Formality in the Deligne-Langlands correspondence

The Deligne-Langlands correspondence parametrizes irreducible representations of the affine Hecke algebra $\mathcal{H}^{\text{aff}}$ 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 $\mathcal{H}^{\text{aff}}$ an equivalence of triangulated categories between a perfect derived category of dg-modules $D_{\text{perf}}(\mathcal{H}^{\text{aff}}/(\text{ker}(\chi)) - \text{dgMod})$ 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 G there is a (specialized) affine Hecke algebra H aff q such that the category of H aff q -modules is equivalent to the category Rep I (G) of smooth G-representations that are generated by their Iwahori-fixed vectors [Bor76].The category Rep I (G) is precisely the principal Bernstein block of G (i.e. the block in the category of smooth Grepresentations that contains the trivial representation) [BD84,Cas80,BK98].Let G be the complex reductive group whose root datum is dual to that of G and assume that G has simply connected derived subgroup.In [KL87] Kazhdan and Lusztig proved the Deligne-Langlands correspondence which parametrizes the irreducible representations of the affine Hecke algebra H aff by geometric data on the group G.We now recall this correspondence in the language of [CG97]: Any irreducible representation of H aff admits a central character.These central characters are parametrized by semisimple conjugacy classes in G× G m .For each semisimple pair (s, q) ∈ G× G m , we denote the corresponding central character by χ (s,q) : Z(H aff ) → C. Then the irreducible representations of H aff with central character χ (s,q) are precisely the irreducible representations of the truncated affine Hecke algebra (1) H aff (s,q) := H aff /(ker(χ (s,q) )).This algebra has a geometric incarnation in the world of constructible sheaves.Let N be the nilpotent cone of G and µ : Ñ → N the Springer resolution.These varieties come with a canonical G × G m -action where G m acts by scaling.Passing to (s, q)-fixed points we obtain a morphism µ (s,q) : Ñ (s,q) → N (s,q) .The corresponding (s, q)-Springer sheaf is defined as (2) S (s,q) := (µ (s,q) ) * C Ñ (s,q) ∈ D b c (N (s,q) ) E-mail address: jonas.antor@maths.ox.ac.uk.Date: March 17, 2023.
Note that this isomorphism induces a grading on the algebra H aff (s,q) which was not visible in its algebraic definition in (1).By the decomposition theorem [BBD82], we can write for certain G(s)-equivariant simple perverse sheaves X j ∈ Perv G(s) (N (s,q) ) and integers k j ∈ Z. From this, one can deduce that there is a bijection Irr(H aff (s,q) ) 1:1 ←→ {X 1 , ..., X n } or equivalently an embedding Irr(H aff (s,q) ) ֒→ Perv G(s) (N (s,q) ).
The simple objects in Perv G(s) (N (s,q) ) are parametrized by geometric data on N (s,q) , namely by the set of irreducible equivariant local systems on the G(s)-orbits of N (s,q) .Hence, we can interpret N (s,q) as a variety of Langlands parameters associated to the pair (s, q).If q ∈ G m is not a root of unity, the local systems that correspond to elements of Irr(H aff (s,q) ) can be characterized more explicitly as those that appear in the cohomology of a certain Springer fiber (c.f.[CG97, 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 χ (s,q) .We denote by D Spr (N (s,q) ) := X 1 , ..., X n ∆ the full triangulated subcategory of D b c (N (s,q) ) generated by the simple constituents of the (s, q)-Springer sheaf S (s,q) .Theorem.(Theorem 6.8)There is an equivalence of triangulated categories D perf (H aff (s,q) − dgMod) ∼ = D Spr (N (s,q) ) op which identifies S (s,q) with the free dg-module H aff (s,q) .Here we consider H aff (s,q) as a dg-algebra with vanishing differential and grading induced by the 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 F (not necessarily of characteristic 0), B ⊂ G a Borel subgroup, V a G-representation and {V i ⊂ V | i ∈ I} a finite collection of B-stable subspaces.Then for each i ∈ I we consider the 'morphism of Springer type' the dg-level, formality will be a consequence of an algebraic result in [PVdB19,Theorem B.1.1].While it would certainly be possible to directly apply the results from [PVdB19] 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 [Rid13,PVdB19,RR21,ES22] or in the context of flag varieties and Koszul duality [BGS96,Sch11] where formality has also been studied for modular coefficients [RSW14,AR16].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 [Lus95a].In terms of geometry, graded Hecke algebras arise as certain Ext-algebras in an equivariant derived category of constructible sheaves [Lus95b,AMS18].Some formality results in this direction can be found in [Sol22].There also is a coherent categorical Deligne-Langlands correspondence [BZCHN20] which works without fixing a central character but replaces constructible sheaves with a certain category of coherent sheaves (see also the conjectures in [Hel20]).The relation between the constructible and the coherent side is discussed in [BZCHN23].
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.

