§7 · Agent 7 · Audit: criterion and bridge · 2026-10-18T19:23:40Z

Audit 7.2 · Audit of A2-3, A2-4: generators $\mathrm{IC}w\ast_I\mathrm{IC}_0$ and $\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D{\mathcal M_I^+}$

proved

The generator lemma (including the two-sided group action and the persistence of spherical summands) and Theorem (I$_I$) of version 1 are correct; repairs only to a shift in a cited identity and to the wording of (e) under disconnected stabilizers.

Audit 7.2.

Holds with a minor repair. A2-3(a)–(c) and A2-4(a)–(e) are correct over $\overline{\mathbb F}_q$ and $\mathbb C$, with the hypotheses stated there; in particular the sandwich $$\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^+}$$ needs no decomposition theorem, and the equivalence $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I$ needs only (DT$_I$). Repairs:

  1. A2-3(c): proposition-I.tex, Step 1, gives $\mathcal G\ast_Ip^M\simeq(q_\mathcal G)\ast_KM$ only up to the shift $[-2d]$ ($p^!=p^*[2d]$); harmless, since summands are taken up to shift. The key lemma used there needs (S1) $|W|$ invertible in the coefficient field, automatic for $\overline{\mathbb Q}_\ell$ and $\mathbb C$.
  2. A2-4(e) under (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.'): the irreducible equivariant local systems on the open piece of a closure $Z$ may grow with the level (the open orbit $o'\subseteq\pi^{-1}(o)$ can have a larger stabilizer component group than $o$), so "at a fine enough level" must read "at every level"; the criterion (every $\mathrm{IC}(Z,\xi)$, at every level, must be reached) is unchanged.
  1. A2-3(a). On a finite union $\mathrm{Fl}{\le}$ of Schubert varieties $I$ acts through $I/I_r$ with $I_r$ pro-unipotent; stabilizers $(I\cap wIw^{-1})I_r/I_r$ are images of connected groups (Chen–Yi, Lemma 71(i): $I\times I$-stabilizers on $LG_w$ are connected; extract cy-lemma71.txt). The finite-type argument of Audit 7.1 steps 2, 5 gives $\mathcal H{c,I}=\mathrm{Thick}\{\mathrm{IC}_w\}$. Holds.
  2. A2-3(b). For an exact functor $\Phi$, $\{\mathcal G:\Phi\mathcal G\in\mathrm{Thick}(\Phi S)\}$ is thick (exactness and additivity), so contains $\mathrm{Thick}(S)$: this is (T2) of proposition-I.tex. Group case: $\widetilde W_G=\widetilde W_H^2$, $\mathrm{IC}{(w_1,w_2)}=\mathrm{IC}{w_1}\boxtimes\mathrm{IC}{w_2}$ (BBD 4.2.7, p. 111, external products commute with the usual operations); the action $[(g_1,g_2),x]\mapsto g_1xg_2^{-1}$ gives $(A\boxtimes B)\ast F=A\ast F\ast\mathrm{inv}^*B$ and $\mathrm{inv}(Iw I)=Iw^{-1}I$, $\ell(w^{-1})=\ell(w)$; $K_H/I_H=\overline{Iw_0I}/I\simeq G/B$ smooth gives $c=\mathrm{IC}{w_0}[-\ell(w_0)]$. Holds (checklist item 2: two-sided).
  3. A2-3(c). $W t^\lambda W$ is a double coset of the finite parabolic $W\times W$ in $W_{\mathrm{aff}}\rtimes\Omega$, so it has a unique element of maximal length $w_\lambda$, and $Kt^\lambda K/I=\bigsqcup_{v\in Wt^\lambda W}IvI/I$ has $Iw_\lambda I/I$ dense: $\overline{Iw_\lambda I}/I=q^{-1}(\mathrm{Gr}{\le\lambda})$. Smooth pull-back of IC (remark after BBD 4.2.6.1) gives $q^*V\lambda=\mathrm{IC}{w\lambda}[-d]$. The key lemma of proposition-I.tex (retraction $\beta_{\mathcal F}$ of $\mathcal F\to q_*q^\mathcal F$ for every $\mathcal F$ on $I\backslash\mathrm{Gr}$, built from $\frac1{|W|}\mathrm{tr}{\mathcal A}$) applies to $\mathcal F=V\lambda$ forgotten to $I$; with Step 1 (up to shift) $p^(V_\lambda\ast_K\mathrm{IC}0)$ is a retract of $G{w_\lambda}$ up to shift. $p^\mathrm{IC}\nu$ is IC of the irreducible $I$-stable $\overline{\mathcal O\nu}$ with an irreducible local system (forgetting equivariance commutes with $j_{!}$; a $K$-equivariant irreducible local system on $K/K_x$ is irreducible as a local system since $\pi_1(K/K_x)\to\pi_0(K_x)$ is onto), and it is simple at a finite level because $\pi^*[d]$ is exact, faithful and kills no non-zero object. Holds (repair 1).
  4. A2-4(a),(b). Step 1: retracts and shifts. Step 2: every $G_w\in\mathcal D_{\mathcal M_I^+}$ by the definition of $\mathcal M_I^+$, and $\mathcal D_{\mathcal M_I^+}$ is thick (Audit 7.1 step 5), so $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{G_w\}\subseteq\mathcal D_{\mathcal M_I^+}$. (b): ${}^pH^0(S)=S$. Holds.
  5. A2-4(c),(d). Under (DT$I$), ${}^pH^i(G_w)=\bigoplus{s_b=-i}S_b$, so constituents are summands: $\mathcal M_I^+=\mathcal M_I$. (d) uses $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$. Holds. No alternating sums are used (checklist item 6), and H1 is never invoked (item 8): only level-wise finiteness.
  6. A2-4(e). With (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) the bijection of A2-2(c). With (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.'): let $o'\subseteq\pi^{-1}(o)$ be the open orbit at a finer level, $y'\mapsto y$; the stabilizer $S'{y'}$ maps to $S_y$ with $\pi_0(S'{y'})\to\pi_0(S_y)$ not necessarily injective, so $\mathrm{Irr}\,\pi_0(S'_{y'})$ may contain local systems not pulled back from $o$; these are new simple objects with the same support $Z$. Repair 2. Holds with the repair.

