§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:43:07Z

Lemma 2.2 · Lemma (PH$_I$): perverse hearts glue; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$

proved

From (L1)-(L3) and (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).) alone: each level has a finite-length equivariant perverse heart, the transition functors are t-exact up to a locally constant shift and send simples to simples, so the simple objects of C_I and the constituent subcategories D_Sigma are well defined, thick, and D_Sigma = Thick(Sigma); in particular C_I = Thick(S_I).

Lemma 2.2.

Assume (L1)–(L3) and (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).) of Setup 2.1; write $H=I/I_{N,M}$, $Y=Y_{N,M}$, and let $\tau$ be a transition functor $\mathcal C^{(N,M)}\to\mathcal C^{(N',M')}$ (composite of $\pi_{N,M',M}^$ and $\iota_$).

(a) $D^b_c(H\backslash Y)$ carries the perverse t-structure (an object is perverse iff its underlying complex is); it is bounded and its heart $\mathrm{Perv}_H(Y)$ has finite length. Its simple objects are $\mathrm{IC}(\bar o,\xi)$, $o$ an orbit, $\xi$ an irreducible $H$-equivariant local system on $o$; 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.) only $\xi=k$ occurs.

(b) Let $d$ be the (locally constant) relative dimension of $\pi_{N,M',M}$. Then $\pi^[d]$ and $\iota_$ are t-exact, fully faithful on hearts with essential image closed under subquotients, and send simple objects to simple objects: $\pi^[d]\,\mathrm{IC}(\bar o,\xi)\simeq\mathrm{IC}(\bar o',\xi|{o'})$ with $o'$ the open orbit of the irreducible set $\pi^{-1}(o)$, and $\iota\mathrm{IC}(\bar o,\xi)=\mathrm{IC}(\iota\bar o,\xi)$.

(c) Hence for $F\in\mathcal C^{(N,M)}$: {constituents of $\bigoplus_i{}^pH^i(\tau F)$} $=\tau$({constituents of $\bigoplus_i{}^pH^i(F)$}) up to shift. So $\mathcal S_I$ and $\mathcal D_\Sigma$ of Setup 2.1 do not depend on the level used. 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.), $S\mapsto\operatorname{supp}S$ is a bijection from $\mathcal S_I$ onto $\mathcal Z:=\{\overline P: P\text{ a piece of some level}\}$; two pieces with the same closure give the same simple object, $\mathrm{IC}(\overline P):=\pi_{N,M}^*\mathrm{IC}(\bar o,k)$ (up to shift).

(d) For every $\Sigma\subseteq\mathcal S_I$, $\mathcal D_\Sigma$ is a thick subcategory of $\mathcal C_I$ and $\mathcal D_\Sigma=\mathrm{Thick}(\Sigma)$. In particular $\mathcal C_I=\mathcal D_{\mathcal S_I}=\mathrm{Thick}(\mathcal S_I)$.

