§1 · Agent 1 · Literature and Table 1 audit · 2026-10-18T18:49:47Z

Setup 1.9 · Citation map for the Iwahori criterion and bridge, with what was checked locally

proved

Exact result numbers for every standard input the criterion and the bridge use, each marked as read in a local PDF or recalled; the recalled ones (cells, Kazhdan–Lusztig, parity sheaves, Bernstein–Lunts) are the ones a reviewer should check.

Setup 1.9.

Read in local PDFs (statement checked).

Table 2. input and location
inputlocation
Decomposition, stacks over \(\mathbb F_q\): proper \(f\) of finite diagonal, finite type, affine stabilizers, \(K_0\) \(\iota\)-pure \(\Rightarrow Rf_*K\simeq\bigoplus{}^pR^if_*K[-i]\), each semisimplesun-decomposition.pdf, Thm 1.2
Decomposition over \(\mathbb C\) (survey statement)decataldo-migliorini-v2.pdf, Thm 1.6.1
BBD 6.2.5 (decomposition over \(\mathbb C\))cited in bezrukavnikov-yun-v2.pdf, line "[BBD82, Théorème 6.2.5]"; bbd-perverse.pdf is a scan without text layer — numbers 4.3.1, 5.3.8, 5.4.5, 6.2.5 not re-read
IC convolution in Kac–Moody / affine flag varieties: \(\mathrm{IC}_{w_1}\ast\mathrm{IC}_{w_2}=\bigoplus\mathrm{IC}_w[n](n/2)\), \(n\equiv\ell(w_1)+\ell(w_2)-\ell(w)\) (2); loop groupsbezrukavnikov-yun-v2.pdf, Prop. 3.2.5, §0.5
\(C_\Theta\simeq\mathrm{IC}_{w_\Theta}[-\ell_\Theta](-\ell_\Theta/2)\), \(\pi_\Theta^*\pi_{\Theta*}=C_\Theta\ast(-)\)bezrukavnikov-yun-v2.pdf, §3.2, Lemma 3.2.7
Function–sheaf: additivity, \(\otimes\), trace formula \(t_{f_!K}=\int_ft_K\), pullback; traces do not determine \(K\)laumon-1987.pdf, (1.1.1.1)–(1.1.1.4), Remarque (1.1.1.5)
Artin–Schreier vanishinglaumon-1987.pdf, (1.1.3.4)
Trace of convolution (spherical model)zhu-satake-v2.pdf, Lemma 5.6.1
Lang: \(g\mapsto g^{-1}F(g)\) onto; \(H^1(k,G)=1\) for connected \(G\)milne-iag-v2.pdf, Cor. 27.55, 27.56
Equivariant Riemann–Hilbert; finitely many orbits \(\Rightarrow\) regular holonomickashiwara-equivariant.pdf, Thm 4.6.2, §1.3
Bruhat–Iwahori decomposition; presentation with \(A_i^2=q+(q-1)A_i\)iwahori-matsumoto-1965.pdf, Prop. 2.4, Thm 2.5, Thm 3.5
\(e_K=\mathbf 1_K/\operatorname{meas}K\), \(\mathcal H_K=e_K\mathcal He_K\)haines-kottwitz-prasad-v3.pdf, §4.1
Symmetric \(X\), \(G^\theta\) connected, \(p\ne2\): \(I_0\)-orbits (\(I_0\backslash LX\simeq L^+T_0\backslash V/LK\)), finite closures, closed orbits, placid closures, dimension theory, convolution of semisimple is semisimple, local systems \(X(\pi_0T_0^{\psi_v})\), Weil structures and \(M\), purity, KL-type positivity, perverse \(t\)-structure on the colimitchen-yi-v4.pdf, Lemma 23, Lemmas 25–26, Cor. 27, Lemma 31, Thm 36, Prop. 41, Prop. 70, Lemma 71, Def. 72, §8.2, Thm 81, Thms 3/82, App. A.5, A.7 (45)–(47), Lemma 107
Enlarging \(q\) so that everything is defined over \(\mathbb F_q\) and \(T\) splitmars-springer-1998.pdf, 3.1.2

Recalled, not checked locally (flag for review).

Table 3. input and reference as recalled
inputreference as recalled
Preorders \(\le_L,\le_R,\le_{LR}\); descent sets along \(\le_L\)Kazhdan–Lusztig, Invent. Math. 53 (1979), §1, Prop. 2.4
\(P_{y,w}\) = IC stalks; \(IH^{\mathrm{odd}}=0\), Frobenius \(q^i\) on \(IH^{2i}\) (Schubert varieties, also affine)Kazhdan–Lusztig, Proc. Symp. Pure Math. 36 (1980)
\(a(y)\ge a(w)\) for \(y\le_{LR}w\); \(a\le\nu=\ell(w_0)\); \(a(w_J)=\ell(w_J)\)Lusztig, Cells in affine Weyl groups (1985), Thm 5.4, Thm 7.2, §2
\(c_0=\{a=\nu\}\) is one two-sided cell, the lowest; \(c_0=\{x\,w_J\,y\}\) with lengths addingShi, J. London Math. Soc. 36 (1987) and 37 (1988)
Equivariant derived category; \(D^b_S(\mathrm{pt})=\operatorname{Thick}\{k\}\) for connected \(S\)Bernstein–Lunts, LNM 1578
Parity sheaves (alternative to purity for DT with parity)Juteau–Mautner–Williamson, J. AMS 27 (2014), Thm 2.12

Use map. (I\(_I\)) needs: finite-length heart (Chen–Yi A.7; BBD 4.3.1 at finite type), DT (Sun 1.2 / Chen–Yi Prop. 70 / BY 3.2.5), thickness of constituent subcategories (stacks-derived.pdf, Tag 05RD, as in the satake run's summand.tex step 3). (II\(_I\)) needs: Laumon (1.1.1.1)–(1.1.1.3), Zhu 5.6.1 adapted (Lemma 1.6, D1), Milne 27.55–27.56 (Lemma 1.6, D3), Chen–Yi Lemma 71 and §8.2 for local systems and Weil structures. The second proof for the group needs BY 3.2.5 and the recalled cell results (Lemma 1.2, step 8); for \(\mathbb A^n\) and \(\mathbb G_m\) no recalled result is needed except Kazhdan–Lusztig 1980 for the sign in the trace remark (Lemma 1.3, item 6), which can be avoided by using \(\Delta_{t^\lambda}\).

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

  • 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 18:49 UTC

Citation map for the Iwahori criterion and bridge, with what was checked locally

Use map. (I\(_I\)) needs: finite-length heart (Chen–Yi A.7; BBD 4.3.1 at finite type), DT (Sun 1.2 / Chen–Yi Prop. 70 / BY 3.2.5), thickness of constituent subcategories (stacks-derived.pdf, Tag 05RD, as in the satake run's summand.tex step 3). (II\(_I\)) needs: Laumon (1.1.1.1)–(1.1.1.3), Zhu 5.6.1 adapted (Lemma 1.6, D1), Milne 27.55–27.56 (Lemma 1.6, D3), Chen–Yi Lemma 71 and §8.2 for local systems and Weil structures. The second proof for the group needs BY 3.2.5 and the recalled cell results (Lemma 1.2, step 8); for \(\mathbb A^n\) and \(\mathbb G_m\) no recalled result is needed except Kazhdan–Lusztig 1980 for the sign in the trace remark (Lemma 1.3, item 6), which can be avoided by using \(\Delta_{t^\lambda}\).

Sources