Coherent categorification of quantum loop algebras: The SL(2) case

Peng Shan; Michela Varagnolo; Eric Vasserot

doi:10.1515/crelle-2022-0046

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Coherent categorification of quantum loop algebras: The SL(2) case

Peng Shan , Michela Varagnolo and Eric Vasserot

Published/Copyright: September 29, 2022

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 792

Abstract

We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type A 1 ( 1 ) to the category of equivariant perverse coherent sheaves on the nilpotent cone of type A. We prove that this equivalence is weakly monoidal. This gives a representation-theoretic categorification of the preprojective K-theoretic Hall algebra considered by Schiffmann and Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type A 1 ( 1 ) given by Kang, Kashiwara, Kim, Oh and Park in terms of quiver-Hecke algebras with the one given by Cautis and Williams in terms of equivariant perverse coherent sheaves on the affine Grassmannians.

A Reminders on Artin stacks and mixed geometry

A.1 Schemes and stacks

Let k = ℚ ¯ ℓ and let F be any field of characteristic prime to ℓ . All F-schemes are assumed to be separated of finite type. Given a F-scheme Z with an action of an affine algebraic F-group G, the quotient F-stack [ Z / G ] is the Artin stack whose set of R-points consists of G-torsors on Spec ⁡ R with G-equivariant map to Z. By a stratification S = { Z w : w ∈ W } of Z we will mean a finite algebraic Whitney stratification.

Let 𝒵 be an Artin F-stack of finite type. Let D b ⁡ ( 𝒵 ) be the bounded constructible derived category of étale sheaves of k -modules on 𝒵 , in the sense of [33]. Let P ⁢ ( 𝒵 ) be the subcategory of perverse sheaves. For each complex ℰ ∈ D b ⁡ ( 𝒵 ) and each integer a, let H a p ⁢ ℰ ∈ P ⁢ ( 𝒵 ) be the a-th perverse cohomology complex. Let C ⁢ ( 𝒵 ) ⊂ D b ⁡ ( 𝒵 ) be the additive full subcategory of semisimple complexes. Let D b ⁡ ( Z , S ) ⊂ D b ⁡ ( Z ) be the full subcategory whose objects have constructible cohomology with respect to S. We write P ⁢ ( Z , S ) = P ⁢ ( Z ) ∩ D b ⁡ ( Z , S ) and C ⁡ ( Z , S ) = C ⁡ ( Z ) ∩ D b ⁡ ( Z , S ) . Unless specified otherwise, we will assume that 𝒵 = [ Z / G ] with Z an F-scheme Z of finite type with a finite number of G-orbits. Then the G-orbits define a stratification S of Z and the categories P ⁢ ( 𝒵 ) , C ⁢ ( 𝒵 ) are G-equivariant analogues of the categories P ⁢ ( Z , S ) , C ⁢ ( Z , S ) . Let For : D b ⁡ ( 𝒵 ) → D b ⁡ ( Z , S ) be the forgetful functor. The categories C ⁢ ( 𝒵 ) and C ⁢ ( Z , S ) are graded. The category grading is given by the cohomological shift functor, i.e., we set

(A.1) 〈 ∙ 〉 = [ ∙ ] .

Let D G b ⁡ ( Z ) denote the G-equivariant derived category of complexes of étale sheaves of k -vector spaces on Z with bounded constructible cohomology, as in [10]. The triangulated category D G b ⁡ ( Z ) depends only on the quotient stack [ Z / G ] , and is equivalent to D b ⁡ ( [ Z / G ] ) .

A.2 Mixed complexes

From now on we assume that F is the algebraic closure of a finite field F 0 of characteristic prime to ℓ . We will use the following convention: objects over F 0 are denoted with a subscript 0, and suppression of the subscript means passing to F by extension of scalars. For instance, we may write Z = F ⊗ F 0 Z 0 for some F 0 -scheme Z 0 . Then we will assume that the stratification S of Z is the extension of scalars of a stratification { Z w , 0 : w ∈ W } of Z 0 . Let D m b ⁡ ( Z 0 ) be the full triangulated subcategory of mixed complexes in D b ⁡ ( Z 0 ) . The extension of scalars yields a triangulated t-exact functor

ω : D m b ⁡ ( Z 0 ) → D b ⁡ ( Z ) .

