§3 · Agent 3 · Symplectic family: fibres · 2026-10-17T02:40:26Z

Proposition 3.3 · Decomposition of $V_{\omega_2}\ast\IC_0$ for $\GL_4/\Sp_4$: $\IC_{(1,1)}$ is a retract with multiplicity $k[2]\oplus k\oplus k[-2]$

proved audit: minor defect

Minor defect (second audit). Degree bookkeeping, the self-duality argument, the list of simple objects and the finite-type reduction are correct (re-derived; trace check 1 + q + q^2 + q^3 matches the lattice count). Citation defect: for the case over C the node points to the final section of Sun, where the decomposition theorem for C-stacks is only stated (Remark 3.14 defers the proof; Theorem 3.15). The node does list the passage to C as an inherited open item, so no claim is affected. (Over C the finite-type statement also follows from BBD 6.2.5 with the equivariant formalism of Bernstein–Lunts; identifying the Betti category C with the finite-type model is part of the workspace's open item.)

The minuscule Satake object of $\GL_4/\Sp_4$ decomposes as $\IC_{(2,0)}\oplus\IC_{(1,1)}\otimes(k[2]\oplus k\oplus k[-2])$. So the standard object of the uncovered closed orbit $(1,1)$ lies in $\Csat$ although H5 fails there, and the closure of $(2,0)$ is rationally smooth. Proved ℓ-adically; the passage to Betti sheaves over $\C$ is the workspace's open transfer item.

Proposition 3.3.

Let $X=\GL_4/\Sp_4$, $Z=\overline{\cO_{(2,0)}}$, $Y=\Gr^{\omega_2}\mathbin{\tilde\times}L^+X=\{(\Lambda',\omega):\omega\text{ perfect on }\Lambda'\}$, and $m:Y\to Z$ the convolution map. Assume:

(i) the decomposition theorem for the proper equivariant map of finite-type models $Y^{(N)}/G(\mathcal O/t^N)\to Z^{(N)}/G(\mathcal O/t^N)$ (Sun, Thm. 1.2 over $\F_q$; Chen–Yi, Prop. 70 as used in the proof of their Thm. 93);
(ii) connected stabilizers (Proposition III, step 4: $Z_{G^\theta}(S)=\SL_2^2$), so the simple equivariant perverse sheaves on $Z$ are $\IC_{(2,0)}:=\IC(Z,k)$ and $\IC_{(1,1)}=k_{\cO_{(1,1)}}[D-4]$;
(iii) Verdier duality: $\mathbb D(m_!k_Y[D])=m_*k_Y[D]$ for smooth $Y$.

Here $D$ is the dimension of the finite-type model of $Y$, equal to that of $Z$. Put $P:=m_!k_Y[D]$, a shift of $V_{\omega_2}\ast\IC_0$. Then $$P\simeq\IC_{(2,0)}\oplus\IC_{(1,1)}[2]\oplus\IC_{(1,1)}\oplus\IC_{(1,1)}[-2].$$ In particular $\IC_{(1,1)}$, which is $\Delta_{(1,1)}$ up to shift, is a retract of $V_{\omega_2}\ast\IC_0$, so $\Delta_{(1,1)}\in\Csat$; and $\IC_{(2,0)}|_{\cO_{(1,1)}}=k[D]$, i.e. $Z$ is rationally smooth along the closed orbit. The transfer of this ℓ-adic decomposition to the Betti category over $\C$ is the workspace's open item, inherited here.

Setting. $V_{\omega_2}=k_{\Gr(2,4)}[4]$ ($\omega_2$ is minuscule and $\langle2\rho,\omega_2\rangle=4$), $\IC_0=k_{L^+X}$, and the twisted product of constant sheaves is constant, so $V_{\omega_2}\ast\IC_0=m_!k_Y[4]$. For $N\ge3$, $Y$ and $Z$ are pulled back along the affine-space bundle $Z\to Z^{(N)}=Z\bmod t^N$: perfectness on $\Lambda'\subseteq t^{-1}\Lambda_0$ depends on $\omega\bmod t^2$, $\val\Pf=2$ is read modulo $t^3$, and the kernel of $L^+G\to G(\mathcal O/t^N)$ acts trivially on $Z^{(N)}$. This is the placid presentation of Chen–Yi, Thm. 36 and Appendix A, on which $\cC_Z\simeq D^b_{G(\mathcal O/t^N)}(Z^{(N)})$ and $m$ becomes a proper map of finite type with smooth source. By Computation 3.2, item 2, $m$ is birational; by Lemma 3.1, the closed stratum has codimension $c=4$.

Proof.

