Abstract
The Deligne-Langlands correspondence parametrizes irreducible representations of the affine Hecke algebra |$\mathcal{H}^{\operatorname{aff}}$| by certain perverse sheaves. We show that this can be lifted to an equivalence of triangulated categories. More precisely, we construct for each central character |$\chi $| of |$\mathcal{H}^{\operatorname{aff}}$| an equivalence of triangulated categories between a perfect derived category of dg-modules |$D_{\operatorname{perf}}(\mathcal{H}^{\operatorname{aff}}/(\ker (\chi )) - \operatorname{dgMod})$| and the triangulated category generated by the corresponding perverse sheaves. The main step in this construction is a formality result that we prove for a wide range of ‘Springer sheaves’.
Introduction
Motivation and main result
Affine Hecke algebras play an important role in the representation theory of |$p$|-adic groups. In fact, for each split |$p$|-adic group |$\textbf{G}$| there is a (specialized) affine Hecke algebra |$\mathcal{H}^{\operatorname{aff}}_{q}$| such that the category of |$\mathcal{H}^{\operatorname{aff}}_{q}$|-modules is equivalent to the category |$\operatorname{Rep}_{I}(\textbf{G})$| of smooth |$\textbf{G}$|-representations that are generated by their Iwahori-fixed vectors [13]. The category |$\operatorname{Rep}_{I}(\textbf{G})$| is precisely the principal Bernstein block of |$\textbf{G}$| (i.e., the block in the category of smooth |$\textbf{G}$|-representations that contains the trivial representation) [10, 14, 15]. Let |$G$| be the complex reductive group whose root datum is dual to that of |$\textbf{G}$| and assume that |$G$| has simply connected derived subgroup. In [28] Kazhdan and Lusztig proved the Deligne-Langlands correspondence which parametrizes the irreducible representations of the affine Hecke algebra |$\mathcal{H}^{\operatorname{aff}}$| by geometric data on the group |$G$|. We now recall this correspondence in the language of [17]:
Any irreducible representation of
|$\mathcal{H}^{\operatorname{aff}}$| admits a central character. These central characters are parametrized by semisimple conjugacy classes in
|$G \times \mathbb{G}_{m}$|. For each semisimple pair
|$(s,q) \in G \times \mathbb{G}_{m} $|, we denote the corresponding central character by
|$\chi _{(s,q)}: Z(\mathcal{H}^{\operatorname{aff}} ) \rightarrow \mathbb{C} $|. Then the irreducible representations of
|$\mathcal{H}^{\operatorname{aff}}$| with central character
|$\chi _{(s,q)}$| are precisely the irreducible representations of the truncated affine Hecke algebra
This algebra has a geometric incarnation in the world of constructible sheaves. Let
|$\mathcal{N}$| be the nilpotent cone of
|$G$| and
|$\mu : \tilde{\mathcal{N}} \rightarrow \mathcal{N}$| the Springer resolution. These varieties come with a canonical
|$G \times \mathbb{G}_{m}$|-action where
|$\mathbb{G}_{m}$| acts by scaling. Passing to
|$(s,q)$|-fixed points we obtain a morphism
The corresponding
|$(s,q)$|-Springer sheaf is defined as
where
|$\mathcal{C}_{\tilde{\mathcal{N}}^{(s,q)}}$| is the constant (perverse) sheaf on
|$\tilde{\mathcal{N}}^{(s,q)}$|. The following isomorphism gives the affine Hecke algebra at
|$\chi _{(s,q)}$| a geometric interpretation [
17,
28]:
Note that this isomorphism induces a grading on the algebra
|$\mathcal{H}^{\operatorname{aff}}_{(s,q)}$| which was not visible in its algebraic definition in (
1). By the decomposition theorem [
6], we can write
for certain
|$G(s)$|-equivariant simple perverse sheaves
|$X_{j} \in \operatorname{Perv}_{G(s)}(\mathcal{N}^{(s,q)})$| and integers
|$k_{j} \in \mathbb{Z}$|. From this, one can deduce that there is a bijection
or equivalently an embedding
The simple objects in
|$ \operatorname{Perv}_{G(s)}(\mathcal{N}^{(s,q)})$| are parametrized by geometric data on
|$\mathcal{N}^{(s,q)}$|, namely by the set of irreducible equivariant local systems on the
|$G(s)$|-orbits of
|$\mathcal{N}^{(s,q)}$|. Hence, we can interpret
|$\mathcal{N}^{(s,q)}$| as a variety of
Langlands parameters associated to the pair
|$(s,q)$|. If
|$q \in \mathbb{G}_{m}$| is not a root of unity, the local systems that correspond to elements of
|$\operatorname{Irr}(\mathcal{H}^{\operatorname{aff}}_{(s,q)})$| can be characterized more explicitly as those that appear in the cohomology of a certain Springer fiber (c.f. [
17, Proposition 8.1.14]). This is known as the
Deligne-Langlands correspondence.
Following the categorical Langlands philosophy, one would like to lift the Deligne-Langlands correspondence to an equivalence of (triangulated) categories. Our main result establishes such an equivalence for each central character
|$\chi _{(s,q)}$|. We denote by
the full triangulated subcategory of
|$D^{b}_{c}(\mathcal{N}^{(s,q)})$| generated by the simple constituents of the
|$(s,q)$|-Springer sheaf
|$\operatorname{\textbf{S}}^{(s,q)}$|.
Theorem.(Theorem
6.8) There is an equivalence of triangulated categories
which identifies
|$\operatorname{\textbf{S}}^{(s,q)}$| with the free dg-module
|$\mathcal{H}^{\operatorname{aff}}_{(s,q)}$|. Here we consider
|$\mathcal{H}^{\operatorname{aff}}_{(s,q)}$| as a dg-algebra with vanishing differential and grading induced by the
|$\operatorname{Hom}^{*}$|-grading in (
3).
Formality of Springer sheaves
The theorem above will be a consequence of a formality result that we prove for a wide range of ‘Springer sheaves’: Let
|$G$| be a reductive group over an algebraically closed field
|$\overline{\mathbb{F}}$| (not necessarily of characteristic
|$0$|),
|$B \subset G$| a Borel subgroup,
|$V$| a
|$G$|-representation and
|$\{V^{i} \subset V \mid i \in I \}$| a finite collection of
|$B$|-stable subspaces. Then for each
|$i \in I$| we consider the ‘morphism of Springer type’
and the associated ‘Springer sheaves’
We will see in Corollary
6.6 that the
|$(s,q)$|-Springer sheaf
|$\textbf{S}^{(s,q)}$| from (
2) is a special case of this construction. We define the ‘Springer category’
|$D_{\operatorname{Spr}}(V)$| to be the full triangulated subcategory of
|$D^{b}_{c}(V)$| generated by the summands of
|$\operatorname{\textbf{S}}$|. Our goal is to establish an equivalence of triangulated categories
The derived category
|$D_{\operatorname{Spr}}(V)^{\operatorname{op}}$| has a natural dg-enhancement (see the remarks below). Hence, by standard dg-techniques, which will be reviewed in Section
4.2, there is an equivalence of triangulated categories
for a dg-algebra
|$R_{\operatorname{\textbf{S}}}$| with
Thus, we need to show that we can replace the dg-algebra
|$R_{\operatorname{\textbf{S}}}$| in the equivalence (
4) with its cohomology, that is, we need to prove that
|$R_{\operatorname{\textbf{S}}}$| is
formal. We will do this by closely following the arguments in [
36, Appendix A,B]. First, we prove formality in the case where
|$\overline{\mathbb{F}}$| is the algebraic closure of a finite field
|$\mathbb{F} = \mathbb{F}_{q}$|. In this case,
|$\operatorname{Hom}^{*}(\operatorname{\textbf{S}}, \operatorname{\textbf{S}})$| can be equipped with a canonical Frobenius action. By analyzing the associated Steinberg variety, we will prove the following strong purity result for this action.
Theorem.(Corollary 3.5) The canonical Frobenius action on |$\operatorname{Hom}^{i}(\textbf{S}, \textbf{S})$| is given by multiplication with |$q^{\frac{i}{2}}$|.
We will then use a general “purity implies formality” result to deduce that |$R_{\operatorname{\textbf{S}}}$| is formal (the idea that purity implies formality goes back to [20, 21]). This completes the proof in the positive characteristic case. Formality in characteristic |$0$| can be deduced from the positive characteristic case using the “De |$\mathbb{F}$| à |$\mathbb{C}$|” technique from [6, §6]. This last step will be discussed in more detail in Section 5.
Some remarks on dg-enhancements
There are several technical problems one has to deal with to obtain a general “purity implies formality” result as we use it: One difficulty that arises is that the constructible derived category |$D^{b}_{c}(X) = D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell })$| is not a genuine derived category, at least in its standard definition [6, 20]. In particular, |$D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell })$| is not naturally dg-enhanced in this setting. It is shown in [36] that this problem can be solved by working with a dg-enhancement of |$D^{b}_{c}(X,\overline{\mathbb{Q}}_{\ell })$| arising from an alternative constructions of the constructible derived category with integral coefficients |$D^{b}_{c}(X, \mathcal{O}_{E})$| [23, 30]. There is yet another construction of |$D^{b}_{c}(X,\overline{\mathbb{Q}}_{\ell })$| coming from the pro-étale topology introduced in [12] which is the most natural setting for dg-purposes. In fact, in the pro-étale topology, the constructible derived category |$D^{b}_{c}(X,\overline{\mathbb{Q}}_{\ell })$| does arise as a full triangulated subcategory of a genuine derived category and thus it is naturally dg-enhanced. For this reason, we choose to work with |$D^{b}_{c}(X,\overline{\mathbb{Q}}_{\ell })$| in its pro-étale realization (which will be reviewed in Section 4.3). To get a “purity implies formality” result, one also has to lift the Frobenius action on morphisms of sheaves to the dg-level as in [36, Appendix A]. We explain a variant of this Frobenius lifting argument in Section 4.4. Once the Frobenius action is lifted to the dg-level, formality will be a consequence of an algebraic result in [36, Theorem B.1.1]. While it would certainly be possible to directly apply the results from [36] in our setting without referring to the pro-étale topology, we hope that our alternative approach clarifies some of the technical difficulties.
Relation to other work
Formality has been discussed in many settings of representation theory such as the Springer correspondence [22, 36, 38, 39] or in the context of flag varieties and Koszul duality [7, 40] where formality has also been studied for modular coefficients [1, 37]. It would also be interesting to prove similar formality results for graded Hecke algebras at central characters. These algebras can be used to study a wider range of representations of |$p$|-adic groups such as unipotent representations [33]. In terms of geometry, graded Hecke algebras arise as certain Ext-algebras in an equivariant derived category of constructible sheaves [3, 34]. Some formality results in this direction can be found in [41]. There also is a coherent categorical Deligne-Langlands correspondence [8] which works without fixing a central character but replaces constructible sheaves with a certain category of coherent sheaves (see also the conjectures in [25]). The relation between the constructible and the coherent side is discussed in [9].
2 Springer Geometry
Let |$\overline{\mathbb{F}}$| be an algebraically closed field and |$\ell $| a prime number which is invertible in |$\overline{\mathbb{F}}$|. For any variety |$X$| over |$\overline{\mathbb{F}}$|, we can consider the constructible derived category of |$\overline{\mathbb{Q}}_{\ell }$|-complexes |$D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell } )$| as defined in [6, 12, 20]. The triangulated category |$D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell } )$| comes with the usual six (derived) operations denoted by |$f^{*},f_{*},f_!,f^!, \otimes ^{L}$| and |$\operatorname{R{\mathcal{H}}\! \textit{om}}$|. Moreover, there is a standard |$t$|-structure on |$D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell } )$| with cohomology functor |$H^{*}$| and heart |$\operatorname{Sh}_{c}(X, \overline{\mathbb{Q}}_{\ell } )$|. The structure map of |$X$| will be denoted by |$a: X \rightarrow \operatorname{pt}$|. Let |$\textbf{1}_{X} \in \operatorname{Sh}_{c}(X, \overline{\mathbb{Q}}_{\ell } )$| be the constant sheaf and |$\omega _{X}:= a^! \textbf{1}_{\operatorname{pt}} \in D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell } )$| the dualizing complex. We denote by |$\operatorname{Perv}(X) \subset D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell })$| the category of perverse sheaves on |$X$| and by |$^{p} H^{*}$| the perverse cohomology functor.
2.1 Borel-Moore homology
In this section we recall a few basic facts about Borel-Moore homology [
31]. The
|$i$|-th Borel-Moore homology is the
|$\overline{\mathbb{Q}}_{\ell }$|-vector space
It can be shown that
|$H_{i}(X, \overline{\mathbb{Q}}_{\ell })$| is concentrated in degrees
|$0,1,...,2 \dim X$|. Moreover, we have the Künneth formula
If
|$i: Y \hookrightarrow X$| is a closed immersion with complement
|$j: U \hookrightarrow X$| and
|$\mathcal{F} \in D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell })$|, there is a canonical distinguished triangle
which is natural in
|$\mathcal{F}$|. For
|$\mathcal{F} = \omega _{X}$| this induces a long exact sequence on Borel-Moore homology
Let
|$p:\tilde{X} \rightarrow X$| be a smooth morphism of relative dimension
|$d$|. The adjoint pair
|$(p^{*}, p_{*})$| gives rise to a canonical morphism
This induces a ‘smooth pullback’ map on Borel-Moore homology
The naturality of the distinguished triangle in (
5) implies that smooth pullback is compatible with the long exact sequence from (
6), that is, if
|$Y \subset X$| is a closed subvariety with open complement
|$U$| and
|$\tilde{Y}:= p^{-1}(Y)$|,
|$\tilde{U}:= p^{-1}(U)$|, the following diagram commutes:
Lemma 2.1.Let |$p:\tilde{X} \rightarrow X$| be a Zariski locally trivial fibration with affine fiber |$\mathbb{A}^{d}$|. Then the smooth pullback map |$ H_{i}(X, \overline{\mathbb{Q}}_{\ell }) \rightarrow H_{i+2d}(\tilde{X}, \overline{\mathbb{Q}}_{\ell })$| is an isomorphism for all |$i \in \mathbb{Z}$|.