We may write a mixed complex with a subscript ∙ m , and abbreviate ℰ = ω ⁢ ( ℰ m ) . Let D m b ⁡ ( Z 0 , S ) be the full triangulated subcategory of D m b ⁡ ( Z 0 ) such that D m b ⁡ ( Z 0 , S ) = ω - 1 ⁢ D b ⁡ ( Z , S ) . Let P m ⁢ ( Z 0 , S ) be the category of mixed perverse sheaves in D m b ⁡ ( Z 0 , S ) . Recall that 𝒵 = [ Z / G ] . Assume further that 𝒵 is isomorphic to F ⊗ F 0 𝒵 0 , with 𝒵 0 = [ Z 0 / G 0 ] and some affine algebraic F 0 -group G 0 such that G = F ⊗ F 0 G 0 . Let D m b ⁡ ( 𝒵 0 ) be the full triangulated subcategory of mixed complexes in D b ⁡ ( 𝒵 0 ) . See [33] and [45] for the definition and the basic properties of the category D m b ⁡ ( 𝒵 0 ) . Let For : D m b ⁡ ( 𝒵 0 ) → D m b ⁡ ( Z 0 , S ) be the forgetful functor, where S is the stratification by G-orbits as above. For each integer w, we consider the full subcategories D ⩽ w b ⁡ ( 𝒵 0 ) , D ⩾ w b ⁡ ( 𝒵 0 ) of D m b ⁡ ( 𝒵 0 ) consisting of the mixed complexes of weight ⩽ w and ⩾ w . The category of pure complexes of weight w is

D w b ⁡ ( 𝒵 0 ) = D ⩽ w b ⁡ ( 𝒵 0 ) ∩ D ⩾ w b ⁡ ( 𝒵 0 ) .

By [7, Proposition 5.1.15], we have

(A.2) Hom D b ⁡ ( 𝒵 0 ) ⁡ ( ℰ , ℱ ) = 0 for all ⁢ ℰ ∈ D < w b ⁡ ( 𝒵 0 ) ⁢ and all ⁢ ℱ ∈ D > w b ⁡ ( 𝒵 0 ) .

We also have the following refinement of (A.2):

(A.3) ω ⁢ ( f ) = 0 for all ⁢ f ∈ Hom D b ⁡ ( 𝒵 0 ) ⁡ ( ℰ , ℱ ) , ℰ ∈ D ⩽ w b ⁡ ( 𝒵 0 ) , ℱ ∈ D > w b ⁡ ( 𝒵 0 ) .

Let P m ⁢ ( 𝒵 0 ) be the category of mixed perverse sheaves in D m b ⁡ ( 𝒵 0 ) . A mixed complex ℰ is pure of weight w, ⩽ w or ⩾ w if and only if the perverse sheaf H a p ⁢ ℰ is pure of weight w + a , ⩽ w + a or ⩾ w + a for each a ∈ ℤ , by [7, Theorem 5.4.1]. A mixed perverse sheaf ℰ has a unique finite increasing weight filtration W a ⁢ ℰ , a ∈ ℤ , such that the subquotient

Gr a W ⁡ ℰ = W a ⁢ ℰ / W a - 1 ⁢ ℰ

is a pure mixed perverse sheaf of weight a, which may not be semisimple, see [7, Theorem 5.3.5].

Proposition A.2.1.

For each E ∈ D m b ⁡ ( Z 0 ) and w ∈ Z , there is a distinguished triangle

(A.4) → ℰ ⩽ w → ℰ → ℰ > w , ℰ ⩽ w ∈ D ⩽ w b ⁡ ( 𝒵 0 ) , ℰ > w ∈ D > w b ⁡ ( 𝒵 0 )

such that E ⩾ w = 0 , E ⩾ - w = E if w ≫ 0 , and

there is a distinguished triangle

→ ℰ w → ℰ ⩾ w → ℰ > w , ℰ w = ( ℰ ⩾ w ) ⩽ w ∈ D w b ⁡ ( 𝒵 0 ) ,
the long exact sequence of perverse cohomologies splits into short exact sequences

0 → H a p ⁢ ℰ ⩽ w → H a p ⁢ ℰ → H a p ⁢ ℰ > w → 0 for all ⁢ a ∈ ℤ .

Proof.

Any perverse sheaf in P m ⁢ ( 𝒵 0 ) has a finite length. The construction of ℰ ⩽ w , ℰ > w is by induction on the total length of ℰ , i.e., on the sum of the lengths of the perverse sheaves H a p ⁢ ℰ , following the lines of [3, Lemma 6.7]. Given w, let a be the smallest integer such that the subobject W w + a ⁢ ( H a p ⁢ ℰ ) of H a p ⁢ ℰ is ≠ 0 . Set 𝒢 = W w + a ⁢ ( H a p ⁢ ℰ ) ⁢ [ - a ] . The inclusion 𝒢 ⊂ H a p ⁢ ℰ ⁢ [ - a ] factors to a distinguished triangle

(A.5) → 𝒢 → 𝑓 ℰ → ℱ ,

see [3, (6.8)] for details, such that H b p ⁢ 𝒢 = 0 for all b ≠ a and

(A.6) 0 → H b p ⁢ 𝒢 → H b p ⁢ ℰ → H b p ⁢ ℱ → 0 for all ⁢ b ∈ ℤ .

