ScalingStacks

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).