Springer geometry
Let F be an algebraically closed field and ℓ a prime number which is invertible in F. For any variety X over F, we can consider the constructible derived category of BBD82,BS15].The triangulated category D b c (X, Q ℓ ) comes with the usual six (derived) operations denoted by f * , f * , f !, f !, ⊗ L and RHom.Moreover, there is a standard t-structure on D b c (X, Q ℓ ) with cohomology functor H * and heart Sh c (X, Q ℓ ).The structure map of X will be denoted by a : X → {pt}.Let 1 X ∈ Sh c (X, Q ℓ ) be the constant sheaf and ω X := a ! 1 pt ∈ D b c (X, Q ℓ ) the dualizing complex.We denote by Perv(X) ⊂ D b c (X, Q ℓ ) 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 [Lau76].The i-th Borel-Moore homology is the Q ℓ -vector space which is natural in F .For F = ω X this induces a long exact sequence on Borel-Moore homology Let p : X → 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), i.e. if Y ⊂ X is a closed subvariety with open complement U and Ỹ := p −1 (Y ), Ũ := p −1 (U ), the following diagram commutes: (7) Lemma 2.1.Let p : X → X be a Zariski locally trivial fibration with affine fiber A d .Then the smooth pullback map 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 : If X is smooth and connected, the dualizing complex is given by The fundamental class of X is the distinguished element More generally, if X is irreducible one can define the fundamental class as follows: Pick a smooth open subset U ⊂ X.Then the long exact sequence (6) induces an isomorphism If elements of this form span the vector space H i (X, Q ℓ ) for each i ∈ Z, we say that H * (X, Q ℓ ) is spanned by fundamental classes.Note that being spanned by fundamental classes implies that H i (X, Q ℓ ) = 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)/ ∼ Rat be the Chow group (c.f.[Ful98]).The fundamental class construction gives rise to a cycle class map which descends to the Chow group by [Lau76,Théorème 6.3].The open-closed exact sequence and smooth pullback map from Borel-Moore homology have analogues for Chow groups: For any Y ⊂ X closed with complement U = X\Y there is an exact sequence and for p : X → 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 [Lau76, Théoreme 6.1].We define Following [DCLP88, 1.7], we say that a variety X has property (S) if By the five lemma, this implies that the map Lemma 2.3.Let p : X → X be a Zariski locally trivial fibration with fiber A d .If X has property (S) then X also has property (S).
Proof.By Lemma 2.1, the pullback map It is also surjective by [Ful98, Proposition 1.9] an thus an isomorphism.This implies that A i+d ( X) 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 F. We fix a maximal torus and a Borel subgroup T ⊂ B ⊂ G with Weyl group W = N G (T )/T .The corresponding flag variety will be denoted by B = G/B.Given a G-representation V and a finite collection {V i ⊂ V | i ∈ I} of B-stable subspaces, we define for each i ∈ I the G-variety This comes with two G-equivariant morphisms where Definition 2.4.We call µ i a morphism of Springer type.For i, j ∈ I, we consider the Steinberg variety This comes with the projection map π i × π j : Z ij → B × B. Consider the orbit partition where ẇ ∈ N G (T ) is a lift of w ∈ W .This induces a partition of Z ij into locally closed subvarieties Lemma 2.6.
(1) The morphism µ i : Ṽ i → V is proper; (2) The morphism π i : Ṽ i → B is a Zariski vector bundle with fiber V i ; (3) For each w ∈ W the morphism π i × π j : Z ij w → Y w is a Zariski vector bundle with fiber V i ∩ ẇV j ; (4) The first projection p 1 : Y w → B is a Zariski locally trivial fibration with fiber A l(w) .Proof.The map µ i can be factored into a closed immersion followed by a projection: Since B is projective, this implies that µ i is proper.The local triviality in (2)-(4) follows from standard results about quotients (c.f.[Jan87, §I.5.16]).The respective fibers are easily computed.
Proof.By Lemma 2.6 the maps Z ij w → Y w → B are locally trivial fibrations with affine fibers.It is well known that B has property (S) (in fact, this follows from Lemma 2.2 and the decomposition of B into Bruhat cells).By Lemma 2.3 this implies that Z ij w has property (S) for all w ∈ W . Pick a total order ≤ on W extending the Bruhat order and define Note that Z ij <w = Z ij ≤w ′ where w ′ ∈ W is the maximal element with w ′ < w.We show by induction along the total order on W that Z ij ≤w has property (S).We have already proved the claim for Z ij ≤e = Z ij e .Now assume we have shown the claim for each y < w.Note that The varieties Z ij <w and Z ij w have property (S), so Z ij ≤w also has property (S) by Lemma 2.2.This completes the induction.Note that Z ij = Z ij ≤w0 where w 0 ∈ W is the longest element.Hence, Z ij has property (S).
Since Ṽ i is smooth, there is the constant perverse sheaf We define the Springer sheaves The morphism µ i is proper by Lemma 2.6.Hence, the decomposition theorem [BBD82] implies that S i (and thus also S) is a semisimple complex.In other words, we have for some simple perverse sheaves X j ∈ Perv(V ) and integers k j ∈ Z.We define the Springer category to be the smallest full triangulated subcategory of D b c (V, Q ℓ ) 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.

