Definitions

Work over \(k=\overline{\mathbb F}_q\), with \(\overline{\mathbb Q}_\ell\) coefficients, \(\ell\ne\operatorname{char}k\). Write \(H_{N,M}=I/I_{N,M}\), \(i_N:Y_N\hookrightarrow LX\), and \(\pi_{N,M}:Y_N\to Y_{N,M}\).

$$\mathcal C_I=\bigcup_{N,M}i_{N,!}\pi_{N,M}^*D^b_c([Y_{N,M}/H_{N,M}]).$$

We use the equivariant perverse convention measured on \(Y_{N,M}\): an equivariant complex is perverse when its underlying complex on that variety is perverse. This fixes the convention independently of shifts for an atlas of the quotient stack.

For an orbit \(o\), let \(a_o:o\hookrightarrow\overline o\) be open and \(b_o:\overline o\hookrightarrow Y_{N,M}\) closed. A piece is \(P=\pi_{N,M}^{-1}(o)\). The function space \(C_I\) is different from the sheaf category \(\mathcal C_I\).

Hypotheses

Use (Lev), I1, I2 and (Dict)(a),(c). In particular the equivariant level embeddings are fully faithful, the levels have finitely many orbits with connected stabilizers, and the affine-bundle ranks \(d(M,M')\) are independent of \(N\).

$$e_M=e_{M'}+d(M,M')\quad(M'\ge M).$$

For the directed presentation, use a cofinal set of common levels where the closed embeddings in (Lev) are defined. The compatibility checks below are required for a different presentation; a bare expression as a union of categories is insufficient.

Lemma IIa · Simple perverse objects over levels

With these hypotheses, the normalized level functors define a bounded perverse t-structure on \(\mathcal C_I\), with heart \(\mathcal P_I\). Its simple objects are

$$\Phi_{N,M}:=i_{N,!}\pi_{N,M}^*[e_M],\qquad \mathrm{IC}(P):=\Phi_{N,M}\big(b_{o,*}a_{o,!*}\overline{\mathbb Q}_{\ell,o}[\dim o]\big)\in\mathcal P_I.$$ $$\mathcal P_\infty:=\Big(\bigsqcup_{N,M}\mathcal P_{N,M}\Big)/{\sim}\ \xrightarrow{\sim}\ \operatorname{Irr}(\mathcal P_I),\qquad[P]\longmapsto[\mathrm{IC}(P)].$$

Here \(\sim\) is generated by passing to the unique dense piece in a finer presentation. Every object of \(\mathcal C_I\) is obtained from finitely many shifted simples by cones.

Use in Proposition II · Affine-space instance

Steps 1–3 · Finite levels · Steps 4–6 · Compatible hearts · Steps 7–9 · Classification

Finite levels

Classify IC extensions and specify their ambient category.

  1. Classify finite-level simples

    At one level, every simple perverse sheaf is an intermediate extension from an irreducible local system.

    $$\mathrm{IC}(\overline o,\xi)=b_{o,*}a_{o,!*}\xi[\dim o].$$
    Proof
    1. A finite orbit stratification makes the support of a simple equivariant perverse sheaf the closure of one orbit.
    2. On its dense orbit its restriction is an irreducible local system in degree \(-\dim o\).
    3. The intermediate extension is the unique perverse extension with no nonzero subobject or quotient supported on the boundary; simplicity forces this extension. This is BBD 4.3.1, with its proof in 4.3.4.
    Sources

  2. Use connected stabilizers

    For \(y\in o\) with stabilizer \(H_y\), equivariant local systems are controlled by its component group.

    $$\operatorname{Loc}_H(o)\simeq\operatorname{Rep}_{\overline{\mathbb Q}_\ell}(\pi_0H_y),\qquad\pi_0H_y=1\ \Longrightarrow\ \xi=\overline{\mathbb Q}_{\ell,o}\text{ for irreducible }\xi.$$

    Example · the two orbits of the affine line

    $$Y_{0,1}=\mathbb A^1,\qquad H_{0,1}=\mathbb G_m,\qquad o_0=\mathbb G_m,\quad o_1=\{0\}.$$

    The stabilizer is trivial on \(o_0\) and is \(\mathbb G_m\) at zero; both are connected.