Hence ℱ has a lower total length than ℰ , and induction yields a distinguished triangle

(A.7) → ℱ ⩽ w → ℱ → ℱ > w , ℱ ⩽ w ∈ D ⩽ w b ⁡ ( 𝒵 0 ) , ℱ > w ∈ D > w b ⁡ ( 𝒵 0 ) .

From (A.5), (A.7) and [7, Lemma 1.3.10], we get distinguished triangles

(A.8) → ℋ → ℰ → ℱ > w , → 𝒢 → ℋ → ℱ ⩽ w , ℋ ∈ D ⩽ w b ( 𝒵 0 ) .

Set ℰ ⩽ w = ℋ and ℰ > w = ℱ > w . The induction hypothesis yields short exact sequences

(A.9) 0 → H b p ⁢ ℱ ⩽ w → H b p ⁢ ℱ → H b p ⁢ ℱ > w → 0 for all ⁢ b ∈ ℤ .

From (A.6) and (A.9), we deduce that the long exact sequence

→ H b p ⁢ ℋ → H b p ⁢ ℰ → H b p ⁢ ℱ > w → H b + 1 p ⁢ ℋ →

splits into short exact sequences, yielding condition (b). Since Hom D b ⁡ ( 𝒵 0 ) ⁡ ( ℰ < w , ℰ > w ) = 0 by (A.2), the map ℰ → ℰ > w factors to a morphism ℰ ⩾ w → ℰ > w . Completing this morphism to a distinguished triangle yields claim (a). ∎

For any mixed complexes ℰ , ℱ on 𝒵 0 , we write

Hom D b ⁡ ( 𝒵 0 ) ∙ ⁡ ( ℰ , ℱ ) = ⊕ a ∈ ℤ Hom D b ⁡ ( 𝒵 0 ) a ⁡ ( ℰ , ℱ ) ⁢ [ - a ] ,

Hom D b ⁡ ( 𝒵 0 ) a ⁡ ( ℰ , ℱ ) = Hom D b ⁡ ( 𝒵 0 ) ⁡ ( ℰ , ℱ ⁢ [ a ] ) .

Let a : 𝒵 0 → Spec ⁡ F 0 be the structure map. We define the geometric Hom functor by

Hom ¯ D b ⁡ ( 𝒵 0 ) ∙ ⁢ ( ℰ , ℱ ) = a * ⁢ R ⁢ ℋ ⁢ o ⁢ m D b ⁡ ( 𝒵 0 ) ⁢ ( ℰ , ℱ ) .

It is a mixed complex on Spec ⁡ F 0 . We define

Hom ¯ D b ⁡ ( 𝒵 0 ) a ⁢ ( ℰ , ℱ ) = H a ⁢ ( Hom ¯ D b ⁡ ( 𝒵 0 ) ∙ ⁢ ( ℰ , ℱ ) ) .

It is a mixed vector space consisting of a graded k -vector space

ω ⁢ Hom ¯ D b ⁡ ( 𝒵 0 ) a ⁢ ( ℰ , ℱ ) = Hom D b ⁡ ( 𝒵 ) a ⁡ ( ω ⁢ ℰ , ω ⁢ ℱ )

and a Frobenius operator Fr . We abbreviate

H ¯ ∙ ⁢ ( 𝒵 0 , ℰ ) = Hom ¯ D b ⁡ ( 𝒵 0 ) ∙ ⁢ ( k 𝒵 0 , ℰ ) .

A.3 Even stratifications, mixed categories and parity sheaves

Let k w denote the mixed constant sheaf in D b ⁡ ( Z w , 0 ) which is pure of weight 0. Let i w be the locally closed embedding of the stratum Z w ⊂ Z . Set dim ⁡ Z w = d w . Let k w = k Z w be the constant sheaf on Z w . We define the following objects in D b ⁡ ( Z , S ) :

(A.10) Δ ⁢ ( w ) = ( i w ) ! ⁢ k w ⁡ [ d w ] ,

∇ ⁡ ( w ) = ( i w ) * ⁢ k w ⁡ [ d w ] ,

I ⁢ C ⁢ ( w ) = ( i w ) ! * ⁢ k w ⁡ [ d w ] .

