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.