    $$\mathrm{IC}(\overline{o_0})=\overline{\mathbb Q}_{\ell,\mathbb A^1}[1],\qquad\mathrm{IC}(\overline{o_1})=\delta_0.$$
    Proof
    1. Pull a local system back along \(H\to H/H_y\). Equivariance trivializes that pullback.
    2. Descent is a representation on the fibre at \(y\); its action is locally constant on \(H_y\), hence factors through \(\pi_0H_y\).
    3. Conversely such a representation supplies descent data. A connected stabilizer therefore admits only the constant irreducible equivariant local system.
    Sources

  3. Transport into the loop category

    The full level functor includes extension by zero.

    $$D^b_c([Y_{N,M}/H_{N,M}])\xrightarrow{\ \pi_{N,M}^*\ }\operatorname{Shv}_I(Y_N)\xrightarrow{\ [e_M]\ }\operatorname{Shv}_I(Y_N)\xrightarrow{\ i_{N,!}\ }\mathcal C_I.$$
    Proof
    1. Inflate equivariance along \(I\to H_{N,M}\), then pull back along the equivariant map \(\pi_{N,M}\).
    2. Shift by the chosen integer \(e_M\).
    3. Extend from the closed ind-subspace \(Y_N\) to \(LX\). The resulting object lies in the specified essential image \(\mathcal C_I^{(N,M)}\).
    4. Full faithfulness of the equivariant functor is part of (Lev). It does not follow merely by forgetting equivariance.
    Sources

Compatible hearts

Check that every common object receives the same perverse normalization.

  1. Refine the truncation

    For \(f:Y_{N,M'}\to Y_{N,M}\), smooth of affine-space rank \(d\),

    $$\Phi_{N,M'}\circ f^*[d]=i_{N,!}\pi_{N,M}^*[d+e_{M'}]=\Phi_{N,M}.$$

    Example · keep one more coefficient

    $$Y_{0,2}=\mathbb A^2\xrightarrow{\ f:(a_0,a_1)\mapsto a_0\ }Y_{0,1}=\mathbb A^1,\qquad d=1,\quad e_1=-1,\quad e_2=-2.$$

    The point-supported IC complex at the coarse level pulls back to the constant perverse sheaf on a line.

    $$f^*\delta_0[1]=\overline{\mathbb Q}_{\ell,\{a_0=0\}}[1],\qquad\Phi_{0,2}(f^*\delta_0[1])=\Phi_{0,1}(\delta_0).$$
    Proof
    1. Smooth pullback shifted by \(d\) is perverse t-exact.
    2. Connected fibres make it fully faithful on perverse sheaves; BBD 4.2.6–4.2.6.2 gives an image closed under subquotients and preservation of simples.
    3. Use the equivariant inflation in (Lev). Since \(d+e_{M'}=e_M\), the displayed equality identifies the two embeddings, not merely their underlying supports.
    Sources

  2. Enlarge the support level

    At a common truncation, the closed inclusion \(j:Y_{N,M}\hookrightarrow Y_{N',M}\) satisfies

    $$j_*:\mathrm{Perv}_{H_{N,M}}(Y_{N,M})\longrightarrow\mathrm{Perv}_{H_{N',M}}(Y_{N',M}),\qquad\Phi_{N',M}j_*\simeq\Phi_{N,M}.$$
    Proof
    1. Use (Lev) to identify \(Y_N\) with the inverse image of the finite closed subspace at a sufficiently fine common level.
    2. Closed push-forward is fully faithful and perverse t-exact. Its image consists of objects supported on the closed subset, hence is closed under subquotients.
    3. The normalizing integer depends on \(M\), not \(N\), so the two extensions to \(LX\) agree.
    Sources

  3. Construct the perverse heart

    Transport finite-level truncations through \(\Phi_{N,M}\).

    $$\mathcal P_I=\bigcup_{N,M}\Phi_{N,M}\mathrm{Perv}_{H_{N,M}}(Y_{N,M})\subset\mathcal C_I.$$
    Proof
    1. Any finite collection of objects and morphisms lies at a common level by I1 and the fully faithful directed presentation.
    2. Steps 4–5 identify the truncations on overlaps. The t-structure axioms can therefore be checked at that common level.
    3. Each object has bounded perverse amplitude and finite-length perverse cohomology, because these properties hold at its finite level.
    4. Images closed under subquotients ensure that a simple finite-level perverse object remains simple after further refinement.
    Sources

Classification

Identify repeated presentations, exhaust all simples, and fix Frobenius.

