§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:48:16Z

Proposition 2.8 · Group: all hypotheses hold ($\Omega$-truncated levels); $\mathcal S_I=\{\mathrm{IC}_v\}$

proved

Verifies every hypothesis of the Iwahori summand criterion for the group case, after repairing the pages' level presentation (which has infinitely many pieces per level when Omega is infinite, e.g. H = GL_n); the simple objects are the IC_v and the generators IC_{w1} * c * IC_{w2} decompose by purity and Sun's theorem.

Proposition 2.8.

$X=H$, $G=H\times H$, $(h_1,h_2)x=h_1xh_2^{-1}$, $W_H\ne1$, $I=I_H$, $\widetilde W=\widetilde W_H=W_{\mathrm{aff}}\rtimes\Omega$, $\mathcal C_I=\mathrm{Shv}c(I\backslash LH/I)$. Fix finite subsets $\Omega_1\subseteq\Omega_2\subseteq\cdots$ exhausting $\Omega$ and put $$Y_N=\bigcup\{\overline{IwI}:\ \ell(w)\le N,\ w\in W_{\mathrm{aff}}\Omega_N\},\quad Y_{N,M}=Y_N/I_M,\quad I_{N,M}=I_{M'}\times I_M,$$ $I_M=\ker(I\to H(\mathcal O/t^M))$, $M\ge1$, $M'=M'(N,M)$ increasing with $x^{-1}I{M'}x\subseteq I_M$ for $x\in Y_N$.

(i) (L1)–(L3) hold (full faithfulness in (L3) as in Setup 2.1, step 1, together with the quotient equivalence for the free right $I/I_M$-action). (ii) (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).): the pieces of level $(N,M)$ are the double cosets $IvI\subseteq Y_N$, finitely many. (iii) (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 stabilizers are images of $I\cap vIv^{-1}$, connected. (iv) $\mathcal S_I=\{\mathrm{IC}v:v\in\widetilde W\}$ up to shift; $\mathrm{IC}\omega=\Delta_\omega$ for $\omega\in\Omega$, in particular $\mathrm{IC}e=\Delta_e$. (v) (DT$_I$): each $P=\mathrm{IC}{w_1}\ast\mathrm{IC}{w_0}\ast\mathrm{IC}{w_2}$ (a shift of the generator $\mathrm{IC}{w_1}\ast c\ast\mathrm{IC}{w_2}$) is, over $\overline{\mathbb F}_q$, pure and a finite direct sum of shifted $\mathrm{IC}_v$ in $\mathcal C_I$; over $\mathbb C$ by the cited equivariant decomposition theorem.

Objection to the pages' Example (hypothesis.html): with $Y_N=\bigcup_{\ell(w)\le N}\overline{IwI}$ the claim $|\mathcal P_{N,M}|<\infty$ fails when $\Omega$ is infinite (e.g. $H=\mathrm{GL}_n$, $\Omega\simeq\mathbb Z$: all $\omega\in\Omega$ have length $0$); truncating by $\Omega_N$ repairs it.

