§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:39:04Z

Setup 3.1 · Set-up over $\mathbb F_q$: levels, the perverse heart of $\mathcal C_I$, Weil structures

proved

Fixes the hypotheses (L1)-(L2) and dictionary inputs (D1)-(D3) used by the bridge, and proves that $\mathcal C_I$ has a bounded t-structure whose heart is a union of finite-length hearts, with simple objects $\mathrm{IC}(P)$ of pieces, each carrying a canonical Weil structure.

Setup 3.1.

Work over $k=\overline{\mathbb F}q$, $p$ odd, $\overline{\mathbb Q}\ell$-coefficients; $G,B,T$ split over $\mathbb F_q$, $X$ smooth affine over $\mathbb F_q$, $\mathrm{Fr}$ the $q$-Frobenius. Assume

(L1) level presentation over $\mathbb F_q$. $Y_N,Y_{N,M},\pi_{N,M},I_{N,M}$ as in the setting, all defined over $\mathbb F_q$, with (a) each transition $Y_{N,M'}\to Y_{N,M}$ ($M'\ge M$) an affine-space bundle of constant rank $d(M,M')$, equivariant for $I/I_{N,M'}\to I/I_{N,M}$; (b) for $N'\ge N$ and $M\gg0$, $Y_N=\pi_{N',M}^{-1}(\bar Y)$ for a closed $I$-stable $\bar Y\subseteq Y_{N',M}$, identified with $Y_{N,M}$; (c) $I/I_{N,M}$ connected of finite type and $I_{N,M}$ connected pro-algebraic; (d) $Y_N(\mathbb F_q)$ is compact open in $X(F)$, $\bigcup_NY_N(\mathbb F_q)=X(F)$, and the fibres of $\pi_{N,M}$ on $Y_N(\mathbb F_q)$, $M\ge0$, form a basis of its topology; (e) $\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)}$.

(L2) finite, connected, rational orbits. Each $Y_{N,M}$ has finitely many $I/I_{N,M}$-orbits; each is $\mathrm{Fr}$-stable and has connected stabilizers.

Put $\Phi_{N,M}:=(\text{extension by zero})\circ\pi_{N,M}^*[e_M]$ with integers $e_M=e_{M'}+d(M,M')$. Then:

(i) The perverse t-structures on $D^b_{I/I_{N,M}}(Y_{N,M})$, transported by $\Phi_{N,M}$, glue to a bounded t-structure on $\mathcal C_I$; every transition functor is t-exact and fully faithful, and $\mathrm{Fr}^*$ is t-exact.

(ii) The heart $\mathcal P_I$ is the increasing union of the finite-length hearts $\mathcal P^{(N,M)}$; its simple objects are the $\mathrm{IC}(P):=\Phi_{N,M}\,\mathrm{IC}(\bar o)$, $P=\pi_{N,M}^{-1}(o)$, with $\mathrm{End}=\overline{\mathbb Q}_\ell$ and $\mathrm{Hom}(\mathrm{IC}(P),\mathrm{IC}(P'))=0$ unless they are isomorphic; at a finer level $\mathrm{IC}(P)=\mathrm{IC}(P^\circ)$, $P^\circ$ the open piece of $P$.

(iii) Every simple object $S$ satisfies $\mathrm{Fr}^*S\simeq S$ and has a unique Weil structure $\Phi_S$ which is the identity on $S|P=\overline{\mathbb Q}\ell[\delta(P)]$, $\delta(P)=\dim o+e_M$.

Dictionary inputs (named once, used in Lemma 3.2–Theorem 3.6):

Proof of (i)–(iii).

  1. Finite level. $Y_{N,M}$ has finitely many orbits of the connected group $I/I_{N,M}$ (L2), so its equivariant perverse heart has finite length with simple objects $\mathrm{IC}(\bar o,\xi)$; connected stabilizers force $\xi$ trivial (equivariant local systems on $o=H/S$ are representations of $\pi_0S=1$).
  2. Change of $M$. $\pi:=Y_{N,M'}\to Y_{N,M}$ is smooth surjective with connected fibres of dimension $d$, so $\pi^[d]$ is t-exact and fully faithful on perverse sheaves, with image closed under subquotients [BBD, 4.2.4–4.2.5]; it sends $\mathrm{IC}(\bar o)$ to $\mathrm{IC}(\overline{\pi^{-1}o})$ [BBD, 4.2.6], and $\pi^{-1}o$ is irreducible with a unique open orbit $o^\circ$ (finitely many orbits). Fibres are affine spaces, so $\pi_\pi^=\mathrm{id}$ and $\pi^$ is fully faithful on $D^b$. The kernel $I_{N,M}/I_{N,M'}$ is unipotent, so the two equivariant categories agree. The choice $e_M=e_{M'}+d(M,M')$ makes $\Phi_{N,M}=\Phi_{N,M'}\circ\pi^*[d]$.
  3. Change of $N$. By (L1b) the inclusion $Y_N\subseteq Y_{N'}$ is, at a common level $M$, a closed immersion of finite-type schemes; closed push-forward is t-exact, fully faithful, and sends simple objects to simple objects. No shift enters because $e_M$ depends only on $M$.
  4. Gluing. By (L1e) $\mathcal C_I$ is the filtered union of the $\mathcal C_I^{(N,M)}$ along t-exact fully faithful functors, so the t-structures glue; boundedness and finite length hold at each level, hence for each object. A simple object of $\mathcal P_I$ lies in some $\mathcal P^{(N,M)}$ and is simple there, and steps 2–3 show it stays simple at all finer levels: this is (ii).
  5. Frobenius. $\mathrm{Fr}^$ is an autoequivalence preserving perversity and sends $\mathrm{IC}(P)$ to $\mathrm{IC}(\mathrm{Fr}^{-1}P)=\mathrm{IC}(P)$ (L2). Restriction to $P$ identifies $\mathrm{Hom}(\mathrm{Fr}^\mathrm{IC}(P),\mathrm{IC}(P))\simeq\mathrm{End}(\mathrm{IC}(P))=\overline{\mathbb Q}\ell$ with $\mathrm{End}(\overline{\mathbb Q}{\ell,P})$, so the normalized $\Phi_S$ exists and is unique; it is automatically $I$-equivariant because $I/I_{N,M}$ is connected (equivariance is a property for perverse sheaves under a connected group). This is (iii). $\square$

