An important insight in geometric representation theory is that algebras and their representations can often be constructed geometrically in terms of convolution of cycles. For example, the Springer correspondence [Spr78] describes how irreducible representations of a Weyl group can be realised in terms of a convolution action on the free vector spaces spanned by irreducible components of Springer fibers, see [CG10]. Similar situations, which we refer to as Springer theories, yield for example the affine Hecke algebra [KL87], the quiver Hecke algebra (KLR algebra) [Rou08] or the quiver Schur algebra [SW14] and their representations.

These constructions are usually employing Borel–Moore homology and constructible sheaves. Our main goal is to establish the foundations of a motivic Springer theory using Chow groups and motivic sheaves instead. Motivic sheaves, see [Ayo07a] and [CD19], are a relative version of Voevodsky’s triangulated category of mixed motives and their Hom-spaces are governed by Chow groups. As established in the setting of flag varieties in [SW18], motivic sheaves can serve as a graded version of constructible sheaves that are technically advantageous over the mixed $\ell$-adic sheaves [BBD82] or mixed Hodge modules [Sai16].

Convolution for Chow groups can be interpreted as composition for $\operatorname{Hom}$-spaces in categories of motivic sheaves. This motivates our definition of a motivic extension algebra. We discuss how the graded affine Hecke algebra as well as quiver Hecke and quiver Schur algebras arise this way. We then prove that purity of certain fibers implies that the perfect derived category of a motivic extension algebra can be realised as a subcategory of equivariant motivic sheaves called Springer motives—a statement we refer to as formality.

To achieve our formality results, we make use of the theory of weight structures from [Bon10] (a concept independently introduced under the name co-$t$-structures in [Pau08] and studied already in the context of silting theory, see e.g. [KY14]) and weight complex functors from [Bon10]. In a geometric context, formality is often obtained using of Deligne’s yoga of weights; for example one makes use of eigenvalues of a Frobenius morphism or weights of a mixed Hodge structure. Various aspects of this yoga are formalised in the notion of the Chow weight structure on categories of motivic sheaves which we extend to the category of Springer motives. In an algebraic context, we show that Koszul duality and Ringel duality, derived equivalences between Koszul and Ringel dual algebras, respectively, can be expressed in terms of a weight complex functor.

We use the term variety for (not necessarily reduced) quasi-projective separated schemes of finite type over a field $k.$ In the introduction and most of the manuscript $k=\overline{\mathbb{F}}_{p}$ and $G$ is a linear algebraic group over $k.$ For rings $R\subset R^{\prime}$ and $R$-modules $M$ we denote by $M_{R^{\prime}}=M\otimes_{R}R^{\prime}$ the extension of scalars. We use a cohomological convention for chain complexes and denote by $C[n]$ the cohomological shift with $(C[n])^{i}=C^{i+n}.$

Let $\mu_{i}:\widetilde{\mathcal{N}}_{i}\to\mathcal{N}$ be a collection of $G$-equivariant proper maps of varieties such that each $\widetilde{\mathcal{N}}_{i}$ is smooth. We consider the motivic extension algebra

which is a motivic version of the $\ell$-adic extension algebra $E^{\acute{e}t}_{\ell}$ defined via the category $\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell})$ of equivariant $\ell$-adic sheaves.

The algebra $E$ can be described in classical terms as the $G$-equivariant Chow groups of the Steinberg varieties $Z_{i,j}=\widetilde{\mathcal{N}}_{i}\times_{\mathcal{N}}\widetilde{\mathcal{N}}_{j}$ equipped with a convolution product. Namely, by Corollary  4.6 there is an isomorphism

The motivic extension algebra is defined using the category $\operatorname{DM}_{G}(\mathcal{N},\mathbb{Q})$ of $G$-equivariant motivic sheaves on $\mathcal{N},$ which was introduced in [SVW18] as a motivic version of $\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell}).$ Similarly to their $\ell$-adic counterpart, equivariant motivic sheaves are equipped with a six-functor-formalism. Their hom-spaces are governed by equivariant higher Chow groups and they admit an autoequivalence $(1)$ called Tate twist, which will serve as a shift of grading functor for us.

We define the full subcategory of Springer motives

see Definition 4.1, and introduce local purity and finiteness conditions (PT) and (FO), see Section 4.1. We will prove the following formality result showing that Springer motives can be described solely in terms of the algebra $E$ carrying no higher structure.

We apply the result in the setting of Lusztig’s graded affine Hecke algebra $\overline{\mathbb{H}}(G)$ associated to the root datum of a reductive group $G,$ see also [Ebe21].

We obtain a similar result for quiver Hecke algebras (KLR algebras) $R_{\mathbf{d}}$ for quivers $Q$ in type $A$ and $\widetilde{A}$ (with cyclic orientation).

