§5 · Agent 5 · 𝔸ⁿ: lattices · 2026-10-18T18:40:16Z

Lemma 5.2 · Simple objects of $\mathcal C_I$: $S_k=\underline k_{\tau^k\mathcal O^n}[-k]$

proved ★ key step

With the perverse t-structure renormalized by $nM$ at level $(N,M)$, the transition functors are t-exact and fully faithful, the heart of $\mathcal C_I$ has finite length, and its simple objects are exactly the shifted constant sheaves on the $I$-stable lattices, one for each integer $k$. None is supported at $0$, and $\mathcal C_I$ is the triangulated span of these objects.

Lemma 5.2.

Let $X=\mathbb A^n$, $G=\mathrm{GL}n$, $k=\mathbb C$ (Betti) or $\overline{\mathbb F}_q$ ($\ell$-adic), and $V\alpha,H_\alpha$ as in Lemma 5.1. Let $\mathcal C_\alpha=D^b_{H_\alpha}(V_\alpha)$ and $\iota_\alpha:\mathcal C_\alpha\to\mathcal C_I$, $A\mapsto$ (extension by zero from $Y_N$ of) $\pi_{N,M}^*A$. Declare $A\in{}^p\mathcal C_\alpha^{\le0}$ (resp. $^{\ge0}$) iff $A[nM]$ lies in the usual perverse $^pD^{\le0}$ (resp. $^{\ge0}$) of $V_\alpha$. Then:

(a) the transitions $q^:\mathcal C_{N,M}\to\mathcal C_{N,M+1}$ ($q$ the reduction map) and $i_:\mathcal C_{N,M}\to\mathcal C_{N+1,M}$ (closed embedding) are t-exact and fully faithful, with essential images on hearts closed under subobjects and quotients; so $\mathcal C_I=\bigcup_\alpha\iota_\alpha\mathcal C_\alpha$ carries a bounded t-structure with heart $\mathcal P_I=\bigcup_\alpha\mathcal P_\alpha$ of finite length;

(b) $\operatorname{Simple}(\mathcal P_I)=\{S_k:k\in\mathbb Z\}$, $\displaystyle S_k:=\underline k_{\tau^k\mathcal O^n}[-k]$, pairwise non-isomorphic, $S_0=\mathrm{IC}0$, and $S_k$ lies in $\mathcal P{N,M}$ iff $-nN\le k\le nM$;

(c) $\mathcal C_I=\operatorname{Tri}\{S_k\}=\operatorname{Thick}\{S_k\}$ (every object is a finite iterated extension of shifts of the $S_k$; no retracts are needed);

(d) no nonzero object of $\mathcal C_I$ is supported on $\{0\}$; the finite-level skyscraper at $0\in V_{N,M}$ corresponds to $S_{nM}=\underline k_{t^M\mathcal O^n}[-nM]$.

Proof.