Fix a square root of the Tate sheaf. For each a ∈ ℤ let ( a / 2 ) be the twist by the ath power of this square root. We abbreviate

(A.11) 〈 ∙ 〉 = ( ∙ / 2 ) [ ∙ ] .

We define the following mixed complexes in D m b ⁡ ( Z 0 , S ) :

(A.12) Δ ⁢ ( w ) m = ( i w ) ! ⁢ k w ⁡ 〈 d w 〉 ,

∇ ( w ) m = ( i w ) * k w 〈 d w 〉 ,

I ⁢ C ⁢ ( w ) m = ( i w ) ! * ⁢ k w ⁡ 〈 d w 〉 .

Let D ◆ , m b ⁡ ( Z 0 , S ) ⊂ D m b ⁡ ( Z 0 , S ) and P ◆ , m ⁢ ( Z 0 , S ) ⊂ P m ⁢ ( Z 0 , S ) be the full triangulated subcategory and the Serre subcategory generated by the set of objects { I ⁢ C ⁢ ( w ) m ⁢ ( a / 2 ) : w ∈ W , a ∈ ℤ } . The triangulated category D ◆ , m b ⁡ ( Z 0 , S ) has a t-structure whose heart is P ◆ , m ⁢ ( Z 0 , S ) .

Definition A.3.1.

The stratification S is

affine if each stratum is isomorphic to an affine space,
even affine if
1. S is affine,
2. H a ⁢ ( ( i u ) * ⁢ I ⁢ C ⁢ ( v ) m ) = 0 for all u , v ∈ W , a ∈ ℤ with a + d v odd,
3. H a ⁢ ( ( i u ) * ⁢ I ⁢ C ⁢ ( v ) m ) is a sum of copies of k u ⁡ ( - a / 2 ) if a + d v is even,
even if
1. there is an even affine stratification T of Z which refines S,
2. the strata of S are connected and simply connected.

Since I ⁢ C ⁢ ( v ) m is Verdier self-dual and Z u is smooth by (1), conditions (2) and (3) are equivalent to the following conditions:

H a ⁢ ( ( i u ) ! ⁢ I ⁢ C ⁢ ( v ) m ) = 0 for all u , v ∈ W , a ∈ ℤ with a + d v odd,
H a ⁢ ( ( i u ) ! ⁢ I ⁢ C ⁢ ( v ) m ) is a sum of copies of k u ⁡ ( a / 2 ) if a + d v is even.

Conditions (2) and (6) imply that the complex I ⁢ C ⁢ ( v ) is a parity sheaf of D b ⁡ ( Z , S ) in the sense of [20]. Conditions (3) and (7) imply that the complex I ⁢ C ⁢ ( v ) is very pure in the sense of [14, Definition 3.1.2]. They tell us in addition that the mixed vector spaces H a ⁢ ( ( i u ) * ⁢ I ⁢ C ⁢ ( v ) m ) and H a ⁢ ( ( i u ) ! ⁢ I ⁢ C ⁢ ( v ) m ) are semisimple.

If the stratification S is even, and T is an even affine stratification which refines S, then there is a full embedding of triangulated categories

D b ⁡ ( Z , S ) ⊂ D b ⁡ ( Z , T ) , D ◆ , m b ⁡ ( Z 0 , S ) ⊂ D ◆ , m b ⁡ ( Z 0 , T ) .

Since each stratum of S contains a unique dense stratum of T, there is a full embedding of additive categories

P ⁢ ( Z , S ) ⊂ P ⁢ ( Z , T ) , P ◆ , m ⁢ ( Z 0 , S ) ⊂ P ◆ , m ⁢ ( Z 0 , T ) .

Note that the Abelian categories P μ ⁢ ( Z , S ) , P ◆ , m ⁢ ( Z 0 , S ) and P ⁢ ( Z , S ) are Krull–Schmidt.

Definition A.3.2.

We define

C m ⁡ ( Z 0 , S ) ⊂ D ◆ , m b ⁡ ( Z 0 , S ) to be the full subcategory of all mixed complexes which are isomorphic to finite direct sums of objects in { I ⁢ C ⁢ ( w ) m ⁢ 〈 a 〉 : w ∈ W , a ∈ ℤ } . It is a graded additive category with the graded shift functor 〈 ∙ 〉 in (A.11).
D μ b ⁡ ( Z , S ) = K b ⁢ ( C m ⁡ ( Z 0 , S ) ) as a graded triangulated category with the graded shift functor 〈 ∙ 〉 in (A.11).
P μ ⁢ ( Z , S ) ⊂ P ◆ , m ⁢ ( Z 0 , S ) to be the full subcategory of all mixed perverse sheaves ℰ such that Gr ∙ W ⁡ ℰ is semisimple. It is a graded Abelian category for the Tate shift functor ( ∙ / 2 ) .