A similar result holds for quiver Schur algebras, see Theorem 6.6. To establish the local purity condition (PT) in this setting, we prove that partial quiver flag varieties in type $\widetilde{A}$ (with cyclic orientation) admit affine pavings.

In Section 2, we recall weight structures, weight complex functors and apply them to Koszul and Ringel duality. In Section 3, we recall the formalism of (equivariant) motivic sheaves and establish important foundational results. In Section 4, we introduce Springer motives, the motivic extension algebra and use the previous results to prove our main theorem on the formality of Springer motives. Finally, in Section 5 and 6 we discuss applications to affine Hecke algebras and quiver Hecke/Schur algebras.

The first author thanks Hans Franzen for extensive discussions on quiver flag varieties. We thank Wolfgang Soergel and Matthias Wendt for their help with equivariant motivic sheaves. This work was supported by the Hausdorff Center of Mathematics (grant EXC 2047).

We start with the definition of a weight structure, see [Bon10, Definition 1.1.1].

Following [Bon10, Theorem 4.3.2] we now show how to define weight structures by specifying their heart.

An object in a triangulated category $\mathcal{C}$ with bounded weight structure can be built from objects which are pure, that is in $\mathcal{C}^{w=n}=\mathcal{C}^{w=0}[-n]$ for $n\in\mathbb{Z},$ using the distinguished triangles in Definition  2.1( 3), see [Bon10, Proposition 1.5.6]. This will imply that any geometric motive can be built from motives of smooth projective varieties. This should be seen as a reflection of Deligne’s yoga of weights in the context of mixed Hodge structures and $\ell$-adic cohomology.

We will now show that, most remarkably, these pure pieces can be assembled into a complex called weight complex. If the category $\mathcal{C}$ admits some enhancement the weight complex gives a well-defined element in the homotopy category of $\mathcal{C}^{w=0}$ in a functorial way.

We will use the following description of weight exact functors due to Sosnilo, see [Sos17, Proposition 3.3(b)].

The result can be used to construct a weight complex functor, see [Sos17, Corollary 3.5]:

Sosnilo [Sos17, Corollary 3.5] and Aoki [Aok20, Theorem 4.3] show that the weight complex functor is compatible with weight exact functors and also with symmetric monoidal structures as follows.

The weight complex functor reflects isomorphisms, but is not necessarily an equivalence. For example, the heart $\mathcal{C}^{w=0}$ is clearly tilting in $\operatorname{K}^{b}(\mathcal{C}^{w=0})$ while in general it is just negative in $\mathcal{C}.$ This is the main obstruction as the following result shows.

It is sometimes convenient to pass from additive categories, such as the heart of a weight structure $\mathcal{C}^{w=0},$ to categories of modules over some algebra with idempotents. We use this perspective to prove a small variation on Corollary 2.12.

For a family of objects $\mathcal{T}$ in an idempotent-closed additive category $\mathcal{C}$ we consider the locally unital algebra with the distinguished idempotents $e_{M}=\operatorname{id}_{M}$ for $M\in\mathcal{T}$

By general nonsense we obtain:

For a graded version, assume that $\mathcal{C}$ is equipped with an autoequivalence $\langle 1\rangle.$ Then one can define the $\mathbb{Z}$-graded locally unital algebra

Again, general nonsense yields

Weight structures and weight complex functors can be used to provide a convenient language for the Koszul duality formalism from [BGS96], and slightly more generally [MOS09].

Let $A$ be a $\mathbb{Z}$-graded algebra which is positively graded, that is, $A^{i}=0$ for $i<0$ and assume that $A_{0}$ is semisimple and finite dimensional. Denote by $\langle 1\rangle$ the shift of grading functor on the category $A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}}$ of finitely generated graded $A$-modules and let $\mathcal{T}(A)$ be a set of representatives of simple objects concentrated in degree $0.$

The algebra $A$ is called Koszul if $\operatorname{Ext}^{i}(L,L^{\prime}\langle j\rangle)=0\text{ for all }i\neq j$ and $L,L^{\prime}\in\mathcal{T}(A).$ Equivalently, $A$ is Koszul if the family $\widehat{\mathcal{T}}(A)=\bigcup_{i}\mathcal{T}(A)\langle i\rangle[i]$ is tilting in $\operatorname{D}^{b}(A\text{-}\operatorname{mod}_{fg}^{\mathbb{Z}}).$ In this case, the Koszul dual $A^{!}$ of $A$ is the graded algebra

where the right hand side is defined as in (2.2) using the autoequivalence $\langle 1\rangle[1].$

Corollary 2.16 implies that the weight complex functor induces an equivalence of categories

In the case that $A$ is finitely generated as an $A_{0}$-left and right module and $A^{!}$ is left Noetherian, (2.4) specializes to the Koszul duality from [BGS96, Theorem 2.12.5]

The heart of the weight structure defined by $\mathcal{T}(A)$ on the left hand side maps to projective modules in cohomological degree $0$ on the right. Moreover, one can show that the heart of the standard $t$-structure on the right hand side corresponds to the category of linear complexes on the left hand side, see [MOS09].