Proof.Using (7) and the five lemma, one can reduce to the case where |$p$| is trivial. By the Künneth formula, it suffices to consider the case where |$p: \mathbb{A}^{d} \rightarrow \operatorname{pt}$|. Note that |$H_{i}(\mathbb{A}^{d}, \overline{\mathbb{Q}}_{\ell }) = 0$| for |$i \neq 2d $| and |$H_{2d}(\mathbb{A}^{d}, \overline{\mathbb{Q}}_{\ell }) = \overline{\mathbb{Q}}_{\ell }$|. The claim now follows since the smooth pullback map |$H_{0}(\operatorname{pt}, \overline{\mathbb{Q}}_{l}) \rightarrow H_{2d}(\mathbb{A}^{d}, \overline{\mathbb{Q}}_{\ell })$| is non-zero.
If
|$X$| is smooth and connected, the dualizing complex is given by
The fundamental class of
|$X$| is the distinguished element
also denoted by
|$[X] \in H_{2 \dim X}(X, \overline{\mathbb{Q}}_{\ell })$|. More generally, if
|$X$| is irreducible one can define the fundamental class as follows: Pick a smooth open subset
|$U \subset X$|. Then the long exact sequence (
6) induces an isomorphism
|$H_{2\dim X}(U, \overline{\mathbb{Q}}_{\ell }) \cong H_{2 \dim X}(X, \overline{\mathbb{Q}}_{\ell }) $|. The image of
|$[U]$| under this isomorphism defines a distinguished element
|$[X] \in H_{2 \dim X}(X, \overline{\mathbb{Q}}_{\ell })$| called the fundamental class of
|$X$|. It can be shown that
|$[X]$| does not depend on the choice of
|$U$|. If
|$Y \subset X$| is an irreducible closed subvariety, the image of the fundamental class of
|$Y$| under
|$H_{2 \dim Y}(Y, \overline{\mathbb{Q}}_{\ell }) \rightarrow H_{2 \dim Y}(X, \overline{\mathbb{Q}}_{\ell })$| also defines a fundamental class
|$[Y] \in H_{2 \dim Y}(X, \overline{\mathbb{Q}}_{\ell })$|. If elements of this form span the vector space
|$H_{i}(X, \overline{\mathbb{Q}}_{\ell })$| for each
|$i \in \mathbb{Z}$|, we say that
|$H_{*}(X, \overline{\mathbb{Q}}_{\ell })$| is
spanned by fundamental classes. Note that being spanned by fundamental classes implies that
|$H_{i}(X, \overline{\mathbb{Q}}_{\ell })= 0$| for
|$i$| odd. Let
|$Z_{i}(X)$| be the free abelian group on the set of
|$i$|-dimensional irreducible closed subvarieties of
|$X$| and let
|$A_{i}(X) = Z_{i}(X)/ \sim _{Rat}$| be the Chow group (c.f. [
24]). The fundamental class construction gives rise to a cycle class map
which descends to the Chow group
by [
31, Théorème 6.3]. The open-closed exact sequence and smooth pullback map from Borel-Moore homology have analogues for Chow groups: For any
|$Y \subset X$| closed with complement
|$U = X\backslash Y$| there is an exact sequence
and for
|$\tilde{p}: \tilde{X} \rightarrow X$| smooth (or more generally flat) of relative dimension
|$d$|, there is a pullback map
The cycle class map
|$cl_{X}$| is functorial with respect to these constructions [
31, Théorème 6.1]. We define
Following [
18, 1.7], we say that a variety
|$X$| has property (S) if
|$H_{i}(X, \overline{\mathbb{Q}}_{\ell }) = 0$| for |$i$| odd;
|$cl_{X}: A_{i}(X)_{\overline{\mathbb{Q}}_{\ell }} \rightarrow H_{2i}(X, \overline{\mathbb{Q}}_{\ell })$| is an isomorphism for all |$i \in \mathbb{Z}$|.
Note that if |$X$| has property (S), then |$H_{*}(X, \overline{\mathbb{Q}}_{\ell })$| is spanned by fundamental classes. The following two lemmas are |$\overline{\mathbb{Q}}_{\ell }$|-versions of [18, Lemma 1.8, 1.9].
Lemma 2.2.Let |$Y \subset X$| be a closed subvariety with complement |$U = X \backslash Y$|. If |$U$| and |$Y$| have property (S) then |$X$| also has property (S).
Proof.The groups |$H_{i}(Y, \overline{\mathbb{Q}}_{\ell })$| and |$H_{i}(U, \overline{\mathbb{Q}}_{\ell })$| vanish for |$i$| odd. Using the long exact sequence (6), we deduce that |$H_{i}(X, \overline{\mathbb{Q}}_{\ell })=0$| for |$i$| odd. Moreover, for any |$i \in \mathbb{Z}$|, we get a commutative diagram with exact rows
By the five lemma, this implies that the map
|$A_{i}(X)_{\overline{\mathbb{Q}}_{\ell }} \rightarrow{H_{2i}(X, \overline{\mathbb{Q}}_{\ell })}$| is an isomorphism.
Lemma 2.3.Let |$p:\tilde{X} \rightarrow X$| be a Zariski locally trivial fibration with fiber |$\mathbb{A}^{d}$|. If |$X$| has property (S) then |$\tilde{X}$| also has property (S).
Proof.By Lemma 2.1, the pullback map |$H_{i}(X, \overline{\mathbb{Q}}_{\ell }) \rightarrow H_{i+2d}(\tilde{X}, \overline{\mathbb{Q}}_{\ell })$| is an isomorphism. In particular, |$H_{i}(X, \overline{\mathbb{Q}}_{\ell }) = 0$| for |$i$| odd. Moreover, for any |$i \in \mathbb{Z}$| we get a commutative diagram
As a consequence, the map
|$A_{i}(X)_{\overline{\mathbb{Q}}_{\ell }} \rightarrow A_{i+d}(\tilde{X})_{\overline{\mathbb{Q}}_{\ell }} $| is injective. It is also surjective by [
24, Proposition 1.9] and thus an isomorphism. This implies that
|$A_{i+d}(\tilde{X})_{\overline{\mathbb{Q}}_{\ell }} \rightarrow H_{2i+2d}(\tilde{X}, \overline{\mathbb{Q}}_{\ell })$| is also an isomorphism.
2.2 Morphisms of Springer type
We now introduce a general setting of ‘Springer geometry’ which we want to study in this paper. Let
|$G$| be a connected reductive group defined over
|$\overline{\mathbb{F}}$|. We fix a maximal torus and a Borel subgroup
|$T \subset B \subset G$| with Weyl group
|$W = N_{G}(T)/T$|. The corresponding flag variety will be denoted by
|$\mathcal{B} = G/B$|. Given a
|$G$|-representation
|$V$| and a finite collection
|$\{V^{i} \subset V \mid i \in I \}$| of
|$B$|-stable subspaces, we define for each
|$i\in I$| the
|$G$|-variety
This comes with two
|$G$|-equivariant morphisms
where
|$\mu _{i}(g,v) = g \cdot v$| and
|$\pi _{i}(g,v) = gB$|.
Definition 2.4.We call |$\mu _{i}$| a morphism of Springer type.
Example 2.5.Let
|$V = \mathfrak{g}$| be the Lie algebra of
|$G$|,
|$\mathfrak{b}$| the Lie algebra of
|$B$| and
|$\mathfrak{n} = [\mathfrak{b}, \mathfrak{b}]$|. Then the morphism of Springer type corresponding to the
|$B$|-stable subspace
|$\mathfrak{n} \subset \mathfrak{g}$| is the Springer resolution
For
|$\mathfrak{b} \subset \mathfrak{g}$| we recover the Grothendieck-Springer alteration
|$G\times ^{B} \mathfrak{b} \rightarrow \mathfrak{g}$|. Other important examples of morphisms of Springer type show up in the representation theory of affine Hecke algebras (see [
17,
28] and Corollary
6.6).
For
|$i,j \in I$|, we consider the Steinberg variety
This comes with the projection map
|$\pi _{i} \times \pi _{j}: Z^{ij} \rightarrow \mathcal{B} \times \mathcal{B}$|. Consider the orbit partition
with
where
|$\dot{w}\in N_{G}(T)$| is a lift of
|$w \in W$|. This induces a partition of
|$Z^{ij}$| into locally closed subvarieties
where
Lemma 2.6.(1) The morphism |$\mu _{i}: \tilde{V}^{i} \rightarrow V$| is proper;
(2) The morphism |$\pi _{i}: \tilde{V}^{i} \rightarrow \mathcal{B}$| is a Zariski vector bundle with fiber |$V^{i}$|;
(3) For each |$w \in W$| the morphism |$\pi _{i} \times \pi _{j}: Z^{ij}_{w} \rightarrow Y_{w}$| is a Zariski vector bundle with fiber |$V^{i} \cap \dot{w} V^{j}$|;
(4) The first projection |$p_{1}:Y_{w} \rightarrow \mathcal{B}$| is a Zariski locally trivial fibration with fiber |$\mathbb{A}^{l(w)}$|.
Proof.The map
|$\mu _{i}$| can be factored into a closed immersion followed by a projection:
Since
|$\mathcal{B}$| is projective, this implies that
|$\mu _{i}$| is proper. The local triviality in (2)-(4) follows from standard results about quotients (c.f. [
26, §I.5.16]). The respective fibers are easily computed.
Proposition 2.7.The variety |$Z^{ij}$| has property (S). In particular, |$H_{*}(Z^{ij}, \overline{\mathbb{Q}}_{\ell })$| is spanned by fundamental classes.
Proof.By Lemma
2.6 the maps
|$Z^{ij}_{w} \rightarrow Y_{w} \rightarrow \mathcal{B}$| are locally trivial fibrations with affine fibers. It is well known that
|$\mathcal{B}$| has property (S) (in fact, this follows from Lemma
2.2 and the decomposition of
|$\mathcal{B}$| into Bruhat cells). By Lemma
2.3 this implies that
|$Z^{ij}_{w}$| has property (S) for all
|$w \in W$|. Pick a total order
|$\le $| on
|$W$| extending the Bruhat order and define
Note that
|$Z^{ij}_{< w} = Z^{ij}_{\le w^{\prime}}$| where
|$w^{\prime} \in W$| is the maximal element with
|$w^{\prime} < w$|. We show by induction along the total order on
|$W$| that
|$Z^{ij}_{\le w}$| has property (S). We have already proved the claim for
|$Z^{ij}_{\le e} = Z^{ij}_{e}$|. Now assume we have shown the claim for each
|$y < w$|. Note that
|$Z^{ij}_{< w} \subset Z^{ij}_{\le w}$| is closed with open complement
|$Z^{ij}_{w}$|. The varieties
|$Z^{ij}_{< w}$| and
|$Z^{ij}_{w}$| have property (S), so
|$Z^{ij}_{\le w}$| also has property (S) by Lemma
2.2. This completes the induction. Note that
|$Z^{ij} = Z^{ij}_{\le w_{0}}$| where
|$w_{0} \in W$| is the longest element. Hence,
|$Z^{ij}$| has property (S).
Since
|$\tilde{V}^{i}$| is smooth, there is the constant perverse sheaf
We define the
Springer sheavesand
The morphism
|$\mu _{i}$| is proper by Lemma
2.6. Hence, the decomposition theorem [
6] implies that
|$\operatorname{\textbf{S}}^{i}$| (and thus also
|$\operatorname{\textbf{S}}$|) is a semisimple complex. In other words, we have
for some simple perverse sheaves
|$X_{j} \in \operatorname{Perv}(V)$| and integers
|$k_{j} \in \mathbb{Z}$|. We define the
Springer categoryto be the smallest full triangulated subcategory of
|$D^{b}_{c}(V, \overline{\mathbb{Q}}_{\ell })$| that is closed under isomorphisms and contains the simple perverse sheaves
|$X_{1},...,X_{n}$|. Our main goal is to give an algebraic description of the Springer category.
3 Purity
In this section, we show that the canonical Frobenius action on the space of morphisms between any two Springer sheaves is pure.