Proposition A.3.3.

Assume that the stratification S is even. Then:

D μ b ⁡ ( Z , S ) has a t-structure and a triangulated t - exact faithful functor

ι : D μ b ⁡ ( Z , S ) → D ◆ , m b ⁡ ( Z 0 , S ) .

The heart of D μ b ⁡ ( Z , S ) is equivalent to P μ ⁢ ( Z , S ) and the restriction of ι to this heart is the full embedding P μ ⁢ ( Z , S ) ⊂ P ◆ , m ⁢ ( Z 0 , S ) .
ζ = ω ∘ ι is a t - exact functor D μ b ⁡ ( Z , S ) → D b ⁡ ( Z , S ) such that

⊕ a ∈ ℤ Hom D μ b ⁡ ( Z ) ⁡ ( ℰ , ℱ ⁢ ( a / 2 ) ) = Hom D b ⁡ ( Z ) ⁡ ( ζ ⁢ ℰ , ζ ⁢ ℱ ) for all ⁢ ℰ , ℱ ∈ D μ b ⁡ ( Z , S ) .
For any inclusion h : Y → Z of a union of strata of S , the functors h * , h ! , h * , h ! between the categories D m b ⁡ ( Y 0 , S ) and D m b ⁡ ( Z 0 , S ) lift to triangulated functors between the categories D μ b ⁡ ( Y , S ) and D μ b ⁡ ( Z , S ) which satisfy the usual adjointness properties.

Proof.

Part (a) is proved in [3, Section 7.2] and [3, Proposition 7.5 (1)–(2)]. Part (b) is proved in [3, Proposition 7.5 (2)]. Part (c) is [3, Theorem 9.5]. ∎

Here, the shift functor ( ∙ / 2 ) on the category D μ b ⁡ ( Z , S ) is defined to be ( ∙ / 2 ) = 〈 ∙ 〉 [ - ∙ ] , where 〈 ∙ 〉 is the grading shift functor on C m ⁡ ( Z 0 , S ) given in (A.11). Note that, if the stratification S is even affine, then the obvious functor P μ ⁢ ( Z , S ) → D μ b ⁡ ( Z , S ) yields an equivalence of triangulated categories K b ⁢ ( P μ ⁢ ( Z , S ) ) → D μ b ⁡ ( Z , S ) by [3, Corollary 7.10]. Under this equivalence ( ∙ / 2 ) coincides with the Tate shift functor on P μ ⁢ ( Z , S ) . See Section A.4 below.

Assume the stratification S is even. We have the graded additive category ( C ⁡ ( Z , S ) , 〈 ∙ 〉 ) such that 〈 ∙ 〉 is as in (A.1), and the graded additive category ( C m ⁡ ( Z 0 , S ) , 〈 ∙ 〉 ) such that 〈 ∙ 〉 is as in (A.11).

Proposition A.3.4.

Assume that the stratification S is even. Then:

Hom D μ b ⁡ ( Z ) ∙ ⁡ ( ℰ , ℱ ) = Hom ¯ D b ⁡ ( Z 0 ) ∙ ⁢ ( ι ⁢ ℰ , ι ⁢ ℱ ) Fr for each ℰ , ℱ ∈ D μ b ⁡ ( Z , S ) .
Hom D μ b ⁡ ( Z ) ∙ ⁡ ( ℰ , ℱ ) = Hom ¯ D b ⁡ ( Z 0 ) ∙ ⁢ ( ι ⁢ ℰ , ι ⁢ ℱ ) for each ℰ , ℱ ∈ C m ⁡ ( Z 0 , S ) .
ζ : ( C m ⁡ ( Z 0 , S ) , 〈 ∙ 〉 ) → ( C ⁡ ( Z , S ) , 〈 ∙ 〉 ) is an equivalence of graded additive categories.

Proof.