Similarly to the last paragraph, weight structures and weight complex functors can be applied to the theory of tilting objects and Ringel duality for highest weight categories and their semi-infinite and stratified generalisation. For the general setup we refer to Brundan–Stroppel [BS21] and freely use the terminology from there. Let $\mathcal{R}$ be a lower finite or essentially finite $\epsilon$-stratified category over an algebraically closed field (in particular it could be a highest weight category).

Then by [BS21, Theorems 4.2, 4.13, 4.18] tilting modules in the sense of highest weight categories exist in $\mathcal{R}$ and form an additive subcategory generated by the family $\mathcal{T}(\mathcal{R})$ of representatives of isomorphism classes of indecomposable tilting modules. For $E=\operatorname{End}_{\mathcal{R}}(\mathcal{T}(\mathcal{R})),$ see (2.2), the category $\mathcal{R}^{\prime}=\operatorname{mod_{lfd}-}E$ of locally finite dimensional right modules is called the Ringel dual of $\mathcal{R}.$

By [BS21, Theorems 3.11, 3.56] the family of tilting modules $\mathcal{T}(\mathcal{R})$ is tilting in $\operatorname{D}^{b}(\mathcal{R})$ in the sense of Definition 2.3. Therefore Corollary 2.16 implies that the weight complex functor induces an equivalence of categories

Now assume that $\mathcal{T}(\mathcal{R})$ generates $\operatorname{D}^{b}(\mathcal{R})$ as a triangulated category and $E$ has finite cohomological dimension. For example, this is the case if $\mathcal{R}$ is a highest weight category. Then (2.6) yields a derived equivalence, called Ringel duality, between $\mathcal{R}$ and its Ringel dual $\mathcal{R}^{\prime}$

This interpretation of Ringel duality in terms of weight complex functors has interesting applications. For example, one can use Proposition 2.10 to show that Ringel duality commutes with functors preserving tilting modules (under the correct technical assumptions).

Let $k$ be a perfect field and $\operatorname{pt}=\operatorname{Spec}(k).$ All varieties are considered to be over $k.$

For a variety $X$ over $k$ we consider the triangulated category $\operatorname{DM}(X,\mathbb{Q})$ of rational motivic sheaves on $X.$¹¹ 1 There are various definitions of $\operatorname{DM}(X,\mathbb{Q})$ that are equivalent, see [CD19, Section C.3]. We leave the choice of definition to the reader. In the literature, objects in $\operatorname{DM}(X,\mathbb{Q})$ are often referred to as relative motives or simply motives. However, we prefer the term motivic sheaf and reserve the term motive for objects in $\operatorname{DM}(k,\mathbb{Q})$. In the special case $X=\operatorname{Spec}(k),$ the category $\operatorname{DM}(k,\mathbb{Q})$ agrees with Voevodsky’s triangulated category of mixed motives over $k.$

The system of categories $\operatorname{DM}(X,\mathbb{Q})$ has remarkable properties, some of which we will recall now.

First, the work of Ayoub [Ayo07a, Ayo07b] and Cisinski–Déglise [CD19] shows that $\operatorname{DM}(X,\mathbb{Q})$ can be equipped with a six-functor-formalism which works very similarly as in the setting of $\ell$-adic sheaves. Important objects are the motive $\operatorname{M}(X)$ and motive with compact support $\operatorname{M}^{c}(X)$ of a variety $f:X\to\operatorname{pt}$ which can be expressed in terms of the six functors as

There is an autoequivalence $(1)=-\otimes\mathbb{Q}(1)$ called Tate twist on $\operatorname{DM}$ which commutes with the six functors and can be defined by splitting the motive of the projective line as

For each prime $\ell$ invertible in $k$, there is an $\ell$-adic realisation functor to the category $\operatorname{D}(X,\mathbb{Q}_{\ell})$ of $\ell$-adic sheaves on $X$

which is compatible with the six functors and the Tate twist, see [Ayo14].

Morphisms in $\operatorname{DM}(X,\mathbb{Q})$ are best understood in terms of higher Chow groups, as defined by Bloch [Blo86]. For $X$ smooth, there is a natural isomorphism

where $\mathbb{Q}_{X}$ denotes the tensor unit and $\operatorname{CH}$ a higher Chow group. In particular, for $n=2m$ one obtains the usual Chow group of codimension-$m$ cycles $\operatorname{CH}^{m}(X,0)_{\mathbb{Q}}=\operatorname{CH}^{m}(X)_{\mathbb{Q}}.$ In this case the realisation functor $\operatorname{Real}_{\ell}$ yields the cycle class map to $\ell$-adic cohomology

For $X=\operatorname{pt}$ and $k=\mathbb{F}_{q}$ or $k=\overline{\mathbb{F}}_{p}$ these hom-groups are particularly simple:

In the following the purity property in (3.1) will be crucial. For this reason, we will from now on restrict to the case $k=\overline{\mathbb{F}}_{p}.$

We will now recall the definition of the category of pure/mixed Tate motives, see [Lev05], which will play for us the role of a graded version of the derived category of sheaves on the point.