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 F q with structure map a : X 0 → Spec(F q ) and let Denote by The category Sh c (Spec(F q ), Q ℓ ) is equivalent to the category of finite-dimensional continuous Gal(F q /F q )-representations.Hence, we can consider Hom i (F 0 , G 0 ) as a Q ℓ -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 Hom i (F , G).Note that the sheaves . This induces a canonical Frobenius action on H * (X, Q ℓ ) 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 ֒→ X 0 is a closed immersion with open complement j : U 0 ֒→ X 0 , there is a distinguished triangle inducing a long exact sequence where all maps commute with the Frobenius action.For any n ∈ Z, let be the n-th Tate twist.This corresponds to the 1-dimensional Q ℓ -vector space on which the (geometric) Frobenius element acts by multiplication with q −n .Moreover, we define If X 0 is smooth and connected, the dualizing complex on X 0 is given by Proof.Let U 0 ⊂ X 0 be a smooth open subset.Then we get a canonical element which is Frobenius invariant and corresponds to the fundamental class [U ] ∈ H 2 dim U (U, Q ℓ ) after forgetting the Frobenius action.Note that the Frobenius action on H 2 dim U (U, Q ℓ ) comes from the Frobenius action on Hom −2 dim U (1 U0 , ω U0 ).Thus, taking into account the Tate twist, we get that Frobenius acts on [U ] by multiplication with q

Hence, Frobenius also acts on [X] by multiplication with q
For F q large enough, we may assume that each of the Y i can be defined over F q .Then Lemma 3.1 implies that Frobenius acts on , this proves the claim.

3.2.
Frobenius and the Springer sheaf.Let µ i : Ṽ i → V (i ∈ I) be a finite collection of morphisms of Springer type (Definition 2.4) defined over F q .Assume that F q is large enough so that there are F q -forms µ i : Ṽ i 0 → V 0 for each of the µ i (i ∈ I).Then Z ij can also be defined over Lemma 3.3.For F q large enough, Frobenius acts on Proof.This follows from Proposition 2.7 and Corollary 3.2.
We fix a square root q 1 2 of q in Q l .This corresponds to fixing a square root 1 Spec(Fq) ( 1 2 ) of the Tate sheaf 1 Spec(Fq) (1).Then we can form the half integer Tate twists F 0 ( n 2 ) for any where d i = dim Ṽ i and let be the corresponding F q -Springer sheaf.We will need the following Frobenius equivariant version of [CG97, Lemma 8.6.1].
Proof.Note that µ i is proper over F q since it is proper over F q (properness can be checked fpqc locally).Hence, we have Using base change with respect to the cartesian diagram ).
Applying H k a * to this, we obtain ).
Corollary 3.5.For F q large enough, Frobenius acts on Hom k (S i 0 , S j 0 ) by multiplication with q k 2 .
Proof.By Lemma 3.3 Frobenius acts on ) by multiplication with The claim now follows from Lemma 3.4.

Formality
4.1.Idempotent complete triangulated categories.Recall that an additive category A is called idempotent complete if any idempotent e : X → X in T splits.This is equivalent to the property that all idempotents have kernels (or cokernels).In an idempotent complete category, every idempotent e : X → X gives rise to a canonical decomposition If A ′ ⊂ A is an additive subcategory that is closed under direct summands and A is idempotent complete, then A ′ is also idempotent complete.Moreover, for any additive category A one can define its idempotent completion Ã.This is an idempotent complete additive category with a fully faithful additive functor ι : A → Ã and the following universal property: Any additive functor A → C with C idempotent complete factors uniquely (up to isomorphism) through ι.For more details about idempotent completeness we refer to [Kar78, 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.[Nee01, Lemma 1.6.8]and [BS01, Lemma 2.4]).
Lemma 4.1.Any triangulated category with countable coproducts is idempotent complete.Moreover, the bounded below derived category D + (A) of an abelian category A is always idempotent complete.
Let T be a triangulated category.For any object X ∈ T , we denote by X T ,∆ = X ∆ the smallest full triangulated subcategory of T that contains X and is closed under isomorphisms.Similarly, we denote by X T ,∆, = X ∆, the smallest full triangulated subcategory of T that contains X and is closed under isomorphisms and direct summands.
Lemma 4.2.Let T be an idempotent complete triangulated category and X ∈ T .Then X ∆, is the idempotent completion of X ∆ .
Proof.By [Kar78, Theorem I.6.12] the idempotent completion of X ∆ can be described as the full additive subcategory C ⊂ T consisting of those objects Y ∈ T which are isomorphic to a direct summand of an object in X ∆ .In particular, we have C ⊂ X ∆, .The cone of a morphism in C is a direct summand of the cone of a morphism in X ∆ .Hence, the category C is closed under cones and thus C = X ∆, .
We will also need the following idempotent complete version of Beilinson's Lemma (see [Bei87] and [Sch11, Lemma 6]) .