Due to the full embedding D b ⁡ ( Z 0 , S ) ⊂ D b ⁡ ( Z 0 , T ) for each affine refinement T of S, we can and will assume that S is even affine. Part (a) follows from [3, Lemma 7.8], which also implies that the mixed complex Hom ¯ D b ⁡ ( Z 0 ) ∙ ⁢ ( ι ⁢ ℰ , ι ⁢ ℱ ) is semisimple for each objects ℰ , ℱ ∈ D μ b ⁡ ( Z , S ) . Hence, for each a , b ∈ ℤ , we have

(A.13) Hom D μ b ⁡ ( Z ) a ⁡ ( ℰ , ℱ ⁢ ( b / 2 ) ) = ( Hom ¯ D b ⁡ ( Z 0 ) a ⁢ ( ι ⁢ ℰ , ι ⁢ ℱ ) ⁢ ( b / 2 ) ) Fr ,

and to prove (b) it is enough to check that Hom ¯ D b ⁡ ( Z 0 ) ∙ ⁢ ( ι ⁢ ℰ , ι ⁢ ℱ ) is pure of weight 0 whenever ℰ , ℱ ∈ C m ⁡ ( Z 0 , S ) . The mixed Abelian category P μ ⁢ ( Z , S ) is Koszul by [9, Theorem 4.4.4]. Hence, if ℰ , ℱ ∈ P μ ⁢ ( Z , S ) are pure of weight zero, we have

(A.14) b ≠ a ⟹ Hom D μ b ⁡ ( Z , S ) a ⁡ ( ℰ , ℱ ⁢ ( b / 2 ) ) = 0 .

So, the mixed vector space Hom ¯ D b ⁡ ( Z 0 ) a ⁢ ( ι ⁢ ℰ , ι ⁢ ℱ ) is pure of weight a, so the mixed complex Hom ¯ D b ⁡ ( Z 0 ) ∙ ⁢ ( ι ⁢ ℰ , ι ⁢ ℱ ) is pure of weight 0, proving part (b) because any object of C m ⁡ ( Z 0 , S ) is a sum of I ⁢ C ⁢ ( w ) m ⁢ 〈 a 〉 . Part (c) follows from (A.14) and Proposition A.3.3, since for any objects ℰ , ℱ ∈ C m ⁡ ( Z 0 , S ) we have

Hom D μ b ⁡ ( Z ) ⁡ ( ℰ , ℱ ) = ⊕ a ∈ ℤ Hom D μ b ⁡ ( Z ) ⁡ ( ℰ , ℱ ⁢ ( a / 2 ) ) = Hom D b ⁡ ( Z ) ⁡ ( ζ ⁢ ℰ , ζ ⁢ ℱ ) . ∎

For each w ∈ W , let I ⁢ C ⁢ ( w ) μ be I ⁢ C ⁢ ( w ) m viewed as an object of D μ b ⁡ ( Z , S ) . Assume that the stratification S is even. By Proposition A.3.4, we have D μ b ⁡ ( Z , S ) = K b ⁢ ( C ⁡ ( Z , S ) ) . This identification takes I ⁢ C ⁢ ( w ) μ to I ⁢ C ⁢ ( w ) . The grading K b ⁢ ( C ⁡ ( Z , S ) ) is given by the shift functor 〈 ∙ 〉 on C ⁡ ( Z , S ) in (A.1).

We define the equivariant mixed category of the stack 𝒵 = [ Z / G ] by

D μ b ⁡ ( 𝒵 ) = K b ⁢ ( C ⁡ ( 𝒵 ) ) .

Let S be the stratification by the G-orbits. We have the forgetful functor

For : D μ b ⁡ ( 𝒵 ) → D μ b ⁡ ( Z , S ) .

We do not know any equivariant analogue of Proposition A.3.4. However, the following holds, see, e.g., [14, Lemma 3.1.5].

Proposition A.3.5.

Assume that the G 0 -orbits in Z 0 are affine. If E , F ∈ D m b ⁡ ( Z 0 ) are very pure of weight 0, then the mixed complex Hom ¯ D b ⁡ ( Z 0 ) ∙ ⁢ ( E , F ) in D + ⁡ ( Spec ⁡ F 0 ) is pure of weight 0 and it is free of finite rank as an H G ∙ -module.

Remark A.3.6.

The following statements hold:

Let S be any stratification of Z. The category C ⁡ ( Z , S ) has split idempotents, and the Verdier duality D yields an equivalence C ( Z , S ) ≃ C ( Z , S ) op . Let ℒ ∈ C ⁡ ( Z , S ) be a graded-generator. Set 𝐑 = End D b ⁡ ( Z , S ) ∙ ( ℒ ) op . The functor ℰ ↦ Hom D b ⁡ ( Z , S ) ∙ ⁡ ( ℒ , ℰ ) gives an equivalence of graded additive categories C ⁡ ( Z , S ) → 𝐑 ⁢ - ⁢ proj . Taking the homotopy categories, we get a graded triangulated equivalence K b ⁢ ( C ⁢ ( Z , S ) ) → D perf ⁡ ( 𝐑 ) .
If h : Y → Z is a closed embedding, the functor h ! = h * : D μ b ⁡ ( Y ) → D μ b ⁡ ( Z ) in Proposition A.3.3 is given by restricting the functor h ! = h * : D b ⁡ ( Y ) → D b ⁡ ( Z ) to C ⁢ ( Y ) → C ⁢ ( Z ) and taking the homotopy categories. If h : Y → Z is an open embedding, the functor h ! = h * is defined in a similar way. We do not know any equivariant analogue of Proposition A.3.3 which would yield functors h * , h ! , h * , h ! between the categories D μ b ⁡ ( 𝒴 ) and D μ b ⁡ ( 𝒵 ) for any inclusion h : 𝒴 → 𝒵 of a union of strata. However the functors h * = h ! are well defined in the equivariant case if h is a closed embedding, so are h * = h ! if h is an open embedding.
If the stratification S is even, then the set { I ⁢ C ⁢ ( w ) ⁢ [ a ] : w ∈ W , a ∈ ℤ } is a complete and irredundant set of indecomposable objects of C ⁡ ( Z , S ) . It is also a complete and irredundant set of parity sheaves of D b ⁡ ( Z , S ) in the sense of [20].
A triangulated functor ϕ m : D ◆ , m b ⁡ ( Y 0 ) → D ◆ , m b ⁡ ( Z 0 ) is geometric if there is a triangulated functor ϕ : D b ⁡ ( Y ) → D b ⁡ ( Z ) with a natural isomorphism ϕ ⁢ ω ⇒ ω ⁢ ϕ m . It is genuine if it is geometric and there is a triangulated functor ϕ μ : D μ b ⁡ ( Y ) → D μ b ⁡ ( Z ) with a natural isomorphism ϕ ⁢ ι ⇒ ι ⁢ ϕ μ .
Let T, V be even affine refinements of even stratifications S, U of F 0 -schemes Y 0 , Z 0 . By [3, Lemma 7.12], there are full embedding of triangulated categories

D ◆ , m b ⁡ ( Y 0 , S ) ⊂ D ◆ , m b ⁡ ( Y 0 , T ) and D ◆ , m b ⁡ ( Z 0 , U ) ⊂ D ◆ , m b ⁡ ( Z 0 , V ) .

By [3, Lemma 7.21], the restriction of a genuine functor D ◆ , m b ⁡ ( Y 0 , T ) → D ◆ , m b ⁡ ( Z 0 , V ) that takes D ◆ , m b ⁡ ( Y 0 , S ) into D ◆ , m b ⁡ ( Z 0 , U ) , is a genuine functor

D ◆ , m b ⁡ ( Y 0 , S ) → D ◆ , m b ⁡ ( Z 0 , U ) .
By Proposition A.3.3, if the stratification S is even, we may view D μ b ⁡ ( Z , S ) as a (non-full) subcategory of D ◆ , m b ⁡ ( Z 0 , S ) consisting of objects whose stalks carry a semisimple action of the Frobenius.
Any object ℰ of C m ⁡ ( Z 0 , S ) is semisimple and pure of weight 0, that is,

ℰ ≃ ⊕ a H a p ⁢ ( ℰ ) ⁢ [ - a ] ,

where each mixed perverse sheaf H a p ⁢ ( ℰ ) is pure of weight a.
An even stratification S is called affable in [3, Definition 7.2]. The category D ◆ , m b ⁡ ( Z 0 , S ) is the same as D S Weil ⁡ ( Z 0 ) in [3, Section 6.1]. If S is even affine, then D ◆ , m b ⁡ ( Z 0 , S ) is the category D ◆ , m ⁢ ( Z 0 ) in [49, Section 2.1].

A.4 Even affine stratifications, projective and tilting objects

Assume that the stratification S is even affine. The objects Δ ⁢ ( w ) m and ∇ ( w ) m have canonical lifts Δ ⁢ ( w ) μ and ∇ ( w ) μ in P μ ⁢ ( Z , S ) by Proposition A.3.3. Further, we have a triangulated t-exact faithful functor ι : D μ b ⁡ ( Z , S ) → D ◆ , m b ⁡ ( Z 0 , S ) such that

ι I C ( w ) μ = I C ( w ) m , ι Δ ( w ) μ = Δ ( w ) m , ι ∇ ( w ) μ = ∇ ( w ) m for all w ∈ W .