3.1 The Frobenius action
Let
|$X_{0}$| be a variety defined over a finite field
|$\mathbb{F}_{q}$| with structure map
|$a: X_{0} \rightarrow \operatorname{Spec}(\mathbb{F}_{q})$| and let
Denote by
|$D^{b}_{c}(X_{0}, \overline{\mathbb{Q}}_{\ell })$| the constructible derived category of
|$\overline{\mathbb{Q}}_{\ell }$|-complexes on
|$X_{0}$|. For any
|$\mathcal{F}_{0} \in D^{b}_{c}(X_{0}, \overline{\mathbb{Q}}_{\ell })$| the pullback of
|$\mathcal{F}_{0}$| along
|$X \rightarrow X_{0}$| will be denoted by
|$\mathcal{F} \in D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell })$|. We also define
for any
|$\mathcal{F}_{0},\mathcal{G}_{0} \in D^{b}_{c}(X_{0}, \overline{\mathbb{Q}}_{\ell })$|. The category
|$\operatorname{Sh}_{c}(\operatorname{Spec}(\mathbb{F}_{q}), \overline{\mathbb{Q}}_{\ell })$| is equivalent to the category of finite-dimensional continuous
|$\operatorname{Gal}(\overline{\mathbb{F}}_{q}/ \mathbb{F}_{q})$|-representations. Hence, we can consider
|$\operatorname{\underline{\operatorname{Hom}}}^{i}(\mathcal{F}_{0}, \mathcal{G}_{0})$| as a
|$\overline{\mathbb{Q}}_{\ell }$|-vector space equipped with a canonical Frobenius action (see Section
4.4 for a construction of this action). Forgetting this action recovers the vector space
|$\operatorname{Hom}^{i}(\mathcal{F}, \mathcal{G})$|. Note that the sheaves
|$\textbf{1}_{X}, \omega _{X} \in D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell })$| have canonical
|$\mathbb{F}_{q}$|-versions
|$\textbf{1}_{X_{0}}$| and
|$\omega _{X_{0}} = a^! \textbf{1}_{\operatorname{Spec}(\mathbb{F}_{q})}$|. This induces a canonical Frobenius action on
|$H_{*}(X, \overline{\mathbb{Q}}_{\ell })$| coming from the Frobenius action on
It can be shown that the constructions on Borel-Moore homology from the previous section are compatible with the Frobenius action. For example, if
|$i: Y_{0} \hookrightarrow X_{0}$| is a closed immersion with open complement
|$j: U_{0} \hookrightarrow X_{0}$|, there is a distinguished triangle
inducing a long exact sequence
where all maps commute with the Frobenius action. For any
|$n \in \mathbb{Z}$|, let
be the
|$n$|-th Tate twist. This corresponds to the
|$1$|-dimensional
|$\overline{\mathbb{Q}}_{\ell }$|-vector space on which the (geometric) Frobenius element acts by multiplication with
|$q^{-n}$|. Moreover, we define
for any
|$\mathcal{F}_{0} \in D^{b}_{c}(X_{0}, \overline{\mathbb{Q}}_{\ell })$| and
|$n \in \mathbb{Z}$|. If
|$X_{0}$| is smooth and connected, the dualizing complex on
|$X_{0}$| is given by
Lemma 3.1.Let |$X$| be irreducible. Then Frobenius acts on |$[X] \in \underline{H}_{2 \dim X}(X, \overline{\mathbb{Q}}_{\ell })$| by multiplication with |$q^{- \dim X}$|.
Proof.Let
|$U_{0} \subset X_{0}$| be a smooth open subset. Then we get a canonical element
which is Frobenius invariant and corresponds to the fundamental class
|$[U] \in H_{2 \dim U}(U, \overline{\mathbb{Q}}_{\ell })$| after forgetting the Frobenius action. Note that the Frobenius action on
|$ \underline{H}_{2 \dim U}(U, \overline{\mathbb{Q}}_{\ell })$| comes from the Frobenius action on
|$\underline{\operatorname{Hom}}^{-2\dim U}(\textbf{1}_{U_{0}},\omega _{U_{0}})$|. Thus, taking into account the Tate twist, we get that Frobenius acts on
|$ [U]$| by multiplication with
|$q^{- \dim U}$|. The fundamental class
|$[X]$| is the inverse image of
|$[U]$| under the Frobenius equivariant restriction isomorphism
|$\underline{H}_{2 \dim X}(X, \overline{\mathbb{Q}}_{\ell }) \overset{\sim }{\rightarrow } \underline{H}_{2 \dim X}(U, \overline{\mathbb{Q}}_{\ell })$|. Hence, Frobenius also acts on
|$[X]$| by multiplication with
|$q^{- \dim U} = q^{- \dim X}$|.
Corollary 3.2.Assume that |$H_{*}(X, \overline{\mathbb{Q}}_{\ell })$| is spanned by fundamental classes. Then for |$\mathbb{F}_{q}$| large enough, Frobenius acts on |$\underline{H}_{i}(X, \overline{\mathbb{Q}}_{\ell })$| by multiplication with |$q^{- \frac{i}{2}}$|.
Proof.Let |$Y_{1},...,Y_{n}$| be irreducible closed subvarieties of |$X$| such that the fundamental classes |$[Y_{1}],...,[Y_{n}]$| span |$H_{*}(X, \overline{\mathbb{Q}}_{\ell })$|. For |$\mathbb{F}_{q}$| large enough, we may assume that each of the |$Y_{i}$| can be defined over |$\mathbb{F}_{q}$|. Then Lemma 3.1 implies that Frobenius acts on |$[Y_{i}] \in \underline{H}_{2 \dim Y_{i}}(X, \overline{\mathbb{Q}}_{\ell })$| by multiplication with |$q^{- \dim Y_{i}}$|. Since the |$[Y_{i}]$| span |$H_{*}(X, \overline{\mathbb{Q}}_{\ell })$|, this proves the claim.
3.2 Frobenius and the Springer sheaf
Let |$\mu _{i}: \tilde{V}^{i} \rightarrow V$| (|$i \in I$|) be a finite collection of morphisms of Springer type (Definition 2.4) defined over |$\overline{\mathbb{F}}_{q}$|. Assume that |$\mathbb{F}_{q}$| is large enough so that there are |$\mathbb{F}_{q}$|-forms |$\mu _{i}: \tilde{V}^{i}_{0} \rightarrow V_{0}$| for each of the |$\mu _{i}$| (|$i \in I$|). Then |$Z^{ij}$| can also be defined over |$\mathbb{F}_{q}$| by considering |$Z^{ij}_{0}:= \tilde{V}^{i}_{0} \times _{V_{0}} \tilde{V}^{j}_{0}$|.
Lemma 3.3.For |$\mathbb{F}_{q}$| large enough, Frobenius acts on |$\underline{H}_{k}(Z^{ij}, \overline{\mathbb{Q}}_{\ell })$| by multiplication with |$q^{-\frac{k}{2}}$|.
Proof.This follows from Proposition 2.7 and Corollary 3.2.
We fix a square root
|$q^{\tfrac{1}{2}}$| of
|$q$| in
|$\overline{\mathbb{Q}}_{l}$|. This corresponds to fixing a square root
|$\textbf{1}_{\operatorname{Spec}(\mathbb{F}_{q})} (\tfrac{1}{2})$| of the Tate sheaf
|$\textbf{1}_{\operatorname{Spec}(\mathbb{F}_{q})} (1)$|. Then we can form the half integer Tate twists
|$\mathcal{F}_{0}(\tfrac{n}{2})$| for any
|$\mathcal{F}_{0} \in D_{c}^{b}(X_{0}, \overline{\mathbb{Q}}_{\ell })$| and
|$n\in \mathbb{Z}$|. Consider the constant perverse sheaf of weight
|$0$| on
|$\tilde{V}^{i}_{0}$|where
|$d_{i} = \dim \tilde{V}^{i}$| and let
be the corresponding
|$\mathbb{F}_{q}$|-Springer sheaf. We will need the following Frobenius equivariant version of [
17, Lemma 8.6.1].
Lemma 3.4.There is a (Frobenius equivariant) isomorphism
Proof.Note that
|$\mu _{i}$| is proper over
|$\mathbb{F}_{q}$| since it is proper over
|$\overline{\mathbb{F}}_{q}$| (properness can be checked fpqc locally). Hence, we have
|$\textbf{S}^{i}_{0} = (\mu _{i})_{*}\mathcal{C}_{\tilde{V}^{i}_{0}}= (\mu _{i})_!\mathcal{C}_{\tilde{V}^{i}_{0}}$|. Using base change with respect to the cartesian diagram
we get
Applying
|$H^{k}a_{*}$| to this, we obtain
Corollary 3.5.For |$\mathbb{F}_{q}$| large enough, Frobenius acts on |$\underline{\operatorname{Hom}}^{k}(\textbf{S}^{i}_{0}, \textbf{S}^{j}_{0})$| by multiplication with |$q^{\frac{k}{2}}$|.
Proof.By Lemma
3.3 Frobenius acts on
|$\underline{H}_{d_{i}+d_{j}-k}(Z^{ij})(\tfrac{-d_{i}-d_{j}}{2})$| by multiplication with
The claim now follows from Lemma
3.4.
4 Formality
4.1 Idempotent complete triangulated categories
Recall that an additive category
|$\mathcal{A}$| is called idempotent complete if any idempotent
|$e: X \rightarrow X$| in
|$\mathcal{A}$| splits. This is equivalent to the property that all idempotents have kernels (or cokernels). In an idempotent complete category, every idempotent
|$e: X \rightarrow X$| gives rise to a canonical decomposition
If
|$\mathcal{A}^{\prime} \subset \mathcal{A}$| is an additive subcategory that is closed under direct summands and
|$\mathcal{A}$| is idempotent complete, then
|$\mathcal{A}^{\prime}$| is also idempotent complete. Moreover, for any additive category
|$\mathcal{A}$| one can define its
idempotent completion|$\tilde{\mathcal{A}}$|. This is an idempotent complete additive category with a fully faithful additive functor
|$\iota : \mathcal{A} \rightarrow \tilde{\mathcal{A}}$| and the following universal property: Any additive functor
|$\mathcal{A} \rightarrow \mathcal{C}$| with
|$\mathcal{C}$| idempotent complete factors uniquely (up to isomorphism) through
|$\iota $|. For more details about idempotent completeness we refer to [
27, I.6].
A triangulated category is called idempotent complete if its underlying additive category is idempotent complete. All triangulated categories we will encounter are idempotent complete thanks to the following well-known result (c.f. [35, Lemma 1.6.8] and [4, Lemma 2.4]).
Lemma 4.1.Any triangulated category with countable coproducts is idempotent complete. Moreover, the bounded below derived category |$D^{+}(\mathcal{A})$| of an abelian category |$\mathcal{A}$| is always idempotent complete.
Let
|$\mathcal{T}$| be a triangulated category. For any object
|$X \in \mathcal{T}$|, we denote by
the smallest full triangulated subcategory of
|$\mathcal{T}$| that contains
|$X$| and is closed under isomorphisms. Similarly, we denote by
the smallest full triangulated subcategory of
|$\mathcal{T}$| that contains
|$X$| and is closed under isomorphisms and direct summands.
Lemma 4.2.Let |$\mathcal{T}$| be an idempotent complete triangulated category and |$X \in \mathcal{T}$|. Then 
is the idempotent completion of |$\langle X \rangle _{\Delta }$|.
Proof.By [27, Theorem I.6.12] the idempotent completion of |$\langle X \rangle _{\Delta }$| can be described as the full additive subcategory |$\mathcal{C} \subset \mathcal{T}$| consisting of those objects |$Y \in \mathcal{T}$| which are isomorphic to a direct summand of an object in |$\langle X \rangle _{\Delta }$|. In particular, we have 
. The cone of a morphism in |$\mathcal{C}$| is a direct summand of the cone of a morphism in |$\langle X \rangle _{\Delta }$|. Hence, the category |$\mathcal{C}$| is closed under cones and thus 
.
We will also need the following idempotent complete version of Beilinson’s Lemma (see [
5] and [
40, Lemma 6]).
Lemma 4.3.Let
|$F: \mathcal{T} \rightarrow \mathcal{T}^{\prime}$| be a triangulated functor between idempotent complete triangulated categories. Let
|$X \in \mathcal{T}$| such that
|$F$| induces an isomorphism
for all
|$i \in \mathbb{Z}$|. Then
|$F$| restricts to an equivalence of triangulated categories
.
Proof.By a standard dévissage argument, |$F$| induces an equivalence |$\langle X \rangle _{ \Delta } \cong \langle F(X) \rangle _{ \Delta }$|. This extends to an equivalence of the respective idempotent completions and thus to an equivalence 
by Lemma 4.2.
4.2 Derived categories and dg-algebras
Let |$k$| be a commutative ring. For any dg-algebra |$R$| over |$k$|, we denote by |$R-\operatorname{dgMod}$| the category of (left) dg-modules over |$R$|. The homotopy category of dg-modules will be denoted by |$K(R-\operatorname{dgMod})$| and the derived category by |$D(R-\operatorname{dgMod})$|. We write
for the perfect derived category. For more details about dg-algebras and dg-modules we refer to [
11,
29].
