Cohomological boundedness of twisted coherent Springer sheaves

TL;DR

This paper proves the cohomological concentration of twisted coherent Springer sheaves and establishes t-exactness properties of the universal trace functor in the context of affine Hecke categories and noncommutative Springer resolutions.

Contribution

It demonstrates the cohomological boundedness of twisted coherent Springer sheaves and constructs explicit complexes for the universal trace functor in monoidal categories over quotient stacks.

Findings

Cohomological degree concentration of Springer sheaves confirmed.

Universal trace functor is right t-exact for the exotic t-structure.

Explicit complexes for the universal trace functor are constructed.

Abstract

We prove that the coherent Springer sheaf and its parabolic analogues are concentrated in cohomological degree $0$, as predicted by Ben-Zvi-Chen-Helm-Nadler, Zhu, Emerton-Gee-Hellmann, Hansen, and others. More generally, we show that the universal trace functor for a mixed partial affine Hecke category is right t-exact with respect to the exotic t-structure given by Bezrukavnikov-Mirkovi\'c's noncommutative Springer resolution, and left t-exact with respect to the monoidally dual t-structure. To this end, we construct an explicit complex computing the universal trace functor for certain monoidal categories over quotient stacks.