Lemma 4.3.Let F : T → T ′ be a triangulated functor between idempotent complete triangulated categories.Let X ∈ T such that F induces an isomorphism for all i ∈ Z. Then F restricts to an equivalence of triangulated categories Proof.By a standard dévissage argument, F induces an equivalence X ∆ ∼ = F (X) ∆ .This extends to an equivalence of the respective idempotent completions and thus to an equivalence X ∆, ∼ = F (X) ∆, 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 − dgMod the category of (left) dg-modules over R. The homotopy category of dg-modules will be denoted by K(R − dgMod) and the derived category by D(R − dgMod).We write for the perfect derived category.For more details about dg-algebras and dg-modules we refer to [Kel94,BL94].
Let A be a k-linear abelian category.Then for any two chain complexes X • , Y • ∈ C(A), we can consider the Hom-complex Hom • dg (X • , Y • ) ∈ C(k).Explicitly, this complex is defined as Taking cohomology of this complex recovers the morphism space in the homotopy category: ) which descends to a triangulated functor on the respective homotopy categories 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 Hom Moreover, this functor induces an isomorphism for all i ∈ Z.Here we have used for the first (resp.last) isomorphism that I • X (resp R X ) is a K-injective complex (resp.K-projective dg-module).Note that the categories D + (A) op and D(R X − dgMod) are idempotent complete by Lemma 4.1 (using that D(R X − 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 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 dgalgebras A → B induces an equivalence D perf (B) ∼ → D 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 (16) The following theorem provides a useful criterion for formality.
Theorem 4.5.[PVdB19] Let R be a dg-algebra over an algebraically closed field k and q ∈ k × not a root of unity.If R can be equipped with a dg-algebra automorphism F : R → 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 = C in [PVdB19, Theorem B.1.1](with a slightly different condition on the action of H i (F )).The same proof works for general k: By [PVdB19, Theorem B.1.2],we may assume that F acts locally finitely on R. Hence, there is a generalized eigenspace decomposition Using this decomposition, we can define a dgg-algebra (i.e. a Z 2 -graded algebra with a differential d homogeneous of degree (1, 0) satisfying the Leibniz rule) via R := i,j∈Z

Ri
q j .Note that the inclusion R ֒→ R is a quasi-isomorphism.Moreover, the cohomology of the dgg-algebra R lives in degrees {(i, i) | i ∈ Z}.Such dgg-algebras are known to be formal (c.f.[Sch11, 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 [BS15].Definition 4.6.A morphism of schemes f : U → X is called weakly étale if both f and its diagonal ∆ f : U → U × X U are flat morphisms.Let X proét be the category of schemes U weakly étale over X.This becomes a site by declaring a family of maps {U i → 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 X proét is weakly étale.We denote the category of sheaves of abelian groups on X proét by Ab(X proét ).Let f : X → Y be a morphism of schemes.Then f induces a morphism of sites X proét → Y proét via the functor There is a corresponding pair of adjoint functors Ab(Y proét ) Ab(X proét ).
Let Λ be a topological ring.By [BS15, Lemma 4.2.12]there is a sheaf of rings Λ X on X proét defined by Λ X (U ) := Hom cont (U, Λ).
If Λ is totally disconnected and U is quasi-compact, we get where π 0 (U ) is the space of connected components.If Λ is discrete, Λ X is the constant sheaf with values in Λ.For any morphism of schemes f : X → Y , there is a canonical map f # : Λ Y → f * Λ X and hence a morphism of ringed sites Thus we get corresponding pairs of adjoint functors Here we adhere to the derived convention, i.e. the functors f * and f * are understood to be derived when applied to complexes.From now on, we fix a prime number ℓ and make the following assumption on the coefficient ring Λ.
Assumption 4.7.The ring Λ is of one of the following forms: (1) Λ = F is a finite field of characteristic ℓ; (2) Λ = O E is the ring of integers in an algebraic extension E/Q ℓ ; (3) Λ = E is an algebraic extension E/Q ℓ .
We equip Λ with the ℓ-adic topology.
Note that the assumption above includes the ring Λ = Q ℓ .For any algebraic extension E/Q ℓ , we denote by O E the ring of integers in E. If E/Q ℓ is finite, we pick a uniformizer ω ∈ O E .
Example 4.8.[BS15, Lemma 6.8.2] gives an alternative description of Λ X under Assumption 4.7: If Λ is a finite ring (e.g.Λ = F a finite field) then Λ X is just the constant sheaf on X proét with values in Λ.In the remaining cases, Λ X can be constructed as follows: Lemma 4.9.For any morphism of schemes X → Y , the canonical map Proof.This result can be found in [Cho20, Proposition 4.6].The proof uses the explicit description of Λ X in Example 4.8.Since pullback commutes with colimits and localization, the main step is to show that f # is an isomorphism when and thus f * Λ Y = Λ X .
Recall that a scheme X is called ℓ-coprime if ℓ is invertible in O X .Following [BS15], the constructible derived category can be defined as follows.