Let
|$\mathcal{A}$| be a
|$k$|-linear abelian category. Then for any two chain complexes
|$X^{\bullet }, Y^{\bullet } \in C(\mathcal{A})$|, we can consider the Hom-complex
|$\operatorname{Hom}_{dg}^{\bullet }(X^{\bullet },Y^{\bullet }) \in C(k)$|. Explicitly, this complex is defined as
with differential
|$d_{\operatorname{Hom}^{\bullet }_{dg}(X^{\bullet },Y^{\bullet })} (f) = d_{Y^{\bullet }} \circ f - (-1)^{|f|} f \circ d_{X^{\bullet }}$|. Taking cohomology of this complex recovers the morphism space in the homotopy category:
Component-wise composition defines a map
for any
|$X^{\bullet },Y^{\bullet },Z^{\bullet } \in C(\mathcal{A})$|. This equips
|$\operatorname{Hom}^{\bullet }_{dg}(Y^{\bullet },Y^{\bullet })$| with the structure of a dg-algebra and
|$\operatorname{Hom}^{\bullet }_{dg}(X^{\bullet },Y^{\bullet })$| with the structure of a (left) dg-module over
|$\operatorname{Hom}^{\bullet }_{dg}(Y^{\bullet },Y^{\bullet })$|. Furthermore, this gives rise to a functor
which descends to a triangulated functor on the respective homotopy categories
If
|$Y^{\bullet }$| is a
|$\mathcal{K}$|-injective complex (e.g., a bounded below complex of injectives), this descends further to a triangulated functor on the corresponding derived categories
Assume now that the abelian category
|$\mathcal{A}$| has enough injectives. Then for each
|$X \in D^{+}(\mathcal{A})$| we can pick a complex of injectives
|$I^{\bullet }_{X} \in C^{+}(\mathcal{A})$| representing
|$X$|. Moreover, we can consider the associated dg-algebra
and the corresponding functor
The following statement is an idempotent complete version of [
40, Proposition 7].
Lemma 4.4.The functor from (15) induces an equivalence of triangulated categories
which sends
|$X$| to the free dg-module
|$R_{X}$|.
Proof.By construction, the functor from (
15) sends
|$X$| to
|$\operatorname{Hom}^{\bullet }_{dg}(I_{X}^{\bullet }, I_{X}^{\bullet }) = R_{X}$|. Moreover, this functor induces an isomorphism
for all
|$i \in \mathbb{Z}$|. Here we have used for the first (resp. last) isomorphism that
|$I^{\bullet }_{X}$| (resp
|$R_{X}$|) is a
|$\mathcal{K}$|-injective complex (resp.
|$\mathcal{K}$|-projective dg-module). Note that the categories
|$ D^{+}(\mathcal{A})^{\operatorname{op}}$| and
|$D(R_{X}-\operatorname{dgMod})$| are idempotent complete by Lemma
4.1 (using that
|$D(R_{X}-\operatorname{dgMod})$| has countable coproducts). By Lemma
4.3 this implies that (
15) restricts to an equivalence of triangulated categories
The dg-algebra
|$R_{X}$| crucially depends on the choice of the injective resolution
|$I_{X}^{\bullet }$| which makes it difficult to compute
|$R_{X}$| explicitly. The cohomology of
|$R_{X}$| on the other hand can be described more concretely as
Hence, we would like to replace the dg-algebra
|$R_{X}$| in Lemma
4.4 with its cohomology. Recall that a dg-algebra
|$R$| is called
formal if there is a chain of quasi-isomorphisms of dg-algebras
Here we consider
|$H^{*}(R)$| as a dg-algebra with vanishing differential. Any quasi-isomorphism of dg-algebras
|$A \rightarrow B$| induces an equivalence
|$D_{\operatorname{perf}}(B) \overset{\sim }{\rightarrow } D_{\operatorname{perf}}(A)$| which identifies the free dg-module
|$B$| with the free dg-module
|$A$|. Hence, if
|$R$| is formal, there is an equivalence of triangulated categories
In particular, if
|$R_{X}$| is formal Lemma
4.4 induces an equivalence
The following theorem provides a useful criterion for formality.
Theorem 4.5.[36] Let |$R$| be a dg-algebra over an algebraically closed field |$k$| and |$q \in k^{\times }$| not a root of unity. If |$R$| can be equipped with a dg-algebra automorphism |$F: R\rightarrow R$| such that |$H^{i}(F)$| acts on |$H^{i}(R)$| by multiplication with |$q^{i}$| then |$R$| is formal.
Proof.This is proved for |$k = \mathbb{C}$| in [36, Theorem B.1.1] (with a slightly different condition on the action of |$H^{i}(F)$|). The same proof works for general |$k$|: By [36, Theorem B.1.2], we may assume that |$F$| acts locally finitely on |$R$|. Hence, there is a generalized eigenspace decomposition |$R^{i}= \bigoplus _{\alpha \in k^{\times }} R^{i}_{\alpha }$| which satisfies |$R^{i}_{\alpha } \cdot R^{j}_{\beta } \subset R^{i+j}_{\alpha \beta }$|. Using this decomposition, we can define a dgg-algebra (i.e., a |$\mathbb{Z}^{2}$|-graded algebra with a differential |$d$| homogeneous of degree |$(1,0)$| satisfying the Leibniz rule) via |$\tilde{R}:= \bigoplus _{i,j \in \mathbb{Z}} R^{i}_{q^{j}}$|. Note that the inclusion |$\tilde{R} \hookrightarrow R$| is a quasi-isomorphism. Moreover, the cohomology of the dgg-algebra |$\tilde{R}$| lives in degrees |$\{ (i,i) \mid i \in \mathbb{Z} \}$|. Such dgg-algebras are known to be formal (c.f. [40, Proposition 4]).
4.3 The pro-étale topology
In this section, we recall the definition of the constructible derived category in the pro-étale topology and a few of its basic properties from [12].
Definition 4.6.A morphism of schemes |$f: U \rightarrow X$| is called weakly étale if both |$f$| and its diagonal |$\Delta _{f}: U \rightarrow U\times _{X} U $| are flat morphisms. Let ![graphic]()
be the category of schemes |$U$| weakly étale over |$X$|. This becomes a site by declaring a family of maps |$\{U_{i} \rightarrow U \}$| to be a covering if it is a covering in the fpqc topology.
The property of being weakly étale is stable under composition and base change. Moreover, any morphism in
![graphic]()
is weakly étale. We denote the category of sheaves of abelian groups on
|$X_{\operatorname{pro\acute{e}t}}$| by
|$\operatorname{Ab}(X_{\operatorname{pro\acute{e}t}})$|. Let
|$f:X \rightarrow Y$| be a morphism of schemes. Then
|$f$| induces a morphism of sites
|$X_{\operatorname{pro\acute{e}t}} \rightarrow Y_{\operatorname{pro\acute{e}t}}$| via the functor
There is a corresponding pair of adjoint functors
Let
|$\Lambda $| be a topological ring. By [
12, Lemma 4.2.12] there is a sheaf of rings
|$\Lambda _{X}$| on
|$X_{\operatorname{pro\acute{e}t}}$| defined by
If
|$\Lambda $| is totally disconnected and
|$U$| is quasi-compact, we get
where
|$\pi _{0}(U)$| is the space of connected components. If
|$\Lambda $| is discrete,
|$\Lambda _{X}$| is the constant sheaf with values in
|$\Lambda $|. For any morphism of schemes
|$f:X \rightarrow Y$|, there is a canonical map
|$f^{\#}: \Lambda _{Y} \rightarrow f_{*}\Lambda _{X}$| and hence a morphism of ringed sites
Thus, we get corresponding pairs of adjoint functors
Here we adhere to the derived convention, that is, the functors
|$f^{*}$| and
|$f_{*}$| are understood to be derived when applied to complexes. From now on, we fix a prime number
|$\ell $| and make the following assumption on the coefficient ring
|$\Lambda $|.
Assumption 4.7.The ring |$\Lambda $| is of one of the following forms:
|$\Lambda = \mathbb{F}$| is a finite field of characteristic |$\ell $|;
|$\Lambda = \mathcal{O}_{E}$| is the ring of integers in an algebraic extension |$E/ \mathbb{Q}_{\ell }$|;
|$\Lambda = E$| is an algebraic extension |$E / \mathbb{Q}_{\ell }$|.
We equip
|$\Lambda $| with the
|$\ell $|-adic topology.
Note that the assumption above includes the ring |$\Lambda = \overline{\mathbb{Q}}_{\ell }$|. For any algebraic extension |$E/\mathbb{Q}_{\ell }$|, we denote by |$\mathcal{O}_{E}$| the ring of integers in |$E$|. If |$E/\mathbb{Q}_{\ell }$| is finite, we pick a uniformizer |$\omega \in \mathcal{O}_{E}$|.
Example 4.8.[
12, Lemma 6.8.2] gives an alternative description of
|$\Lambda _{X}$| under Assumption
4.7: If
|$\Lambda $| is a finite ring (e.g.,
|$\Lambda = \mathbb{F}$| a finite field) then
|$\Lambda _{X}$| is just the constant sheaf on
|$X_{\operatorname{pro\acute{e}t}}$| with values in
|$\Lambda $|. In the remaining cases,
|$\Lambda _{X}$| can be constructed as follows:
Lemma 4.9.For any morphism of schemes |$X \rightarrow Y$|, the canonical map |$f^{\#}: f^{-1}\Lambda _{Y} \rightarrow \Lambda _{X}$| is an isomorphism. In particular |$f^{*} = f^{-1}$| is exact.
Proof.This result can be found in [
16, Proposition 4.6]. The proof uses the explicit description of
|$\Lambda _{X}$| in Example
4.8. Since pullback commutes with colimits and localization, the main step is to show that
|$f^{\#}$| is an isomorphism when
|$\Lambda = \mathcal{O}_{E}$| for a finite extension
|$E/\mathbb{Q}_{\ell }$|. In this case,
|$\Lambda _{Y} = \varprojlim (\mathcal{O}_{E}/\omega ^{n}\mathcal{O}_{E})_{Y} $| is represented by the scheme
|$ \varprojlim (\bigsqcup _{\mathcal{O}_{E}/\omega ^{n}\mathcal{O}_{E}} Y)$|. Hence,
|$f^{*}\Lambda _{Y}$| is represented by the scheme
and thus
|$f^{*}\Lambda _{Y} = \Lambda _{X}$|.
Recall that a scheme
|$X$| is called
|$\ell $|-coprime if
|$\ell $| is invertible in
|$\mathcal{O}_{X}$|. Following [
12], the constructible derived category can be defined as follows.
Definition 4.10.Let |$X$| be a noetherian |$\ell $|-coprime scheme.
A sheaf |$\mathcal{F} \in \operatorname{Sh}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| is called locally constant of finite presentation if it is locally isomorphic to |$M \otimes _{\Lambda } \Lambda _{X}$| for a finitely-presented |$\Lambda $|-module |$M$|.
A sheaf |$\mathcal{F} \in \operatorname{Sh}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| is called constructible if there exists a finite stratification |$\{X_{i} \rightarrow X\}$| such that |$\mathcal{F}|_{X_{i}}$| is locally constant of finite presentation.
A complex |$\mathcal{F} \in D(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| is called constructible if it is bounded and all its cohomology sheaves are constructible.
We denote by
|$\operatorname{Sh}_{c}(X, \Lambda ) \subset \operatorname{Sh}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| the category of all constructible sheaves. The
constructible derived category is the full subcategory
|$D^{b}_{c}(X, \Lambda ) \subset D(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| consisting of constructible complexes.
Remark 4.11.The assumptions on |$X$| in the definition above can be weakened to topologically noetherian and qcqs. For our purposes, noetherian is good enough since all schemes we will encounter are noetherian.
The six functor formalism can also be conveniently described in the pro-étale topology: The functors
|$\otimes ^{L}, \operatorname{R{\mathcal{H}}\! \textit{om}}$| and
|$f^{*}$| preserve constructible complexes and thus they descend to the constructible derived category. The same is true for
|$f_{*}$| if
|$f:X \rightarrow Y$| is of finite type and
|$Y$| satisfies some mild assumptions (e.g., for
|$Y$| of finite type over a field or more generally for
|$Y$| quasi-excellent). If
|$j: U \hookrightarrow X$| is an open immersion, the pullback functor
|$j^{*}$| has a left adjoint
|$j_!$| which preserves constructible complexes. For a general
|$f:X \rightarrow Y$| (of finite type with
|$Y$| quasi-excellent), we define
|$f_!:= \overline{f}_{*} \circ j_!$| where we factor
|$f$| as an open immersion followed by a proper map
When restricted to the constructible derived category, the functor
|$f_!$| admits a right adjoint denoted by
|$f^!$|. These are the usual six functors
|$f^{*},f_{*},f_!,f^!, \otimes ^{L}$| and
|$\operatorname{R{\mathcal{H}}\! \textit{om}}$|.
Remark 4.12.In [12] a slightly different pullback functor |$f_{comp}^{*}$| is used which is defined as |$f^{*}$| followed by a certain completion operation [12, Lemma 6.5.9]. It turns out that in our situation (i.e., under Assumption 4.7) the two functors agree on the constructible derived category. To see this, recall that |$f^{*}$| is exact by Lemma 4.9. Using this, it is straightforward to check that |$f^{*}$| preserves constructible complexes. Since constructible complexes are complete ([12, Proposition 6.8.11(3), Definition 6.5.1]), it follows that |$f^{*} = f^{*}_{comp}$| on |$D^{b}_{c}(X, \Lambda )$|.
If
|$\mathbb{F}$| is a finite field of characteristic
|$\ell $|, we denote by
|$D^{b}_{c}(X_{{\acute{\textrm e}\textrm t}}, \mathbb{F})$| the constructible derived category in the étale topology with coefficients in
|$\mathbb{F}$|. The standard compatibility and base change results for the six functors on
|$D^{b}_{c}(X, \Lambda )$| can be deduced from the corresponding results in
|$D^{b}_{c}(X_{{\acute{\textrm e}\textrm t}}, \mathbb{F})$| using the following reduction steps.
