§1 · Agent 1 · Literature and Table 1 audit · 2026-10-18T18:41:48Z

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

proved

Fixes the conventions of the run and checks that the pages' coherent D-module categories and this run's constructible categories agree level by level; the first half of the pages' 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). becomes a definition here, and only the transfer of summand statements between characteristic 0 and p remains an input.

Setup 1.1.

Let \(X\) be a row of Table 1 with a level presentation \(LX=\varinjlim_N Y_N\), \(Y_N=\varprojlim_M Y_{N,M}\) in which every \(Y_{N,M}\) has finitely many \(I/I_{N,M}\)-orbits.

(a) Over \(k=\mathbb C\) the de Rham functor gives, at each level, an equivalence between \(I/I_{N,M}\)-equivariant coherent \(\mathcal D\)-modules on \(Y_{N,M}\) (the pages' \(\mathcal D_c\)) and \(D^b_{I/I_{N,M},c}(Y_{N,M}^{an})\), compatible with smooth pullback (up to the relative-dimension shift), closed and proper pushforward and \(\boxtimes\). Hence the pages' \(\mathcal C_I\), \(\mathcal H_{c,I}\), \(\ast_I\) and \(\mathrm{IC}_0\) correspond to the Betti ones of this run (\(\mathcal O_{L^+X}\leftrightarrow \underline k_{L^+X}\) up to a shift).

(b) With \(\mathcal C_I:=\bigcup_{N,M}\mathcal C_I^{(N,M)}\) (this run's definition), the first half of the pages' 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). ("every object is pulled back from a finite level") holds by definition.

(c) Conventions used by every node of A1: \(\operatorname{vol}(I)=1\), \(T_w=\mathbf 1_{IwI}\), \(T_s^2=(q-1)T_s+q\); the pages' \(\Delta_w=j_{w!}\underline k\) is unshifted, the brief's \(\Delta_w=j_{w!}\underline k[\ell(w)]\) is shifted; \(\mathrm{IC}_w|_{IwI/I}=\underline k[\ell(w)]\); \(\operatorname{tr}(K[1])=-\operatorname{tr}K\).

(d) Gap: a statement "\(\mathrm{IC}_P[s]\) is a direct summand of \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\)" proved over \(\overline{\mathbb F}_q\) transfers to Betti sheaves over \(\mathbb C\) only through a spreading-out comparison (the satake run's D5); for rows 1–3 it is not needed, because the decompositions are computed identically in both settings (Lemma 1.2, Lemma 1.3, Lemma 1.4).

Proof.

  1. Regular holonomicity. At a finite level the group \(I/I_{N,M}\) is affine algebraic and acts on \(Y_{N,M}\) with finitely many orbits, so every coherent equivariant \(\mathcal D\)-module is regular holonomic (kashiwara-equivariant.pdf, §1.3, citing [15] there). For singular \(Y_{N,M}\) (the group and symmetric rows) embed \(Y_{N,M}\) equivariantly in a smooth finite-type scheme (a jet scheme \(X(\mathcal O/t^{M'})\) or \(LH/I_{M}\) restricted to a bounded piece) and use Kashiwara's equivalence.
  2. Riemann–Hilbert. kashiwara-equivariant.pdf, Theorem 4.6.2: \(\mathrm{DR}_X: D^b_{G,rh}(\mathcal D_X)\xrightarrow{\sim}D^b_{G^{an},\mathbb C\text{-}c}(\mathbb C_{X^{an}})\) for an affine algebraic \(G\) acting on a quasi-projective manifold. Apply it to \(G=I/I_{N,M}\).
  3. Transition functors and convolution. \(\mathrm{DR}\) commutes with proper pushforward, with smooth pullback up to the shift by relative dimension, and with external products. The transition functors of the colimit are smooth pullback along \(Y_{N,M+1}\to Y_{N,M}\) and closed pushforward along \(Y_N\subset Y_{N+1}\); the convolution \(\mathcal G\ast_I M=m_!(\mathcal G\mathbin{\widetilde\boxtimes}M)\) is computed on a finite-type model (a bounded union of cells times a finite level), where \(m\) is proper. So both sides are matched functor by functor. \(\mathrm{IC}_0\) is pulled back from the smooth scheme \(X(\mathcal O/t^M)\), on which \(\mathcal O\) corresponds to \(\underline k[\dim]\).
  4. (b). By definition of \(\mathcal C_I\) in the setting. The pages assume this clause (hypothesis.html, 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). discussion; case-affine.html, "Assumed") because their \(\mathcal D_c(I\backslash LX)\) is defined as coherent objects of \(\mathcal D\)-mod\((LX)^I\); for this run that comparison is not needed. Consequence: the pages' Proposition IV.1, "given 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).", is unconditional for this run's \(\mathcal C_I\) (its proof uses only base change and explicit translates, valid in all three settings).
  5. (c). Iwahori–Matsumoto, Theorem 3.5 (iwahori-matsumoto-1965.pdf): the kernel of \(\mathbb Z\langle A_i\rangle\to\mathcal H(G,B)\) is generated by \(A_i^2-(q\cdot1+(q-1)A_i)\) and the braid relations; this is \(T_s^2=(q-1)T_s+q\) with \(\operatorname{vol}(I)=1\), the convention of classical.html.

