A categorical action of the shifted $q=0$ affine algebra

TL;DR

This paper introduces the shifted q=0 affine algebra, explores its categorical action on derived categories of coherent sheaves on flag varieties, and constructs a categorical action of the q=0 affine Hecke algebra.

Contribution

It defines a new algebra called the shifted q=0 affine algebra and establishes its categorical action on derived categories of coherent sheaves, linking it to the affine Hecke algebra.

Findings

Defined the shifted q=0 affine algebra from geometric contexts.

Proved the existence of a categorical action on derived categories.

Constructed a categorical action of the q=0 affine Hecke algebra.

Abstract

We introduce a new algebra called the shifted $q=0$ affine algebra, which arises naturally from the study of coherent sheaves on Grassmannians and n-step partial flag varieties via a natural correspondence. It has similar presentation as the shifted quantum affine algebra defined by Finkelberg-Tsymbaliuk arXiv:1708.01795v6. We then give a definition of its categorical action and prove that there is a categorical action on the bounded derived categories of coherent sheaves on n-step partial flag varieties. Finally, as an application, we use it to construct a categorical action of the $q=0$ affine Hecke algebra on the bounded derived category of coherent sheaves on the full flag variety.