By (i) and (ii), $P\simeq\IC_{(2,0)}\otimes A\oplus\IC_{(1,1)}\otimes B$ with graded vector spaces $A,B$. Over $\cO_{(2,0)}$ the fibre is a point, so $P=k[D]$ there and $A=k$.
At $x\in\cO_{(1,1)}$, proper base change gives $P_x=H^*(LG(2,4))[D]$, one-dimensional in degrees $-D,-D+2,-D+4,-D+6$.
The support condition for an intermediate extension gives $\mathcal H^i(\IC_{(2,0)})_x=0$ for $i\ge-\dim\cO_{(1,1)}=-D+4$, while $\mathcal H^{-D}(\IC_{(2,0)})_x\ne0$: it is the stalk of the direct image of the constant sheaf from the open orbit, whose rank is the number of local branches. The summand $\IC_{(1,1)}[-j]$ contributes exactly in degree $-D+4+j$.
Degrees $-D+4$ and $-D+6$ can only come from $B$, so $j=0$ and $j=2$ occur, once each. By (iii) and the self-duality of $\IC_{(2,0)}$, $B$ is symmetric under $j\mapsto-j$; so $j=-2$ occurs and exhausts degree $-D+2$, and $j=\pm4$ cannot occur since degree $-D+8$ is empty. Hence $B=k[2]\oplus k\oplus k[-2]$ and $(\IC_{(2,0)})_x=k[D]$. $\square$

Consistency. (a) The slice of Lemma 3.1 is the cone over $Q^3$; by the Gysin sequence (multiplication by $h$ is injective on $H^*(Q^3;\Q)$, since $h^3\ne0$) the link of its vertex has the rational cohomology of $S^7$, so the cone is rationally smooth, matching $(\IC_{(2,0)})_x=k[D]$. (b) Frobenius traces: $1+(q+q^2+q^3)=(1+q)(1+q^2)=\#LG(2,4)(\F_q)$. (c) Only even shifts occur, as in Chen–Yi, Thm. 93.

For the case over $\C$ the original cites the final section of Sun; the node inherits the transfer of the ℓ-adic decomposition to Betti sheaves as the criterion page's open item.

Builds on

Lemma 3.1 · The loop space of $\GL_{2m}/\Sp_{2m}$: orbits, closure order, scalar shift and codimensions
Computation 3.2 · Convolution fibres over $x_{(2,0)}$ and $x_{(1,1)}$ for $\lambda=\omega_2$ and $\lambda=(2,0,0,0)$

Built on by

Proposition 3.4 · Where Proposition I uses H5, and the summand criterion $\mathcal M=\Lambda_S^+$
Summary 3.11 · Verdict for the symplectic family: no disagreement between the levels; $\GL_4/\Sp_4$ settled at both
Conjecture 3.8 · Relative-Satake prediction of every multiplicity; general uncovered labels and the exact gap
Audit 7.12 · Check of the decomposition of $V_{\omega_2}\ast\IC_0$: it holds ℓ-adically; Sun's version over $\C$ is only stated
Re-derivation X1.6 · The verdict inputs downstream of A1-4, A4-10, A5-6, A7-11: what changes

Original record, as the agent wrote it at 02:40 UTC

Decomposition: IC of the closed orbit (1,1) is a retract of V_omega2 * IC_0 with multiplicity k[2]+k+k[-2]

Setting. $Z=\overline{\mathcal O_{(2,0)}}$, $Y=\mathrm{Gr}^{\omega_2}\widetilde\times L^+X=\{(\Lambda',\omega):\omega\text{ perfect on }\Lambda'\}$, $m:Y\to Z$. $V_{\omega_2}=\underline k_{\mathrm{Gr}(2,4)}[4]$ (minuscule, $\langle2\rho,\omega_2\rangle=4$), $\mathrm{IC}_0=\underline k_{L^+X}$; the twisted product of constant sheaves is constant (torsor descent), so $V_{\omega_2}\ast\mathrm{IC}_0=m_!\underline k_Y[4]$. Finite-type model: for $N\ge3$ both $Y$ and $Z$ are pulled back along the affine-space bundle $Z\to Z^{(N)}=Z\bmod t^N$ ($K_N$ acts trivially on $Z^{(N)}$); this is the placid presentation of Chen–Yi Thm. 36 / App. A, on which $\mathcal C_Z\simeq D^b_{G(\mathcal O/t^N),c}(Z^{(N)})$ and $m$ becomes the proper map $m^{(N)}:Y^{(N)}\to Z^{(N)}$ of smooth-source finite type (same device as symmetric.html, H2 step 1–2). Put $D=\dim Y^{(N)}=\dim Z^{(N)}$ (birational by A3-2.2), $c=4$ the codimension of the closed stratum $\mathcal O_{(1,1)}$.