Remark (heuristic; reading of \(\mathcal D_c\)). In the \(\infty\)-category \(\mathcal D\)-mod\((LX)^I\) the compact objects and the coherent objects need not agree when stabilizers contain a torus (Drinfeld–Gaitsgory's distinction between safe and coherent objects). The comparison in (a) holds for \(\mathcal D_c\) read as coherent (holonomic) objects, which is how the pages use it (constructible \(\leftrightarrow\) coherent); it is Bernstein–Lunts' bounded constructible category on the Betti side.

Remark (fields). Chen–Yi state that their proofs are geometric and apply to D-modules over \(\mathbb C\) via Saito's mixed Hodge modules (chen-yi-v4.pdf, §1.2.6); this is an assertion of that paper, not a proof checked here. Statements about symmetric rows below are recorded over \(\overline{\mathbb F}_q\), \(p\) odd, \(\overline{\mathbb Q}_\ell\)-coefficients.

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Original record, as the agent wrote it at 18:41 UTC

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

Proof.

  1. Regular holonomicity. At a finite level the group \(I/I_{N,M}\) is affine algebraic and acts on \(Y_{N,M}\) with finitely many orbits, so every coherent equivariant \(\mathcal D\)-module is regular holonomic (kashiwara-equivariant.pdf, §1.3, citing [15] there). For singular \(Y_{N,M}\) (the group and symmetric rows) embed \(Y_{N,M}\) equivariantly in a smooth finite-type scheme (a jet scheme \(X(\mathcal O/t^{M'})\) or \(LH/I_{M}\) restricted to a bounded piece) and use Kashiwara's equivalence.
  2. Riemann–Hilbert. kashiwara-equivariant.pdf, Theorem 4.6.2: \(\mathrm{DR}_X: D^b_{G,rh}(\mathcal D_X)\xrightarrow{\sim}D^b_{G^{an},\mathbb C\text{-}c}(\mathbb C_{X^{an}})\) for an affine algebraic \(G\) acting on a quasi-projective manifold. Apply it to \(G=I/I_{N,M}\).
  3. Transition functors and convolution. \(\mathrm{DR}\) commutes with proper pushforward, with smooth pullback up to the shift by relative dimension, and with external products. The transition functors of the colimit are smooth pullback along \(Y_{N,M+1}\to Y_{N,M}\) and closed pushforward along \(Y_N\subset Y_{N+1}\); the convolution \(\mathcal G\ast_I M=m_!(\mathcal G\mathbin{\widetilde\boxtimes}M)\) is computed on a finite-type model (a bounded union of cells times a finite level), where \(m\) is proper. So both sides are matched functor by functor. \(\mathrm{IC}_0\) is pulled back from the smooth scheme \(X(\mathcal O/t^M)\), on which \(\mathcal O\) corresponds to \(\underline k[\dim]\).
  4. (b). By definition of \(\mathcal C_I\) in the setting. The pages assume this clause (hypothesis.html, I1 discussion; case-affine.html, "Assumed") because their \(\mathcal D_c(I\backslash LX)\) is defined as coherent objects of \(\mathcal D\)-mod\((LX)^I\); for this run that comparison is not needed. Consequence: the pages' Proposition IV.1, "given I1", is unconditional for this run's \(\mathcal C_I\) (its proof uses only base change and explicit translates, valid in all three settings).
  5. (c). Iwahori–Matsumoto, Theorem 3.5 (iwahori-matsumoto-1965.pdf): the kernel of \(\mathbb Z\langle A_i\rangle\to\mathcal H(G,B)\) is generated by \(A_i^2-(q\cdot1+(q-1)A_i)\) and the braid relations; this is \(T_s^2=(q-1)T_s+q\) with \(\operatorname{vol}(I)=1\), the convention of classical.html.

Remark (heuristic; reading of \(\mathcal D_c\)). In the \(\infty\)-category \(\mathcal D\)-mod\((LX)^I\) the compact objects and the coherent objects need not agree when stabilizers contain a torus (Drinfeld–Gaitsgory's distinction between safe and coherent objects). The comparison in (a) holds for \(\mathcal D_c\) read as coherent (holonomic) objects, which is how the pages use it (constructible \(\leftrightarrow\) coherent); it is Bernstein–Lunts' bounded constructible category on the Betti side.

Remark (fields). Chen–Yi state that their proofs are geometric and apply to D-modules over \(\mathbb C\) via Saito's mixed Hodge modules (chen-yi-v4.pdf, §1.2.6); this is an assertion of that paper, not a proof checked here. Statements about symmetric rows below are recorded over \(\overline{\mathbb F}_q\), \(p\) odd, \(\overline{\mathbb Q}_\ell\)-coefficients.

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