§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:42:36Z

Setup 2.1 · Set-up: level hypotheses, simple objects, $\mathcal M_I\subseteq\mathcal M_I^+$

proved

Fixes the hypotheses (L1)-(L3), (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).), (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.)/(I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') on a level presentation of LX, and defines the simple objects S_I of C_I, the summand set M_I and the constituent set M_I^+ used in the Iwahori summand criterion.

Setup 2.1.

Field. Every statement is over $k=\overline{\mathbb F}q$ with $\overline{\mathbb Q}\ell$-coefficients unless marked; the Betti case $k=\mathbb C$ is marked separately (D-modules via Riemann–Hilbert, each finite level having finitely many orbits).

Hypotheses on a level presentation $LX=\varinjlim_N Y_N$, $Y_N=\varprojlim_M Y_{N,M}$ (hypothesis.html, Levels, pieces):

Definitions.

  1. $\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)}$, a directed union of full triangulated subcategories (the brief's definition).
  2. A simple object of $\mathcal C_I$ is an object isomorphic to $\pi_{N,M}^*\bar S$ (extended by zero) for some level and some simple $\bar S$ of the equivariant perverse heart $\mathrm{Perv}{I/I{N,M}}(Y_{N,M})$. $\mathcal S_I$ is the set of simple objects up to isomorphism and shift.
  3. For $w\in\widetilde W$ put $G_w:=\mathrm{IC}w\ast_I\mathrm{IC}_0$ (for the group $X=H$, $G=H\times H$: $G{w_1,w_2}=\mathrm{IC}{w_1}\ast c\ast\mathrm{IC}{w_2}$, see Lemma 2.3).
$$\mathcal M_I:=\{S\in\mathcal S_I:\ S[s]\ \text{is a direct summand of some }G_w\},\qquad \mathcal M_I^+:=\{S\in\mathcal S_I:\ S\ \text{is a constituent of some }{}^pH^i(G_w)\},$$

the perverse cohomology taken at any level containing $G_w$.

  1. For $\Sigma\subseteq\mathcal S_I$: $\mathcal D_\Sigma:=\{F\in\mathcal C_I:$ every constituent of every ${}^pH^i(F)$, at a level containing $F$, lies in $\Sigma\}$.
  2. (DT_I) each $G_w$ lies in $\mathcal C_I$ and is isomorphic in $\mathcal C_I$ to a finite direct sum of shifted simple objects.

Clearly $\mathcal M_I\subseteq\mathcal M_I^+$ (a summand $S[s]$ of $G_w$ gives a summand $S$ of ${}^pH^{-s}(G_w)$); under (DT_I) they are equal.

  1. Role of (L3). It is the reading of the brief's "$\mathcal C_I^{(N,M)}$ = complexes pulled back from $Y_{N,M}$" as an equivalence $D^b_c(I/I_{N,M}\backslash Y_{N,M})\simeq\mathcal C_I^{(N,M)}$; the pages assume the same (hypothesis.html, remark after 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).: "assumed, not proved"). Under (L1) it reduces to: affine-space fibrations have $\pi_*\pi^*\simeq\mathrm{id}$ (homotopy invariance) and inflation along a unipotent kernel is fully faithful on equivariant derived categories (Bernstein–Lunts, Equivariant sheaves and functors, LNM 1578, §2–3). These two facts are standard background, not re-proved here; for $\mathbb A^n$ and the group the inputs (affine-space fibres, unipotent kernels) are checked in Proposition 2.6, Proposition 2.8.
  2. Idempotents. Each $D^b_c(I/I_{N,M}\backslash Y_{N,M})$ is idempotent complete, so by (L3) a retract (in $\mathcal C_I$) of an object of $\mathcal C_I^{(N,M)}$ lies in $\mathcal C_I^{(N,M)}$: the idempotent lives in the full subcategory and splits there. Hence $\mathcal C_I$ is idempotent complete and "Thick" may be computed level by level.
  3. Two sets. $\mathcal M_I$ is the brief's set (summands). $\mathcal M_I^+$ (constituents) is introduced because the inclusion $\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^+}$ needs no decomposition theorem (Theorem 2.4), which is what makes the group case work with semisimplicity of perverse cohomology alone (Theorem 2.9).

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Original record, as the agent wrote it at 18:42 UTC

Set-up: level hypotheses, simple objects, $\mathcal M_I\subseteq\mathcal M_I^+$

  1. Role of (L3). It is the reading of the brief's "$\mathcal C_I^{(N,M)}$ = complexes pulled back from $Y_{N,M}$" as an equivalence $D^b_c(I/I_{N,M}\backslash Y_{N,M})\simeq\mathcal C_I^{(N,M)}$; the pages assume the same (hypothesis.html, remark after I1: "assumed, not proved"). Under (L1) it reduces to: affine-space fibrations have $\pi_*\pi^*\simeq\mathrm{id}$ (homotopy invariance) and inflation along a unipotent kernel is fully faithful on equivariant derived categories (Bernstein–Lunts, Equivariant sheaves and functors, LNM 1578, §2–3). These two facts are standard background, not re-proved here; for $\mathbb A^n$ and the group the inputs (affine-space fibres, unipotent kernels) are checked in Proposition 2.6, Proposition 2.8.
  2. Idempotents. Each $D^b_c(I/I_{N,M}\backslash Y_{N,M})$ is idempotent complete, so by (L3) a retract (in $\mathcal C_I$) of an object of $\mathcal C_I^{(N,M)}$ lies in $\mathcal C_I^{(N,M)}$: the idempotent lives in the full subcategory and splits there. Hence $\mathcal C_I$ is idempotent complete and "Thick" may be computed level by level.
  3. Two sets. $\mathcal M_I$ is the brief's set (summands). $\mathcal M_I^+$ (constituents) is introduced because the inclusion $\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^+}$ needs no decomposition theorem (Theorem 2.4), which is what makes the group case work with semisimplicity of perverse cohomology alone (Theorem 2.9).

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