Lemma 4.13.[12] Let |$X$| be noetherian and |$\ell $|-coprime.
(1) Let
|$\mathbb{F}$| be a finite field of characteristic
|$\ell $|. Then the canonical morphism of sites
|$\nu : X_{\operatorname{pro\acute{e}t}} \rightarrow X_{{\acute{\textrm e}\textrm t}}$| induces an equivalence of categories
(2) Let
|$E/\mathbb{Q}_{\ell }$| be a finite extension and
|$\kappa $| the residue field of
|$\mathcal{O}_{E}$|. Then the functor
is well-defined (i.e., preserves constructibility) and conservative (i.e., reflects isomorphisms);
(3) Let
|$E / \mathbb{Q}_{\ell }$| be algebraic. Then the canonical functor
is an equivalence of categories;
(4) Let
|$E / \mathbb{Q}_{\ell }$| be algebraic. Then the canonical functor
is an equivalence of categories. Here
|$D^{b}_{c}(X, \mathcal{O}_{E})[\ell ^{-1}]$| is the category with the same objects as
|$D^{b}_{c}(X, \mathcal{O}_{E})$| and
Moreover, the functors in (1)-(4) are compatible with the six functors
|$f^{*},f_{*},f_!,f^!,\otimes ^{L}$| and
|$\operatorname{R{\mathcal{H}}\! \textit{om}}$|.
Proof.This follows from various results in [
12]: (1) follows from [
12, Corollary 5.1.5, 5.1.6]. The well-definedness in (2) follows from [
12, Proposition 6.8.11(3), Definition 6.5.1]. To show that the functor
|$\kappa _{X} \otimes ^{L}_{\mathcal{O}_{E,X}} -$| is conservative, it suffices to prove that it reflects zero objects. Let
|$\mathcal{F} \in D^{b}_{c}(X, \mathcal{O}_{E})$| with
|$ \kappa _{X} \otimes ^{L}_{\mathcal{O}_{E,X}} \mathcal{F} = 0$|. We need to show that
|$\mathcal{F} = 0$|. Let
|$\mathfrak{m} \subset \mathcal{O}_{E}$| be the maximal ideal. Then there is a short exact sequence of
|$\mathcal{O}_{E}$|-modules
with
|$ \mathfrak{m}^{n}/\mathfrak{m}^{n+1} \cong \kappa $|. This induces a distinguished triangle
By an induction argument, this implies that
|$(\mathcal{O}_{E}/\mathfrak{m}^{n})_{X} \otimes ^{L}_{\mathcal{O}_{E,X}} \mathcal{F} = 0$| for all
|$n\ge 1$|. Moreover,
|$\mathcal{F}$| is
|$\mathfrak{m}$|-adically complete by [
12, Proposition 6.8.11(3), Definition 6.5.1], which means that
This completes the proof of (2). The claims in (3) and (4) are [
12, Proposition 6.8.14]. It remains to prove compatibility with the six functors in (1)-(4). Note for this that the category
|$\operatorname{colim}_{F} D^{b}_{c}(X, \mathcal{O}_{F})$| in (3) inherits the six functors from the
|$D^{b}_{c}(X, \mathcal{O}_{F})$| (a standard argument shows that the transition maps in the colimit are compatible with the six functors using that
|$\mathcal{O}_{F^{\prime},X} \cong \mathcal{O}_{F,X}^{\oplus [F^{\prime}:F]} $| as sheaves of
|$\mathcal{O}_{F,X}$|-modules for any finite extension
|$F^{\prime}/F$|). Similarly, the category
|$D^{b}_{c}(X, \mathcal{O}_{E})[\ell ^{-1}]$| in (4) inherits the six functors from
|$D^{b}_{c}(X, \mathcal{O}_{E})$|. All functors in (1)-(4) are induced by pullbacks along morphisms of ringed sites (e.g., the functor in (2) is the (derived) pullback along the morphism of ringed sites
|$(X_{\operatorname{pro\acute{e}t}}, \kappa _{X}) \rightarrow (X_{\operatorname{pro\acute{e}t}}, \mathcal{O}_{E,X})$|). As such, they commute with
|$f^{*}$| and
|$\otimes ^{L}$| by standard results about ringed sites [
42, Tag 0D6D, 07A4]. Since the functors in (1),(3) and (4) are equivalences, they also commute with the corresponding (right) adjoints
|$f_{*}$| and
|$\operatorname{R{\mathcal{H}}\! \textit{om}}$|. Similarly, they commute with
|$j_!$| for
|$j$| an open immersion which is left adjoint to
|$j^{*}$|. Hence, they also commute with
|$f_! = \overline{f}_{*} \circ j_! $| (see (
18)) and thus also with its right adjoint
|$f^!$|. It remains to show that the functor in (2) commutes with
|$f_{*}, \operatorname{R{\mathcal{H}}\! \textit{om}}, f_!$| and
|$f^!$|. This follows from [
12, Lemma 6.5.11(3), 6.7.13, 6.7.14, 6.7.19].
We conclude this section by collecting a few useful properties about pro-étale sheaves. In the pro-étale topology, so-called
|$w$|-contractible affine schemes play a distinguished role.
Definition 4.14.An affine scheme |$U$| is called |$w$|-contractible if every faithfully-flat weakly étale map |$V\rightarrow U$| has a section.
Lemma 4.15.Let |$U$| be a |$w$|-contractible affine scheme.
(1) The global sections functor |$\Gamma (U,-)$| is exact on |$\operatorname{Sh}(U_{\operatorname{pro\acute{e}t}}, \Lambda _{U})$|;
(2) Any locally constant sheaf of finite presentation |$\mathcal{F} \in \operatorname{Sh}_{c}(U, \Lambda )$| is already constant, that is, |$\mathcal{F} \cong M \otimes _{\Lambda } \Lambda _{U}$| for a finitely-presented |$\Lambda $|-module |$M$|;
(3) If |$R$| is a strictly Henselian local ring (e.g., an algebraically closed field) then |$\operatorname{Spec}(R)$| is |$w$|-contractible.
Proof.The exactness in (1) is mentioned in the introduction of [
12] (see also [
42, Tag 098H, 0946]). Let
|$\mathcal{F} \in \operatorname{Sh}_{c}(U, \Lambda )$| be locally constant of finite presentations. Then we can find a weakly étale cover
|$V = \bigsqcup V_{i} \rightarrow U$| with
|$\mathcal{F}|_{V} = M \otimes _{\Lambda } \Lambda _{V}$| for some finitely-presented
|$\Lambda $|-module
|$M$|. Since
|$U$| is weakly contractible, there is a section
|$s:U \rightarrow V$| and thus
This proves (2). (3) follows from [
12, Theorem 1.8].
Lemma 4.16.The triangulated category |$D^{b}_{c}(X, \Lambda ) \subset D(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| is closed under direct summands. In particular, |$D^{b}_{c}(X, \Lambda )$| is idempotent complete.
Proof.Since taking cohomology commutes with direct sums, it suffices to prove that |$\operatorname{Sh}_{c}(X, \Lambda ) \subset \operatorname{Sh}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| is closed under direct summands. In fact, by [12, Lemma 6.8.7, Proposition 6.8.11] the category |$\operatorname{Sh}_{c}(X, \Lambda )$| is abelian so it is certainly closed under direct summands.
Lemma 4.17.If |$f:X \rightarrow Y$| is weakly étale, the pullback functor |$f^{*}$| preserves injectives.
Proof.Compositions of weakly étale maps are weakly étale and morphisms between weakly étale maps are weakly étale. Together with Lemma 4.9 this implies that |$(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| is the localization of the ringed site |$(Y_{\operatorname{pro\acute{e}t}}, \Lambda _{Y})$| at |$X \rightarrow Y$|. By general results on ringed sites [42, Tag 04IX] this implies that |$f^{*}$| has an exact left adjoint |$f_!$|. In particular, |$f^{*}$| preserves injectives.
Let
|$\mathcal{F} \in D^{b}_{c}(X, \Lambda )$| and pick a complex of injectives
|$I_{\mathcal{F}}^{\bullet } \in C^{+}(\operatorname{Sh}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X}))$| representing
|$\mathcal{F}$|. Let
|$R_{\mathcal{F}} = \operatorname{Hom}_{dg}^{\bullet }(I_{\mathcal{F}}^{\bullet },I_{\mathcal{F}}^{\bullet })$| be the dg-algebra from (
14). Thanks to the pro-étale formalism, we obtain the following algebraic description of the category
.
Corollary 4.18.There is an equivalence of triangulated categories
which send
|$\mathcal{F}$| to the free dg-module
|$R_{\mathcal{F}}$|.
Proof.By Lemma 4.16 we have 
. The result now follows from Lemma 4.4.
4.4 The Frobenius action on dg-algebras
In this section we show that the Frobenius action on |$\operatorname{\underline{\operatorname{Hom}}}$| from (10) is compatible with the dg-techniques from Section 4.2. These results are similar to the ones in [36, Appendix A] but some of the arguments simplify because the canonical morphism |$\operatorname{Spec}(\overline{\mathbb{F}}_{q}) \rightarrow \operatorname{Spec}(\mathbb{F}_{q})$| is a weakly étale (but not étale).
Let |$X_{0}$| be a variety defined over a finite field |$\mathbb{F}_{q}$|. Recall that we define |$X:= X_{0} \otimes _{\mathbb{F}_{q}} \overline{\mathbb{F}}_{q}$|. Let |$\pi :X \rightarrow X_{0}$| be the canonical map and define |$\mathcal{F}:= \pi ^{*}(\mathcal{F}_{0}) \in \operatorname{Sh}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| for any |$\mathcal{F}_{0} \in \operatorname{Sh}(X_{0, \operatorname{pro\acute{e}t}}, \Lambda _{X_{0}})$|.
Lemma 4.19.The pullback functor |$\pi ^{*}: \operatorname{Sh}(X_{0, \operatorname{pro\acute{e}t}}, \Lambda _{X_{0}}) \rightarrow \operatorname{Sh}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| preserves injectives.
Proof.The morphism |$\operatorname{Spec}(\overline{\mathbb{F}}_{q}) \rightarrow \operatorname{Spec}(\mathbb{F}_{q})$| is weakly étale. By base change, |$\pi ^{*}: X \rightarrow X_{0}$| is also weakly étale. Hence, |$\pi ^{*}$| preserves injectives by Lemma 4.17.
The geometric
|$q$|-Frobenius
|$F \in \operatorname{Gal}(\overline{\mathbb{F}}_{q} / \mathbb{F}_{q})$| defines a morphism
|$\operatorname{Spec}(\overline{\mathbb{F}}_{q}) \rightarrow \operatorname{Spec}(\overline{\mathbb{F}}_{q})$| in the category
|$\operatorname{Spec}(\mathbb{F}_{q})_{\operatorname{pro\acute{e}t}}$|. Thus, for any
|$\mathcal{F}_{0} \in \operatorname{Sh}( \operatorname{Spec}(\mathbb{F}_{q})_{\operatorname{pro\acute{e}t}}, \Lambda _{\operatorname{Spec}(\mathbb{F}_{q})} )$| there is a corresponding restriction map on sections
This map is an isomorphism with inverse given by restriction along the arithmetic Frobenius
|$F^{-1} \in \operatorname{Gal}(\overline{\mathbb{F}}_{q} / \mathbb{F}_{q})$|. Hence, we can equip
|$\Gamma (\operatorname{Spec}(\overline{\mathbb{F}}_{q}), \mathcal{F}_{0} )$| with the structure of a
|$\Lambda [F,F^{-1}]$|-module. Moreover, for any morphism of sheaves
|$\mathcal{F}_{0} \rightarrow \mathcal{G}_{0}$|, we get a commutative diagram
Hence, we obtain a functor
Remark 4.20.Recall from Section 3.1 that there is an equivalence between the category of constructible sheaves |$\operatorname{Sh}_{c}(\operatorname{Spec}(\mathbb{F}_{q}), \overline{\mathbb{Q}}_{\ell })$| and the category of finite-dimensional continuous |$\operatorname{Gal}(\overline{\mathbb{F}}_{q} / \mathbb{F}_{q})$|-representations over |$\overline{\mathbb{Q}}_{\ell }$|. On |$\operatorname{Sh}_{c}(\operatorname{Spec}(\mathbb{F}_{q}), \overline{\mathbb{Q}}_{\ell })$| the functor |$\Gamma _{F}$| simply corresponds to restricting the |$\operatorname{Gal}(\overline{\mathbb{F}}_{q} / \mathbb{F}_{q})$|-action to the (geometric) Frobenius element |$F \in \operatorname{Gal}(\overline{\mathbb{F}}_{q} / \mathbb{F}_{q})$|.
Note that
where
is the forgetful functor.
Lemma 4.21.The functors |$\operatorname{For}_{F}, \Gamma _{F}$| and |$ \Gamma (\operatorname{Spec}(\overline{\mathbb{F}}_{q}), -)$| are exact.
Proof.The forgetful functor |$\operatorname{For}_{F}$| is exact and reflects exact sequences. Hence, by (19) it suffices to prove that |$\Gamma (\operatorname{Spec}(\overline{\mathbb{F}}_{q}), -)$| is exact. This follows from Lemma 4.15.
