_1\tau+a_2\tau^2&b_1\tau+b_2\tau^2\\ \cdot&1-a_1\tau+d_2\tau^2\end{pmatrix}$, $\tau=t^{-1}$, equations $a_2+d_2=a_1^2+b_1^2$, $a_1(d_2-a_2)=2b_1b_2$, $a_2d_2=b_2^2$. Solving $(a_2,b_2,d_2)=\lambda(p^2,pr,r^2)$: for $p^2+r^2\ne0$ one gets $Y_{\mathrm{main}}=\{(2\mu pr,\mu(r^2-p^2),\mu^2(p^2+r^2)(p^2,pr,r^2))\}\simeq\{xz=y^2\}$ ($A_1$ cone, normal, unibranch); for $r=\pm ip$ the two planes $P_\pm=\{(a_1,\pm ia_1,\nu,\pm i\nu,-\nu)\}$. Three components through $x_0$, each with $\mathrm{IC}$ stalk $k$: total $k^3=k^{s_1(0)}$. ✓ (And the normalisation of this slice is $Z_S$: one point of $F_{x_0}$ per component.)
Function level (A2-8), split form over $\mathbb F_q$: $\operatorname{Tr}(V_{\alpha^\vee}\ast\mathrm{IC}_0)=(T+1)f_0=3f_0+2f_1+f_2$, the three fibre cardinalities. ✓

Remark. The decomposition theorem is not used; rational smoothness of $\mathrm{Gr}^{\le n\alpha^\vee}$ is not needed (though true for $\mathrm{SL}_2$). The special feature making $m_n$ finite is that $\mathcal O_0=L^+X$ is 'a line in a tree': the fibres are vertices of a line inside a ball.

Sources

sources/chen-yi-v4.pdf §6.1 (W^λ, S^λ), Lemma 45, Prop 47
sources/bbd-perverse.pdf §1.4 (intermediate extension)
criterion.html step 3

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Lemma 2.6 · Each Satake object is a single IC sheaf: \(V_{n\alpha^\vee}\ast\IC_0\simeq\IC(\overline{\mathcal O_{2n}})\)

Builds on

Built on by

Each Satake object is a single IC sheaf: V_n * IC_0 equals IC of the closure of O_2n

Sources