(1) A sheaf F ∈ Sh(X proét , Λ X ) is called locally constant of finite presentation if it is locally isomorphic to M ⊗ Λ Λ X for a finitely-presented Λ-module M .(2) A sheaf F ∈ Sh(X proét , Λ X ) is called constructible if there exists a finite stratification {X i → X} such that F | Xi is locally constant of finite presentation.(3) A complex F ∈ D(X proét , Λ X ) is called constructible if it is bounded and all its cohomology sheaves are constructible.We denote by Sh c (X, Λ) ⊂ Sh(X proét , Λ X ) the category of all constructible sheaves.The constructible derived category is the full subcategory D b c (X, Λ) ⊂ D(X proét , Λ 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 ⊗ L , RHom and f * preserve constructible complexes and thus they descend to the constructible derived category.The same is true for f * if f : X → 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 ֒→ X is an open immersion, the pullback functor j * has a left adjoint j ! which preserves constructible complexes.For a general f : X → Y (of finite type with Y quasi-excellent), we define f !:= f * • j !where we factor f as an open immersion followed by a proper map (18) 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 !, ⊗ L and RHom.
Remark 4.12.In [BS15] a slightly different pullback functor f * comp is used which is defined as f * followed by a certain completion operation ([BS15, 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 ([BS15, Proposition 6.8.11(3), Definition 6.5.1]), it follows that If F is a finite field of characteristic ℓ, we denote by D b c (X ét , F) the constructible derived category in the étale topology with coefficients in F. The standard compatibility and base change results for the six functors on D b c (X, Λ) can be deduced from the corresponding results in D b c (X ét , F) using the following reduction steps.Lemma 4.13.[BS15] Let X be noetherian and ℓ-coprime.
(1) Let F be a finite field of characteristic ℓ.Then the canonical morphism of sites ν : X proét → X ét induces an equivalence of categories (2) Let E/Q ℓ be a finite extension and κ the residue field of O E .Then the functor ) is well-defined (i.e.preserves constructibility) and conservative (i.e.reflects isomorphisms); (3) Let E/Q ℓ be algebraic.Then the canonical functor is an equivalence of categories; (4) Let E/Q ℓ be algebraic.Then the canonical functor Moreover, the functors in (1)-(4) are compatible with the six functors f * , f * , f !, f !, ⊗ L and RHom.
Proof.This follows from various results in [BS15]: (1) follows from [BS15, Corollary 5.1.5,5.1.6].The well-definedness in (2) follows from [BS15, Proposition 6.8.11(3), Definition 6.5.1].To show that the functor κ X ⊗ L OE,X − is conservative, it suffices to prove that it reflects zero objects.Let By an induction argument, this implies that (O E /m n ) X ⊗ L OE,X F = 0 for all n ≥ 1.Moreover, F is m-adically complete by [BS15, Proposition 6.8.11(3), Definition 6.5.1] which means that OE,X F ) = 0.This completes the proof of (2).The claims in (3) and ( 4) are [BS15, Proposition 6.8.14].It remains to prove compatibility with the six functors in (1)-(4).Note for this that the category colim F D b c (X, O F ) in (3) inherits the six functors from the D b c (X, O F ) (a standard argument shows that the transition maps in the colimit are compatible with the six functors using that as sheaves of O F,X -modules for any finite extension F ′ /F ).Similarly, the category 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 proét , κ X ) → (X proét , O E,X )).As such, they commute with f * and ⊗ L by standard results about ringed sites [Sta23, Tag 0D6D, Tag 07A4].Since the functors in (1),( 3) and ( 4) are equivalences, they also commute with the corresponding (right) adjoints f * and RHom.Similarly, they commute with j ! for j an open immersion which is left adjoint to j * .Hence, they also commute with f != f * • j !(see ( 18)) and thus also with its right adjoint f ! .It remains to show that the functor in (2) commutes with f * , RHom, f ! and f ! .This follows from [BS15, 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 → U has a section.Lemma 4.15.Let U be a w-contractible affine scheme.
Proof.The exactness in (1) is mentioned in the introduction of [BS15] (see also [Sta23, Tag 098H, Tag 0946]).Let F ∈ Sh c (U, Λ) be locally constant of finite presentations.Then we can find a weakly-étale cover Since U is weakly contractible, there is a section s : U → V and thus This proves (2).(3) follows from [BS15, Theorem 1.8].
Lemma 4.16.The triangulated category D b c (X, Λ) ⊂ D(X proét , Λ X ) is closed under direct summands.In particular, D b c (X, Λ) is idempotent complete.Proof.Since taking cohomology commutes with direct sums, it suffices to prove that Sh c (X, Λ) ⊂ Sh(X proét , Λ X ) is closed under direct summands.In fact, by [BS15, Lemma 6.8.7, Proposition 6.8.11] the category Sh c (X, Λ) is abelian so it is certainly closed under direct summands.
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 proét , Λ X ) is the localization of the ringed site (Y proét , Λ Y ) at X → Y .By general results on ringed sites [Sta23, Tag 04IX] this implies that f * has an exact left adjoint f ! .In particular, f * preserves injectives.