  1. Identify repeated presentations

    A coarse piece and its unique dense fine piece define the same simple object.

    $$P\sim P^\circ,\qquad\mathrm{IC}(P)\simeq\mathrm{IC}(P^\circ),\qquad\delta(P):=\dim o+e_M.$$

    Example · a piece need not be one loop orbit

    $$\pi_{0,1}^{-1}(0)=t\mathcal O,\qquad\pi_{0,2}^{-1}(\{a_0=0,\ a_1\ne0\})=t\mathcal O^\times.$$

    These subsets differ; their IC extensions have the same closed support.

    $$\Phi_{0,1}(\delta_0)=\overline{\mathbb Q}_{\ell,t\mathcal O}[-1],$$ $$\Phi_{0,2}(\overline{\mathbb Q}_{\ell,\{a_0=0\}}[1])=\overline{\mathbb Q}_{\ell,t\mathcal O}[1-2]=\overline{\mathbb Q}_{\ell,t\mathcal O}[-1].$$
    Proof
    1. The inverse image of an irreducible orbit under the affine-space fibration is irreducible. It is a union of finitely many finer orbits, so exactly one is dense and open.
    2. The smooth pullback of the coarse IC complex is the IC complex of that inverse image, by the intermediate-extension characterization and Step 4.
    3. If the fine orbit has dimension \(\dim o+d\), its normalized degree is \(\dim o+d+e_{M'}=\dim o+e_M\).
    4. At a common refinement distinct dense supports give nonisomorphic simples. This proves injectivity of the stated indexing map.
    Sources

  2. Exhaust and generate

    Every simple occurs at a finite level, and every object is built from such simples.

    $$\operatorname{Irr}(\mathcal P_I)=\{\mathrm{IC}(P)\}/{\cong},\qquad\mathcal C_I=\operatorname{Thick}\{\mathrm{IC}(P)\}.$$

    Example · all valuation indices

    $$\mathcal O=k[\![t]\!],\qquad Y_{N,M}=t^{-N}\mathcal O/t^M\mathcal O,\qquad e_M=-M.$$

    For \(-N\le j<M\), the orbit with valuation \(j\) has dimension \(M-j\); the zero orbit has dimension zero.

    $$\mathrm{IC}(t^j\mathcal O^\times)=\overline{\mathbb Q}_{\ell,t^j\mathcal O}[(M-j)-M]=\overline{\mathbb Q}_{\ell,t^j\mathcal O}[-j],$$ $$\mathrm{IC}(t^M\mathcal O)=\overline{\mathbb Q}_{\ell,t^M\mathcal O}[-M],\qquad\operatorname{Irr}(\mathcal P_I)\simeq\mathbb Z.$$

    The singleton \(\{0\}\) is not a finite-level piece: its indicator is not locally constant on \(F\), and its skyscraper is not obtained by pullback from a fixed truncation.

    Proof
    1. A global simple descends to one level; exact full faithfulness and the subquotient property make its representative simple there.
    2. Steps 1–2 identify that representative as an IC extension with constant local system.
    3. For an arbitrary complex, use its finitely many perverse truncation triangles, then a Jordan–Hölder filtration of each perverse cohomology object. Each quotient is a simple; short exact sequences in the heart give triangles.
    Sources

  3. Normalize Frobenius

    For a Frobenius-stable piece, choose the Weil structure that is the identity on its constant open restriction.

    $$\theta_P:\mathrm{Fr}^*\mathrm{IC}(P)\xrightarrow{\sim}\mathrm{IC}(P),\qquad\theta_P|_P=\mathrm{id},\qquad\mathrm{IC}(P)|_P=\overline{\mathbb Q}_{\ell,P}[\delta(P)].$$
    Proof
    1. By (Dict)(c), Frobenius preserves the finite orbit and its constant local system.
    2. Intermediate extension uniquely extends the identity on that local system; restriction identifies the endomorphism ring of the simple IC extension with \(\overline{\mathbb Q}_\ell\).
    3. The refinement isomorphisms agree with the identity on the common dense piece and preserve \(\delta\), so the normalized Weil structure is independent of the level.
    4. Consequently its trace on its own piece is \((-1)^{\delta(P)}\); values on lower pieces come from the IC extension.
    Sources