Motivic Springer Theory

TL;DR

This paper develops a motivic Springer theory framework, connecting convolution algebra representations with equivariant motivic sheaves called Springer motives, and explores dualities and geometric structures in affine type A.

Contribution

It establishes foundational results for motivic Springer theory, linking algebraic representations to motivic sheaves and proving formality and paving results in affine type A.

Findings

Realization of algebra representations via Springer motives

Formality results using weight structures

Affine paving of partial quiver flag varieties in affine type A

Abstract

We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves called Springer motives. To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures. As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in affine type A (with cyclic orientation) admit an affine paving.