Builds on

  • Lemma 2.3 · Lemma: generators $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; spherical summands persist
  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$

Built on by

  • Summary 7.11 · Verdict of the audit: (I$_I$) over $\overline{\mathbb F}_q$ and $\mathbb C$, (II$_I$) over $\overline{\mathbb F}_q$ only, converses as stated
Original record, as the agent wrote it at 19:23 UTC

Audit of A2-3, A2-4: generators $\mathrm{IC}w\ast_I\mathrm{IC}_0$ and $\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D{\mathcal M_I^+}$

  1. A2-3(a). On a finite union $\mathrm{Fl}{\le}$ of Schubert varieties $I$ acts through $I/I_r$ with $I_r$ pro-unipotent; stabilizers $(I\cap wIw^{-1})I_r/I_r$ are images of connected groups (Chen–Yi, Lemma 71(i): $I\times I$-stabilizers on $LG_w$ are connected; extract cy-lemma71.txt). The finite-type argument of Audit 7.1 steps 2, 5 gives $\mathcal H{c,I}=\mathrm{Thick}\{\mathrm{IC}_w\}$. Holds.
  2. A2-3(b). For an exact functor $\Phi$, $\{\mathcal G:\Phi\mathcal G\in\mathrm{Thick}(\Phi S)\}$ is thick (exactness and additivity), so contains $\mathrm{Thick}(S)$: this is (T2) of proposition-I.tex. Group case: $\widetilde W_G=\widetilde W_H^2$, $\mathrm{IC}{(w_1,w_2)}=\mathrm{IC}{w_1}\boxtimes\mathrm{IC}{w_2}$ (BBD 4.2.7, p. 111, external products commute with the usual operations); the action $[(g_1,g_2),x]\mapsto g_1xg_2^{-1}$ gives $(A\boxtimes B)\ast F=A\ast F\ast\mathrm{inv}^*B$ and $\mathrm{inv}(Iw I)=Iw^{-1}I$, $\ell(w^{-1})=\ell(w)$; $K_H/I_H=\overline{Iw_0I}/I\simeq G/B$ smooth gives $c=\mathrm{IC}{w_0}[-\ell(w_0)]$. Holds (checklist item 2: two-sided).
  3. A2-3(c). $W t^\lambda W$ is a double coset of the finite parabolic $W\times W$ in $W_{\mathrm{aff}}\rtimes\Omega$, so it has a unique element of maximal length $w_\lambda$, and $Kt^\lambda K/I=\bigsqcup_{v\in Wt^\lambda W}IvI/I$ has $Iw_\lambda I/I$ dense: $\overline{Iw_\lambda I}/I=q^{-1}(\mathrm{Gr}{\le\lambda})$. Smooth pull-back of IC (remark after BBD 4.2.6.1) gives $q^*V\lambda=\mathrm{IC}{w\lambda}[-d]$. The key lemma of proposition-I.tex (retraction $\beta_{\mathcal F}$ of $\mathcal F\to q_*q^\mathcal F$ for every $\mathcal F$ on $I\backslash\mathrm{Gr}$, built from $\frac1{|W|}\mathrm{tr}{\mathcal A}$) applies to $\mathcal F=V\lambda$ forgotten to $I$; with Step 1 (up to shift) $p^(V_\lambda\ast_K\mathrm{IC}0)$ is a retract of $G{w_\lambda}$ up to shift. $p^\mathrm{IC}\nu$ is IC of the irreducible $I$-stable $\overline{\mathcal O\nu}$ with an irreducible local system (forgetting equivariance commutes with $j_{!}$; a $K$-equivariant irreducible local system on $K/K_x$ is irreducible as a local system since $\pi_1(K/K_x)\to\pi_0(K_x)$ is onto), and it is simple at a finite level because $\pi^*[d]$ is exact, faithful and kills no non-zero object. Holds (repair 1).
  4. A2-4(a),(b). Step 1: retracts and shifts. Step 2: every $G_w\in\mathcal D_{\mathcal M_I^+}$ by the definition of $\mathcal M_I^+$, and $\mathcal D_{\mathcal M_I^+}$ is thick (Audit 7.1 step 5), so $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{G_w\}\subseteq\mathcal D_{\mathcal M_I^+}$. (b): ${}^pH^0(S)=S$. Holds.
  5. A2-4(c),(d). Under (DT$I$), ${}^pH^i(G_w)=\bigoplus{s_b=-i}S_b$, so constituents are summands: $\mathcal M_I^+=\mathcal M_I$. (d) uses $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$. Holds. No alternating sums are used (checklist item 6), and H1 is never invoked (item 8): only level-wise finiteness.
  6. A2-4(e). With (I2) the bijection of A2-2(c). With (I2'): let $o'\subseteq\pi^{-1}(o)$ be the open orbit at a finer level, $y'\mapsto y$; the stabilizer $S'{y'}$ maps to $S_y$ with $\pi_0(S'{y'})\to\pi_0(S_y)$ not necessarily injective, so $\mathrm{Irr}\,\pi_0(S'_{y'})$ may contain local systems not pulled back from $o$; these are new simple objects with the same support $Z$. Repair 2. Holds with the repair.

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