Let F ∈ D b c (X, Λ) and pick a complex of injectives ) be the dg-algebra from (14).Thanks to the pro-étale formalism, we obtain the following algebraic description of the category F ∆, .
Corollary 4.18.There is an equivalence of triangulated categories 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 Hom from (10) is compatible with the dg-techniques from Section 4.2.These results are similar to the ones in [PVdB19, Appendix A] but some of the arguments simplify because the canonical morphism Spec(F q ) → Spec(F q ) is a weakly-étale (but not étale).
The geometric q-Frobenius F ∈ Gal(F q /F q ) defines a morphism Spec(F q ) → Spec(F q ) in the category Spec(F q ) proét .Thus, for any F 0 ∈ Sh(Spec(F q ) proét , Λ Spec(Fq) ) there is a corresponding restriction map on sections This map is an isomorphism with inverse given by restriction along the arithmetic Frobenius F −1 ∈ Gal(F q /F q ).Hence, we can equip Γ(Spec(F q ), F 0 ) with the structure of a Λ[F, F −1 ]-module.Moreover, for any morphism of sheaves F 0 → 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 Sh c (Spec(F q ), Q ℓ ) and the category of finite-dimensional continuous Gal(F q /F q )representations over Q ℓ .On Sh c (Spec(F q ), Q ℓ ) the functor Γ F simply corresponds to restricting the Gal(F q /F q )-action to the (geometric) Frobenius element F ∈ Gal(F q /F q ).Note that where Lemma 4.21.The functors For F , Γ F and Γ(Spec(F q ), −) are exact.
Proof.The forgetful functor For F is exact and reflects exact sequences.Hence, by (19) it suffices to prove that Γ(Spec(F q ), −) is exact.This follows from Lemma 4.15.
We define a functor Hence, Hom(F 0 , G 0 ) is just the vector space Hom(F , G) together with a canonical Frobenius action.

The composition map
Hom(G 0 , K 0 ) ⊗ Hom(F 0 , G 0 ) → Hom(F 0 , K 0 ) gives rise to a map and thus to a map of Λ[F, Similarly, the unit morphism by adjunction and thus a morphism of Λ[F, Forgetting the Frobenius action in (20) and (21) recovers the standard composition map Hom(G, K) ⊗ Λ Hom(F , G) → Hom(F , K) and unit map Λ → Hom(F , F ).
Similarly, one can define pre-and post-composition maps when forgetting the Frobenius action.
We also define a functor on derived categories (recall that we adhere to the derived convention, i.e.Γ F and a * are understood to be derived when applied to complexes).Moreover, we set Remark 4.22.This definition of Hom i extends our previous definition (10) for constructible complexes to arbitrary complexes in D(X 0,proét , Λ X0 ) (see also Remark 4.20).
For any two chain complexes F • 0 , G • 0 ∈ C(Sh(X 0,proét , Λ X0 )) we define a chain complex Hom Explicitly, this complex is given by where we use ( 22)).The composition map on Hom(−, −) from (20) can be applied component-wise to define a composition map with the structure of a dg-algebra over Λ together with a dg-algebra automorphism induced by the Frobenius action.Forgetting this Frobenius action recovers the dg-algebra Hom Lemma 4.23.Let F 0 , G 0 ∈ D + (X 0,proét , Λ X0 ) and pick bounded below complexes of injectives Proof.By definition, RHom(F 0 , G 0 ) is represented by the complex Hom • dg (I • F0 , I • G0 ) defined as . Note that Hom(I −l F0 , I m G0 ) is acyclic for a * for all l, m ∈ Z (see [AGV72, V-(4.10),V-(5.2)]).Hence, ).This is precisely the complex Hom • dg (I • F0 , I • G0 ).Hence, The constructions of this section can be summarized as follows.
Proposition 4.24.For any F 0 ∈ D b c (X 0 , Λ), there is a bounded below complex of injectives ) is a complex of injectives representing F by Lemma 4.19.The dg-algebra Hom ) equipped with a dg-algebra automorphism F By Lemma 4.23, we have is the canonical Frobenius action on Hom i (F 0 , F 0 ).4.5.Formality for the Springer category.Let µ i : Ṽ i → V be a finite collection of morphisms of Springer type defined over F q and let S = i∈I S i ∈ D b c (V, Q ℓ ) 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 S. Lemma 4.25.We have D Spr (V, Q ℓ ) = S ∆, .
Proof.Since X 1 , ..., X n ∈ S ∆, , we have D Spr (V, Q ℓ ) ⊂ S ∆, .Hence, it suffices to show that D Spr (V, Q ℓ ) is closed under direct summands.Let Perv Spr (V ) ⊂ 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 for all i ∈ Z}.Note that as a Serre subcategory, Perv Spr (V ) is closed under direct summands in Perv(V ).Hence, D Spr (V, Q ℓ ) 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 S with the free dg-module Hom * (S, S).
