§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:50:21Z

Audit 4.9 · Transfer to $\mathbb C$ and checklist audit for the group

proved

Transfers the vanishing of every Δ_ω-multiplicity to complex coefficients by comparing stalk dimensions, which determine decomposition multiplicities through a unitriangular matrix, and runs the nine-point checklist over the group argument.

Audit 4.9.

(a) For $k=\mathbb C$ (Betti sheaves, or regular holonomic D-modules via Riemann–Hilbert on each $\mathrm{Fl}_{\le u}$), for all $x,y\in\widetilde W$, $\omega\in\Omega$, $n$: $E^{\mathbb C}_{\omega,n}(x,y)=0$; hence $\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}$ over $\mathbb C$, and Proposition 4.7(1)–(3) hold over $\mathbb C$. Proved modulo the standard $\overline{\mathbb F}_p\leftrightarrow\mathbb C$ comparison for complexes on schemes of finite type over $\mathbb Z[1/N]$ (generic base change and Artin comparison, bbd-perverse.pdf §6.1). (b) Checklist items 1–9 are satisfied by Setup 4.1–Proposition 4.7.

Multiplicities are read off from stalks. Let $d_{z,y}(t)=\sum_j\dim\mathcal H^j(\mathrm{IC}_z)_{\dot y}\,t^j$. Then $d_{z,z}=t^{-\ell(z)}$ and $d_{z,y}=0$ unless $y\le z$: the matrix is unitriangular up to units on every finite Bruhat lower set. If $M\simeq\bigoplus_{z,n}\mathrm{IC}_z[-n]^{\oplus e_{z,n}}$, then $\sum_j\dim\mathcal H^j(M)_{\dot y}t^j=\sum_z\big(\sum_ne_{z,n}t^{n}\big)d_{z,y}(t)$, so the $e_{z,n}$ are determined by the stalk dimensions of $M$ and of the $\mathrm{IC}_z$.
Spreading out. Fix $x,y$. $H$, $\mathrm{Fl}_{\le u}$ ($u$ the Demazure product of $x,w_0,y$), its cells $\mathbb A^{\ell(z)}$, the twisted product and $m$ are defined over $\mathbb Z$, the points $\dot y$ are $\mathbb Z$-points. For $N$ large, $j_{!*}$ and $m_*$ commute with base change to every geometric point of $\operatorname{Spec}\mathbb Z[1/N]$ (generic base change; bbd-perverse.pdf §6.1), so the stalk dimensions of $\mathrm{IC}_z$ and $M_{x,y}$ at $\dot y$ agree over $\mathbb C$ (Artin comparison) and over $\overline{\mathbb F}_p$, $p\nmid 2N$.
Conclusion. $M_{x,y}$ is semisimple over $\mathbb C$ (decomposition theorem, decataldo-migliorini-v2.pdf Thm 1.6.1) and over $\overline{\mathbb F}_p$ (Setup 4.1(e)); by steps 1–2 the multiplicities agree, and over $\overline{\mathbb F}_p$ the $\Omega$-multiplicities vanish (Theorem 4.5, valid for every $p$). One $p$ per pair $(x,y)$ suffices. Proposition 4.2 over $\mathbb C$ gives $\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}$. The same transfer applies to Proposition 4.7(1)–(3). Riemann–Hilbert (finitely many cells on $\mathrm{Fl}_{\le u}$) is t-exact and compatible with proper pushforward, giving the D-module form. The pages' Proposition II (group.html) is an independent proof over $\mathbb C$; it is not used.
Checklist. (1) Bruhat intervals are finite; no infinite-codimension orbit occurs (Setup 4.1(a)). (2) $\mathcal H_{c,I}=\operatorname{Thick}\{\mathrm{IC}_x\boxtimes\mathrm{IC}_y\}$ and the two-sided formula $\mathcal A\ast\mathcal M\ast\iota^*\mathcal B$ (Setup 4.1(d)). (3) Finite-type proper model and decomposition (Setup 4.1(e)). (4) Connected stabilizers, only constant local systems (Setup 4.1(b)). (5) Cells are $\mathrm{Fr}$-stable (split $H$), Lang for connected stabilizers, unitriangular IC-trace basis, $\operatorname{vol}(I)=1$, trace of convolution (Lemma 4.3, Corollary 4.6). (6) No cancellation by weights (Theorem 4.5 step 4). (7) $\overline{\mathbb F}_q$ proved, $\mathbb C$ by (a). (8) No spherical-level hypothesis ((R), (C), H1–H3 of the satake run) is transferred; the Iwahori-level ones are checked directly. (9) Neither the affine Whittaker functor (Proposition II) nor the I3 dévissage (IV.1, IV.2) is used.

