# Coherent IC-sheaves on type $A_{n}$ affine Grassmannians and dual canonical basis of affine type $A_{1}$

## TL;DR

This paper establishes a connection between the basis of irreducible equivariant perverse coherent sheaves on affine Grassmannians of type A and the dual canonical basis of a quantum unipotent cell of affine type A, deepening the understanding of geometric representation theory.

## Contribution

It identifies the basis of irreducible equivariant perverse coherent sheaves with the dual canonical basis of a quantum unipotent cell, linking geometric and algebraic structures.

## Key findings

- Basis of irreducible equivariant perverse coherent sheaves matches the dual canonical basis.
- Identifies convolution ring with a quantum unipotent cell of loop group $LSL_2$.
- Provides a geometric realization of the dual canonical basis.

## Abstract

The convolution ring $K^{GL_n(\mathcal{O})\rtimes\mathbb{C}^\times}(\mathrm{Gr}_{GL_n})$ was identified with a quantum unipotent cell of the loop group $LSL_2$ in [Cautis-Williams, J. Amer. Math. Soc. 32 (2019), pp. 709-778]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.