Algebras and their representations in geometric representation theory can often be constructed geometrically in terms of convolution of cycles using Borel-Moore homology and constructible sheaves, e.g., the Springer correspondence describes how irreducible representations of a Weyl group can be realized in terms of a convolution action on the free vector spaces spanned by irreducible components of Springer fibers.
The authors establish the foundations of a motivic Springer theory using Chow groups and motivic sheaves (Theorem 4.9, page 209). Motivic sheaves are a relative version of triangulated category of mixed motives and their Hom-spaces are governed by Chow groups. As established in the setting of flag varieties, motivic sheaves are a graded version of constructible sheaves that are mathematically advantageous over the mixed \(\ell\)-adic sheaves or mixed Hodge modules.
They show that representations of convolution algebras, such as Lusztig’s graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in type \(A\) and \(\widetilde{A}\), can be realized in terms of Springer motives (Theorem 5.2, page 211 and Theorem 6.5, page 215). The authors lay foundations to a motivic Springer theory and prove formality results using weight structures. They also express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in type \(\widetilde{A}\) with cyclic orientation admit an affine paving (Theorem 6.3, page 213).