We define a functor
via
Note that we have
Hence,
|$\operatorname{\underline{\operatorname{Hom}}}(\mathcal{F}_{0}, \mathcal{G}_{0})$| is just the vector space
|$\operatorname{Hom}( \mathcal{F}, \mathcal{G})$| together with a canonical Frobenius action. The composition map
gives rise to a map
and thus to a map of
|$\Lambda [F,F^{-1}]$|-modules
Similarly, the unit morphism
induces a map
by adjunction and thus a morphism of
|$\Lambda [F,F^{-1}]$|-modules
Forgetting the Frobenius action in (
20) and (
21) recovers the standard composition map
and unit map
Similarly, one can define pre- and post-composition maps
for any
|$\alpha : \mathcal{F}_{0} \rightarrow \mathcal{F}_{0}^{\prime}$| and
|$\beta : \mathcal{G}_{0} \rightarrow \mathcal{G}_{0}^{\prime}$| which recover the maps
when forgetting the Frobenius action.
We also define a functor on derived categories
via
(recall that we adhere to the derived convention, that is,
|$\Gamma _{F}$| and
|$a_{*}$| are understood to be derived when applied to complexes). Moreover, we set
Remark 4.22.This definition of |$\operatorname{\underline{\operatorname{Hom}}}^{i}$| extends our previous definition (10) for constructible complexes to arbitrary complexes in |$D(X_{0,\operatorname{pro\acute{e}t}}, \Lambda _{X_{0}})$| (see also Remark 4.20).
For any two chain complexes
|$\mathcal{F}_{0}^{\bullet }, \mathcal{G}_{0}^{\bullet } \in C(\operatorname{Sh}(X_{0, \operatorname{pro\acute{e}t}}, \Lambda _{X_{0}}))$| we define a chain complex
Explicitly, this complex is given by
with differential
|$d_{\operatorname{\underline{\operatorname{Hom}}}^{\bullet }_{dg}(\mathcal{F}_{0}^{\bullet },\mathcal{G}_{0}^{\bullet })} (f) = d_{\mathcal{G}_{0}^{\bullet }} \circ f - (-1)^{|f|} f \circ d_{\mathcal{F}_{0}^{\bullet }}$| (where we use (
22)). The composition map on
|$\operatorname{\underline{\operatorname{Hom}}}(-,-)$| from (
20) can be applied component-wise to define a composition map
This equips
|$ \operatorname{\underline{\operatorname{Hom}}}^{\bullet }_{dg}(\mathcal{F}_{0}^{\bullet },\mathcal{F}_{0}^{\bullet })$| with the structure of a dg-algebra over
|$\Lambda $| together with a dg-algebra automorphism induced by the Frobenius action. Forgetting this Frobenius action recovers the dg-algebra
|$\operatorname{Hom}^{\bullet }_{dg}(\mathcal{F}^{\bullet },\mathcal{F}^{\bullet })$|.
Lemma 4.23.Let
|$\mathcal{F}_{0}, \mathcal{G}_{0} \in D^{+}(X_{0, \operatorname{pro\acute{e}t}}, \Lambda _{X_{0}})$| and pick bounded below complexes of injectives
|$I_{\mathcal{F}_{0}}^{\bullet }, I_{\mathcal{G}_{0}}^{\bullet } \in C^{+}(\operatorname{Sh}(X_{0, \operatorname{pro\acute{e}t}}, \Lambda _{X_{0}}))$| representing
|$\mathcal{F}_{0}$| and
|$\mathcal{G}_{0}$|. Then
as
|$\Lambda [F,F^{-1}]$|-modules.
Proof.By definition,
|$\operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}_{0}, \mathcal{G}_{0})$| is represented by the complex
|$\operatorname{{\mathcal{H}}om}_{dg}^{\bullet } (I_{\mathcal{F}_{0}}^{\bullet },I_{\mathcal{G}_{0}}^{\bullet } )$| defined as
with differential
|$d_{\operatorname{{\mathcal{H}}om}^{\bullet }_{dg}(I_{\mathcal{F}_{0}}^{\bullet },I_{\mathcal{G}_{0}}^{\bullet })} (f) = d_{I_{\mathcal{G}_{0}}^{\bullet }} \circ f - (-1)^{|f|} f \circ d_{I_{\mathcal{F}_{0}}^{\bullet }}$|. Note that
|$\operatorname{{\mathcal{H}}om}(I_{\mathcal{F}_{0}}^{-l}, I_{\mathcal{G}_{0}}^{m})$| is acyclic for
|$a_{*}$| for all
|$l,m \in \mathbb{Z}$| (see [
2, V-(4.10), V-(5.2)]). Hence,
|$\operatorname{{\mathcal{H}}om}_{dg}^{i} (\mathcal{F}_{0}^{\bullet }, \mathcal{G}_{0}^{\bullet })$| is acyclic for
|$a_{*}$| for all
|$i \in \mathbb{Z}$|. Since
|$\Gamma _{F}$| is exact,
|$\operatorname{{\mathcal{H}}om}_{dg}^{i} (\mathcal{F}_{0}^{\bullet }, \mathcal{G}_{0}^{\bullet })$| is also acyclic for
|$\Gamma _{F} \circ a_{*}$| for all
|$i \in \mathbb{Z}$|. In particular,
|$\operatorname{R\underline{\operatorname{Hom}}}(\mathcal{F}_{0}, \mathcal{G}_{0}) = \Gamma _{F}\circ a_{*} \circ \operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}_{0}, \mathcal{G}_{0})$| is represented by the complex obtained by applying
|$\Gamma _{F} \circ a_{*}$| to each component in
|$\operatorname{{\mathcal{H}}om}_{dg}^{\bullet } (I_{\mathcal{F}_{0}}^{\bullet },I_{\mathcal{G}_{0}}^{\bullet } )$|. This is precisely the complex
|$\operatorname{\underline{\operatorname{Hom}}}_{dg}^{\bullet } (I_{\mathcal{F}_{0}}^{\bullet },I_{\mathcal{G}_{0}}^{\bullet } )$|. Hence,
The constructions of this section can be summarized as follows.
Proposition 4.24.For any |$\mathcal{F}_{0} \in D^{b}_{c}(X_{0}, \Lambda )$|, there is a bounded below complex of injectives |$I_{\mathcal{F}}^{\bullet } \in C^{+}(\operatorname{Sh}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X}))$| representing |$\mathcal{F}$| and dg-algebra automorphism |$F$| of |$R_{\mathcal{F}}= \operatorname{Hom}_{dg}^{\bullet }(I_{\mathcal{F}}^{\bullet }, I_{\mathcal{F}}^{\bullet })$| such that the action of |$H^{i}(F)$| on |$H^{i}(R_{\mathcal{F}}) = \operatorname{Hom}^{i}(\mathcal{F}, \mathcal{F})$| is the Frobenius action coming from |$\operatorname{\underline{\operatorname{Hom}}}^{i}(\mathcal{F}_{0}, \mathcal{F}_{0})$|.
Proof.Pick a complex of injectives |$I_{\mathcal{F}_{0}}^{\bullet } \in C^{+}(\operatorname{Sh}(X_{0, \operatorname{pro\acute{e}t}}, \Lambda _{X_{0}}))$| representing |$\mathcal{F}_{0}$|. Then |$I_{\mathcal{F}}^{\bullet }:= \pi ^{*}(I_{\mathcal{F}_{0}}^{\bullet })$| is a complex of injectives representing |$\mathcal{F}$| by Lemma 4.19. The dg-algebra |$\operatorname{\underline{\operatorname{Hom}}}_{dg}^{\bullet }(I_{\mathcal{F}_{0}}^{\bullet }, I_{\mathcal{F}_{0}}^{\bullet })$| is just |$R_{\mathcal{F}} = \operatorname{Hom}_{dg}^{\bullet }(I_{\mathcal{F}}^{\bullet }, I_{\mathcal{F}}^{\bullet })$| equipped with a dg-algebra automorphism |$F$|. By Lemma 4.23, we have |$H^{i}(\operatorname{\underline{\operatorname{Hom}}}_{dg}^{\bullet }(I_{\mathcal{F}_{0}}^{\bullet }, I_{\mathcal{F}_{0}}^{\bullet })) = \operatorname{\underline{\operatorname{Hom}}}^{i}(\mathcal{F}_{0}, \mathcal{F}_{0})$|. Hence, |$H^{i}(F)$| is the canonical Frobenius action on |$\operatorname{\underline{\operatorname{Hom}}}^{i}(\mathcal{F}_{0}, \mathcal{F}_{0})$|.
4.5 Formality for the Springer category
Let
|$\mu _{i}: \tilde{V}^{i} \rightarrow V$| be a finite collection of morphisms of Springer type defined over
|$\overline{\mathbb{F}}_{q}$| and let
|$\operatorname{\textbf{S}} = \bigoplus _{i \in I} \operatorname{\textbf{S}}^{i} \in D^{b}_{c}(V, \overline{\mathbb{Q}}_{\ell })$| be the associated Springer sheaf. Recall from (
9) that the Springer category is defined as
where the
|$X_{i}$| are the simple perverse sheaves appearing in
|$\operatorname{\textbf{S}}$|.
Lemma 4.25.We have 
.
Proof.Since
, we have
. Hence, it suffices to show that
|$D_{\operatorname{Spr}}(V, \overline{\mathbb{Q}}_{\ell })$| is closed under direct summands. Let
|$\operatorname{Perv}_{\operatorname{Spr}}(V) \subset \operatorname{Perv}(V)$| be the Serre subcategory generated by the
|$X_{1},...,X_{n}$|. Then by a standard dévissage argument, the Springer category can be described as
Note that as a Serre subcategory,
|$\operatorname{Perv}_{\operatorname{Spr}}(V)$| is closed under direct summands in
|$\operatorname{Perv}(V)$|. Hence,
|$D_{\operatorname{Spr}}(V, \overline{\mathbb{Q}}_{\ell })$| is also closed under direct summands.
We obtain the following general formality result for the Springer category.
Theorem 4.26.There is an equivalence of triangulated categories
which identifies
|$\operatorname{\textbf{S}}$| with the free dg-module
|$\operatorname{Hom}^{*}(\operatorname{\textbf{S}}, \operatorname{\textbf{S}})$|.
Proof.Combining Corollary
4.18 and Lemma
4.25 we get an equivalence
which sends
|$\operatorname{\textbf{S}}$| to the free dg-module
|$ R_{\operatorname{\textbf{S}}}$|. By Corollary
3.5 and Proposition
4.24 the dg-algebra
|$R_{\operatorname{\textbf{S}}}$| can be equipped with a dg-algebra automorphism
|$F: R_{\operatorname{\textbf{S}}} \rightarrow R_{\operatorname{\textbf{S}}}$| such that
|$H^{i}(F)$| acts by multiplication with
|$q^{\tfrac{i}{2}}$|. This implies that
|$R_{\operatorname{\textbf{S}}}$| is formal by Theorem
4.5. In particular, by (
16) we get an equivalence
which sends
|$\operatorname{\textbf{S}}$| to the free dg-module
|$\operatorname{Hom}^{*}(\operatorname{\textbf{S}},\operatorname{\textbf{S}})$|.
5 De |$\mathbb{F}$| à |$\mathbb{C}$|
In this section we prove a formality result for Springer sheaves on varieties over a field of characteristic |$0$| by reduction to the positive characteristic case. This is a standard application of the “De |$\mathbb{F}$| à |$\mathbb{C}$|” technique from [6, §6]. We will explain how these arguments work in the pro-étale setting. If |$X_{A}$| is a scheme defined over a ring |$A$| and |$A \rightarrow R$| is a ring homomorphism, we denote by |$X_{R}$| the base change of |$X_{A}$| to |$R$|. Similarly, if |$S \rightarrow \operatorname{Spec}(A)$| is a morphism of schemes, we denote by |$X_{S}$| the corresponding base change to |$S$|. Furthermore, for |$\mathcal{F} \in D^{b}_{c}(X_{A}, \overline{\mathbb{Q}}_{\ell })$|, we denote by |$\mathcal{F}_{R} \in D^{b}_{c}(X_{R}, \overline{\mathbb{Q}}_{\ell })$| (resp. |$\mathcal{F}_{S} \in D^{b}_{c}(X_{S}, \overline{\mathbb{Q}}_{\ell })$|) the corresponding pullback to |$X_{R}$| (resp. |$X_{S}$|).
5.1 Generic base change
The main tool that we need to compare constructible sheaves on varieties in characteristic |$p$| and characteristic |$0$| is the generic base change theorem. Let us first explain what we mean by generic base change.
Definition 5.1.Let |$S$| be a scheme and |$f:X\rightarrow Y$| a morphism of |$S$|-schemes. Let |$\mathcal{F}, \mathcal{G} \in D(X_{{\acute{\textrm e}\textrm t}}, \Lambda )$| for a noetherian ring |$\Lambda $| (resp. |$\mathcal{F}, \mathcal{G} \in D(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| for |$\Lambda $| as in Assumption 4.7).
We say that the formation |$f_{*}\mathcal{F}$| commutes with generic base change if there is a dense open subscheme |$U \subset S$| such that for each morphism of schemes |$g:S^{\prime} \rightarrow U \subset S$| with corresponding pullback diagram
the canonical map |$(g^{\prime})^{*}f_{*} \mathcal{F} \rightarrow f^{\prime}_{*}(g^{\prime\prime})^{*}\mathcal{F}$| is an isomorphism.We say that the formation |$\operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}, \mathcal{G})$| commutes with generic base change if there is a dense open subscheme |$U \subset X$| such that for each morphism of schemes |$g: S^{\prime} \rightarrow U \subset S$| the canonical map |$g^{*}\operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}, \mathcal{G}) \rightarrow \operatorname{R{\mathcal{H}}\! \textit{om}}(g^{*}\mathcal{F}, g^{*}\mathcal{G})$| is an isomorphism.