We equip the triangulated category D b ⁡ ( P μ ⁢ ( Z , S ) ) with the grading shift functors

(A.15) 〈 ∙ 〉 = ( ∙ / 2 ) [ ∙ ] ,

where ( ∙ / 2 ) is the Tate shift functor on P μ ⁢ ( Z , S ) and [ ∙ ] is the cohomological shift. By [3, Corollary 7.10], there is an equivalence of graded triangulated categories

D b ⁡ ( P μ ⁢ ( Z , S ) ) → D μ b ⁡ ( Z , S )

which identifies the grading shift functors (A.11) and (A.15). We will use two refinements of this equivalence which involve projective and tilting objects of P μ ⁢ ( Z , S ) .

Proposition A.4.1.

Assume that the stratification S is even affine. Then:

P μ ⁢ ( Z , S ) , P ⁢ ( Z , S ) have enough projectives and finite cohomological dimension. The sets of indecomposable objects in P ⁢ ( Z , S ) proj and P μ ⁢ ( Z , S ) proj are { P ⁢ ( w ) : w ∈ W } and { P ⁢ ( w ) μ ⁢ ( a / 2 ) : w ∈ W , a ∈ ℤ } , where P ⁢ ( w ) , P ⁢ ( w ) μ are the projective covers of Δ ⁢ ( w ) , Δ ⁢ ( w ) μ in P ⁢ ( Z , S ) , P μ ⁢ ( Z , S ) .
An object ℰ ∈ P μ ⁢ ( Z , S ) is projective if and only if ζ ⁢ ℰ ∈ P ⁢ ( Z , S ) proj .
P μ ⁢ ( Z , S ) proj ⊂ D μ b ⁡ ( Z , S ) extends to a graded triangulated equivalence

K b ⁢ ( P μ ⁢ ( Z , S ) proj ) → D μ b ⁡ ( Z , S ) .

Proof.

Part (a) is [3, Theorem 7.7 (1)–(2)], part (b) is [3, Theorem 7.7 (2)], and part (c) is proved [3, Corollary 7.10, Proposition 7.11]. ∎

Definition A.4.2 ([3, 8, 49]).

Assume that the stratification S is even affine. A mixed perverse sheaf ℰ ∈ P ◆ , m ⁢ ( Z 0 , S ) is tilting if either of the following equivalent conditions hold:

( i w ) * ⁢ ℰ and ( i w ) ! ⁢ ℰ are perverse for each w ∈ W .
ℰ has both a filtration by Δ ⁢ ( w ) m ⁢ ( a / 2 ) ’s and by ∇ ( w ) m ( a / 2 ) ’s, with w ∈ W and a ∈ ℤ .

We define a tilting object in P ⁢ ( Z , S ) and P μ ⁢ ( Z , S ) tilt in a similar way.

Let P ⁢ ( Z , S ) tilt ⊂ P ⁢ ( Z , S ) , P ◆ , m ⁢ ( Z 0 , S ) tilt ⊂ P ◆ , m ⁢ ( Z 0 , S ) and P μ ⁢ ( Z , S ) tilt ⊂ P μ ⁢ ( Z , S ) be the full additive subcategories of tilting objects.

Proposition A.4.3.

Assume that the stratification S is even affine.

For each w ∈ W , there are unique indecomposable objects T ⁢ ( w ) , T ⁢ ( w ) μ in P ⁢ ( Z , S ) tilt , P μ ⁢ ( Z , S ) tilt supported on X ¯ w whose restriction to X w are k w ⁡ [ d w ] , k w ⁡ 〈 d w 〉 respectively. The sets of indecomposable objects in P ⁢ ( Z , S ) tilt , P μ ⁢ ( Z , S ) tilt are { T ⁢ ( w ) : w ∈ W } and { T ⁢ ( w ) μ ⁢ ( a / 2 ) : w ∈ W , a ∈ ℤ } .
The functor ι takes P μ ⁢ ( Z , S ) tilt into P ◆ , m ⁢ ( Z 0 , S ) tilt .
The functor ζ takes P μ ⁢ ( Z , S ) tilt into P ⁢ ( Z , S ) tilt .
P μ ⁢ ( Z , S ) tilt ⊂ D μ b ⁡ ( Z , S ) extends to a graded triangulated equivalence

K b ⁢ ( P μ ⁢ ( Z , S ) tilt ) → D μ b ⁡ ( Z , S ) .

Proof.

Part (a) is [3, Proposition 10.3], [8]. Parts (b) and (c) are obvious. Part (d) is [3, Proposition 10.5]. ∎

Acknowledgements

Initial stages of this work were partly based on discussions with R. Rouquier. We would like to thank him for all these discussions. We are also grateful to S. Riche and O. Schiffmann for answering various questions.

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Received: 2020-11-25

Revised: 2022-06-09

Published Online: 2022-09-29

Published in Print: 2022-11-01

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