Proof.Combining Corollary 4.18 and Lemma 4.25 we get an equivalence which send S to the free dg-module R S .By Corollary 3.5 and Proposition 4.24 the dg-algebra R S can be equipped with a dg-algebra automorphism F : R S → R S such that H i (F ) acts by multiplication with q i 2 .This implies that R S is formal by Theorem 4.5.In particular, by (16) we get an equivalence which sends S to the free dg-module Hom * (S, S).

De F à 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 F a C" technique from [BBD82, §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 → R is a ring homomorphism, we denote by X R the base change of X A to R. Similarly, if S → Spec(A) is a morphism of schemes, we denote by X S the corresponding base change to S. Furthermore, for Definition 5.1.Let S be a scheme and f : X → Y a morphism of S-schemes.Let F , G ∈ D(X ét , Λ) for a noetherian ring Λ (resp.F , G ∈ D(X proét , Λ X ) for Λ as in Assumption 4.7).
(1) We say that the formation f * F commutes with generic base change if there is a dense open subscheme U ⊂ S such that for each morphism of schemes g : S ′ → U ⊂ S with corresponding pullback diagram ) We say that the formation RHom(F , G) commutes with generic base change if there is a dense open subscheme U ⊂ X such that for each morphism of schemes g : There is the following generic base change theorem for étale sheaves.where By proper base change we have u * (S i A ) = S i R and u * (S A ) = S R .Moreover, if R = k is an algebraically closed field, the morphism µ i : Ṽ i k → V k is of Springer type and S i k is the associated Springer sheaf (c.f.Section 2.2).Here we use that Corollary 5.6.Let µ i : Ṽ i → V be a finite collection of morphisms of Springer type over Q ℓ .Then there is an equivalence of triangulated categories which identifies S with the free dg-module Hom * (S, S).
Proof.Let A, Ṽ i A , V A , ... be as in Lemma 5.5 and pick a closed point s → Spec(A) with algebraic closure s.Note that since A is finitely-generated over Z[ℓ −1 ], we have s = Spec(F q ) for a finite field F q with ℓ invertible in F q .By Lemma 5.4 there is an equivalence of triangulated categories D Spr (V, Q ℓ ) ∼ = D Spr (V s , , Q ℓ ) which identifies S with S s .In particular, we get Hom * (S, S) ∼ = Hom * (S s , S s ).Hence, by Theorem 4.26, we have

A derived Deligne-Langlands correspondence
Let G be a connected reductive group over Q ℓ ( ∼ = C) with simply connected derived subgroup and let (X * , Φ, X * , Φ ∨ ) be the associated root datum.Fix a torus and a Borel subgroup T ⊂ B ⊂ G. Let Π ⊂ Φ 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 H aff of G is the Q ℓ [q, q −1 ]-algebra with generators {T w , θ x | w ∈ W, x ∈ X * } and relations We collect a few well-known algebraic properties of the affine Hecke algebra which can be found in [Lus89,CG97].Lemma 6.2.
(1) H aff is a free Q ℓ [q, q −1 ]-module with basis {θ x T w | x ∈ X * , w ∈ W }; (2) The θ x (x ∈ X * ) span a subalgebra of H aff isomorphic to the group algebra Q ℓ [q, q −1 ][X * ]; (3) The center of the affine Hecke algebra is Z(H be the decomposition into connected components.The variety Ñ (s,q) is smooth (c.f.[CG97, Lemma 5.11.1]) and hence its connected components Ñ (s,q),i are also smooth.Thus, we can consider the constant perverse sheaves C Ñ (s,q),i := 1 Ñ (s,q),i [dim Ñ (s,q),i ] ∈ D b c ( Ñ (s,q),i , Q ℓ ) and the corresponding (s, q)-Springer sheaves S (s,q),i := (µ (s,q) ) * C Ñ (s,q),i ∈ D b c (N (s,q) , Q ℓ ) S (s,q) := i∈I S (s,q),i .Theorem 6.4.[CG97] There is an isomorphism of Q ℓ -algebras H aff (s,q) ∼ = Hom * D b c (N (s,q) ,Q ℓ ) (S (s,q) , S (s,q) ).Proof.This is proved in [CG97, Proposition 8.1.5,Lemma 8.6.1].The only difference to our situation is that we work with the constructible derived category of Q ℓ -sheaves D b c (X, Q ℓ ) coming from the (pro-)étale topology whereas loc.cit.works with the constructible derived category D b c (X(C), C) coming from the associated complex analytic space X(C).It turns out that the two approaches are equivalent: By [BBD82, p.146] there is a fully faithful functor D b c (X, Q ℓ ) → D b c (X(C), C) (which involves choosing an isomorphism C ∼ = Q ℓ ).For X = N (s,q) this identifies the Q ℓ -Springer sheaf S (s,q) with its analytic version and thus we get an isomorphism of graded algebras Hom * D b c (X,Q ℓ ) (S (s,q) , S (s,q) ) ∼ = Hom * D b c (X(C),C) (S (s,q) , S (s,q) ).However, since we have avoided analytic arguments so far, it is probably more naturally to prove the theorem directly in the Q ℓ -setting.It turns out that this can be done by essentially the same argument as in the analytic setting in [CG97]: Let A ⊂ G×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(H aff )-representation corresponding to χ (s,q) .Then there is a chain of algebra isomorphisms c (N (s,q) ,Q ℓ ) (S (s,q) , S (s,q) ).The first four algebra isomorphisms are exactly as in [CG97,(8.1.6)](note that these are statements about equivariant algebraic K-theory of varieties over C ∼ = Q ℓ , which does not involve the analytic topology).The last algebra isomorphism is proved exactly as [CG97, 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 [Ful98, 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 ⊗ (resp.∩) constructions that go into the definition of convolution are preserved.This can be found in [Ful98,Theorem 18.3] and [Lau76, Théorème 6.1,7.2].