  1. Equivariant hearts. $H_\alpha$ is connected (Lemma 5.1(d)). For a connected group the perverse heart $\operatorname{Perv}{H}(V)$ of $D^b_H(V)$ maps fully faithfully to $\operatorname{Perv}(V)$ with image closed under subquotients (apply BBD 4.2.5–4.2.6 to the smooth maps $\mathrm{act},\mathrm{pr}:H\times V\to V$, which have connected fibres). Hence $\mathcal P\alpha$ has finite length (BBD 4.3.1) and its simple objects are $\mathrm{IC}(\bar o,\xi)$, $o$ an orbit, $\xi$ an irreducible $H_\alpha$-equivariant local system on $o=H_\alpha/S$; these correspond to representations of $\pi_0(S)=1$ (Lemma 5.1(d), step 5), so $\xi$ is constant.
  1. Simples of one level. The orbit closures are the linear subspaces $V_{\alpha,k}$, $-nN\le k\le nM$ ($V_{\alpha,nM}=\{0\}$), smooth of dimension $nM-k$, so $\mathrm{IC}(V_{\alpha,k})=\underline k_{V_{\alpha,k}}[nM-k]$. In the renormalized convention $\underline k_{V_{\alpha,k}}[e]\in\mathcal P_\alpha$ iff $e+nM=nM-k$, i.e. $e=-k$; pulled back, this is $\underline k_{\tau^k\mathcal O^n}[-k]=S_k$. So $\operatorname{Simple}(\mathcal P_\alpha)=\{S_k:-nN\le k\le nM\}$, $n(N+M)+1$ objects, one per piece of level $\alpha$.
  1. Transitions. $q:V_{N,M+1}\to V_{N,M}$ is a smooth surjection with fibres $t^M\mathcal O^n/t^{M+1}\mathcal O^n\simeq\mathbb A^n$. $q^*[n]$ is t-exact (BBD 4.2.4), conservative ($q$ surjective), fully faithful on perverse sheaves with image closed under subquotients (BBD 4.2.5, 4.2.6; connected fibres). Since $(q^A)[n(M+1)]=(q^[n])(A[nM])$, the plain $q^$ is t-exact for the renormalized structures. Enlarging the group $H_{N,M}\leftarrow H_{N,M+1}$ (kernel acting trivially, both connected) changes nothing on hearts by step 1. $i_$ for $V_{N,M}\hookrightarrow V_{N+1,M}$ is t-exact and fully faithful with image the objects supported on the closed subset, and the shift $nM$ does not depend on $N$. The two kinds of transition commute (cartesian squares, base change).
  1. Colimit. Put $\mathcal C_I^{\le0}=\bigcup\iota_\alpha\mathcal C_\alpha^{\le0}$ and likewise $^{\ge0}$; by step 3 membership does not depend on the level used, and Hom-vanishing and truncation triangles are computed at a common level (the transitions are fully faithful also on derived categories: smooth pull-back with affine-space fibres, change of group along a surjection with unipotent kernel acting trivially, closed push-forward; this is what makes $\mathcal C_I$ a union in the setting). The heart is $\bigcup\mathcal P_\alpha$. A fully faithful exact functor between finite-length abelian categories whose image is closed under subquotients sends simple objects to simple objects, so a composition series at level $\alpha$ remains one at every later level. Hence $\mathcal P_I$ has finite length and $\operatorname{Simple}(\mathcal P_I)=\bigcup_\alpha\operatorname{Simple}(\mathcal P_\alpha)=\{S_k\}{k\in\mathbb Z}$; distinct $k$ give distinct supports; $S_0=\underline k{\mathcal O^n}=\mathrm{IC}_0$.
  1. (c) An object of $\mathcal C_\alpha$ has finitely many nonzero perverse cohomology objects, each of finite length; truncation triangles and composition series exhibit it as an iterated extension of shifts of the $S_k$.
  1. (d) An object $\pi_{N,M}^*A$ has support a union of fibres $v+t^M\mathcal O^n$, which $\{0\}$ is not. The skyscraper $\underline k_0$ on $V_{N,M}$ is $\mathrm{IC}(\{0\})$; renormalized it is $\underline k_0[-nM]$, and its pullback is $\underline k_{t^M\mathcal O^n}[-nM]=S_{nM}$ (Lemma 5.1(a): $t^M\mathcal O^n=\tau^{nM}\mathcal O^n$).

Remarks. The shift is the dimension theory $d(\Lambda)=\dim\Lambda/(\Lambda\cap\mathcal O^n)-\dim\mathcal O^n/(\Lambda\cap\mathcal O^n)$, with $d(\tau^k\mathcal O^n)=-k$ and $S_k=\underline k_{\tau^k\mathcal O^n}[d]$. Any other normalization compatible with the transitions differs by one global shift, which does not change the set of simple objects up to shift. The point $0$ and the tails $T_M=t^M\mathcal O^n$: a tail is the support of the simple object $S_{nM}$, and the point $0$ is never a stratum of any level (checklist items 1 and 6).

Builds on

  • Lemma 5.1 · Orbits of $I$ on $L\mathbb A^n$: closures $\tau^k\mathcal O^n$, $\tau\in\Omega$

Built on by

  • Computation 5.3 · $\mathrm{IC}_{\tau^k}\ast_I\mathrm{IC}_0=S_k[k]$ and the simple reflections
  • Proposition 5.4 · (I$_I$) The Iwahori summand criterion: generation iff $\mathcal M_I$ is all
  • Theorem 5.5 · $\mathbb A^n$: $\mathcal M_I=\{S_k\}$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$
  • Corollary 5.8 · Reverse route for $\mathbb A^n$: (IG) and (II$_I$) give sheaf generation
  • Audit 8.8 · Audit of $\mathbb A^n$ orbits, levels and simple objects (A5-1, A5-2, A2-6)
  • 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 18:40 UTC

Simple objects of $\mathcal C_I$: $S_k=\underline k_{\tau^k\mathcal O^n}[-k]$

Proof.