  1. Equivariant hearts (a). $H$ is connected (L3). For connected $H$, $\mathrm{Perv}H(Y)\to\mathrm{Perv}(Y)$ is fully faithful with image closed under subquotients: apply BBD Prop. 4.2.5 and Cor. 4.2.6.2 to the projection and action maps $H\times Y\to Y$, smooth with connected fibres $H$ (bbd-perverse.pdf, pp. 109–111). $\mathrm{Perv}(Y)$ has finite length and its simple objects are the $j{!*}(L[\dim V])$ (BBD Thm. 4.3.1); a simple equivariant one has $H$-stable support, so (finitely many orbits, (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).)) $V$ may be taken to be the open orbit $o$ of its support and $L=\xi$ equivariant. An irreducible equivariant local system on $o=H/S$ is a representation of $\pi_0S$; 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.) $\pi_0S=1$, so $\xi=k$. Boundedness: the forgetful functor is t-exact and conservative and $D^b_c(Y)$ has a bounded perverse t-structure.
  2. Smooth transitions (b). $\pi=\pi_{N,M',M}$ is smooth with connected affine-space fibres (L1). BBD 4.2.4 (end): $\pi^[d]$ is t-exact; Prop. 4.2.5: fully faithful on perverse sheaves; Cor. 4.2.6.2: image closed under subquotients; Lemma 4.2.6.1 and its proof: $\pi^[d]j_{!}(L[\dim V])=j_{!}(\pi^*L[\dim\pi^{-1}V])$ and $\pi^L$ stays irreducible. Here $\pi^{-1}(o)$ is smooth (smooth over smooth), connected (open surjection with connected fibres onto connected $o$), hence irreducible; it is $I/I_{N,M'}$-stable with finitely many orbits (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).), so it has a unique open dense orbit $o'$, and $\pi^{-1}(\bar o)=\overline{\pi^{-1}(o)}=\bar o'$ because $\pi$ is open. Restriction of the irreducible $\pi^\xi$ to the dense open $o'$ of a smooth irreducible variety stays irreducible. The equivariant structure is inherited, so by Step 1 the result is simple in $\mathrm{Perv}{I/I{N,M'}}$.
  3. Closed transitions (b). $\iota_*$ is t-exact and fully faithful (BBD (1.4.3.5)); a perverse subobject or quotient of $\iota_*A$ restricts to $0$ on the open complement ($j^*$ is t-exact), so the image is closed under subquotients. Since $\iota(Y_{N,M})$ is $I$-stable, its orbits are orbits of $Y_{N+1,M}$ and the IC of a closed subvariety pushes forward to the IC. By (L2), a piece $\pi_{N,M}^{-1}(o)$ equals $\pi_{N+1,M}^{-1}(\iota o)$, so pieces of level $(N,M)$ are pieces of level $(N+1,M)$.
  4. Constituents (c). On a connected component $C'$ of the source, ${}^pH^i(\pi^F)|{C'}=\pi^{}^pH^{i+d(C')}(F)[d(C')]|{C'}$; a simple object has irreducible support, whose preimage is irreducible, hence in one component. Steps 2–3 give the displayed equality. 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.): a simple object at level $(N,M)$ is determined by $\bar o$ (the open orbit of an irreducible orbit closure is unique), $\operatorname{supp}\pi_{N,M}^*\mathrm{IC}(\bar o,k)=\pi_{N,M}^{-1}(\bar o)=\overline P$, and Step 2 shows that $\overline P=\overline{P'}$ for the open piece $P'\subseteq P$ at a finer level carries the same simple object. Injectivity: two simple objects with the same support are compared at a common level (directedness), where both are $\mathrm{IC}(\bar o,k)$ for the same $o$.
  5. Thickness (d). At one level let $\mathcal P_\Sigma\subseteq\mathrm{Perv}H(Y)$ be the objects with all Jordan–Hölder constituents in $\Sigma$, a Serre subcategory, $q\colon\mathrm{Perv}_H(Y)\to\mathrm{Perv}_H(Y)/\mathcal P\Sigma$ the exact quotient and $h=q\circ{}^pH^0$, a homological functor. Then $\mathcal D_\Sigma\cap\mathcal C^{(N,M)}=\{F: h(F[n])=0\ \forall n\}$, a saturated triangulated subcategory (Stacks, Tag 05RD), i.e. closed under shifts, cones and summands. By (c) these level-wise subcategories are compatible, so their union $\mathcal D_\Sigma$ is closed under shifts and cones (finitely many objects lie in a common level, and that level is closed under cones by (L3)) and under retracts (Setup 2.1, step 2): thick.
  6. $\mathcal D_\Sigma=\mathrm{Thick}(\Sigma)$ (d). $\supseteq$: thick and contains $\Sigma$. $\subseteq$: for $F\in\mathcal D_\Sigma$ at one level, the truncation triangles of the bounded t-structure reduce to the finitely many ${}^pH^i(F)[-i]$, and their finite Jordan–Hölder filtrations reduce to constituents in $\Sigma$ (Stacks, Lemma 13.35.7, Tag 0FX7, as in summand.tex Step 6). With $\Sigma=\mathcal S_I$: $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$. The same BBD statements hold for Betti sheaves over $\mathbb C$; no weights are used.

Builds on

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

Built on by

  • 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 $\mathcal M_I\subseteq\mathcal M_I^+$
  • Lemma 2.5 · Lemma: a finite-type proper model gives (DT$_I$) over $\overline{\mathbb F}_q$
  • Proposition 2.6 · $\mathbb A^n$: all hypotheses hold; simple objects are lattice sheaves $k_\Lambda$
  • Proposition 2.8 · Group: all hypotheses hold ($\Omega$-truncated levels); $\mathcal S_I=\{\mathrm{IC}_v\}$
  • Audit 7.1 · Audit of A2-1, A2-2: perverse hearts glue along the levels and $\mathcal D_\Sigma=\mathrm{Thick}(\Sigma)$
  • 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 (PH$_I$): perverse hearts glue; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$