For example, the decompostion of a projective space into affine spaces

induces a decomposition of its motive into a direct sum of Tate motives

which shows that $\operatorname{M}(\mathbb{P}^{n})$ is pure Tate.

We will use the following generalization.

As the notation suggests, the category of pure Tate motives is the heart of a weight structure:

Using that $k=\overline{\mathbb{F}}_{p}$ we obtain a simple decription of the category of Tate motives:

We note that the weight structure on mixed Tate motives considered here is just the shadow of the Chow weight structure on $\operatorname{DM}^{gm}(k,\mathbb{Q}),$ see Example 2.2.

Soergel–Virk–Wendt [SVW18] introduced an equivariant version of the above formalism. For a variety $X$ with an action of a linear algebraic group $G$ they define the category $\operatorname{DM}_{G}(X,\mathbb{Q})$ of $G$-equivariant motivic sheaves on $X.$ This system of categories still carries a six-functor-formalism³³ 3 For technical reasons, Soergel–Virk–Wendt construct a six-functor-formalism for the full subcategory $\operatorname{DM}^{+}_{G}(X,\mathbb{Q})\subset\operatorname{DM}_{G}(X,\mathbb{Q})$ of objects that are bounded below with respect to the homotopy $t$-structure, see [SVW18, Section I.6]. All objects that we consider here are automatically in $\operatorname{DM}^{+}_{G}(X,\mathbb{Q})$ and we simply ignore this technicality. and induction/restriction functors changing the group.

Similarly to the non-equivariant case, there is a realization functor to the equivariant derived category of $\ell$-adic sheaves $\operatorname{D}_{G}(X,\mathbb{Q}_{\ell})$ of Bernstein–Lunts [BL94]

Moreover, for $X$ smooth there is a natural isomorphism

to the equivariant higher Chow groups as defined by Totaro [Tot99] and Edidin–Graham [EG98].

There is a forgetful functor from equivariant to non-equivariant motivic sheaves

commuting with the six functors.

An important property of equivariant (motivic) sheaves is the induction equivalence, which allows to describe $G$-equivariant motivic sheaves on a $G$-orbit $G/H$ in terms of $H$-equivariant motives on a point:

There is also an equivariant version of the category of mixed Tate motives defined in Section 3.2. For the remainder of the section let $k=\overline{\mathbb{F}}_{p}.$

We now explain how the explicit description of this category in [SVW18, Theorem II.3.1] can be obtained using the weight complex functor.

Denote by $G^{0}\subset G$ the connected component of the identity. Then one can consider the object $\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})\in\operatorname{DM}_{G}(k)$ which plays the role of the local system on $BG$ with free action of the component group $\pi_{0}(G)=G/G^{0}.$ The heart of $w$ admits the following explicit description by [SVW18, Proposition II.4.5].

For the moment, assume that $G=G_{0}$ is connected. By [SVW18, Proposition I.7.6] the categories $\operatorname{DM}_{G}$ just depend on the quotient $G/R_{u}(G)$ of $G$ by its unipotent radical. Hence, we can assume that $G$ is reductive. Denote by $T$ a maximal torus in $G$ and by $W=N_{G}(T)/T$ the Weyl group. Let $S=\operatorname{Sym}(X(T)_{\mathbb{Q}})$ be the symmetric algebra of the rationalized character lattice of $T.$ The algebra $S$ is isomorphic to a polynomial ring in $\operatorname{rank}(T)$ many variables and graded where we put $X(T)$ in degree one.

Since we are only considering rational coefficients, the $T$- and $G$-equivariant Chow rings agree with equivariant cohomology rings. In particular, the Chern class map induces isomorphisms of graded algebras

see [Tot99] or [EG98, Section 3.2]. Similarly, for higher Chow groups

using [Kri17, Theorem 1.5] and [Kri13, Theorem 5.7].

If $G$ is not connected, we consider the extension algebra

which can be thought of as the Chow ring of $BG$ with coefficients in the local system given by the regular representation of $\pi_{0}(G).$ For an explicit description denote by $S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)]$ the twisted group algebra, see [SVW18, A.2.3], which as a graded vector space is just the tensor product $S^{W}\otimes\mathbb{Q}[\pi_{0}(G)]$ with $\mathbb{Q}[\pi_{0}(G)]$ concentrated in degree zero.