  1. Finiteness. $\ell(w'\sigma)=\ell(w')$ for $w'\in W_{\mathrm{aff}}$, $\sigma\in\Omega$; $W_{\mathrm{aff}}$ is a Coxeter group with finitely many generators, so it has finitely many elements of length $\le N$, and $\Omega_N$ is finite. $LH=\bigsqcup_{w\in\widetilde W}IwI$ (Iwahori–Matsumoto, Thm. 2.16), and $\overline{IwI}=\bigsqcup_{v\le w}IvI$ for the Bruhat order (standard; finite intervals, and only elements with the same $\Omega$-part are comparable). So $Y_N$ is a finite union of double cosets, closed and $I\times I$-stable, and $LH=\varinjlim Y_N$.
  2. (L1)–(L3). $Y_N$ is bounded, so in a faithful representation $H\subseteq\mathrm{GL}r$ the entries of $x^{\pm1}$, $x\in Y_N$, lie in $t^{-c}\mathcal O$, and $M'=M+2c$ works. $Y_N/I{M+1}\to Y_N/I_M$ is a torsor under the vector group $I_M/I_{M+1}$ ($M\ge1$), Zariski-locally trivial since $H^1$ of an affine scheme with coefficients in $\mathbb G_a$ vanishes: an affine-space fibration (L1). $Y_N$ is right $I$-stable, so $Y_N/I_M\subseteq Y_{N+1}/I_M$ is closed with $Y_N$ its full preimage (L2). $I/I_M$ is connected (an extension of $B$ by a unipotent group) and $I_{N,M}/I_{N',M'}$ is unipotent (L3).
  3. (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).). Each $IvI$ is right $I_M$-stable, so it is the preimage of its image, an $I\times I$-orbit; by Step 1 there are finitely many. Every piece is an orbit, closures are the Schubert varieties $\overline{IvI}$.
  4. (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.). $(g_1,g_2)\cdot vI_M=g_1vg_2^{-1}I_M$ equals $vI_M$ iff $g_2\in v^{-1}g_1vI_M$; since $g_2\in I$ this forces $g_1\in I\cap vIv^{-1}$. So the stabilizer is the image of $I\cap vIv^{-1}$ under $g\mapsto(g\bmod I_{M'},\,v^{-1}gv\bmod I_M)$. $I\cap vIv^{-1}$ is connected: Chen–Yi, Lemma 71(i) (points of $I\backslash LG/I$-orbits have connected $I\times I$-stabilizers; there $G$ is any reductive group, here $H$); concretely it is $T(\mathcal O)$ times a product of affine root subgroups (Iwahori factorization). The image of a connected group is connected.
  5. (iv). By Lemma 2.2(c) the simple objects are the IC's of the piece closures with constant coefficients: the $\mathrm{IC}v$, pairwise distinct. For $\omega\in\Omega$, $I\omega I=\omega I$ is a point, so $\mathrm{IC}\omega=\Delta_\omega$.
  6. (v), model. By Lemma 2.3(b) the generators are $\mathrm{IC}{w_1}\ast c\ast\mathrm{IC}{w_2}$ with $c=\mathrm{IC}{w_0}[-\ell(w_0)]$. Let $Z=\overline{Iw_1I}\times^I\overline{Iw_0I}\times^I\overline{Iw_2I}/I$ and $m[g_1,g_2,g_3]=g_1g_2g_3I$; $Z$ is an iterated fibration with projective fibres (Schubert varieties), so it is projective and $m$ is proper, and $P=m!(\mathrm{IC}{w_1}\tilde\boxtimes\mathrm{IC}{w_0}\tilde\boxtimes\mathrm{IC}_{w_2})$, the twisted product defined as in Chen–Yi (20).
  7. (v), purity and splitting. Over $\overline{\mathbb F}q$ (everything defined over $\mathbb F_q$, $H$ split): each $\mathrm{IC}_v$ is pure of weight $\ell(v)$ (BBD 5.3.2). The twisted product pulls back, along smooth surjections, to an external product of pure complexes, which is pure (BBD 5.1.14; Sun, Prop. 3.9(iv)); purity descends along smooth surjections because the weight conditions are pointwise. $m$ is proper, so $P$ is pure (Sun, Prop. 3.9(iii)). Sun, Thm. 1.2 applied to $[H''\backslash Z]\to[H''\backslash Y{N,M}]$ ($H''$ a finite-dimensional quotient of $I$ acting on the left) gives $P\simeq\bigoplus_i{}^pH^i(P)[-i]$ with semisimple ${}^pH^i(P)$, i.e. (DT$_I$) by Step 5. Independent citation: the group is the symmetric variety $(H\times H)/\Delta H$, and Chen–Yi, Prop. 70(ii) gives semisimplicity of $M\star F$ directly ($p$ odd). Over $\mathbb C$: Lemma 2.5, Step 5 (cited).

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; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$
  • Lemma 2.3 · Lemma: generators $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; spherical summands persist
  • Lemma 2.5 · Lemma: a finite-type proper model gives (DT$_I$) over $\overline{\mathbb F}_q$

Built on by

  • Theorem 2.9 · Group: $\Omega\cap\mathcal M_I^+=\varnothing$ by sign character and weights
  • Audit 8.1 · Audit of the group set-up (A4-1, A2-8): pieces, stabilizers, generators, levels
Original record, as the agent wrote it at 18:48 UTC

Group: all hypotheses hold ($\Omega$-truncated levels); $\mathcal S_I=\{\mathrm{IC}_v\}$