There is the following generic base change theorem for étale sheaves.
Theorem 5.2.[19] Let |$S$| be a noetherian scheme, |$f:X \rightarrow Y$| a morphism of |$S$|-schemes of finite type and |$\Lambda $| a noetherian ring annihilated by an integer invertible in |$\mathcal{O}_{S}$|. Then for any |$\mathcal{F}, \mathcal{G} \in D^{b}_{c}(X_{{\acute{\textrm e}\textrm t}}, \Lambda )$| the formations |$f_{*} \mathcal{F}$| and |$\operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}, \mathcal{G})$| commute with generic base change.
Proof.This is proved for constructible sheaves |$\mathcal{F} \in \operatorname{Sh}_{c}(X_{{\acute{\textrm e}\textrm t}}, \Lambda )$| in [19, Th. finitude 1.9, 2.10]. The case of an arbitrary complex |$\mathcal{F} \in D^{b}_{c}(X_{{\acute{\textrm e}\textrm t}}, \Lambda )$| can be deduced from this by a dévissage argument.
We get a similar result for non-torsion coefficients in the pro-étale topology. For the rest of this section, we assume that
|$\Lambda $| is as in Assumption
4.7.
Lemma 5.3.Let |$S$| be a noetherian, quasi-excellent, |$\ell $|-coprime scheme and |$f:X \rightarrow Y$| a morphism of |$S$|-schemes of finite type. Then for any |$\mathcal{F},\mathcal{G} \in D^{b}_{c}(X_{\operatorname{pro\acute{e}t}}, \Lambda _{X})$| the formations |$f_{*} \mathcal{F}$| and |$\operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}, \mathcal{G})$| commute with generic base change.
Proof.Using the reduction steps for constructible sheaves from Lemma 4.13 this can be reduced to the étale generic base change theorem (Theorem 5.2). The assumption that |$S$| is quasi-excellent ensures that |$f_{*} \mathcal{F}$| is constructible.
The following lemma can also be deduced from [
6, Lemme 6.1.9] or [
36, A.5].
Lemma 5.4.Let |$X_{A}$| be a scheme of finite type over a finitely-generated |$\mathbb{Z}[\ell ^{-1}]$|-algebra |$A \subset \overline{\mathbb{Q}}_{\ell }$| and |$s \in \operatorname{Spec}(A)$| a closed point. For any |$\mathcal{F}_{A} \in D^{b}_{c}(X_{A}, \Lambda )$| there exists a strictly Henselian local ring |$R$| such that
(1) |$A \subset R \subset \overline{\mathbb{Q}}_{\ell }$|;
(2) the residue field of |$R$| is the algebraic closure |$\overline{s}$| of |$s$|;
(3) Pulling back along the morphisms in the diagram |$X_{\overline{\mathbb{Q}}_{\ell }} \overset{u}{\rightarrow } X_{R} \overset{v}{\leftarrow } X_{s}$| induces equivalences of categories
Proof.By constructibility, we may enlarge
|$A$| so that the cohomology sheaves of
|$a_{*}\operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}_{A}, \mathcal{F}_{A})$| are locally constant of finite presentation, that is, each
|$H^{i}(a_{*}\operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}_{A}, \mathcal{F}_{A}))$| is locally isomorphic to
|$M_{i} \otimes _{\Lambda } \Lambda _{\operatorname{Spec}(A)}$| for some finitely-presented
|$\Lambda $|-module
|$M_{i}$|. By Lemma
5.3 we may enlarge
|$A$| further so that
|$a_{*}\operatorname{R{\mathcal{H}}\! \textit{om}}(\mathcal{F}_{A}, \mathcal{G}_{A})$| commutes with arbitrary base change, that is, for any
|$g: S \rightarrow \operatorname{Spec}(A)$|, we have
Let
|$S$| be a
|$w$|-contractible affine scheme. Then any locally constant sheaf of finite presentation on
|$S_{\operatorname{pro\acute{e}t}}$| is constant by Lemma
4.15 and hence
Note that the global sections functor
|$\Gamma _{S} = \Gamma (S,-)$| is exact by Lemma
4.15. Hence, for
|$S$| affine, connected and
|$w$|-contractible, we get
Here the first equality follows from general results about sites (see [
2, V-(4.10), V-(5.2)]). The second uses exactness of
|$\Gamma _{S}$| and the third follows from (
23). To see the last equality, pick a presentation
|$\Lambda ^{n} \rightarrow \Lambda ^{m} \rightarrow M_{i} \rightarrow 0$|. This induces a presentation
|$ \Lambda _{S}^{n} \rightarrow \Lambda _{S}^{m} \rightarrow M_{i} \otimes _{\Lambda } \Lambda _{S} \rightarrow 0$|. Note that
|$\Gamma _{S}(\Lambda _{S}) = \Lambda $| by (
17) and the fact that
|$S$| is connected. Since
|$\Gamma _{S}$| is exact, this implies
|$\Gamma _{S} (M_{i} \otimes _{\Lambda } \Lambda _{S}) = \operatorname{coker}( \Lambda ^{n} \rightarrow \Lambda ^{m}) = M_{i}$|. Now let
|$R$| be any strictly Henselian local ring satisfying (1) and (2) (it is explained in [
6, p.156] that such a ring always exists and one can even assume that
|$R$| is also a discrete valuation ring). Then the schemes
|$\operatorname{Spec}(\overline{\mathbb{Q}}_{\ell }), \operatorname{Spec}(R), $| and
|$\overline{s}$| are all connected
|$w$|-contractible affine schemes by Lemma
4.15. Using (
24), we get
for all
|$i \in \mathbb{Z}$|. By Lemma
4.3 this implies that there are equivalences of triangulated categories
5.2 Lifting Springer sheaves
Any variety |$X$| over |$\overline{\mathbb{Q}}_{\ell }$| can be defined over a finitely-generated |$\mathbb{Z}[\ell ^{-1}]$|-algebra |$A \subset \overline{\mathbb{Q}}_{\ell }$|, that is, there exists a scheme |$X_{A}$| (of finite type) over |$\operatorname{Spec}(A)$| such that |$X = X_{A} \otimes _{A} \overline{\mathbb{Q}}_{\ell }$|. Choosing a maximal ideal of |$A$| gives rise to a ring homomorphism |$A \rightarrow \mathbb{F}_{q}$| for a finite field |$\mathbb{F}_{q}$|. Hence, it makes sense to consider |$X_{\mathbb{F}_{q}}$| relating |$X$| to a scheme over a field of positive characteristic. We first check that this procedure preserves morphisms of Springer type.
Let |$G$| be a connected reductive group over |$\overline{\mathbb{Q}}_{\ell }$|. There always exists a split reductive group scheme |$G_{\mathbb{Z}}$| over |$\mathbb{Z}$| with |$G = G_{\mathbb{Z}} \otimes _{\mathbb{Z}} \overline{\mathbb{Q}}_{\ell }$|. Fix a Borel subgroup |$B_{\mathbb{Z}} \subset G_{\mathbb{Z}}$| and let |$B:= B_{\mathbb{Z}} \otimes _{\mathbb{Z}} \overline{\mathbb{Q}}_{\ell } $| be the corresponding Borel subgroup of |$G$|. Let |$V$| be a |$G$|-representation and |$\{V^{i} \subset V \mid i \in I\}$| a finite collection of |$B$|-stable subspaces. Recall that a representation of an affine group scheme |$H$| is the same as a comodule over the Hopf algebra |$\mathcal{O}(H)$|.
Lemma 5.5.There is a finitely generated |$\mathbb{Z}[\ell ^{-1}]$|-algebra |$A \subset \overline{\mathbb{Q}}_{\ell }$|, a |$G_{A}$|-representation |$V_{A}$| and |$B_{A}$|-stable |$A$|-submodules |$V_{A}^{i} \subset V_{A}$| which recover the |$V^{i} \subset V$| when extending scalars to |$\overline{\mathbb{Q}}_{\ell }$|.
Proof.Pick a basis
|$v_{1},..., v_{n} \in V$|. Then we can write the comultiplication on the
|$v_{i}$| as
for some
|$f_{ij} \in \mathcal{O}(G)$|. Since
|$\mathcal{O}(G) = \mathcal{O}(G_{\mathbb{Z}}) \otimes _{\mathbb{Z}} \overline{\mathbb{Q}}_{\ell }$|, we can write each
|$f_{ij}$| as
for some
|$g_{ijk} \in \mathcal{O}(G_{\mathbb{Z}})$| and
|$c_{ijk} \in \overline{\mathbb{Q}}_{\ell }$|. Let
|$A \subset \overline{\mathbb{Q}}_{\ell }$| be the
|$\mathbb{Z}[\ell ^{-1}]$|-algebra generated by all the
|$c_{ijk}$|. Then
|$V_{A}:= \operatorname{Span}_{A} \{v_{1},..., v_{n}\}$| is an
|$\mathcal{O}(G_{A})$|-comodule and thus a
|$G_{A}$|-representation which recovers
|$V$| when extending scalars to
|$\overline{\mathbb{Q}}_{\ell }$|. We can use the same argument (potentially enlarging
|$A$| further) to show that there are
|$B_{A}$|-representations
|$V^{i}_{A}$| which are spanned by finitely many vectors
|$w^{i}_{j} \in V^{i}$| and recover
|$V^{i}$| when extending scalars to
|$\overline{\mathbb{Q}}_{l}$|. After enlarging
|$A$| further, we may assume that
|$V_{A}$| contains all the
|$w^{i}_{j}$| and thus
|$V^{i}_{A} \subset V_{A}$|.
Consider the variety
|$\tilde{V}^{i}_{A}:= G_{A} \times ^{B_{A}} V^{i}_{A}$| and the canonical map
|$\mu _{i}: \tilde{V}^{i}_{A} \rightarrow V_{A}$| induced by the action map
|$G_{A} \times V^{i}_{A} \rightarrow V_{A}$|. For any ring homomorphism
|$A \rightarrow R$|, we get a pullback diagram
where
|$\tilde{V}^{i}_{R} = G_{R} \times ^{B_{R}} V^{i}_{R} $|. We define
By proper base change we have
|$u^{*}(\operatorname{\textbf{S}}^{i}_{A}) = \operatorname{\textbf{S}}^{i}_{R}$| and
|$u^{*}(\operatorname{\textbf{S}}_{A}) = \operatorname{\textbf{S}}_{R}$|. Moreover, if
|$R = \overline{k}$| is an algebraically closed field, the morphism
|$\mu _{i}: \tilde{V}^{i}_{\overline{k}} \rightarrow V_{\overline{k}}$| is of Springer type and
|$\operatorname{\textbf{S}}^{i}_{\overline{k}}$| is the associated Springer sheaf (c.f. Section
2.2). Here we use that
Corollary 5.6.Let
|$\mu _{i}: \tilde{V}^{i} \rightarrow V$| be a finite collection of morphisms of Springer type over
|$\overline{\mathbb{Q}}_{\ell }$|. Then there is an equivalence of triangulated categories
which identifies
|$\operatorname{\textbf{S}}$| with the free dg-module
|$\operatorname{Hom}^{*}(\operatorname{\textbf{S}}, \operatorname{\textbf{S}})$|.
Proof.Let
|$A$|,
|$\tilde{V}^{i}_{A}, V_{A},...$| be as in Lemma
5.5 and pick a closed point
|$s \rightarrow \operatorname{Spec}(A)$| with algebraic closure
|$\overline{s}$|. Note that since
|$A$| is finitely-generated over
|$\mathbb{Z}[\ell ^{-1}]$|, we have
|$s = \operatorname{Spec}(\mathbb{F}_{q})$| for a finite field
|$\mathbb{F}_{q}$| with
|$\ell $| invertible in
|$\mathbb{F}_{q}$|. By Lemma
5.4 there is an equivalence of triangulated categories
|$D_{\operatorname{Spr}}(V, \overline{\mathbb{Q}}_{\ell }) \cong D_{\operatorname{Spr}}(V_{\overline{s}},, \overline{\mathbb{Q}}_{\ell })$| which identifies
|$\operatorname{\textbf{S}}$| with
|$\operatorname{\textbf{S}}_{\overline{s}}$|. In particular, we get
|$\operatorname{Hom}^{*}(\operatorname{\textbf{S}}, \operatorname{\textbf{S}}) \cong \operatorname{Hom}^{*}(\operatorname{\textbf{S}}_{\overline{s}}, \operatorname{\textbf{S}}_{\overline{s}}) $|. Hence, by Theorem
4.26, we have
6 A Derived Deligne-Langlands Correspondence
Let |$G$| be a connected reductive group over |$\overline{\mathbb{Q}}_{\ell }$|(|$\cong \mathbb{C}$|) with simply connected derived subgroup and let |$(X^{*},\Phi , X_{*},\Phi ^{\vee })$| be the associated root datum. Fix a torus and a Borel subgroup |$T\subset B \subset G$|. Let |$\Pi \subset \Phi $| be the associated set of simple roots and |$W$| the Weyl group. We recall the definition of the affine Hecke algebra in its Bernstein presentation.