  1. Equivariant hearts (a). $H$ is connected (L3). For connected $H$, $\mathrm{Perv}H(Y)\to\mathrm{Perv}(Y)$ is fully faithful with image closed under subquotients: apply BBD Prop. 4.2.5 and Cor. 4.2.6.2 to the projection and action maps $H\times Y\to Y$, smooth with connected fibres $H$ (bbd-perverse.pdf, pp. 109–111). $\mathrm{Perv}(Y)$ has finite length and its simple objects are the $j{!*}(L[\dim V])$ (BBD Thm. 4.3.1); a simple equivariant one has $H$-stable support, so (finitely many orbits, (I1)) $V$ may be taken to be the open orbit $o$ of its support and $L=\xi$ equivariant. An irreducible equivariant local system on $o=H/S$ is a representation of $\pi_0S$; under (I2) $\pi_0S=1$, so $\xi=k$. Boundedness: the forgetful functor is t-exact and conservative and $D^b_c(Y)$ has a bounded perverse t-structure.
  2. Smooth transitions (b). $\pi=\pi_{N,M',M}$ is smooth with connected affine-space fibres (L1). BBD 4.2.4 (end): $\pi^[d]$ is t-exact; Prop. 4.2.5: fully faithful on perverse sheaves; Cor. 4.2.6.2: image closed under subquotients; Lemma 4.2.6.1 and its proof: $\pi^[d]j_{!}(L[\dim V])=j_{!}(\pi^*L[\dim\pi^{-1}V])$ and $\pi^L$ stays irreducible. Here $\pi^{-1}(o)$ is smooth (smooth over smooth), connected (open surjection with connected fibres onto connected $o$), hence irreducible; it is $I/I_{N,M'}$-stable with finitely many orbits (I1), so it has a unique open dense orbit $o'$, and $\pi^{-1}(\bar o)=\overline{\pi^{-1}(o)}=\bar o'$ because $\pi$ is open. Restriction of the irreducible $\pi^\xi$ to the dense open $o'$ of a smooth irreducible variety stays irreducible. The equivariant structure is inherited, so by Step 1 the result is simple in $\mathrm{Perv}{I/I{N,M'}}$.
  3. Closed transitions (b). $\iota_*$ is t-exact and fully faithful (BBD (1.4.3.5)); a perverse subobject or quotient of $\iota_*A$ restricts to $0$ on the open complement ($j^*$ is t-exact), so the image is closed under subquotients. Since $\iota(Y_{N,M})$ is $I$-stable, its orbits are orbits of $Y_{N+1,M}$ and the IC of a closed subvariety pushes forward to the IC. By (L2), a piece $\pi_{N,M}^{-1}(o)$ equals $\pi_{N+1,M}^{-1}(\iota o)$, so pieces of level $(N,M)$ are pieces of level $(N+1,M)$.
  4. Constituents (c). On a connected component $C'$ of the source, ${}^pH^i(\pi^F)|{C'}=\pi^{}^pH^{i+d(C')}(F)[d(C')]|{C'}$; a simple object has irreducible support, whose preimage is irreducible, hence in one component. Steps 2–3 give the displayed equality. Under (I2): a simple object at level $(N,M)$ is determined by $\bar o$ (the open orbit of an irreducible orbit closure is unique), $\operatorname{supp}\pi_{N,M}^*\mathrm{IC}(\bar o,k)=\pi_{N,M}^{-1}(\bar o)=\overline P$, and Step 2 shows that $\overline P=\overline{P'}$ for the open piece $P'\subseteq P$ at a finer level carries the same simple object. Injectivity: two simple objects with the same support are compared at a common level (directedness), where both are $\mathrm{IC}(\bar o,k)$ for the same $o$.
  5. Thickness (d). At one level let $\mathcal P_\Sigma\subseteq\mathrm{Perv}H(Y)$ be the objects with all Jordan–Hölder constituents in $\Sigma$, a Serre subcategory, $q\colon\mathrm{Perv}_H(Y)\to\mathrm{Perv}_H(Y)/\mathcal P\Sigma$ the exact quotient and $h=q\circ{}^pH^0$, a homological functor. Then $\mathcal D_\Sigma\cap\mathcal C^{(N,M)}=\{F: h(F[n])=0\ \forall n\}$, a saturated triangulated subcategory (Stacks, Tag 05RD), i.e. closed under shifts, cones and summands. By (c) these level-wise subcategories are compatible, so their union $\mathcal D_\Sigma$ is closed under shifts and cones (finitely many objects lie in a common level, and that level is closed under cones by (L3)) and under retracts (Setup 2.1, step 2): thick.
  6. $\mathcal D_\Sigma=\mathrm{Thick}(\Sigma)$ (d). $\supseteq$: thick and contains $\Sigma$. $\subseteq$: for $F\in\mathcal D_\Sigma$ at one level, the truncation triangles of the bounded t-structure reduce to the finitely many ${}^pH^i(F)[-i]$, and their finite Jordan–Hölder filtrations reduce to constituents in $\Sigma$ (Stacks, Lemma 13.35.7, Tag 0FX7, as in summand.tex Step 6). With $\Sigma=\mathcal S_I$: $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$. The same BBD statements hold for Betti sheaves over $\mathbb C$; no weights are used.

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