Terms. Piece = preimage of a finite-level orbit, not an $I$-orbit (for $\mathbb A^n$ the closed piece $t^M\mathcal O^n$ contains infinitely many orbits and the orbit $\{0\}$ is never a piece). $\mathrm{IC}0$ is the unshifted constant sheaf on $L^+X$, so $\mathrm{tr}\,\mathrm{IC}_0=\varphi_0$. $\Delta_P=j{P!}k_P$ is unshifted. Thick = closed under shifts, cofibres, retracts.

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Original record, as the agent wrote it at 18:39 UTC

Set-up over $\mathbb F_q$: levels, the perverse heart of $\mathcal C_I$, Weil structures

Dictionary inputs (named once, used in Lemma 3.2–Theorem 3.6):

Proof of (i)–(iii).

  1. Finite level. $Y_{N,M}$ has finitely many orbits of the connected group $I/I_{N,M}$ (L2), so its equivariant perverse heart has finite length with simple objects $\mathrm{IC}(\bar o,\xi)$; connected stabilizers force $\xi$ trivial (equivariant local systems on $o=H/S$ are representations of $\pi_0S=1$).
  2. Change of $M$. $\pi:=Y_{N,M'}\to Y_{N,M}$ is smooth surjective with connected fibres of dimension $d$, so $\pi^[d]$ is t-exact and fully faithful on perverse sheaves, with image closed under subquotients [BBD, 4.2.4–4.2.5]; it sends $\mathrm{IC}(\bar o)$ to $\mathrm{IC}(\overline{\pi^{-1}o})$ [BBD, 4.2.6], and $\pi^{-1}o$ is irreducible with a unique open orbit $o^\circ$ (finitely many orbits). Fibres are affine spaces, so $\pi_\pi^=\mathrm{id}$ and $\pi^$ is fully faithful on $D^b$. The kernel $I_{N,M}/I_{N,M'}$ is unipotent, so the two equivariant categories agree. The choice $e_M=e_{M'}+d(M,M')$ makes $\Phi_{N,M}=\Phi_{N,M'}\circ\pi^*[d]$.
  3. Change of $N$. By (L1b) the inclusion $Y_N\subseteq Y_{N'}$ is, at a common level $M$, a closed immersion of finite-type schemes; closed push-forward is t-exact, fully faithful, and sends simple objects to simple objects. No shift enters because $e_M$ depends only on $M$.
  4. Gluing. By (L1e) $\mathcal C_I$ is the filtered union of the $\mathcal C_I^{(N,M)}$ along t-exact fully faithful functors, so the t-structures glue; boundedness and finite length hold at each level, hence for each object. A simple object of $\mathcal P_I$ lies in some $\mathcal P^{(N,M)}$ and is simple there, and steps 2–3 show it stays simple at all finer levels: this is (ii).
  5. Frobenius. $\mathrm{Fr}^$ is an autoequivalence preserving perversity and sends $\mathrm{IC}(P)$ to $\mathrm{IC}(\mathrm{Fr}^{-1}P)=\mathrm{IC}(P)$ (L2). Restriction to $P$ identifies $\mathrm{Hom}(\mathrm{Fr}^\mathrm{IC}(P),\mathrm{IC}(P))\simeq\mathrm{End}(\mathrm{IC}(P))=\overline{\mathbb Q}\ell$ with $\mathrm{End}(\overline{\mathbb Q}{\ell,P})$, so the normalized $\Phi_S$ exists and is unique; it is automatically $I$-equivariant because $I/I_{N,M}$ is connected (equivariance is a property for perverse sheaves under a connected group). This is (iii). $\square$

Terms. Piece = preimage of a finite-level orbit, not an $I$-orbit (for $\mathbb A^n$ the closed piece $t^M\mathcal O^n$ contains infinitely many orbits and the orbit $\{0\}$ is never a piece). $\mathrm{IC}0$ is the unshifted constant sheaf on $L^+X$, so $\mathrm{tr}\,\mathrm{IC}_0=\varphi_0$. $\Delta_P=j{P!}k_P$ is unshifted. Thick = closed under shifts, cofibres, retracts.

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