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The convolution ring$K^{GL_n(\mathcal{O})\rtimes\mathbb{C}^\times}(\mathrm{Gr}_{GL_n})$ wasidentified 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 thebasis formed by the classes of irreducible equivariant perverse coherentsheaves with the dual canonical basis of the quantum unipotent cell.

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