As the notation suggests, the sheaf 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 [CG97, Proposition 8.8.7]).Lemma 6.5.
(i) The centralizer G(s) is connected and reductive; (ii) Each connected component of the fixed-point variety B s is G(s)-equivariantly isomorphic to the flag variety of G(s).
We pick the Borel subgroup B ⊂ G such that s ∈ B, i.e.B ∈ B s .Note that B(s) = G(s) ∩ B is the stabilizer of B ∈ B s in G(s).By Lemma 6.5(ii) this implies that B(s) ⊂ G(s) is a Borel subgroup.Let B s = j∈J C j be the decomposition of B s into connected components.Then for each j ∈ J there is a unique element B j ∈ C j ∼ = G(s)/B(s) whose stabilizer in G(s) is B(s) (i.e.B j (s) = B(s)).Let b j be the Lie algebra of B j and n j = [b j , b j ].The fiber of B j under π (s,q) is given by (π (s,q) ) −1 (B j ) ∼ = N (s,q) ∩ π −1 (B j ) = N (s,q) ∩ b j = n (s,q) j .
Note that the variety G(s) × B(s) n (s,q) j is connected, so the (π (s,q) ) −1 (C j ) are already the connected components Ñ (s,q),i of 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 ∈ I, the morphism µ (s,q) : Ñ (s,q),i → g (s,q) is of Springer type with Springer sheaf S (s,q),i .
Hence, we can consider the corresponding Springer category D Spr (N (s,q) , Q ℓ ) := D Spr (g (s,q) , Q ℓ ) = X 1 , ..., X n D b c (g (s,q) ,Q ℓ ),∆ where the X i are the simple perverse constituents of S (s,q) .Remark 6.7.It might be more natural to consider the Springer category D Spr (N (s,q) , Q ℓ ) as a subcategory of D b c (N (s,q) , Q ℓ ) instead of D b c (g (s,q) , Q ℓ ).Note that this is not much of a difference since the canonical functor D b c (N (s,q) , Q ℓ ) → D b c (g (s,q) , Q ℓ ) induced by the closed immersion N (s,q) ֒→ g (s,q) is fully faithful.In particular, we also have D Spr (N (s,q) , Q ℓ ) = X 1 , ..., X n D b c (N (s,q) ,Q ℓ ),∆ .
Example 2.5.Let V = g be the Lie algebra of G, b the Lie algebra of B and n = [b, b].Then the morphism of Springer type corresponding to the B-stable subspace n ⊂ g is the Springer resolution Ñ = G × B n → N ⊂ g.For b ⊂ g we recover the Grothendieck-Springer alteration G × B b → g.Other important examples of morphisms of Springer type show up in the representation theory of affine Hecke algebras (see [KL87, CG97] and Corollary 6.6).
g. a bounded below complex of injectives), this descends further to a triangulated functor on the corresponding derived categories(13) Hom • dg (−, Y • ) : D(A) op → D(Hom • dg (Y • , Y • ) − dgMod).Assume now that the abelian category A has enough injectives.Then for each X ∈ D + (A) we can pick a complex of injectives I • X ∈ C + (A) representing X.Moreover, we can consider the associated dg-algebra (14) R X := Hom • dg (I • X , I • X ) and the corresponding functor (15) Hom • dg (−, I • X ) : D(A) op → D(R X − dgMod).The following statement is an idempotent complete version of [Sch11, Proposition 7].

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.
Theorem 5.2.[Del77]Let S be a noetherian scheme, f : X → Y a morphism of S-schemes of finite type and Λ a noetherian ring annihilated by an integer invertible in O S .Then for any F , G ∈ D b c (X ét , Λ) the formations f * F and RHom(F , G) commute with generic base change.
is an isomorphism for all i ∈ Z.The groups H i (Y, Q ℓ ) and H i (U, Q ℓ ) vanish for i odd.Using the long exact sequence (6), we deduce that H i (X, Q ℓ ) = 0 for i odd.Moreover, for any i ∈ Z, we get a commutative diagram with exact rows This equips Hom • dg (Y • , Y • ) with the structure of a dg-algebra and Hom • dg (X • , Y • ) with the structure of a (left) dg-module over Hom • dg (Y • , Y • ).