  1. Finiteness. $\ell(w'\sigma)=\ell(w')$ for $w'\in W_{\mathrm{aff}}$, $\sigma\in\Omega$; $W_{\mathrm{aff}}$ is a Coxeter group with finitely many generators, so it has finitely many elements of length $\le N$, and $\Omega_N$ is finite. $LH=\bigsqcup_{w\in\widetilde W}IwI$ (Iwahori–Matsumoto, Thm. 2.16), and $\overline{IwI}=\bigsqcup_{v\le w}IvI$ for the Bruhat order (standard; finite intervals, and only elements with the same $\Omega$-part are comparable). So $Y_N$ is a finite union of double cosets, closed and $I\times I$-stable, and $LH=\varinjlim Y_N$.
  2. (L1)–(L3). $Y_N$ is bounded, so in a faithful representation $H\subseteq\mathrm{GL}r$ the entries of $x^{\pm1}$, $x\in Y_N$, lie in $t^{-c}\mathcal O$, and $M'=M+2c$ works. $Y_N/I{M+1}\to Y_N/I_M$ is a torsor under the vector group $I_M/I_{M+1}$ ($M\ge1$), Zariski-locally trivial since $H^1$ of an affine scheme with coefficients in $\mathbb G_a$ vanishes: an affine-space fibration (L1). $Y_N$ is right $I$-stable, so $Y_N/I_M\subseteq Y_{N+1}/I_M$ is closed with $Y_N$ its full preimage (L2). $I/I_M$ is connected (an extension of $B$ by a unipotent group) and $I_{N,M}/I_{N',M'}$ is unipotent (L3).
  3. (I1). Each $IvI$ is right $I_M$-stable, so it is the preimage of its image, an $I\times I$-orbit; by Step 1 there are finitely many. Every piece is an orbit, closures are the Schubert varieties $\overline{IvI}$.
  4. (I2). $(g_1,g_2)\cdot vI_M=g_1vg_2^{-1}I_M$ equals $vI_M$ iff $g_2\in v^{-1}g_1vI_M$; since $g_2\in I$ this forces $g_1\in I\cap vIv^{-1}$. So the stabilizer is the image of $I\cap vIv^{-1}$ under $g\mapsto(g\bmod I_{M'},\,v^{-1}gv\bmod I_M)$. $I\cap vIv^{-1}$ is connected: Chen–Yi, Lemma 71(i) (points of $I\backslash LG/I$-orbits have connected $I\times I$-stabilizers; there $G$ is any reductive group, here $H$); concretely it is $T(\mathcal O)$ times a product of affine root subgroups (Iwahori factorization). The image of a connected group is connected.
  5. (iv). By Lemma 2.2(c) the simple objects are the IC's of the piece closures with constant coefficients: the $\mathrm{IC}v$, pairwise distinct. For $\omega\in\Omega$, $I\omega I=\omega I$ is a point, so $\mathrm{IC}\omega=\Delta_\omega$.
  6. (v), model. By Lemma 2.3(b) the generators are $\mathrm{IC}{w_1}\ast c\ast\mathrm{IC}{w_2}$ with $c=\mathrm{IC}{w_0}[-\ell(w_0)]$. Let $Z=\overline{Iw_1I}\times^I\overline{Iw_0I}\times^I\overline{Iw_2I}/I$ and $m[g_1,g_2,g_3]=g_1g_2g_3I$; $Z$ is an iterated fibration with projective fibres (Schubert varieties), so it is projective and $m$ is proper, and $P=m!(\mathrm{IC}{w_1}\tilde\boxtimes\mathrm{IC}{w_0}\tilde\boxtimes\mathrm{IC}_{w_2})$, the twisted product defined as in Chen–Yi (20).
  7. (v), purity and splitting. Over $\overline{\mathbb F}q$ (everything defined over $\mathbb F_q$, $H$ split): each $\mathrm{IC}_v$ is pure of weight $\ell(v)$ (BBD 5.3.2). The twisted product pulls back, along smooth surjections, to an external product of pure complexes, which is pure (BBD 5.1.14; Sun, Prop. 3.9(iv)); purity descends along smooth surjections because the weight conditions are pointwise. $m$ is proper, so $P$ is pure (Sun, Prop. 3.9(iii)). Sun, Thm. 1.2 applied to $[H''\backslash Z]\to[H''\backslash Y{N,M}]$ ($H''$ a finite-dimensional quotient of $I$ acting on the left) gives $P\simeq\bigoplus_i{}^pH^i(P)[-i]$ with semisimple ${}^pH^i(P)$, i.e. (DT$_I$) by Step 5. Independent citation: the group is the symmetric variety $(H\times H)/\Delta H$, and Chen–Yi, Prop. 70(ii) gives semisimplicity of $M\star F$ directly ($p$ odd). Over $\mathbb C$: Lemma 2.5, Step 5 (cited).

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