§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:43:42Z

Lemma 2.3 · Lemma: generators $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; spherical summands persist

proved

Reduces the generators of the Iwahori-generated part to the countably many objects IC_w *_I IC_0 (two-sided for the group), and shows that every simple summand of a Satake object, forgotten to I, lies in M_I.

Lemma 2.3.

(a) $\mathcal H_{c,I}=\mathrm{Thick}\{\mathrm{IC}_w:w\in\widetilde W\}$, where $\mathrm{IC}_w$ is the intersection complex of $\overline{IwI}/I$ (perverse normalization).

(b) If every $G_w=\mathrm{IC}w\ast_I\mathrm{IC}_0$ lies in $\mathcal C_I$, then $\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{G_w:w\in\widetilde W\}$. For the group $X=H$ under $G=H\times H$, $(h_1,h_2)x=h_1xh_2^{-1}$, with $c:=\mathrm{IC}_0=k{K_H/I_H}=\mathrm{IC}_{w_0}[-\ell(w_0)]$: $$\mathcal C_I^{\mathrm{Iwa}}=\mathrm{Thick}\{\mathrm{IC}_{w_1}\ast c\ast\mathrm{IC}_{w_2}:\ w_1,w_2\in\widetilde W_H\}\subseteq\mathcal C_I=\mathcal H_{c,I_H}.$$

(c) (comparison with the spherical level, $k=\overline{\mathbb F}q$ or $\mathbb C$). Let $V\lambda=\mathrm{IC}(\mathrm{Gr}{\le\lambda})$, $\lambda\in\Lambda_T^+$, and assume $V\lambda\ast_K\mathrm{IC}0$ (computed in $\mathcal C_K$) is a finite sum of shifted simple objects and that the forgetful functor $p^\colon\mathcal C_K\to\mathcal C_I$ lands in $\mathcal C_I$. Then every simple summand of $p^(V\lambda\ast_K\mathrm{IC}0)$ lies in $\mathcal M_I$. In particular, if $\mathrm{IC}\nu$ lies in the spherical summand set $\mathcal M$ of summand.tex, then $p^*\mathrm{IC}\nu=\mathrm{IC}(\overline{\mathcal O\nu})\in\mathcal M_I$.

  1. (a) An object of $\mathcal H_{c,I}=\mathrm{Shv}c(I\backslash\mathrm{Fl})$ is supported on a finite union $\mathrm{Fl}{\le}$ of Schubert varieties, on which $I$ acts through a finite-dimensional connected quotient with finitely many orbits, the cells $IwI/I\simeq\mathbb A^{\ell(w)}$, with connected stabilizers $I\cap wIw^{-1}$ (Chen–Yi, Lemma 71(i), chen-yi-v4.pdf §8.1.3). Steps 1, 5, 6 of Lemma 2.2 apply verbatim to this finite-type stack: bounded perverse t-structure, finite-length heart with simple objects $\mathrm{IC}_w$, hence $\mathrm{Thick}\{\mathrm{IC}_w\}$.
  2. (b) $-\ast_I\mathrm{IC}0$ is exact, so it maps $\mathrm{Thick}\{\mathrm{IC}_w\}=\mathcal H{c,I}$ into $\mathrm{Thick}\{G_w\}$ (property (T2) of proposition-I.tex); the reverse inclusion is trivial. For $G=H\times H$: $\widetilde W_G=\widetilde W_H\times\widetilde W_H$, $\mathrm{Fl}G=\mathrm{Fl}_H\times\mathrm{Fl}_H$ and $\mathrm{IC}{(w_1,w_2)}=\mathrm{IC}{w_1}\boxtimes\mathrm{IC}{w_2}$ (BBD Prop. 4.2.8: $\boxtimes$ is t-exact, and 4.2.7: it commutes with the usual operations; hence the support characterization of $j_{!*}$, BBD Cor. 1.4.24, is preserved). The action map $[(g_1,g_2),x]\mapsto g_1xg_2^{-1}$ gives $(A\boxtimes B)\ast_I F=A\ast F\ast\mathrm{inv}^B$, and $\mathrm{inv}\colon g\mapsto g^{-1}$ carries $Iw_2I$ to $Iw_2^{-1}I$, so $\mathrm{inv}^\mathrm{IC}{w_2}=\mathrm{IC}{w_2^{-1}}$. Finally $K_H/I_H=\overline{Iw_0I}/I\simeq G/B$ is smooth of dimension $\ell(w_0)$, so $c=\mathrm{IC}_{w_0}[-\ell(w_0)]$. (Checklist item 2: the action is two-sided.)
  3. (c), the retraction. Let $q\colon I\backslash\mathrm{Fl}\to I\backslash\mathrm{Gr}$ and let $w_\lambda$ be the longest element of $Wt^\lambda W$, so $\overline{Iw_\lambda I}/I=q^{-1}(\mathrm{Gr}{\le\lambda})$, a smooth $G/B$-bundle over $\mathrm{Gr}{\le\lambda}$; by BBD Lemma 4.2.6.1, $q^*V_\lambda=\mathrm{IC}{w\lambda}[-d]$, $d=\dim G/B$. By the key lemma of proposition-I.tex (Lemma 1; $|W|\in k^\times$), $V_\lambda$ is a retract of $q_*q^V_\lambda$; by Step 1 of proposition-I.tex, $(q_\mathcal G)\ast_K\mathrm{IC}_0\simeq\mathcal G\ast_I\mathrm{IC}_0$. Hence
$$p^*(V_\lambda\ast_K\mathrm{IC}_0)\ \text{is a retract of}\ (q^*V_\lambda)\ast_I\mathrm{IC}_0=G_{w_\lambda}[-d].$$
  1. (c), summands. A simple summand $S[s]$ of a retract of $G_{w_\lambda}[-d]$ is a summand of $G_{w_\lambda}$ up to shift, so $S\in\mathcal M_I$. If $\mathrm{IC}\nu[s]$ is a summand of $V\lambda\ast_K\mathrm{IC}0$, then $p^\mathrm{IC}\nu[s]$ is a summand of $p^(V_\lambda\ast_K\mathrm{IC}0)$; $p^\mathrm{IC}\nu$ is the IC of the irreducible $I$-stable closed set $\overline{\mathcal O_\nu}$ with constant coefficients (forgetting equivariance commutes with $j_{!}$), which is simple in $\mathcal C_I$ by Lemma 2.2(a),(c).