Builds on

Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
Proposition 4.7 · $\mathcal M_I$ for the group: top cells in, $\Omega$ out, exact in rank one

Built on by

Summary 4.11 · Verdict for $X=H$: both levels fail, no divergence

Original record, as the agent wrote it at 18:50 UTC

Transfer to $\mathbb C$ and checklist audit for the group

Multiplicities are read off from stalks. Let $d_{z,y}(t)=\sum_j\dim\mathcal H^j(\mathrm{IC}_z)_{\dot y}\,t^j$. Then $d_{z,z}=t^{-\ell(z)}$ and $d_{z,y}=0$ unless $y\le z$: the matrix is unitriangular up to units on every finite Bruhat lower set. If $M\simeq\bigoplus_{z,n}\mathrm{IC}_z[-n]^{\oplus e_{z,n}}$, then $\sum_j\dim\mathcal H^j(M)_{\dot y}t^j=\sum_z\big(\sum_ne_{z,n}t^{n}\big)d_{z,y}(t)$, so the $e_{z,n}$ are determined by the stalk dimensions of $M$ and of the $\mathrm{IC}_z$.
Spreading out. Fix $x,y$. $H$, $\mathrm{Fl}_{\le u}$ ($u$ the Demazure product of $x,w_0,y$), its cells $\mathbb A^{\ell(z)}$, the twisted product and $m$ are defined over $\mathbb Z$, the points $\dot y$ are $\mathbb Z$-points. For $N$ large, $j_{!*}$ and $m_*$ commute with base change to every geometric point of $\operatorname{Spec}\mathbb Z[1/N]$ (generic base change; bbd-perverse.pdf §6.1), so the stalk dimensions of $\mathrm{IC}_z$ and $M_{x,y}$ at $\dot y$ agree over $\mathbb C$ (Artin comparison) and over $\overline{\mathbb F}_p$, $p\nmid 2N$.
Conclusion. $M_{x,y}$ is semisimple over $\mathbb C$ (decomposition theorem, decataldo-migliorini-v2.pdf Thm 1.6.1) and over $\overline{\mathbb F}_p$ (Setup 4.1(e)); by steps 1–2 the multiplicities agree, and over $\overline{\mathbb F}_p$ the $\Omega$-multiplicities vanish (Theorem 4.5, valid for every $p$). One $p$ per pair $(x,y)$ suffices. Proposition 4.2 over $\mathbb C$ gives $\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}$. The same transfer applies to Proposition 4.7(1)–(3). Riemann–Hilbert (finitely many cells on $\mathrm{Fl}_{\le u}$) is t-exact and compatible with proper pushforward, giving the D-module form. The pages' Proposition II (group.html) is an independent proof over $\mathbb C$; it is not used.
Checklist. (1) Bruhat intervals are finite; no infinite-codimension orbit occurs (Setup 4.1(a)). (2) $\mathcal H_{c,I}=\operatorname{Thick}\{\mathrm{IC}_x\boxtimes\mathrm{IC}_y\}$ and the two-sided formula $\mathcal A\ast\mathcal M\ast\iota^*\mathcal B$ (Setup 4.1(d)). (3) Finite-type proper model and decomposition (Setup 4.1(e)). (4) Connected stabilizers, only constant local systems (Setup 4.1(b)). (5) Cells are $\mathrm{Fr}$-stable (split $H$), Lang for connected stabilizers, unitriangular IC-trace basis, $\operatorname{vol}(I)=1$, trace of convolution (Lemma 4.3, Corollary 4.6). (6) No cancellation by weights (Theorem 4.5 step 4). (7) $\overline{\mathbb F}_q$ proved, $\mathbb C$ by (a). (8) No spherical-level hypothesis ((R), (C), H1–H3 of the satake run) is transferred; the Iwahori-level ones are checked directly. (9) Neither the affine Whittaker functor (Proposition II) nor the I3 dévissage (IV.1, IV.2) is used.

Sources

bbd-perverse.pdf, §6.1
decataldo-migliorini-v2.pdf, Thm 1.6.1
group.html, Proposition II (comparison only)

Table 1. Cumulative token usage and estimated cost.
Work	Tokens	US $
Original search, audits and page construction	384.00 M	≈ $171.00
Presentation edits and follow-up corrections	9.64 M	≈ $4.99
Presentation edits and follow-up corrections	9.82 M	≈ $4.29
Total	≈ 403.46 M	≈ $180.28