Definition 6.1.The affine Hecke algebra
|$\mathcal{H}^{\operatorname{aff}}$| of
|$G$| is the
|$\overline{\mathbb{Q}}_{\ell }[q,q^{-1}]$|-algebra with generators
|$\{T_{w}, \theta _{x} \mid w \in W, x \in X^{*}\}$| and relations
We collect a few well-known algebraic properties of the affine Hecke algebra which can be found in [
17,
32].
Lemma 6.2.|$\mathcal{H}^{\operatorname{aff}}$| is a free |$\overline{\mathbb{Q}}_{\ell }[q,q^{-1}]$|-module with basis |$\{\theta _{x}T_{w} \mid x\in X^{*}, w \in W\}$|;
The |$\theta _{x}$| (|$x \in X^{*}$|) span a subalgebra of |$\mathcal{H}^{\operatorname{aff}}$| isomorphic to the group algebra |$\overline{\mathbb{Q}}_{\ell }[q,q^{-1}][X^{*}]$|;
The center of the affine Hecke algebra is |$ Z(\mathcal{H}^{\operatorname{aff}}) = \overline{\mathbb{Q}}_{\ell }[q,q^{-1}][X^{*}]^{W}$|;
|$\mathcal{H}^{\operatorname{aff}}$| is a free |$Z(\mathcal{H}^{\operatorname{aff}})$|-module of rank |$|W|^{2}$|.
The center of
|$\mathcal{H}^{\operatorname{aff}}$| also has a geometric interpretation:
Hence, the central characters
|$\chi : Z(\mathcal{H}^{\operatorname{aff}}) \rightarrow \overline{\mathbb{Q}}_{\ell }$| are parametrized by the points of
|$T/W \times \mathbb{G}_{m} $| or equivalently by semisimple conjugacy classes in
|$G\times \mathbb{G}_{m}$|. For
|$(s,q) \in G \times \mathbb{G}_{m}$| semisimple we denote the corresponding central character by
|$\chi _{(s,q)}$|. By (the countable dimension version of) Schur’s lemma [
17, Lemma 2.1.3], any simple
|$\mathcal{H}^{\operatorname{aff}}$|-module admits a central character. Together with Lemma
6.2 this implies that the simple
|$\mathcal{H}^{\operatorname{aff}}$|-modules are finite-dimensional. Moreover, the simple
|$\mathcal{H}^{\operatorname{aff}}$|-modules with central character
|$\chi _{(s,q)}$| are precisely the simple modules of the finite-dimensional algebra
Here
|$(\ker (\chi _{(s,q)})) = \mathcal{H}^{\operatorname{aff}} \cdot \ker (\chi _{(s,q)}) $| is the two-sided ideal in
|$\mathcal{H}^{\operatorname{aff}}$| generated by
|$ \ker (\chi _{(s,q)})$|. The affine Hecke algebra also has a geometric incarnation which we recall next. Let
|$\mathfrak{g}$| (resp.
|$\mathfrak{b}$|) be the Lie algebra of
|$G$| (resp.
|$B$|),
|$\mathfrak{n} = [\mathfrak{b}, \mathfrak{b}]$| and
|$\mathcal{N} \subset \mathfrak{g}$| the nilpotent cone. The Springer resolution is the space
where we identify
|$\mathcal{B}$| with the set of all Borel subgroups
|$B^{\prime}\subset G$| and denote by
|$\mathfrak{b}^{\prime}$| the Lie algebra of
|$B^{\prime}$|. This comes with the two projections
The maps
|$\pi $| and
|$\mu $| are
|$G\times \mathbb{G}_{m}$|-equivariant where
|$t \in \mathbb{G}_{m}$| acts by scaling with
|$t^{-1}$| on
|$\mathcal{N}$| (resp.
|$\mathfrak{n}$|) and trivially on
|$\mathcal{B}$|. For any
|$G\times \mathbb{G}_{m}$|-variety
|$X$|, we denote by
|$ K^{G\times \mathbb{G}_{m}}(X)$| the equivariant algebraic
|$K$|-theory of
|$X$| and we define
Let
|$Z = \tilde{\mathcal{N}} \times _{\mathcal{N}} \tilde{\mathcal{N}}$| be the Steinberg variety. By [
17, (5.2.21)] there is a convolution operation
which equips
|$K^{G \times \mathbb{G}_{m}} (Z)_{\overline{\mathbb{Q}}_{\ell }}$| with the structure of a
|$\overline{\mathbb{Q}}_{\ell }[q,q^{-1}]$|-algebra (here multiplication by
|$q$| corresponds to tensoring with the irreducible weight
|$1$| representation of
|$\mathbb{G}_{m}$|).
Theorem 6.3.[
17,
28] There is an isomorphism of
|$\overline{\mathbb{Q}}_{\ell }[q,q^{-1}]$|-algebras
Under this isomorphism, the center
|$Z(\mathcal{H}^{\operatorname{aff}})$| corresponds to
|$K^{G \times \mathbb{G}_{m}}(\operatorname{pt})_{\overline{\mathbb{Q}}_{\ell }}$|.
The truncated Hecke algebra
|$\mathcal{H}^{\operatorname{aff}}_{(s,q)}$| also has a geometric description which we discuss next. Passing to
|$(s,q)$|-fixed points in (
25), we obtain a diagram
Let
be the decomposition into connected components. The variety
|$\tilde{\mathcal{N}}^{(s,q)}$| is smooth (c.f. [
17, Lemma 5.11.1]) and hence its connected components
|$\tilde{\mathcal{N}}^{(s,q),i}$| are also smooth. Thus, we can consider the constant perverse sheaves
and the corresponding
|$(s,q)$|-Springer sheaves
Theorem 6.4.[17] There is an isomorphism of |$\overline{\mathbb{Q}}_{\ell }$|-algebras |$\mathcal{H}^{\operatorname{aff}}_{(s,q)} \cong \operatorname{Hom}_{D^{b}_{c}(\mathcal{N}^{(s,q)}, \overline{\mathbb{Q}}_{\ell })}^{*}(\operatorname{\textbf{S}}^{(s,q)}, \operatorname{\textbf{S}}^{(s,q)})$|.
Proof.This is proved in [
17, Proposition 8.1.5, Lemma 8.6.1]. The only difference to our situation is that we work with the constructible derived category of
|$\overline{\mathbb{Q}}_{\ell }$|-sheaves
|$D^{b}_{c}(X, \overline{\mathbb{Q}}_{\ell })$| coming from the (pro-)étale topology whereas
loc.cit. works with the constructible derived category
|$D^{b}_{c}(X(\mathbb{C}), \mathbb{C})$| coming from the associated complex analytic space
|$X(\mathbb{C})$|. It turns out that the two approaches are equivalent: By [
6, p.146] there is a fully faithful functor
(which involves choosing an isomorphism
|$\mathbb{C} \cong \overline{\mathbb{Q}}_{\ell }$|). For
|$X = \mathcal{N}^{(s,q)}$| this identifies the
|$\overline{\mathbb{Q}}_{\ell }$|-Springer sheaf
|$\operatorname{\textbf{S}}^{(s,q)}$| with its analytic version and thus we get an isomorphism of graded algebras
However, since we have avoided analytic arguments so far, it is probably more natural to prove the theorem directly in the
|$\overline{\mathbb{Q}}_{\ell }$|-setting. It turns out that this can be done by essentially the same argument as in the analytic setting in [
17]: Let
|$A \subset G \times \mathbb{G}_{m}$| be the closed subgroup generated by
|$(s,q)$|. Then
|$Z^{(s,q)} = Z^{A}$|. Denote by
|$L_{(s,q)}$| the
|$1$|-dimensional
|$Z(\mathcal{H}^{\operatorname{aff}})$|-representation corresponding to
|$\chi _{(s,q)}$|. Then there is a chain of algebra isomorphisms
The first four algebra isomorphisms are exactly as in [
17, (8.1.6)] (note that these are statements about equivariant algebraic
|$K$|-theory of varieties over
|$\mathbb{C} \cong \overline{\mathbb{Q}}_{\ell }$|, which does not involve the analytic topology). The last algebra isomorphism is proved exactly as [
17, Theorem 8.6.7] which only relies on the six functor formalism. The remaining (fifth) isomorphism is given by the composition of the “Riemann-Roch map” for singular varieties and the cycle class map from (
8):
These maps are isomorphisms by [
24, Corollary 18.3.2] and Proposition
2.7 (together with Corollary
6.6). To prove that the isomorphism from (
26) is compatible with convolution, it suffices to show that the push, pull and
|$\otimes $| (resp.
|$\cap $|) constructions that go into the definition of convolution are preserved. This can be found in [
24, Theorem 18.3] and [
31, Théorème 6.1,7.2].
As the notation suggests, the sheaf
|$\operatorname{\textbf{S}}^{(s,q)}$| is a Springer sheaf in the sense of Section
2.2. To check this, we need the following standard results about centralizers (see [
17, Proposition 8.8.7]).
Lemma 6.5.The centralizer |$G(s)$| is connected and reductive;
Each connected component of the fixed-point variety |$\mathcal{B}^{s}$| is |$G(s)$|-equivariantly isomorphic to the flag variety of |$G(s)$|.
We pick the Borel subgroup
|$B \subset G$| such that
|$s \in B$|, that is,
|$B \in \mathcal{B}^{s}$|. Note that
|$B(s) = G(s) \cap B$| is the stabilizer of
|$B \in \mathcal{B}^{s}$| in
|$G(s)$|. By Lemma
6.5 this implies that
|$B(s) \subset G(s)$| is a Borel subgroup. Let
be the decomposition of
|$\mathcal{B}^{s}$| into connected components. Then for each
|$j \in J$| there is a unique element
|$B_{j} \in C_{j} \cong G(s)/B(s)$| whose stabilizer in
|$G(s)$| is
|$B(s)$| (i.e.,
|$B_{j}(s) = B(s)$|). Let
|$\mathfrak{b}_{j}$| be the Lie algebra of
|$B_{j}$| and
|$\mathfrak{n}_{j} = [\mathfrak{b}_{j}, \mathfrak{b}_{j}]$|. The fiber of
|$B_{j}$| under
|$\pi ^{(s,q)}$| is given by
Hence, we get an isomorphism (c.f. [
17, Corollary 8.8.9])
Note that the variety
|$G(s) \times ^{B(s)} \mathfrak{n}_{j}^{(s,q)}$| is connected, so the
|$(\pi ^{(s,q)})^{-1}( C_{j} )$| are already the connected components
|$\tilde{\mathcal{N}}^{(s,q), i}$| of
|$\mathcal{N}^{(s,q)}$|. We have thus shown that the
|$(s,q)$|-Springer resolution is of Springer type (c.f. Definition
2.4).
Corollary 6.6.For each |$i \in I$|, the morphism |$\mu ^{(s,q)}: \tilde{\mathcal{N}}^{(s,q),i} \rightarrow \mathfrak{g}^{(s,q)}$| is of Springer type with Springer sheaf |$\operatorname{\textbf{S}}^{(s,q),i}$|.
Hence, we can consider the corresponding Springer category
where the
|$X_{i}$| are the simple perverse constituents of
|$\operatorname{\textbf{S}}^{(s,q)}$|.
Remark 6.7.It might be more natural to consider the Springer category
|$D_{\operatorname{Spr}}(\mathcal{N}^{(s,q)},\overline{\mathbb{Q}}_{\ell }) $| as a subcategory of
|$D^{b}_{c}(\mathcal{N}^{(s,q)}, \overline{\mathbb{Q}}_{\ell })$| instead of
|$D^{b}_{c}(\mathfrak{g}^{(s,q)}, \overline{\mathbb{Q}}_{\ell })$|. Note that this is not much of a difference since the canonical functor
|$D^{b}_{c}(\mathcal{N}^{(s,q)}, \overline{\mathbb{Q}}_{\ell }) \rightarrow D^{b}_{c}(\mathfrak{g}^{(s,q)}, \overline{\mathbb{Q}}_{\ell })$| induced by the closed immersion
|$\mathcal{N}^{(s,q)} \hookrightarrow \mathfrak{g}^{(s,q)}$| is fully faithful. In particular, we also have
By Corollary
5.6 and Corollary
6.6 there is an equivalence of triangulated categories
which sends
|$\textbf{S}^{(s,q)}$| to the free dg-module
|$\operatorname{Hom}^{*}(\textbf{S}^{(s,q)},\textbf{S}^{(s,q)})$|. Combining this with Theorem
6.4, we arrive at the following “derived Deligne-Langlands correspondence”.
Theorem 6.8.There is an equivalence of triangulated categories
which sends
|$\operatorname{\textbf{S}}^{(s,q)}$| to the free dg-module
|$\mathcal{H}^{\operatorname{aff}}_{(s,q)}$|. Here we consider
|$\mathcal{H}^{\operatorname{aff}}_{(s,q)}$| as a dg-algebra with vanishing differential and grading induced by the
|$\operatorname{Hom}^{*}$|-grading in Theorem
6.4.
Funding
This work was supported by the Lincoln–Kingsgate Graduate Scholarship.
Acknowledgments
I am very grateful to my supervisor Kevin McGerty for his guidance and support and I would like to thank him for many enlightening discussions and conversations. I would also like to thank Dan Ciubotaru for many useful conversations and Ruben La and Emile Okada for many useful discussions on affine Hecke algebras and perverse sheaves.
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