Builds on

  • Setup 2.1 · Set-up: level hypotheses, simple objects, $\mathcal M_I\subseteq\mathcal M_I^+$
  • Lemma 2.2 · Lemma (PH$_I$): perverse hearts glue; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$

Built on by

  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$
  • Proposition 2.8 · Group: all hypotheses hold ($\Omega$-truncated levels); $\mathcal S_I=\{\mathrm{IC}_v\}$
  • Proposition 2.10 · Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction
  • Audit 7.2 · Audit of A2-3, A2-4: generators $\mathrm{IC}w\ast_I\mathrm{IC}_0$ and $\mathrm{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D{\mathcal M_I^+}$
  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 18:43 UTC

Lemma: generators $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; spherical summands persist

  1. (a) An object of $\mathcal H_{c,I}=\mathrm{Shv}c(I\backslash\mathrm{Fl})$ is supported on a finite union $\mathrm{Fl}{\le}$ of Schubert varieties, on which $I$ acts through a finite-dimensional connected quotient with finitely many orbits, the cells $IwI/I\simeq\mathbb A^{\ell(w)}$, with connected stabilizers $I\cap wIw^{-1}$ (Chen–Yi, Lemma 71(i), chen-yi-v4.pdf §8.1.3). Steps 1, 5, 6 of Lemma 2.2 apply verbatim to this finite-type stack: bounded perverse t-structure, finite-length heart with simple objects $\mathrm{IC}_w$, hence $\mathrm{Thick}\{\mathrm{IC}_w\}$.
  2. (b) $-\ast_I\mathrm{IC}0$ is exact, so it maps $\mathrm{Thick}\{\mathrm{IC}_w\}=\mathcal H{c,I}$ into $\mathrm{Thick}\{G_w\}$ (property (T2) of proposition-I.tex); the reverse inclusion is trivial. For $G=H\times H$: $\widetilde W_G=\widetilde W_H\times\widetilde W_H$, $\mathrm{Fl}G=\mathrm{Fl}_H\times\mathrm{Fl}_H$ and $\mathrm{IC}{(w_1,w_2)}=\mathrm{IC}{w_1}\boxtimes\mathrm{IC}{w_2}$ (BBD Prop. 4.2.8: $\boxtimes$ is t-exact, and 4.2.7: it commutes with the usual operations; hence the support characterization of $j_{!*}$, BBD Cor. 1.4.24, is preserved). The action map $[(g_1,g_2),x]\mapsto g_1xg_2^{-1}$ gives $(A\boxtimes B)\ast_I F=A\ast F\ast\mathrm{inv}^B$, and $\mathrm{inv}\colon g\mapsto g^{-1}$ carries $Iw_2I$ to $Iw_2^{-1}I$, so $\mathrm{inv}^\mathrm{IC}{w_2}=\mathrm{IC}{w_2^{-1}}$. Finally $K_H/I_H=\overline{Iw_0I}/I\simeq G/B$ is smooth of dimension $\ell(w_0)$, so $c=\mathrm{IC}_{w_0}[-\ell(w_0)]$. (Checklist item 2: the action is two-sided.)
  3. (c), the retraction. Let $q\colon I\backslash\mathrm{Fl}\to I\backslash\mathrm{Gr}$ and let $w_\lambda$ be the longest element of $Wt^\lambda W$, so $\overline{Iw_\lambda I}/I=q^{-1}(\mathrm{Gr}{\le\lambda})$, a smooth $G/B$-bundle over $\mathrm{Gr}{\le\lambda}$; by BBD Lemma 4.2.6.1, $q^*V_\lambda=\mathrm{IC}{w\lambda}[-d]$, $d=\dim G/B$. By the key lemma of proposition-I.tex (Lemma 1; $|W|\in k^\times$), $V_\lambda$ is a retract of $q_*q^V_\lambda$; by Step 1 of proposition-I.tex, $(q_\mathcal G)\ast_K\mathrm{IC}_0\simeq\mathcal G\ast_I\mathrm{IC}_0$. Hence
$$p^*(V_\lambda\ast_K\mathrm{IC}_0)\ \text{is a retract of}\ (q^*V_\lambda)\ast_I\mathrm{IC}_0=G_{w_\lambda}[-d].$$
  1. (c), summands. A simple summand $S[s]$ of a retract of $G_{w_\lambda}[-d]$ is a summand of $G_{w_\lambda}$ up to shift, so $S\in\mathcal M_I$. If $\mathrm{IC}\nu[s]$ is a summand of $V\lambda\ast_K\mathrm{IC}0$, then $p^\mathrm{IC}\nu[s]$ is a summand of $p^(V_\lambda\ast_K\mathrm{IC}0)$; $p^\mathrm{IC}\nu$ is the IC of the irreducible $I$-stable closed set $\overline{\mathcal O_\nu}$ with constant coefficients (forgetting equivariance commutes with $j_{!}$), which is simple in $\mathcal C_I$ by Lemma 2.2(a),(c).

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