  1. Equivariant hearts. $H_\alpha$ is connected (Lemma 5.1(d)). For a connected group the perverse heart $\operatorname{Perv}{H}(V)$ of $D^b_H(V)$ maps fully faithfully to $\operatorname{Perv}(V)$ with image closed under subquotients (apply BBD 4.2.5–4.2.6 to the smooth maps $\mathrm{act},\mathrm{pr}:H\times V\to V$, which have connected fibres). Hence $\mathcal P\alpha$ has finite length (BBD 4.3.1) and its simple objects are $\mathrm{IC}(\bar o,\xi)$, $o$ an orbit, $\xi$ an irreducible $H_\alpha$-equivariant local system on $o=H_\alpha/S$; these correspond to representations of $\pi_0(S)=1$ (Lemma 5.1(d), step 5), so $\xi$ is constant.
  1. Simples of one level. The orbit closures are the linear subspaces $V_{\alpha,k}$, $-nN\le k\le nM$ ($V_{\alpha,nM}=\{0\}$), smooth of dimension $nM-k$, so $\mathrm{IC}(V_{\alpha,k})=\underline k_{V_{\alpha,k}}[nM-k]$. In the renormalized convention $\underline k_{V_{\alpha,k}}[e]\in\mathcal P_\alpha$ iff $e+nM=nM-k$, i.e. $e=-k$; pulled back, this is $\underline k_{\tau^k\mathcal O^n}[-k]=S_k$. So $\operatorname{Simple}(\mathcal P_\alpha)=\{S_k:-nN\le k\le nM\}$, $n(N+M)+1$ objects, one per piece of level $\alpha$.
  1. Transitions. $q:V_{N,M+1}\to V_{N,M}$ is a smooth surjection with fibres $t^M\mathcal O^n/t^{M+1}\mathcal O^n\simeq\mathbb A^n$. $q^*[n]$ is t-exact (BBD 4.2.4), conservative ($q$ surjective), fully faithful on perverse sheaves with image closed under subquotients (BBD 4.2.5, 4.2.6; connected fibres). Since $(q^A)[n(M+1)]=(q^[n])(A[nM])$, the plain $q^$ is t-exact for the renormalized structures. Enlarging the group $H_{N,M}\leftarrow H_{N,M+1}$ (kernel acting trivially, both connected) changes nothing on hearts by step 1. $i_$ for $V_{N,M}\hookrightarrow V_{N+1,M}$ is t-exact and fully faithful with image the objects supported on the closed subset, and the shift $nM$ does not depend on $N$. The two kinds of transition commute (cartesian squares, base change).
  1. Colimit. Put $\mathcal C_I^{\le0}=\bigcup\iota_\alpha\mathcal C_\alpha^{\le0}$ and likewise $^{\ge0}$; by step 3 membership does not depend on the level used, and Hom-vanishing and truncation triangles are computed at a common level (the transitions are fully faithful also on derived categories: smooth pull-back with affine-space fibres, change of group along a surjection with unipotent kernel acting trivially, closed push-forward; this is what makes $\mathcal C_I$ a union in the setting). The heart is $\bigcup\mathcal P_\alpha$. A fully faithful exact functor between finite-length abelian categories whose image is closed under subquotients sends simple objects to simple objects, so a composition series at level $\alpha$ remains one at every later level. Hence $\mathcal P_I$ has finite length and $\operatorname{Simple}(\mathcal P_I)=\bigcup_\alpha\operatorname{Simple}(\mathcal P_\alpha)=\{S_k\}{k\in\mathbb Z}$; distinct $k$ give distinct supports; $S_0=\underline k{\mathcal O^n}=\mathrm{IC}_0$.
  1. (c) An object of $\mathcal C_\alpha$ has finitely many nonzero perverse cohomology objects, each of finite length; truncation triangles and composition series exhibit it as an iterated extension of shifts of the $S_k$.
  1. (d) An object $\pi_{N,M}^*A$ has support a union of fibres $v+t^M\mathcal O^n$, which $\{0\}$ is not. The skyscraper $\underline k_0$ on $V_{N,M}$ is $\mathrm{IC}(\{0\})$; renormalized it is $\underline k_0[-nM]$, and its pullback is $\underline k_{t^M\mathcal O^n}[-nM]=S_{nM}$ (Lemma 5.1(a): $t^M\mathcal O^n=\tau^{nM}\mathcal O^n$).

Remarks. The shift is the dimension theory $d(\Lambda)=\dim\Lambda/(\Lambda\cap\mathcal O^n)-\dim\mathcal O^n/(\Lambda\cap\mathcal O^n)$, with $d(\tau^k\mathcal O^n)=-k$ and $S_k=\underline k_{\tau^k\mathcal O^n}[d]$. Any other normalization compatible with the transitions differs by one global shift, which does not change the set of simple objects up to shift. The point $0$ and the tails $T_M=t^M\mathcal O^n$: a tail is the support of the simple object $S_{nM}$, and the point $0$ is never a stratum of any level (checklist items 1 and 6).

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