Inputs. (i) Decomposition theorem for the proper equivariant map $Y^{(N)}/G(\mathcal O/t^N)\to Z^{(N)}/G(\mathcal O/t^N)$: Sun, Thm. 1.2 (stacks over $\mathbb F_q$; over $\mathbb C$ his final section), or Chen–Yi Prop. 70 as used in the proof of Thm. 93, which states that $\mathrm{IC}_V\star e_{L^+X}$ is a direct sum of $\mathrm{IC}_{\chi,\lambda}$ shifted by $[2n](n)$. (ii) Connected stabilizers (Prop. III step 4, $Z_{G^\theta}(S)=\mathrm{SL}_2^2$): the simple equivariant perverse sheaves on $Z$ are $\mathrm{IC}_{(2,0)}:=\mathrm{IC}(Z,\underline k)$ and $\mathrm{IC}_{(1,1)}=\underline k_{\mathcal O_{(1,1)}}[D-4]$. (iii) Verdier self-duality: $\mathbb D(m_!\underline k_Y[D])=m_*\underline k_Y[D]$, $Y$ smooth.

Claim. $P:=m_!\underline k_Y[D]\simeq\mathrm{IC}_{(2,0)}\ \oplus\ \mathrm{IC}_{(1,1)}[2]\oplus\mathrm{IC}_{(1,1)}\oplus\mathrm{IC}_{(1,1)}[-2]$. In particular $\mathrm{IC}_{(1,1)}=\Delta_{(1,1)}$ (up to shift) is a retract of $V_{\omega_2}\ast\mathrm{IC}_0$, so $\Delta_{(1,1)}\in\mathcal C^{\mathrm{Sat}}$. Moreover $\mathrm{IC}_{(2,0)}|_{\mathcal O_{(1,1)}}=\underline k[D]$: $Z$ is rationally smooth along the closed orbit.

Proof.

By (i),(ii): $P\simeq\mathrm{IC}_{(2,0)}\otimes A\oplus\mathrm{IC}_{(1,1)}\otimes B$ with graded multiplicity spaces $A,B$. Over $\mathcal O_{(2,0)}$, $P=\underline k[D]$ (point fibre), so $A=k$.
Stalk at $x\in\mathcal O_{(1,1)}$: $P_x=H^*(LG(2,4))[D]$, one-dimensional in degrees $-D,-D+2,-D+4,-D+6$.
Support condition for the intermediate extension: $\mathcal H^i(\mathrm{IC}_{(2,0)})_x=0$ for $i\ge-\dim\mathcal O_{(1,1)}=-D+4$, and $\mathcal H^{-D}(\mathrm{IC}_{(2,0)})_x\neq0$ (number of local branches $\ge1$). $\mathrm{IC}_{(1,1)}[-j]$ contributes exactly degree $-D+4+j$.
Degrees $-D+4,-D+6$ must come from $B$: $j=0,2\in B$. By (iii) and self-duality of $\mathrm{IC}_{(2,0)}$, $B$ is self-dual, so $j=-2\in B$, which exhausts degree $-D+2$. Hence $B=k[2]\oplus k\oplus k[-2]$ and $(\mathrm{IC}_{(2,0)})_x=k[D]$. $\square$

Consistency. (a) The slice is the cone over $Q^3$ (A3-1.6); the link of the vertex has the rational cohomology of $S^7$ (Gysin with $h,h^2,h^3\ne0$ on $H^*(Q^3)$), so the cone is rationally smooth, matching $(\mathrm{IC}_{(2,0)})_x=k[D]$. (b) Trace of Frobenius: $1+(q+q^2+q^3)=(1+q)(1+q^2)=\#LG(2,4)(\mathbb F_q)$. (c) Even shifts only, as in Chen–Yi Thm. 93.

Transfer of the $\ell$-adic decomposition to the Betti category over $\mathbb C$ is the page's open item 1 (criterion.html, Open); this node inherits it.

Sources

sources/sun-decomposition.pdf Thm. 1.2
sources/chen-yi-v4.pdf Prop. 70, Thm. 93 (proof), Thm. 36, App. A
criterion.html step 3, step 4, step 6
symmetric.html H2 steps 1–2
sources/bbd-perverse.pdf 5.4.5–5.4.6

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

Proposition 3.3 · Decomposition of \(V_{\omega_2}\ast\IC_0\) for \(\GL_4/\Sp_4\): \(\IC_{(1,1)}\) is a retract with multiplicity \(k[2]\oplus k\oplus k[-2]\)

Builds on

Built on by

Decomposition: IC of the closed orbit (1,1) is a retract of V_omega2 * IC_0 with multiplicity k[2]+k+k[-2]

Sources